Handbook of Teichmuller theory, Vol.4 3037191171, 978-3-03719-117-0

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IRMA Lectures in Mathematics and Theoretical Physics 19 Edited by Christian Kassel and Vladimir G. Turaev

Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 rue René Descartes 67084 Strasbourg Cedex France

IRMA Lectures in Mathematics and Theoretical Physics Edited by Christian Kassel and Vladimir G. Turaev This series is devoted to the publication of research monographs, lecture notes, and other material arising from programs of the Institut de Recherche Mathématique Avancée (Strasbourg, France). The goal is to promote recent advances in mathematics and theoretical physics and to make them accessible to wide circles of mathematicians, physicists, and students of these disciplines. Previously published in this series: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20

Deformation Quantization, Gilles Halbout (Ed.) Locally Compact Quantum Groups and Groupoids, Leonid Vainerman (Ed.) From Combinatorics to Dynamical Systems, Frédéric Fauvet and Claude Mitschi (Eds.) Three courses on Partial Differential Equations, Eric Sonnendrücker (Ed.) Infinite Dimensional Groups and Manifolds, Tilman Wurzbacher (Ed.) Athanase Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature Numerical Methods for Hyperbolic and Kinetic Problems, Stéphane Cordier, Thierry Goudon, Michaël Gutnic and Eric Sonnendrücker (Eds.) AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, Oliver Biquard (Ed.) Differential Equations and Quantum Groups, D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah and R. Schäfke (Eds.) Physics and Number Theory, Louise Nyssen (Ed.) Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.) Quantum Groups, Benjamin Enriquez (Ed.) Handbook of Teichmüller Theory, Volume II, Athanase Papadopoulos (Ed.) Michel Weber, Dynamical Systems and Processes Renormalization and Galois Theories, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis (Eds.) Handbook of Pseudo-Riemannian Geometry and Supersymmetry, Vicente Cortés (Ed.) Handbook of Teichmüller Theory, Volume III, Athanase Papadopoulos (Ed.) Strasbourg Master Class on Geometry, Athanase Papadopoulos (Ed.) Singularities in Geometry and Topology – Strasbourg 2009, Vincent Blanlœil and Toru Ohmoto (Eds.)

Volumes 1–5 are available from Walter de Gruyter (www.degruyter.de)

Handbook of Teichmüller Theory Volume IV Athanase Papadopoulos Editor

Editor: Athanase Papadopoulos Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 Rue René Descartes 67084 Strasbourg Cedex France

2010 Mathematics Subject Classification: Primary 30-00, 32-00, 57-00, 32G13, 32G15, 30F60; Secondary 11F06, 11F75, 14D20, 14H15, 14H60, 14H55, 14J60, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10, 22E46, 30-03, 30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 32-03, 32S30, 37-99, 53A35, 53B35, 53C35, 53C50, 53C80, 53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16.

ISBN 978-3-03719-117-0 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2014 European Mathematical Society Contact address: European Mathematical Society Publishing House ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

In memory of William P. Thurston

Foreword

Teichmüller theory is today one of the most active research areas in mathematics, with a very wide range of applications in several fields, including Riemann surface theory, hyperbolic geometry, low-dimensional topology, several complex variables, algebraic geometry, arithmetic, partial differential equations, dynamical systems, representation theory, symplectic geometry, geometric group theory and mathematical physics. This Handbook project in several volumes arose from an attempt to present the various aspects of Teichmüller theory with its relations to all the other research fields mentioned. The present volume, Number IV in the series, is divided into five parts, namely: Part A: The metric and the analytic theory, 4 Part B: Representation spaces and generalized structures, 2 Part C: Dynamics Part D: The quantum theory, 2 Part E: Sources Parts A, B and D are sequels of parts carrying the same name in previous volumes of this Handbook. Part E, entitled Sources, has a new character in the series; it contains the translation together with a commentary of an important paper by Teichmüller which is almost unknown even to specialists of the subject. We hope that making available this translation together with the commentary will give an impulse to new ideas and will help putting the theory in a broader perspective. Most of the chapters in the present volume are expository, and written by experts who have a broad view on the theory, but several chapters also contain new and important results. Together with all the other subjects that were treated in the previous volumes, this constitutes an overview of quite a large number of beautiful ideas. The topics presented in this volume involve several areas of mathematics and I do not exclude any other area in future volumes. This volume contains surveys on the Weil–Petersson metric, on the geometry associated to simple closed curves on surfaces, on the curve complex and on its relations to buildings, on the arc complex and the related operad structure, on extremal length, on holomorphic families, on various boundary structures, on infinite-dimensional Teichmüller spaces, on moduli spaces of affine structures, on higher Teichmüller theory, on quasi-conformal mappings in higher dimensions, on the Teichmüller theory of iterations of rational maps of the two-sphere, on the dynamics of the mapping class group actions on Teichmüller spaces of surfaces of infinite type and on quantization. Once more, I am grateful to Irene Zimmermann for her excellent work, and to Manfred Karbe and Vladimir Turaev for their permanent interest and support. I would

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Foreword

also like to thank the 24 contributors of this volume for a fruitful collaboration. We all hope that this series will be a useful reference to the whole mathematical community. Strasbourg, April 2014

Athanase Papadopoulos

Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Introduction to Teichmüller theory, old and new, IV by Athanase Papadopoulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part A. The metric and the analytic theory, 4 Chapter 1. Local and global aspects of Weil–Petersson geometry by Sumio Yamada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Chapter 2. Simple closed geodesics and the study of Teichmüller spaces by Hugo Parlier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Chapter 3. Curve complexes versus Tits buildings: structures and applications by Lizhen Ji . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Chapter 4. Extremal length geometry by Hideki Miyachi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Chapter 5. Compactifications of Teichmüller spaces by Ken’ichi Ohshika. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235 Chapter 6. Arc geometry and algebra: foliations, moduli spaces, string topology and field theory by Ralph M. Kaufmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Chapter 7. The horoboundary and isometry group of Thurston’s Lipschitz metric by Cormac Walsh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Chapter 8. The horofunction compactification of the Teichmüller metric by Lixin Liu and Weixu Su . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

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Contents

Chapter 9. Lipschitz algebras and compactifications of Teichmüller space by Hideki Miyachi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Chapter 10. On the geodesic geometry of infinite-dimensional Teichmüller spaces by Zhong Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Chapter 11. Holomorphic families of Riemann surfaces and monodromy by Hiroshige Shiga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 Part B. Representation spaces and generalized structures, 2 Chapter 12. The deformation of flat affine structures on the two-torus by Oliver Baues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 Chapter 13. Higher Teichmüller spaces: from SL(2, R) to other Lie groups by Marc Burger, Alessandra Iozzi, and Anna Wienhard . . . . . . . . . . . . . . . . . . . . . . . . 539 Chapter 14. The theory of quasiconformal mappings in higher dimensions, I by Gaven J. Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 Part C. Dynamics Chapter 15. Infinite-dimensional Teichmüller spaces and modular groups by Katsuhiko Matsuzaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 Chapter 16. Teichmüller spaces and holomorphic dynamics by Xavier Buff, Guizhen Cui, and Lei Tan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 Part D. The quantum theory, 2 Chapter 17. A survey of quantum Teichmüller space and Kashaev algebra by Ren Guo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759 Part E. Sources Chapter 18. Variable Riemann surfaces (translated from the German by Annette A’Campo-Neuen) by Oswald Teichmüller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787

Contents

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Chapter 19. A commentary on Teichmüller’s paper “Veränderliche Riemannsche Flächen” by Annette A’Campo-Neuen, Norbert A’Campo, Lizhen Ji, and Athanase Papadopoulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837

Introduction to Teichmüller theory, old and new, IV Athanase Papadopoulos

Contents 1

2

3

4 5

Part A. The metric and the analytic theory, 4 . . . . . . . . . . . . 1.1 Weil–Petersson geometry . . . . . . . . . . . . . . . . . . 1.2 Simple closed geodesics in Teichmüller theory . . . . . . . 1.3 Curve complexes and buildings . . . . . . . . . . . . . . . 1.4 Extremal length . . . . . . . . . . . . . . . . . . . . . . . 1.5 Compactifications of Teichmüller space . . . . . . . . . . . 1.6 Operad theory and Teichmüller space . . . . . . . . . . . . 1.7 The horofunction boundary of Thurston’s metric . . . . . . 1.8 The horofunction boundary of the Teichmüller metric . . . . 1.9 The Lipschitz algebra on Teichmüller space . . . . . . . . . 1.10 Geodesics in infinite-dimensional Teichmüller spaces . . . . 1.11 Holomorphic families . . . . . . . . . . . . . . . . . . . . Part B. Representation spaces and generalized structures, 2 . . . . 2.1 Flat affine structures . . . . . . . . . . . . . . . . . . . . . 2.2 Higher Teichmüller theory . . . . . . . . . . . . . . . . . . 2.3 Quasiconformal mappings in higher dimensions . . . . . . Part C. Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Dynamics on Teichmüller spaces of surfaces of infinite type 3.2 Teichmüller theory and complex dynamics . . . . . . . . . Part D. The quantum theory, 2 . . . . . . . . . . . . . . . . . . . Part E. Sources . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 2 3 9 11 12 13 15 17 18 19 21 22 22 23 27 30 30 34 38 38

Like the introductions to the three previous volumes of this Handbook, the present introduction will give the reader an overview of the content of the present volume, and at the same time it will present some classical and recent developments of Teichmüller theory. One of the most fascinating features of this theory is the rich interaction that it establishes between different branches of mathematics: analysis, geometry, dynamics, arithmetic, algebraic geometry, algebraic topology, theoretical physics, etc., and this is exemplified in the various chapters of the volume. Readers will sometimes find here a systematic exposition of a subject that was only sketched in the preceding volumes, and sometimes they will find an exposition of a new subject, or of an application or a connection between Teichmüller theory and some field in mathematics that was or was not treated in the preceding volumes. The chapters differ in length and level of

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Athanase Papadopoulos

difficulty. Some of them contain proofs and others contain only references to papers that contain proofs, but all the chapters contain motivating material and numerous examples. Several chapters also contain open problems and programs of research. In the following sections, we consider the chapters separately, we give an overview of each of them and we comment on each subject treated. I would like to thank Manfred Karbe who read this introduction and made several useful stylistic remarks.

1 Part A. The metric and the analytic theory, 4 1.1 Weil–Petersson geometry The first chapter, written by Sumio Yamada, concerns the Weil–Petersson metric of Teichmüller space. It complements the chapter by Wolpert on the same subject in Volume II of this Handbook. In this chapter, Yamada starts by studying the local Riemannian structure of the Weil–Petersson metric and he then gives a proof of the Weil–Petersson convexity of the energy of harmonic maps which updates a version that he published a few years ago. He also carefully describes the Weil–Petersson metric as the induced metric from the L2 pairing of symmetric .0; 2/ tensors, which constitute the tangent space of the space of all smooth metrics. This work was initiated by Fischer–Marsden and Fischer–Tromba, and Yamada gives a careful description of the L2 -decomposition result, which he calls a Hodge-type theorem because the harmonic part is picked as the intersection of kernels of two linear elliptic operators. A well-known result of Masur says that the Weil–Petersson completion of the Teichmüller space of a surface S coincides, as a set, with the augmented Teichmüller space, that is, the space obtained by adding to Teichmüller space spaces of nodal marked surfaces obtained by pinching to a point a certain collection of disjoint essential simple closed curves on S. It is also known that the Weil–Petersson completion of the Weil–Petersson metric (which is a Riemannian metric) is not Riemannian at points on the boundary but that it has a CAT.0/-geometry. Chapter 1 contains a survey on this CAT.0/-geometry. The CAT.0/ study of augmented Teichmüller space was initiated by Yamada, and the CAT.0/ geometric techniques in some sense remedy the fact that the Weil–Petersson metric is not complete. The author then discusses metric incompleteness of Teichmüller space versus its geodesic incompleteness. While we have a nice model for the metric completion of the Weil–Petersson metric of Teichmüller space, namely, the augmented Teichmüller space, a model for the geodesic Weil– Petersson completion is proposed by Yamada’s Weil–Petersson–Coxeter–Teichmüller complex (which we call for simplicity the Coxeter complex), a complex whose basic simplex is the Weil–Petersson metric completion. This Coxeter complex is obtained by reflecting the Weil–Petersson metric completion along some of its boundary strata.

Introduction to Teichmüller theory, old and new, IV

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The construction is based on a result of Wolpert stating that any two intersecting strata of the same dimension in the boundary of augmented space make right angles at their intersection, the angles being understood here in the sense of Alexandrov. The chapter also contains a proof of the fact that the Teichmüller space of the torus (as a parameter space of marked flat structures) equipped with its Weil-Petersson metric is isometric to the 2-dimensional hyperbolic plane, a result which was inexistent in the literature. The proof given here uses harmonic maps. The author also surveys the Weil–Petersson geometry of the universal Teichmüller space, based on works of Nag–Verjovsky and Takhtajian–Teo. There is an embedding of the Weil–Petersson Coxeter complex in the universal Teichmüller space. In the context of the universal space, the Weil–Petersson metric tensor is defined as a Hessian N of the @-energy of harmonic maps. This result was known for the compact surface case (shown by M. Wolf). Chapter 1 also contains a description of the metric completion of Teichmüller space as a stratified space, and of the expansion of the Weil–Petersson tensor near the boundary strata. This description involves the study of the behavior of the surface when a curve is pinched to a point. In the last section, Yamada provides a description of Teichmüller space as a Weil– Petersson convex body in the Coxeter complex. Once this is done, he defines an asymmetric metric1 (respectively a metric) on Teichmüller space, via a construction analogous to the one of the Funk asymmetric metric (respectively the Hilbert metric, which is a symmetrization of the Funk asymmetric metric) associated to an open convex subset of a Euclidean space. In this definition of the “Weil–Petersson–Funk metric”, the boundary faces of the Weil–Petersson completion act as supporting hyperplanes to the convex body. The author points out relations between this Weil–Petersson– Funk metric, the Funk metric on convex sets, and Thurston’s asymmetric metric on Teichmüller space. The embedding of the Weil–Petersson Coxeter complex in the universal Teichmüller space provides an interesting example of a geometric subspace of the universal space which is not the Teichmüller space of a surface.

1.2 Simple closed geodesics in Teichmüller theory Chapter 2 by Hugo Parlier is the first of a series of chapters whose subject is the study of homotopy classes of simple closed curves on surfaces and how they appear in Teichmüller theory. The chapter concerns more particularly simple closed geodesics on hyperbolic surfaces. The study of the set of (homotopy classes of) simple closed curves is a vast subject, related in several ways to Teichmüller theory, and a lot of work on this theme was done by various people over a span of several decades. We start by recalling some of these works. The relation between simple closed geodesics and Teichmüller space can be traced back to the works of Fricke, Klein and Vogt, done around the end of the 19th century. Further work on the subject was done by Dehn in the first quarter of the 20th century. 1An

asymmetric metric is a structure that satisfies all the axioms of a metric except the symmetry action.

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More recently, the relation between simple closed geodesics and Teichmüller theory was highlighted by Thurston in the 1970s, when he circulated his manuscript On the geometry and dynamics of diffeomorphisms of surfaces (written in 1976). With this work, hyperbolic geometry appeared at the forefront of Teichmüller theory, at the same level of importance as complex analysis. Simple closed geodesics play several roles in this and another manuscript of Thurston, Minimal stretch maps between hyperbolic surfaces, written in 1985. We now review some precise aspects of the relation between simple closed geodesics and Teichmüller theory. The first important fact that comes to mind in this respect is the fact that geodesic length functions can be used as local parameters for Teichmüller spaces. For surfaces of finite topological type, a finite set of curves suffices, and this is implicit in the work of Vogt, Fricke and Klein that we mentioned, although these authors did not formulate their results in terms of lengths of simple closed curves but in terms of traces of 2  2 matrices acting by isometries on some models of the hyperbolic plane. In these works, a set of parameters originating from these matrix actions was referred to as a set of “moduli”. This point of view is treated by Goldman in Chapter 15 of Volume II of this Handbook. The question of finding the minimal cardinality of a set of simple closed curves whose lengths parametrize (locally or globally) the Teichmüller space of a surface of finite type was thoroughly investigated in the 1990s, and one should mention here the names of Buser, Wolpert, Schmutz Schaller, Seppala-Sorvali, Hamenstädt, and there are certainly others. The question is now settled. In particular, it is known that the cardinality of a set of homotopy classes of simple closed curves whose associated length functions give global parameters for Teichmüller space must be at least one unit larger than the dimension of the space, and that such a set with this cardinality exists. Let us now review other relations between geodesic length functions and Teichmüller spaces. In 1985, Thurston gave the definition of a metric on Teichmüller space which is based on the length of a random closed geodesic. The idea behind Thurston’s definition was that a geodesic length function on Teichmüller space is convex along earthquake paths and behaves in some sense like the square of a distance function. The second derivative at the minimum point of such a function can then be thought of as a metric tensor. Motivated by this idea, Thurston defined a metric on Teichmüller space using the concept of length of a random geodesic. Soon after that, Wolpert showed that this metric is nothing else than the Weil–Petersson metric. Length functions of simple closed curves also play an important role in the description of the Weil–Petersson symplectic structure of Teichmüller space (works of Wolpert and of Goldman). In some sense, since length functions can be used to define the real and complex part of the Weil–Petersson Kähler metric, they can be used to define the complex structure of Teichmüller space since the complex structure is completely determined by the real and complex part of the metric.

Introduction to Teichmüller theory, old and new, IV

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There are other metrics and generalized metrics on Teichmüller space whose definitions use length functions of simple closed geodesics. Two important examples are the asymmetric metric which Thurston introduced in his manuscript Minimal stretch maps between hyperbolic surfaces, which we already mentioned, and its symmetrization, the length spectrum metric. Thurston’s asymmetric metric appears in several chapters of this Handbook and, for the convenience of the reader, we now recall the definition. If g and h are two complete hyperbolic metrics of finite area on a surface S , we set K.g; h/ D log sup ˛2Ã

lh .˛/ ; lg .˛/

(1.1)

where  is the set of homotopy classes of simple closed curves on S which are essential (that is, not homotopic to a point or to a puncture) and where, for each ˛ in , lg .˛/ denotes the length of the unique closed geodesic in the class ˛ for the metric g. The function K on the set of (homotopy classes of) pairs of hyperbolic metrics induces a function on T .S/T .S/ which satisfies all the axioms of a metric except the symmetry axiom. This is Thurston’s asymmetric metric on Teichmüller space. Several results on this metric have been recently obtained by various people, in particular, on its geodesics, and on its isometries. Chapter 2 of Volume I of this Handbook, by Théret and the author of this introduction, contains some basic facts about this metric. New results are presented by Walsh in Chapter 7 of the present volume. There is another asymmetric metric on Teichmüller space, the “dual” K 0 of K, defined by K 0 .x; y/ D K.y; x/ for x and y in . The metrics K and K 0 have different characters, and much more is known about Thurston’s asymmetric metric than about its dual. Finally, we mention the length spectrum metric, which is a symmetrization of Thurston’s asymmetric metric, defined by the formula ²

d.g; h/ D log sup ˛2Ã

³

lh .˛/ lg .˛/ ; : lg .˛/ lh .˛/

For surfaces of finite type, this metric has been studied by Sorvali, Li, Shiga, Liu– Sun–Wei and others. It seems that the name “length spectrum metric” is due to Zhong Li. Let us now mention other instances of the role played by the set  of homotopy classes of essential simple closed curves in Teichmüller theory. This set is in natural one-to-one correspondence with the set of simple closed geodesics as soon as a hyperbolic metric is chosen on the surface. Kerckhoff found (1980) the following formula for the Teichmüller metric that is based on extremal length: d.g; h/ D log sup ˛2Ã

Exth .˛/ : Ext g .˛/

(1.2)

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We recall that given a conformal structure H on the surface S , the extremal length function ˛ 7! ExtH .˛/ is defined on the set  of homotopy classes of simple closed curves. (In fact, it is defined on more general families of curves.) This function was introduced by Ahlfors and Beurling in 1960, and the initial function on the space  has a continuous extension to the space of measured laminations on S . It has been used in several contexts, including in the definition of a boundary for Teichmüller space, namely, the Gardiner–Masur boundary. This boundary is considered in Chapter 8 of this volume by Liu and Su and in Chapter 9 by Miyachi. The extremal length geometry of Teichmüller space is also studied by Miyachi in Chapter 4. Length functions of simple closed curves are also involved in the description of several boundary structures for Teichmüller space, e.g. Thurston’s boundary and the Weil–Petersson completion, which, as we saw, appears in Chapter 1 of this Handbook. Thurston’s boundary is defined via intersection functions associated to simple closed curves. It is reviewed in Chapter 5 by Ohshika. The Weil–Petersson completion is a stratified space whose strata are encoded by the curve complex. Length functions of simple closed geodesics also played a crucial role in the work of Kerckhoff on the Nielsen Realization Problem, and in the more recent work done around the quantization theories of Teichmüller space. Thurston circulated, again in the 1980s, a short manuscript entitled A spine for Teichmüller space, in which he outlined the construction of a mapping class groupequivariant spine for Teichmüller space. In that manuscript, the length of simple closed geodesics is used to define a stratification of Teichmüller space, the strata being subsets consisting of surfaces on which the set of geodesics of minimal length satisfy certain equalities. The idea behind that construction was further investigated by various people, including Schmutz Schaller, Parlier and Ji. Another impulse to the study of simple closed geodesics on surfaces was given by a famous formula discovered by G. McShane (in his Warwick PhD thesis, 1991), saying that for any complete hyperbolic metric of finite area on the once-punctured torus, we have X 1 1 D ; `. / 2 1Ce  where the sum is taken over all simple closed geodesics  and where l. / is the length of  for a fixed hyperbolic metric of finite area. For the experts in the field, the formula came as a surprise because of its simplicity, regardless of the fact that it does not depend on the chosen hyperbolic metric. This formula soon led to interesting improvements and generalizations by a number of people and groups of people, including McShane himself (to other punctured hyperbolic surfaces), Mirzakhani (to surfaces with boundary), Norbury (extending Mirzakhani’s work to non-orientable surfaces with boundary), Bowditch (who gave a new proof of McShane’s identity using Markov triples, that is, triples of real numbers that are > 2 and that satisfy the equation x 2 C y 2 C z 2 D wyz; Bowditch later on generalized the formula to some quasi-Fuchsian groups and to punctured surface bundles over the circle), Akiyoshi–

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Miyachi–Sakuma (extending some work by Bowditch), Tan–Wong–Zhang (to hyperbolic surfaces with cone singularities with cone angles  , possibly with cups or geodesic boundaries), Tan–Luo (who produced McShane-type identities for hyperbolic closed surfaces whose terms depend on the dilogarithm function), and there are other works, some of which we mention below. Motivated by McShane’s formula, new estimates were obtained for the asymptotics of the number of simple closed geodesics on a hyperbolic surface whose length is bounded above by a given constant L as L ! 1. In particular, Mirzakhani showed that for a surface of genus g with n cusps, the asymptotics is of the order L6g6C2n . It should be noted that the exponent in this formula is precisely the dimension of Teichmüller space. Mirzakhani also gave precise values for the other constants involved in the asymptotics. A by-product of Mirzakhani’s work is a method for computing Weil–Petersson volumes of moduli spaces using counting functions of lengths of simple closed geodesics. Mirzakhani’s volume formulae also involve identities that generalize McShane’s identity. All this shows that it is natural to have in this Handbook this survey on simple closed geodesics and on their relation to Teichmüller theory. The survey is divided into two parts. In the first part, Parlier considers a natural question, namely, how the set of simple closed geodesics differs from the set of all closed geodesics, and in particular, he examines the question of the sparseness of the former compared to the latter. To illustrate this theme, Parlier studies the behavior of these sets from three different points of view, namely: (1) the thickness of the set of points on a hyperbolic surface that belong to some simple closed geodesic compared to the thickness of the set of points that belong to some (not necessarily simple) closed geodesic; (2) the asymptotic growth of the number of simple closed geodesics with a given bound on their length compared to the corresponding growth of the number of all closed geodesics; (3) multiplicity in the simple geodesic length spectrum compared to multiplicity in the full geodesic length spectrum. Talking about the length spectrum and of multiplicity in this length spectrum, one might recall that the length spectrum of the set of all closed geodesics on a hyperbolic surface is related to the theory of the Laplace operator, but that the existence of a similar relation is unclear in the case of the length spectrum of simple closed geodesics. Let us now point out a few milestones in the work done on these three questions. We start with the first question. It is known that on any closed hyperbolic surface (and, more generally, on any closed Riemannian manifold of negative curvature) the set of points that lie on closed (non-necessarily simple) geodesics is dense. This follows from the fact that in the unit tangent bundle of compact manifolds of negative curvature, periodic orbits of the geodesic flow are dense. In contrast, Birman and Series showed in 1985 that on a closed hyperbolic surface the set of points that lie on simple closed geodesics (and more generally, on complete simple geodesics, closed or not) is nowhere dense and has Hausdorff measure zero. At the same time, the same

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authors worked out algorithms for recognizing conjugacy classes of simple closed curves in fundamental groups of surfaces. Further work on this topic was done later on by various authors. In particular, Buser and Parlier showed in 2007 that any compact hyperbolic surface of genus g contains a disk of radius cg which is disjoint from all the simple closed geodesics, where cg is a positive constant that depends only on g. Now we consider the second question. It follows from works of Huber and of Selberg in the 1950s on the spectrum of the Laplace operator that the asymptotic growth of the number of closed geodesics of length  L on a closed hyperbolic surface of genus  2 is of order e L =L. Results on the asymptotic growth rate of simple closed geodesics were obtained much later. We already mentioned Mirzakhani’s 2007 result stating that the asymptotic growth of the number of simple closed geodesics of length  L on a closed surface of genus g is of order cS L6g6 , where cS is a constant that depends only on the surface S. Mirzakhani’s work was preceded in the special case of the punctured torus by work by McShane and Rivin. The proof of the growth estimates in this special case uses homology considerations that do not apply to surfaces of higher genera. Now we consider the third question. Randol proved in 1980 that on a hyperbolic surface, the multiplicity in the set of lengths of closed geodesics is unbounded. The study of multiplicity in the length spectrum and in the simple length spectrum was further extended by Buser, and more recently by Rivin, and in joint work of McShane and Parlier. In particular, McShane and Parlier proved in 2009 that the set of surfaces in which all simple closed geodesics have distinct lengths is dense in Teichmüller space, and that the complement of this set is Baire meagre. Studying geodesics of shortest lengths naturally leads to the systole function defined on Teichmüller space, and we must say a few words on systoles. A systole on a surface is a closed geodesic of shortest length. The systole length, as a function on Teichmüller space, has been thoroughly studied in the last few decades, and it was probably Schmutz Schaller who made most of the advertising regarding this theory. Other authors who studied systoles on hyperbolic surfaces include Bavard, Buser, Buser–Sarnak, Babenko, Balacheff–Saboureau, and Massart. In the second part of Chapter 2, Parlier makes a review of systoles and of a related function which can be described as the shortest pair of pants length function. This function can be considered a higher-order analogue of the systole function, pairs of pants being maximal systems of disjoint simple closed curves. The length of a pair of pants decomposition is defined as the length of the largest curve in the decomposition. The systole function and its pants decomposition analogue are bounded by constants that depend only on the topological type of the surface. Parlier addresses several questions on systoles and their higher-dimensional analogues. As in the study of closed geodesics, there are questions on individual systoles, and others on the systole function defined on Teichmüller space. There are also investigations on the asymptotic behavior of the systole function in terms of the genus of the surface, and there are other interesting growth problems. Furthermore, there are analogous questions regarding the shortest pair of pants function.

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An object whose study is closely related to the study of the shortest pair of pants decomposition function is the Bers constant. This constant (which depends only on the genus, the number of cusps and of boundary components of the surface) is defined as the supremum of the length of shortest pants decompositions over all hyperbolic surfaces of the given genus and number of boundary components. Bers showed that the supremum of the Bers constants of all surfaces of some fixed finite hyperbolic type is finite. Buser worked out explicit bounds for this supremum. The Bers constant was used as an essential ingredient in the proof of a result of Brock (2003) stating that the Teichmüller space of a surface of finite type is quasi-isometric to the pants complex of the surface. Balacheff and Parlier (2009) obtained monotonicity results for the Bers constant in terms of the genus. There are no known analogues of these monotonicity results for systoles.

1.3 Curve complexes and buildings Chapter 3 by Lizhen Ji is also concerned with simple closed curves on surfaces, this time from the point of view of the comparison between curve complexes and buildings. Before describing the content of this chapter, let us start by recalling a few facts about Tits buildings. Tits buildings are simplicial complexes associated to semisimple Lie groups and to linear algebraic groups over the field Q of rational numbers, or over other fields. To such groups are also related symmetric and locally symmetric spaces, and Tits buildings appear as a kind of structure at infinity. They can be used to define (partial) compactifications of these symmetric spaces and locally symmetric spaces. For example, the asymptotic cone of a locally symmetric space associated to an arithmetic group  is the cone over the quotient by  of the Tits building over Q. Tits buildings provide valuable information about the asymptotic geometry of the symmetric spaces, and, in several instances, their rigidity properties have been used to prove rigidity results for locally symmetric spaces. One of the most famous rigidity results in this area is the so-called Mostow rigidity theorem stating that if 1 and 2 are two isomorphic irreducible lattices in semisimple Lie groups of noncompact type with trivial center (but not Fuchsian groups), then the Lie groups are isometric and the lattices are conjugate. Another famous result is the so-called arithmeticity of lattices result for higher rank symmetric spaces stating that any irreducible lattice in a higher-rank semisimple Lie group is arithmetic. This result is a consequence of the so-called super-rigidity of lattices in real semisimple and p-adic semisimple Lie groups. In the original proof by Mostow of his rigidity theorem the isomorphism between the two lattices is used to induce an equivariant quasi-isometry between the symmetric spaces, and then it is shown that in rank  2 this induces an isomorphism between Tits buildings. Rigidity of Tits buildings is used to show that the locally symmetric spaces are isometric. Hyperbolic geometers are familiar with a special case of Mostow rigidity through a proof by Thurston contained in his famous Princeton lectures, which says that a finite-volume hyperbolic 3-manifold is determined by its fundamental group.

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The other major actors in this chapter, namely, curve complexes, are flag simplicial complexes associated to surfaces whose vertices are the homotopy classes of essential simple closed curves (i.e., the elements of the set we denoted by ) and where for each k  1, a collection of k C1 vertices forms a k-simplex if and only if the corresponding homotopy classes can be represented by disjoint simple closed curves on the surface. Curve complexes were used to get information about boundary structure of Teichmüller spaces, about mapping class groups through their actions on these complexes and on related spaces, and about the quotient spaces of these actions, namely, moduli spaces. For instance, as we already mentioned, curve complexes parametrize the strata of the Weil–Petersson completion. They are used in the proofs of several rigidity results of mapping class group actions. A well-known instance is Ivanov’s use of the action of the mapping class group on the curve complex to obtain a geometric proof of the famous theorem of Royden stating that (except for a small number of surfaces) the isometry group of Teichmüller space coincides with the natural image of the extended mapping class group in that group. Several relations between curve complexes and buildings are analyzed in detail in Chapter 3. Geometrically, a real Tits building may be defined in terms of a classification of geodesics of the symmetric space associated to the semisimple Lie groups to which they are attached. It also admits a description in terms of proper parabolic subgroups. Buildings are classified into three types: Euclidean, spherical and hyperbolic. The adjectives spherical, Euclidean and hyperbolic refer to the geometry of apartments, which are special sub-simplicial complexes, built in the definitions of the buildings. In the spherical Tits building case, each apartment is a triangulation of a sphere. Euclidean buildings can be described in terms of Euclidean reflection groups and they are associated to Euclidean Coxeter complexes. An important class of Euclidean buildings is the class of Bruhat–Tits buildings that appear in the study of linear semisimple algebraic groups. Such buildings also play a role in the theory of p-adic Lie groups, as analogues of symmetric spaces on which these groups act. Examples of spherical buildings include the spherical Tits buildings that are associated to semisimple Lie groups; they are related to the geodesic compactifications of the associated symmetric spaces and they play an important role in the compactification of the associated locally symmetric spaces. Hyperbolic buildings are described in terms of reflection groups of hyperbolic spaces, and they are associated to hyperbolic Coxeter complexes. A spherical (respectively Euclidean, hyperbolic) building admits a natural complete geodesic metric whose restriction to every top-dimensional apartment is isometric to a unit sphere (respectively a Euclidean space, a hyperbolic space) of appropriate dimension, which depends on the rank of the building. Curve complexes have properties in common with each of the three classes of buildings, and as Ji puts it, from the point of view of their geometric and their topological properties, curve complexes are in some sense combinations of spherical, Euclidean and hyperbolic buildings. Furthermore, Euclidean and hyperbolic buildings are CAT.0/ and CAT.1/ spaces respectively, their geometries can be studied in the setting of CAT.0/ and CAT.1/ geometry, and as such they have natural geodesic compactifications. The natural simplicial metric

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of the curve complex is Gromov-hyperbolic and has a natural compactification. The Gromov boundary of the curve complex, which was identified by E. Klarreich as the space of minimal and filling (unmeasured) laminations, was used in the proof of Thurston’s ending lamination conjecture by Minsky, Brock and Canary, which is one of the major recent results in geometric 3-manifold theory. A large part of Chapter 3 concerns applications of curve complexes. The author discusses several results where the curve complex is used as an ingredient. These results include the following: the ending lamination conjecture, which is put in parallel with Mostow strong rigidity; quasi-isometric rigidity of mapping class groups (i.e. the fact that quasi-isometries are uniformly close to isometries induced by left-multiplication by elements of the group, a result obtained by Hamenstädt and others); the relation with the Novikov conjectures for certain classes of S-arithmetic subgroups of algebraic groups; the finiteness of the asymptotic dimension of the mapping class group (works of Bestvina, Bromberg and Fujiwara); the non-hyperbolicity of the Weil–Petersson metric (works of Brock and Farb); the work on the so-called Hempel distance on Heegaard splittings of 3-manifolds (works of Hempel, and, more recently, of Moore and Rathbun); the construction of partial compactifications of Teichmüller space (works of Harvey and others); cohomological properties of mapping class groups (works of Harer, Ivanov and Ji); the description of asymptotic cones of moduli space (works of Leuzinger, Farb and Masur); the computation of the simplicial volume of moduli space (work of Ji), and there are others applications. The main goal of Chapter 3 is to put these applications in parallel with analogous results on symmetric spaces.

1.4 Extremal length Chapter 4 by Hideki Miyachi is a survey on the geometry of Teichmüller space which is based on extremal length. We already recalled that the notion of extremal length, as a conformal invariant of a family of curves on a Riemann surface, was introduced by Ahlfors and Beurling in 1960. Several techniques and results on that theory have been obtained starting in the 1970s, by Thurston, Kerckhoff, Gardiner, Masur and Minsky, and more recently by Miyachi. In the work of Kerckhoff in the late 1970s, the extremal length of a homotopy class of curves, for a given marked Riemann surface, was extended to the extremal length of the equivalence class of a measured foliation. In this way, extremal length became a function on the product of Teichmüller space with measured foliation space. In Chapter 4, the author starts by recalling some basic material and then he discusses the recent developments. We already recalled in this introduction (Formula (1.2)) that extremal length can be used to measure distances between conformal structures, and that Kerckhoff obtained a formula for the Teichmüller distance which is based on extremal length comparison. We also mentioned the Gardiner–Masur compactification of Teichmüller space, which uses the extremal length function. Gardiner and Masur showed that this boundary is strictly larger than the Thurston boundary when the complex dimension of Teichmüller space is greater than one (and in the case where

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the dimension is one, the two boundaries coincide). There is a horofunction boundary of Teichmüller space, which has been studied by Liu and Su and which coincides with the Gardiner–Masur boundary. This is reported on in Chapter 8 of this volume. Gardiner and Masur also discovered a relation between extremal length theory and lines of minima. Masur and Minsky found a combinatorial model of a space they called the “electrified Teichmüller space”, obtained by modifying the geometry of some “thin sets” of Teichmüller space in order to make them of bounded diameter, and these thin sets are defined by the fact that the extremal lengths of some curves are small. All these results show that the extremal length geometry of Teichmüller space very rich. One unifying framework in Chapter 4 is the notion of intersection number applied to the setting of extremal length. In analogy with work done by Bonahon, Miyachi uses the intersection number function to obtain a hyperboloid model of Teichmüller space using extremal length. In this setting, the extremal length of a measured foliation is regarded as the intersection number between a marked Riemann surface and the given measured foliation. The picture parallels the one obtained by Bonahon using the theory of geodesic currents. A relation between intersection number and the Gromov product with respect to the Teichmüller metric is also established. Miyachi shows as a corollary that the Gromov product extends to the Gardiner–Masur boundary. There is an infinitesimal distance on Teichmüller space which is induced by the intersection number seen as a quadratic form in the new hyperboloid model, which is analogous to an infinitesimal distance that is defined in Bonahon’s setting. Miyachi formulates open questions concerning this infinitesimal distance.

1.5 Compactifications of Teichmüller space In Chapter 5, Ken’ichi Ohshika surveys three compactifications of Teichmüller space: Thurston’s compactification, the compactification by Teichmüller rays, and the Bers compactification. The first is purely topological, the second is metrical, and the third one is group-theoretical. In each case, the author discusses the mapping class group action on the compactified space, in particular, whether it is continuous or not. Thurston’s compactification is certainly the most useful one and the most wellknown. It is obtained by adjoining to Teichmüller space the space P MF of projective measured laminations, which is usually described in terms of convergence of geodesic length functions to intersection functions of measured foliations, in the projective sense. This compactification is natural in the sense that the mapping class group action on Teichmüller space extends continuously to the compactified space. Ohshika gives a description of this compactification in terms of limits of earthquake flows and in terms of the action of the surface fundamental group on R-trees. The Teichmüller compactification is a geodesic ray compactification, obtained by choosing a basepoint in Teichmüller space and adjoining to this space, as boundary at infinity, the space of endpoints of geodesic rays (for the Teichmüller metric) starting at that point. The mapping class group action on Teichmüller space does not extend continuously to this compactification, by a result due to Kerckhoff (1980). However, it

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was observed by Kerckhoff that if we consider the quotient of this Teichmüller boundary obtained by identifying to one point any pair of projective classes of measured foliations whenever they are topologically equivalent (that is, by forgetting the transverse measure), then the mapping class group action extends continuously to this new compactification. The new boundary obtained is the space UMF of unmeasured foliations. This space is closely related to the space UML of unmeasured laminations, which appears later in this chapter. The third compactification that is described in Chapter 5 is the Bers compactification. It also depends on the choice of a basepoint in Teichmüller space, and it is obtained by embedding this space in the space of faithful and discrete representations of the fundamental group of the surface in the Lie group PSL.2; C/. The image of such an embedding (which strongly depends on the choice of a basepoint) is called a Bers slice. A Kleinian group in the image is called a quasi-Fuchsian group: the limit set of such a group action is a Jordan curve in the complex plane and its domain of discontinuity has two components. Bers, who introduced this compactification (1970), showed that each slice is relatively compact and that the Kleinian groups which are in the image are b-groups: their domain of discontinuity is connected. A Bers compactification is the closure of such a slice. Ohshika mentions some recent developments of this compactification, namely, the work started by Thurston on parametrizing the boundary of a Bers slice using an invariant consisting of a conformal structure on the surface and an ending lamination, and the culmination of this work in the proof of Thurston’s Ending Lamination Conjecture by Brock, Canary and Minsky. It was proved by Kerckhoff and Thurston (1990) that the mapping class group action does not extend continuously to the Bers boundary. An analysis of the proof of this result shows that the reason why the mapping class group action does not extend continuously to the boundary is the existence of quasi-conformal deformation components in the Bers boundary. Ohshika showed (2011) that if we collapse each such deformation component to a point then the action of the mapping class group extends to the resulting space, which is called a reduced Bers boundary. This confirms a conjecture made by Thurston and quoted by McMullen in his talk at the Kyoto ICM (1990). The Bers compactification as well as the result by Ohshika are reported on in Chapter 5. A Kleinian group in the reduced Bers boundary is parametrized by an element of unmeasured lamination space UML, that is, the quotient space of measured lamination space ML obtained by forgetting the transverse measure. There are subtleties involving two different topologies on UML, the one obtained as the quotient of the topology of ML and the one induced on the reduced Bers boundary. The chapter ends with a survey of rigidity results of the actions of the mapping class group on UML and a review of the reduced Bers boundary.

1.6 Operad theory and Teichmüller space Operads first appeared as a tool in homotopical algebra. They can be traced back to works done in the 1960s by Stasheff, Boardmann–Vogt and May in homotopy theory,

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and more particularly in the study of loop spaces and iterated loop spaces.2 Roughly speaking, an operad is a set of relations that are “composable”, each of them having as an input some finite number of arguments (instead of only two, as in the familiar case of a binary relation). Such relations appear for instance when one considers multilinear self-maps of vector spaces. The set of relations defining an operad is equipped with the action of the symmetric group by permuting the variables, and the relations are required to satisfy certain conditions, such as the associativity of composition and the existence of a unit. With such relations one defines a category of algebras that generalize the one that arises in the composition of endomorphisms of vector spaces. Operad theory is also in some sense an analogue, in the setting of representation theory of associative algebras, of the representation theory of groups. For this reason, representations of operads are called operad algebras. It was realized in the 1990s that operad theory provides a combinatorial setting for the study of moduli spaces of algebraic curves. As a matter of fact, moduli spaces can be thought of in some sense as operad algebras. This establishes a strong relation between operads and Teichmüller theory. A basic and particularly simple example of an operad is the rooted tree operad, whose objects are rooted trees and whose compositions are defined by grafting a root to a leaf. Another famous operad is the little-dics or little n-dics operad, defined in terms of ordered collections of disjoint n-discs in the unit disc in Rn , with a certain rule for composition defined by taking disjoint unions and then scaling down to get again an ordered collection of disjoint n-discs in the unit n-disc. An ancestor of this operad is the little n-cube operad, which was defined in a similar way by Boardman and Vogt, taking configurations of disjoint n-cubes in the unit cube in Rn . May defined more generally the little convex bodies operad. Since their introduction, operads appeared in several fields of mathematics including, besides homotopy theory, homological algebra, K-theory, category theory, complex algebraic geometry, real algebraic geometry, mathematical physics (string theory and vertex operator algebras) and Teichmüller theory. In the 1990s, Kontsevich used operads in his work on formal deformation theory and of formal moduli spaces. In a joint paper with Soibelman (2000), he developed the deformation theory of operads and algebras over operads, proving a conjecture of Deligne stating that the Hochschild complex of an associative algebra is equipped with a canonical action of the operad obtained by taking chains of the little discs operad. In their proof, Kontsevich and Soibelman constructed a geometric operad that acts on the Hochschild complex. With this and related works, it became clear that operads are useful in understanding the internal structures that occur in theoretical physics. The work of Kontsevich and Soibelman also pointed out a relationship between operads and the Grothendieck–Teichmüller group. It was realized later on that this group acts 2 The word “operad” was coined by May, in the early 1970s, and it appears in his book The Geometry of Iterated Loop Spaces (1972).

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(homotopically) on the moduli space of structures of 2-algebras on the Hochschild complex. In the work of Kaufmann and collaborators, operads emerge in an essential way in surface theory and mapping class group theory. This is surveyed in Chapter 6 of the present volume, written by Ralph Kaufmann. One setting to start with is the following: Consider a surface with nonempty boundary, with subsets of the boundary called windows. There is also a version with an “in boundary” and an “out boundary” of the surface. The surface can be equipped with a system of arcs going from window to window. A collection of such surfaces can be glued together in various ways, matching the windows, like in the case of the rooted tree operad. The result of this gluing (regarded as a composition relation) is an operad. In fact, there are several operads that one can define in this and similar ways, obtained by requiring special properties for the arcs considered. A PROP is an object more general than an operad in the sense that it admits several inputs but also several outputs. PROPs appear in the 1960s in the work of Mac Lane. The theory of PROPs is closely related to the theory of operads. PROPs are also mentioned in Chapter 6. In 2003, Kaufmann, Livernet and Penner defined a particularly interesting operad, denoted by Arc, using weighted arc systems such that arcs emanate from each boundary component. Such a weighted arc system gives rise to a (partial) measured foliations on a surface. This operad makes a link between combinatorial compactifications of moduli spaces and a conjecture that was formulated by Penner in 1996, called the “sphericity conjecture”, which says that the arc complex of the surface is homeomorphic to a sphere of a certain dimension. Furthermore, the arc operads, with their foliation description, provide geometric models for several algebraic constructions, including a new model of the little discs operad and its framed version, and they can be used in the study of loop spaces to define natural actions on Hochschild cohomology of associative or Frobenius algebras. There are also obvious relations between these arc operads and string topology, conformal field theory and string field theory. In conclusion, the combinatorics of arcs on surfaces provides a new example but also a very powerful geometric setting for operad theory. It also sheds new light on several basic constructions in mathematics.

1.7 The horofunction boundary of Thurston’s metric Chapter 7 by Cormac Walsh concerns Thurston’s asymmetric metric on Teichmüller space, introduced by Thurston in his 1986 preprint Minimal stretch maps between hyperbolic surfaces, which was mentioned in this introduction. The asymmetric metric was already surveyed in Chapter 2 of Volume I of this Handbook. The present chapter contains new results. We gave the definition of this metric in (1.1), and it will be useful for what follows to recall another definition.

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Let S D Sg;n be a surface of finite type and negative Euler characteristic, of genus g  0 with n  0 punctures. We consider hyperbolic structures on S that are complete and of finite area. Given two such hyperbolic structures g and h, and given a homeomorphism ' W S ! S which is isotopic to the identity, the Lipschitz constant Lip.'/ of ' is defined by   dh '.x/; '.y/   : Lip.'/ D sup dg x; y x¤y2S The infimum of such constants over the set of all homeomorphisms ' W S ! S isotopic to the identity is denoted by L.g; h/ D log inf Lip.'/: 'IdS

(1.3)

Replacing g and h by homotopic metrics does not change the value of L.g; h/. Therefore the function L induces a function on T .S /  T .S /. By a result of Thurston, this function coincides with the asymmetric metric defined by Equation (1.1). The metric is also called Thurston’s metric (being understood that it is not a genuine metric) on Teichmüller space. We shall denote it by the same letter L: L W T .S/  T .S / ! RC : Thurston’s metric defined using the function L is reminiscent of Kerckhoff’s formula for the Teichmüller metric based on extremal length which we recalled in (1.2) above, with the Lipschitz constant of maps replacing their quasiconformal dilatation. This remark is at the basis of a connection that is presented in Chapter 8, on which we comment later on in this introduction. In Chapter 7, Walsh presents several results on Thurston’s metric. The first result concerns its horofunction boundary. Before stating the result, we recall that the horofunction bordification X.1/ of a (possibly asymmetric) metric space .X; d / is the closure of a set of normalized distance functions in the space of all continuous functions on X. More precisely, we choose a basepoint b in X and we associate to each point z in X the function z W X ! R defined by z .x/

D d.x; z/  d.b; z/:

This gives a map from X to the set C.X/ of continuous functions on X . We equip the latter with the topology of uniform convergence on bounded sets with respect to the topology defined by d .3 The map W X ! C.X/ defined by z 7! z is continuous and injective and the horofunction boundary (or horoboundary) of X is the complement of the image of X in the closure of this image in C.X/. The elements of the horoboundary are called horofunctions. 3 Some care is needed to define the topology associated to d because the metric d is not necessarily symmetric. The topology on X is (by definition) the topology associated to its symmetrized metric d.x; y/ C d.y; x/, which is a genuine metric on X.

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The horoboundary of a metric space was introduced in 1978 by Gromov, in the setting of symmetric metric spaces. There is a simple transformation which carries horofunctions on X with respect to one basepoint to horofunctions with respect to another basepoint, and the resulting map between horoboundaries is a homeomorphism. Thus, one can talk about the horoboundary of .X; d / without reference to a basepoint. This space is denoted by X.1/. In the case where the topology on X (defined by the symmetrized metric) is proper (that is, if all closed balls are compact), the closure of the image of X in C.X / is compact, and the horoboundary X.1/ is a horofunction compactification. The results by Walsh that are presented in Chapter 7 include the following: (1) There is a natural identification between the horoboundary of Teichmüller space for Thurston’s asymmetric metric and Thurston’s boundary of that space, defined topologically. (2) Any almost-geodesic for Thurston’s asymmetric metric converges in the forward direction to a point in Thurston’s boundary. (3) If S is not a torus with at most four punctures or a torus with at most two punctures, then any isometry of Thurston’s asymmetric metric is induced by an element of the extended mapping class group of S . Item (2) contrasts with the situation of Teichmüller geodesics. We note in this respect that Kerkhoff (1980) and then Lenzhen (2008) showed that there are Teichmüller geodesics which do not have limits in Thurston’s boundary. Item (3) is an analogue of Royden’s result (1971) concerning the isometries of the Teichmüller metric, and of a result of Masur and Wolf (2002) concerning the Weil–Petersson metric. It answers a question that was raised in Chapter 2 (Problem IV, p. 198) of Volume I of this Handbook. Walsh also obtained a concrete formula that describes horofunctions for Thurston’s asymmetric metric.

1.8 The horofunction boundary of the Teichmüller metric Chapter 8 by Lixin Liu and Weixu Su is in some sense parallel to Chapter 7 by Walsh. Instead of the horofunction boundary for Thurston’s asymmetric metric, the authors study here the horofunction boundary of Teichmüller’s metric. For a surface of finite type, they show the following: (1) There is a natural identification between the horofunction compactification of the Teichmüller space equipped with the Teichmüller metric and the Gardiner–Masur compactification of that space. (2) Every (almost-)geodesic ray for the Teichmüller metric converges to a point in the Gardiner–Masur boundary. (3) The action of the mapping class group on Teichmüller space extends continuously to the Gardiner–Masur boundary.

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The Gardiner–Masur boundary compactification was introduced by Gardiner and Masur in 1991, in a paper called Extremal length geometry of Teichmüller space. They showed in particular that this new boundary contains the Thurston boundary. Later on, this compactification was studied by several authors. For instance, Miyachi studied the asymptotic behavior of Teichmüller geodesic rays under the Gardiner– Masur embedding, and he also showed that the action of the mapping class group on Teichmüller space extends continuously to the Gardiner–Masur boundary. Some other work of Miyachi is surveyed in Chapter 9 of this volume. There is a formal analogy between the Gardiner–Masur compactification and Thurston’s compactification which is apparent right in the definitions: Thurston’s compactification is defined by embedding Teichmüller space in the space of functions on the set of homotopy classes of essential simple closed curves on the surface, while using the hyperbolic length functions, and taking the closure of the image, the Gardiner–Masur boundary is defined in a similar way, using the (square root) of the extremal length functions instead of the hyperbolic length functions. Thus, whereas the developments of the Teichmüller compactification use hyperbolic geometry, those of the Gardiner–Masur boundary use its conformal geometry, and it is useful to compare in detail the results and the formal statements that hold in both settings. The horofunction boundary of the Teichmüller metric is also considered in the next chapter.

1.9 The Lipschitz algebra on Teichmüller space In Chapter 9, Hideki Miyachi initiates a new object of study in Teichmüller theory, namely, the Lipschitz algebra associated to the Teichmüller metric. It turns out that this Lipschitz algebra is closely related to the notion of extremal length, a relation which stems again from Kerckhoff’s formula for the Teichmüller metric which we already recalled (1.2). The results that are presented in Chapter 9 include a proof of a Stone-Weierstrass type theorem for this Lipschitz algebra and a definition of a new boundary structure of Teichmüller space that arises from this Lipschitz algebra. The Stone-Weierstrass type theorem asserts that any norm-closed subalgebra of the algebra of real- or complex-valued Lipschitz functions on Teichmüller space which vanish at a certain point and which satisfy certain algebraic conditions coincides with the whole algebra. Following a construction of Loeb, Miyachi introduces the notion of a Q-compactification of a non-compact Hausdorff metric space M associated to a nonempty set Q of bounded continuous functions on M . This is a compactification which has the property that every function in Q extends to a continuous function on the compactification. Miyachi then introduces the notion of a Lipschitz compactification of M , whose definition depends on the choice of a basepoint. Applying this notion to the Lipschitz algebra of Teichmüller space, he gets the Lipschitz compactification of that space.

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There is a strong relation between the Lipschitz compactification and the Gardiner– Masur compactification, a relation which originates in the fact that the Lipschitz algebra can be defined in terms of extremal length. More precisely, Miyachi shows that there is a canonical continuous surjection from the Lipschitz compactification to the Gardiner–Masur compactification. He then describes a certain subalgebra Q of the Lipschitz algebra on Teichmüller space for which the Gardiner–Masur compactification becomes a Q-compactification. As a corollary, he obtains the following naturality property of the Gardiner–Masur compactification: Any Lipschitz map from a metric space M to Teichmüller space extends continuously from the Lipschitz compactification of M to the Gardiner–Masur compactification. Making a relation with more classical notions, Miyachi notes that in general Lipschitz mappings on Teichmüller space do not extend continuously to the horofuction boundary. In some sense, this makes the Gardiner–Masur boundary of Teichmüller space more canonical. He obtains the following corollary, which concerns the complex geometry of Teichmüller space: If M is a complex manifold which is Kobayashi hyperbolic (that is, if the Kobayashi pseudo-distance on M is a genuine distance), then any holomorphic mapping from M to Teichmüller space extends continuously to a map from the Lipschitz compactification of M to the Gardiner–Masur compactification of Teichmüller space.

1.10 Geodesics in infinite-dimensional Teichmüller spaces The Teichmüller space of a surface of infinite type, equipped with its Teichmüller metric, is an infinite-dimensional non-separable Banach manifold. The metric is complete and it is geodesically convex. Chapter 10 by Zhong Li concerns properties of geodesics in general infinite-dimensional Teichmüller spaces, with a stress on properties that do not hold in finite-dimensional ones. The author notes that the following properties, in which distances, geodesics, etc. refer to the Teichmüller metric, hold in any infinite-dimensional Teichmüller space but not in any finite-dimensional Teichmüller space. (1) There exist local geodesics that are not global geodesics. (2) There exist self-intersecting geodesics. (3) There exist pairs of points with more than one straight line joining them. (4) There exist closed geodesics of arbitrarily short length. (5) The Teichmüller distance function is not differentiable. (6) The Finsler norm function of the Teichmüller metric is not of class C 1 . (7) There exist pairs of points with infinitely many geodesic disks containing them. (A geodesic disk in Teichmüller space is an isometric image of the hyperbolic plane.) Other properties of infinite-dimensional Teichmüller spaces that are presented in Chapter 10 include the fact that these spaces admit isometric holomorphic embeddings

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of the infinite-dimensional polydisk D 1 (a result of Earle and Li). Furthermore, it is known that in the infinite-dimensional case, no sphere in Teichmüller space is geodesically strictly convex, whereas in the finite-dimensional case the response is unknown. Several of the above properties can be deduced from each other, although originally they were proved by independent means. Another theme that arises in Chapter 10 is that there is a characterization of the points in infinite-dimensional Teichmüller spaces which satisfy the above properties. Indeed, the author gives several characterizations of points and pairs of points where Properties (3) and (7) hold. In this setting, the points in Teichmüller space are seen as equivalence classes Œ of Beltrami differentials on the base surface. In particular, there is a characterization of the set of points Œ for which the geodesic joining such a point to the base surface (that is, the equivalence class of the zero Beltrami differential) is unique. Earle and Li (1999) gave such a characterization in terms of a property that involves the boundary dilatation of the Beltrami differential . Lakic proved (1997) that in an infinite-dimensional Teichmüller space the “good points” (i.e., the points for which the geodesic joining them to the origin is unique) form an open and dense subset. These good points coincide with the so-called Strebel points, or Strebel differentials, a terminology introduced by Earle and Li in their 1999 paper.4 In the case of finitedimensional Teichmüller spaces, all points are good points (a result that follows from the work of Teichmüller). A theorem by Gardiner and Lakic states that the set of Strebel points is dense and open in Teichmüller space. One of the facts that the reader can learn in Chapter 10 is that several of the geometric features of infinite-dimensional Teichmüller spaces – non-uniqueness of geodesics joining two points, existence of closed geodesics, non-strict convexity of spheres, etc. – have infinitesimal analogues (and in fact they can be deduced from their infinitesimal analogues) in the tangent space to Teichmüller space at the basepoint, that is, the space dual to the space of holomorphic quadratic differentials on the base surface, equipped with the metric induced from the sup norm. The notion of extremality is an important notion for finite and infinite-dimensional Teichmüller spaces, but in the latter setting it is much more intricate. In this respect, Li also presents a result by Bozin, Lakic, Markovic and Mateljevi´c (1998) establishing a characterization of good points Œ in terms of extremality of the Beltrami differential . Here, one says that  is extremal in its class Œ if for any other 0 in the same class, one has jj1  j0 j1 . In the case of surfaces of finite type, there exists a unique extremal Beltrami differential in each class, and it is defined in terms of a quadratic differential. In an infinite-dimensional Teichmüller space, a theorem due to Reich, Strebel, Kra and Krushkal’states a criterion for extremality of a quadratic differential in terms of a notion of infinitesimal extremality. This result reduces a question of finding extremal objects up to homotopy to a question of extremality in the tangent space. Another important notion is the one of unique extremality. A Beltrami differential is 4 This notion is not to be confused with the notion of a Strebel or Jenkins–Strebel quadratic differential, which is well known to Teichmüller theorists.

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said to be uniquely extremal if it is the unique extremal differential in its class. The theorem of Bozin, Lakic, Markovic and Mateljevi´c says that a Beltrami differential  is uniquely extremal if and only if it is infinitesimally uniquely extremal. Lets us mention that one of the most important recent results in this theory is the theorem of Markovic stating that if two infinite-dimensional Teichmüller spaces are isometric, then the corresponding base surfaces are quasi-conformally equivalent.

1.11 Holomorphic families Chapter 11 by Hiroshige Shiga is about holomorphic families of Riemann surfaces. A holomorphic family of Riemann surfaces is a family of Riemann surfaces parametrized by some holomorphic parameter. More precisely, a holomorphic family of type .g; n/ is a triple .M; ; B/ where B is a complex manifold called the base, M is a complex manifold, and  W M ! B is a surjective holomorphic map of maximal rank at each point of M such that for every t in the base, S t D  1 .t / is a conformally finite type surface of genus n with p punctures and the map t 7! S t is holomorphic. To a holomorphic family of Riemann surfaces .M; ; B/ is associated a holomorphic “classifying map” from the base into the moduli space of Riemann surfaces and a homomorphism from the fundamental group of the base to the mapping class group of a fiber S t called the monodromy of the family. The monodromy is well defined up to conjugation. Holomorphic families of Riemann surfaces, in the particular case where the base space is the punctured disk, were already studied by Imayoshi in Chapter 3 of Volume II of this Handbook. In Chapter 11, of the present volume, the author considers the case where B is an arbitrary Riemann surface. The association t 7! S t is a map  from B to the moduli space of Riemann surfaces of type .g; n/. There is a natural notion of isomorphic holomorphic families of Riemann surfaces (biholomorphisms of the total space respecting the projections), and a rigidity theorem states that two holomorphic families over the same base surface are isomorphic if and only if their monodromies are conjugate. A result by Imayoshi and Shiga says that for a fixed base, there are only finitely many holomorphic families (except the locally trivial ones). The proof is based on a generalized form of the Schwarz Lemma, on the fact that a holomorphic family is determined by its monodromy and on a result of Shiga stating that the image of the monodromy is always an irreducible subgroup (in the sense of Thurston’s classification) of the mapping class group of the fibre. In Chapter 11, the author considers applications of holomorphic families in several geometrical settings. One setting is the problem of solving Diophantine equations over function fields. This problem consists in finding meromorphic functions satisfying some homogeneous polynomial equations with coefficients in some function field of meromorphic functions on some Riemann surface. The rigidity result by Imayoshi and Shiga that we mentioned above is considered as the geometric Shafarevich conjecture. Relations are made with the Mordell conjecture asserting that some diophantine equations have a finite number of solutions. The author then discusses

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the relation between holomorphic families and holomorphic motions, and with the finiteness properties of Teichmüller curves (in particular Veech surfaces) in moduli space.

2 Part B. Representation spaces and generalized structures, 2 2.1 Flat affine structures Chapter 12 by Oliver Baues is a survey on flat affine structures on surfaces. A flat affine structure is a geometric structure defined by an atlas with values in the Euclidean plane R2 and coordinate changes in the group Aff.2/ of affine transformations of the plane. By a theorem of Benzécri (1960), a closed orientable surface which carries a flat affine structure is necessarily the two-torus. Thus, this chapter concerns only the torus. Despite this restriction, the theory is rich and has many interesting facets. In a way that parallels the corresponding notions for Teichmüller space, one studies the holonomy maps of flat affine structures, the representation space as a subset of the character variety Hom.Z2 ; Aff.2//=Aff.2/, the surface fundamental group action on this representation space, the resulting moduli space, one can study discreteness criteria for such an action, and so on. The work on flat affine structures can be traced back at least to 1953, when Kuiper started a classification of these structures on surfaces. This classification was completed in the 1970s by Nagano–Yagi, and by Furness–Arrowsmith. More recent activity on the subject was carried out in the 1990s by Baues, Baues–Goldman, and Benoist. The main results obtained so far include a complete classification of affine structures on the torus, a description of the deformation space and of the moduli space of such structures (the analogues of the Teichmüller space and the moduli space of a surface of negative Euler characteristic), with a study of various group actions and flows associated to these spaces. Chapter 12 contains a detailed account of the classical results as well as some new results on flat affine structures and their deformation spaces. The survey starts with a proof of Benzécri’s theorem and of the classification theory of flat affine structures on the torus, together with a rich variety of examples and methods of construction of such structures: taking quotients of affine Lie groups, gluing patches of affine space along their boundary (in particular, gluing polygons and annuli along their sides), examples obtained from linear algebra, and others. An affine version of the classical Poincaré polygon gluing theorem is given. The author considers families of flat affine two-tori obtained as quotients of quadrilaterals glued along their boundaries by affine maps. He studies the dependence of such flat affine tori on the shape of the quadrilaterals and on the annuli that are used to define them. He shows that the flat affine tori that are obtained by gluing affine quadrilaterals along their sides are all homogeneous (that is, their groups of affine automorphisms act transitively on these spaces). He then

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describes in detail a construction of affine tori with developing image in A2  f0g. The chapter also contains a review of more recent results on discontinuous affine actions with compact quotients, and a survey of the structure of the deformation and of the moduli spaces. The reader will learn that even though the definition of the deformation space of flat affine structures on the two-torus is comparable to that of the Teichmüller space of a surface of higher genus, the basic properties of the two spaces differ in a substantial manner. For instance, while the latter is a Hausdorff and metrizable space, the former is not Hausdorff. An important feature of the action of Aff.2/ on R2 that makes things different from the group action that defines Teichmüller space is that the Aff.2/-action has non-compact stabilizers. The author shows that the development map of a flat affine structure on the twotorus is always a covering map. He gives a description of the topological local structure of the space of deformations of flat affine structures, proving that the holonomy map defined on that space is a local homeomorphism onto an open connected subset of the character variety. He then provides examples that show that this property does not always hold for other locally homogeneous structures. In other words, there exist deformation spaces of locally homogeneous structures on the torus whose holonomy maps at some points are not locally injective. Furthermore, Chapter 12 contains all the necessary introductory material on general locally homogeneous structures on manifolds and their deformation spaces, and on Thurston’s treatment of development maps and holonomy homomorphisms. The theory of affine structures can be developed in dimension greater than two. One of the main conjectures in this respect is the so-called Chern conjecture, stating that the Euler characteristic of a compact flat affine manifold is zero. (This would be the analogue of Benzécri’s Theorem in dimension two.)

2.2 Higher Teichmüller theory Chapter 13 by Marc Burger, Alessandra Iozzi and Anna Wienhard is a survey on higher Teichmüller theory. This is an extension of Teichmüller theory to representations of fundamental groups of surfaces into Lie groups which are not PSL.2; R/ (which is the setting of the classical theory). In the classical case, one considers the space of homotopy classes of marked hyperbolic structures on, say, a closed surface S, as a subset of the character variety Hom.1 .S/; PSL.2; R//=PSL.2; R/. This is obtained by considering PSL.2; R/ as the isometry group of the upper half-space model of the hyperbolic plane and associating to each hyperbolic structure its holonomy homomorphism, an element of Hom.1 .S/; PSL.2; R//. The group PSL.2; R/ acts by conjugation on the space Hom.1 .S/; PSL.2; R//, and the image of Teichmüller space in Hom.1 .S/; PSL.2; R//=PSL.2; R/ coincides with a connected component of this variety. This component consists entirely of discrete and faithful representations. It is also well known that one can obtain a description of Teichmüller space by working with the projective special unitary group PSU.1; 1/ instead of the projective

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special linear group PSL.2; R/, identifying PSU.1; 1/ with the orientation-preserving isometry group of the Poincaré disc. The theory obtained is identical. Higher Teichmüller theory was developed as a generalization of the representation theory of 1 .S/ to other simple Lie groups. The natural questions that appear right at the beginning of the theory are whether the results that hold in the classical representation theory still hold in the generalized setting. It turns out that for some groups the situation is quite different from the one in the case of PSL.2; R/. There are still several interesting questions that remain unanswered in the general theory. A higher Teichmüller space is defined accordingly as a connected component (or a union of connected components) of a space Hom.1 .S /; G/=G that consists of equivalence classes of discrete and faithful representations of 1 .S / into some simple Lie group G. Some of the natural questions that one would like to answer in this theory are the following: (1) Study individual representations  W 1 .S / ! G, find invariants of these representations and detect when such a representation is discrete and faithful. (2) Figure out whether there exist connected components in the representation variety that consist entirely of discrete and faithful representations. (3) Study the geometry of such components, equipping them with various structures (metric, symplectic, complex, combinatorial, etc.), in analogy with the structures that are known to exist on Teichmüller space. (4) Study the dynamics of the surface mapping class group actions on such components. (5) Find geometric structures on surfaces that correspond to these new conjugacy classes of representations, possibly via the holonomy representation of the fundamental group of the surface, and in analogy with the fact that homotopy classes of hyperbolic structures correspond to conjugacy classes of representations in PSL.2; R/ or in PSU.1; 1/. Some of these questions are already settled and some others are under thorough investigation, by several people. For instance, it is known that the answer to (2) is no for some Lie groups. We shall comment below on the other questions. A few words on the development of higher Teichmüller theory may be appropriate here. It is usually acknowledged that higher Teichmüller theory started with the work of Hitchin (1992), who studied representations of fundamental groups of closed surfaces in PSL.n; R/ (generalizing the classical case where n D 2) and, more generally, in the adjoint group of a split real simple Lie group, namely, the groups which are in the following list: PSL.n; R/, PSp.2n; R/, PSO.n; n/, PSO.n; nC1/ (and there are some other such groups, called exceptional). Using techniques of Higgs bundles, Hitchin showed the existence of a component in the representation variety of such a group that has properties which resemble those of Teichmüller space. Higgs bundles are holomorphic vector bundles, equipped with so-called “Higgs fields” that appeared in

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works of Hitchin and of Simpson. Their study brought new tools in surface group representation theory. In the special case where S is a closed surface of genus g  2 and where G D PSL.n; R/, the component highlighted by Hitchin is homeomorphic 2 to R.2g2/.n 1/ ; it is now called Hitchin component. Hitchin also studied the other components. One year later (1993), Choi and Goldman showed that in the case where G D PSL.3; R/, the Hitchin component consists precisely of the holonomy representations of convex real projective structures on the surface, thus answering Questions (2) and (5) above in the particular case of PSL.3; R/. This component contains the usual Teichmüller space as a subset. Following the work of Hitchin in the case of PSL.n; R/, an extensive research on higher Teichmüller theory was conducted in the last decade by several authors, including Burger, Iozzi, Wienhard, Labourie, Fock, Goncharov, Guichard, Loftin, Bradlow, García-Prada, Mundet i Riera, Gothen, and others. In particular, Fock and Goncharov developed a higher Teichmüller theory which is parallel to the theory of Hitchin representations, using a notion of positivity of representations of the fundamental group of the surface into a split semisimple algebraic group G with trivial center. A similar notion of positivity was already introduced by G. Lusztig in his theory of canonical bases that appeared in the 1990s. Fock and Goncharov gave a more geometric version of that theory, where positivity of a representation of a surface fundamental group into a real split simple Lie group is defined in terms of the familiar Thurston shear coordinates on the edges of surface hyperbolic ideal triangulations. They proved that these positive representations are faithful and discrete, and that their moduli space is an open cell in the space of all representations, generalizing the classical case where G D PSL.2; R/. Furthermore, they developed a relation between these positive representations and cluster algebras and with the quantization theory of Teichmüller space. The work of Fock and Goncharov was motivated in part by the theory of quantum representations of mapping class groups. Labourie, in 2006, associated to every representation in a Hitchin component a curve in a projective space, which he called a hyperconvex Frenet curve, generalizing the Veronese embedding from P .R2 / into P .Rn /. He also showed that a Hitchin component in the case of PSL.n; R/ consists entirely of discrete and faithful representations, and that a representation in such a component is a quasi-isometric embedding. He also showed that the mapping class group acts properly discontinuously on the Hitchin component. He introduced the concept of Anosov representations. These are also discrete and faithful representations, and they are quasi-isometric embeddings. An Anosov representation generalizes the notion of a convex cocompact representation. Anosov representations form a set on which the mapping class group acts property discontinuously. In the case where G D PSL.n; R/, Hitchin representations are Anosov representations, but the converse does not hold. In its original version, an Anosov representation arises as the holonomy of an Anosov structure on the underlying surface. This concept became the basis of a new dynamical framework for the study of representations in the Hitchin components.

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In 2008, Guichard and Wienhard gave a characterization of the representations in the Hitchin components in the case where G D PSL.4; R/ as properly convex foliated projective structures on the unit tangent bundle of the surface, thus answering Question (5) in that particular case. Guichard and Wienhard gave a characterization of convex Anosov representations in the case of irreducible representations. They also developed an analogue of the Labourie notion of Anosov representation theory of fundamental groups of surfaces in the context of representation theory of hyperbolic groups. Labourie, Guichard and Wienhard showed that in the case where G is a rank one semisimple Lie group, a representation is Anosov if and only if it is a quasiisometric embedding, and moreover that this representation is Anosov if and only if its image is a convex cocompact group. To sum up, there are, so far, two large classes of simple Lie groups G for which higher Teichmüller theory has been developed: • the class of split real simple Lie groups that were mentioned above, i.e., SL.n; R/, Sp.2n; R/, SO.n; n C 1/, SO.n; n/; • the class of Lie groups of Hermitian type, that is, those that carry an invariant complex structure, or, equivalently, those whose associated symmetric space is Hermitian. (Note that the Lie groups Sp.2n; R/ belong to both classes.) In the case of Lie groups of Hermitian type, there is a bounded integer-valued function defined on the character variety, called the Toledo invariant. D. Toledo, in 1989, studied such a function in the particular setting of representations of fundamental groups of surfaces into the groups of isometries of complex hyperbolic spaces. Burger, Iozzi and Wienhard introduced the Toledo invariant in the general setting of representations into Lie groups of Hermitian type. The Toledo invariant is reminiscent of the Euler number, an integer-valued function defined on the character variety Hom.1 .S /; PSL.2; R//=PSL.2; R/ and which is constant on the connected components of this variety. The classical Teichmüller space is precisely a connected component with maximal Euler number (Goldman’s thesis, 1980). In the setting of Lie groups of Hermitian type, the level set of the maximal value of the modulus of the Toledo invariant is a union of connected components of the representation variety whose elements are called maximal representations, and such a component is an instance of a higher Teichmüller space. Global properties of the components of the representation variety that have maximal Toledo invariant were also studied (in the case of closed surfaces) by Bradlow, García-Prada and Gothen, using Higgs bundles. The geometric properties of the representations that lie in these components were studied by Burger, Iozzi and Wienhard, who also considered the case of surfaces with boundary. These authors obtained a structure theorem for maximal representations as well as information about the representation variety. They associated to maximal representations boundary maps with monotonicity (positivity) properties expressed in terms of maximal triples of points in the Shilov boundary of

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the symmetric space associated to the underlying Lie group G, and they gave a formula for the Toledo invariant in terms of the bounded Euler class of a group action on a circle, a notion that generalizes the classical Poincaré rotation number for circle homeomorphisms. Chapter 13 is a survey of all these results.

2.3 Quasiconformal mappings in higher dimensions It is well known that the development of the theory of quasiconformal mappings in dimension two was largely motivated by Teichmüller theory. Conversely, Teichmüller theory (especially the complex-analytic part of it) relies substantially on twodimensional quasiconformal mapping theory. In this sense, the theory of quasiconformal mappings in higher dimensions can be considered as a higher Teichmüller theory. Therefore it is not unreasonable to include a survey on that theory in the present section of the Handbook. This is the subject of Chapter 14 of this volume, by Gaven Martin. In this chapter, the author outlines the major ideas and results in higher-dimensional quasiconformal theory, starting from its developments in the early 1960s, in particular in works of Reshetnyak, Gehring and Väisälä. The exposition also highlights the connections of quasiconformal theory with other theories such as geometric function theory, the calculus of variations, nonlinear partial differential equations, differential and geometric topology, and the dynamics of iteration of holomorphic functions in higher dimensions. There are common features of quasiconformal mappings in dimension two and in higher dimensions. For instance, quasiconformal mappings, in any dimension, are Hölder continuous, they are solutions of a Beltrami-type partial differential equation, they satisfy a generalized version of the Schwarz Lemma and a normal family-type compactness property. But it is also important to know that the theory of quasiconformal mappings in higher dimensions is not simply a generalization of quasiconformal theory in dimension two; there are severe differences between the two theories, and even at the level of the most basic principles, the theory in dimension  3 has its proper features. As a matter of fact, one might first ponder on the difference in the case of conformal mappings, where in dimension  3 there are no conformal – sufficiently smooth – mappings from a domain  Rn to Rn , except restrictions on of Möbius transformations. This is the so-called Liouville theorem, a phenomenon that is very different from what occurs the case of dimension two where there are many conformal mappings. We recall by the way that the word quasiconformal, in dimension two, first appeared in Ahlfors’ 1935 paper Zur Theorie der Überlagerungsflächen5 (On the theory of covering surfaces). Before that, in papers published in 1929 and 1932, Grötzsch had used a similar notion for maps that deviate from conformality (without giving them a name). In 1935, the year where Ahlfors’ paper appeared, Lavrentieff published a 5 This is the paper for which Ahlfors, in 1936, was awarded the Fields medal; it concerns a generalization of Nevanlinna theory, which will be mentioned later in this introduction.

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paper in which he introduced a weaker notion of functions deviating from conformality, which he called almost-analytic (the paper is in French and Lavrentieff called these functions “fonctions presque-analytiques”). Now we pass to dimension  3. There are several definitions of quasiconformal mappings in higher dimensions which generalize the definitions in dimension two, and each of them highlights some important aspect of the theory. We now recall some of them. If f W ! 0 is a diffeomorphism between open subsets of Rn , then one possible measure of the deviation of f from conformality is the quantity sup k.f 0 .x//; x2 0

where for each x in , f .x/ is the derivative (infinitesimal linear map) of f at x, and where k.f 0 .x// is the value of the ratio of the major axis to the minor axis of the ellipsoid f 0 .Sxn /, with Sxn being the unit sphere in the tangent space Tx of at x. (Recall that the image by a linear map of a sphere centered at the origin is an ellipsoid.) This notion of quasiconformality was used by Grötzsch and by Teichmüller. It is geometrically very appealing, but it has the disadvantage of applying only to C 1 maps, or to maps which are C 1 except on a very small set. We recall a second definition. It involves moduli of families of curves. Given a family of curves  in a domain , the modulus M. / of  is defined by the formula Z n .x/jdxj; M./ D inf 



where the infimum is taken over all Borel functions on satisfying Z ds  1 for all  2 : 

A homeomorphism f W ! 0 is then said to be K-quasiconformal for some K 2 Œ0; 1 if we have 1 M./  M.f .//  KM./ (2.1) K for every family of curves  in . Such a definition of quasiconformal mappings was given by Ahlfors in his paper On quasiconformal mappings (1954). Grötzsch had already shown that Property (2.1) is satisfied by quadrilaterals under the first definition of K-quasiconformal maps, that is, when the letter  in (2.1) denotes the family of curves joining the two vertical sides of a quadrilateral (and in this case the quantity M./ is called the modulus of the quadrilateral). Thus, Ahlfors used Grötzsch’s property to get a new definition of quasiconformality. He defined a homeomorphism f W ! 0 to be K-quasiconformal if Property (2.1) holds for every quadrilateral  in . The modulus of a family of curves is a conformal invariant, and it follows directly from this second definition that the notion of K-quasiconformality of a function is

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invariant by pre- and post-composition by conformal mappings. In this sense, quasiconformality can be seen as a conformal invariant. A third definition, valid in an arbitrary metric space, requires that a homeomorphism f W ! 0 between two domains in Rn has bounded infinitesimal distortion. Here, one defines the infinitesimal distortion H.x; f / of f at a point x in by max jf .x C h/  f .x/j ; jhj!0 min jf .x C h/  f .x/j

H.x; f / D lim sup

and the map f is said to be quasiconformal in if its infinitesimal distortion is bounded, that is, if sup H.x; f / < 1. x2

It turns out that in higher dimensions there is an important generalization of quasiconformal mappings to non-injective mappings, called quasiregular mappings.6 Quasiconformal mappings and quasiregular mappings are solutions of partial differential equations which are analogous to the Beltrami equation satisfied by quasiconformal mappings in the plane. The bases of the theory of quasiregular mappings was developed by Reshetnyak, Martio, Rickman and Väisälä. In Chapter 14, after surveying the general theories of quasiconformal and quasiregular mappings in domains in Rn , the author presents the important results on the quasiconformal theory of manifolds. There are several interesting results in this setting. It is known that the topological and the smooth categories of manifolds in dimension four are very different. The theory of quasiconformal 4-manifolds lies between the two. The author in Chapter 14 surveys in particular Sullivan’s important uniformization theorem stating that except in dimension 4, every topological manifold admits a unique quasiconformal structure. Donaldson and Sullivan, in joint work, showed that several results of Donaldson in the smooth category hold in the quasiconformal one. The results they obtained shed new light on the fact that the (quasiconformal) theory of manifolds of 4-manifolds is very different from the theory in other dimensions. The following are two of their main results: I. There are topological 4-manifolds that do not admit any quasiconformal structure. This results is in line with a result of Freedman. (In contrast, a result by Sullivan says that any topological manifold in dimension 6D 4 admits a quasiconformal structure.) II. There are smooth (and in particular quasiconformal) compact 4-manifolds that are homeomorphic but not quasiconformally homeomorphic. The author in Chapter 14 also touches upon the Donaldon–Sullivan approach to Yang–Mills theory on quasiconformal 4-manifolds. He also presents an outline of Nevanlinna theory on the growth rate of meromorphic function, a theory which is very powerful in complex analysis. He then reviews the appearance of quasiconformal mappings in non-linear potential theory and the dynamics of quasiregular mappings, 6 It seems that this notion is due to Reshetnyak, who called these maps “maps of bounded distortion”. The name “quasiregular mappings was introduced by Martio, Rickman and Väisälä. (See Vuorinen’s review of Reshetnyak’s book Space mappings with bounded distortion, Bulletin of the AMS, 1991.) In some precise sense, quasiconformal mappings in Euclidean n-space generalize plane conformal mappings whereas quasiregular mappings generalize analytic functions of one complex variable.

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highlighting the analogies between this theory and the theory of iteration of rational maps on the Riemann sphere. The survey contains a section on quasiconformal group actions, introduced by Gehring and Palka, and also developed by Sullivan, Tukia and others. The author then presents the solution of the Hilbert-Smith conjecture for quasiconformal actions (Martin,1999) stating that any locally compact group of quasiconformal homeomorphisms acting effectively on a Riemannian manifold is a Lie group. He formulates a “Lichnerowicz problem” for rational maps of manifolds, asking for a classification of closed n-manifolds that admit a non-injective rational map; for injective mappings, the problem was solved in the 1970s. The author also reports on a “Negative Curvature Theorem” implying that a branched quasiregular mapping between closed hyperbolic manifolds cannot induce an injection at the level of fundamental groups. He mentions works of Gromov, Varopoulos, Saloff-Coste and Coulhon in this connection. He also presents several topological rigidity results for quasiregular mappings between hyperbolic manifolds, works on quasiconformal groups by Tukia, Gromov, Sullivan, Gehring, Freedman, Skora and others, as well as the theory of quasiregular semigroups developed by himself and Mayer.

3 Part C. Dynamics 3.1 Dynamics on Teichmüller spaces of surfaces of infinite type Chapter 15 by Katsuhiko Matsuzaki is a survey on the dynamics of actions of mapping class groups of surfaces of topologically infinite type on the corresponding Teichmüller spaces. Let us recall some facts on Teichmüller spaces of surfaces of infinite type. Like in the case of surfaces of finite type, we take a base Riemann surface R and define its Teichmüller space T .R/ as the space of homotopy classes of quasiconformal homeomorphisms from R to another (varying) Riemann surface. Unlike the definition in the case of surfaces of finite type, the definition here depends on the choice of the base surface, for several reasons. First of all, not all surfaces of infinite type are homeomorphic, and therefore the associated spaces are a priori different. Secondly, it is known that there exist pairs of Riemann surfaces of infinite type that are homeomorphic but not related by any quasiconformal homeomorphism. Therefore, the Teichmüller space T .R/ of R is restricted by the choice of the base conformal structure R. The Teichmüller space of any Riemann surface of infinite type is infinite-dimensional. A result by Fletcher, which is reported on in Volume II of this Handbook, says however that all these spaces, equipped with their Teichmüller metric, are locally bi-Lipschitz equivalent; in fact, they are locally bi-Lipschitz equivalent to the Banach space l 1 .7 7 Fletcher’s result is stated in the setting of the so-called non-reduced Teichmüller theory, that is, the theory where the equivalence relation that defines the elements of the space is homotopy that is the identity on the ideal boundary of the marked surfaces.

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We also recall that in the theory of mapping class group actions on Teichmüller spaces, there are severe differences between the cases of surfaces of finite type and surfaces of infinite type. First of all, while in the case of surfaces of finite type there is one commonly used definition of the mapping class group, a definition which is purely topological (with some variations, regarding the actions on the boundary components, or the fact that the mapping classes considered preserve the punctures pointwise, and so on), in the case of surfaces of infinite type there are many possibilities. For instance, for the actions on Teichmüller spaces defined using quasiconformal maps that we just recalled, one usually considers only homotopy classes of quasiconformal homeomorphisms.8 The mapping class group of R, which is also called in Chapter 15 the Teichmüller modular group (because it depends on the choice of the Teichmüller space), is defined as the space of homotopy classes of quasiconformal homeomorphisms of R. This chapter is a survey of the dynamics of the action of the mapping class group of R on the Teichmüller space T .R/. Another difference between the case considered here and the case of surfaces of topologically finite type is that the mapping class group of a surface of infinite type is generally uncountable. Furthermore, in the case of surfaces of finite type, the action of the mapping class group on Teichmüller space is properly discontinuous, and therefore the study of the dynamical properties of such an action has a rather limited scope. In the setting of surfaces of infinite type, this is not the case, and there are different types of orbit behavior under the corresponding actions.9 Besides the dynamical properties of the action of the mapping class group itself, several dynamical properties of actions of subgroups of the mapping class groups of surfaces of infinite type on the corresponding Teichmüller spaces are highlighted in Chapter 15. Such actions can be classified at a given point, with respect to the behavior of the orbit of that point. For instance, such an action might be (in the terminology used by Matsuzaki): • discontinuous: the orbit of the point is discrete and its stabilizer is finite; • weakly discontinuous: the orbit of the point is discrete; • stable: the orbit of the point is closed and the stabilizer is finite; • weakly stable: the orbit of the point is closed. There are other dynamical properties that enter the scene. For instance, Matsuzaki makes a distinction between subgroups of bounded type (i.e., those for which orbits 8 There are other settings than the quasiconformal one, and they are not considered in this chapter. For instance, one can study mapping class group actions on Teichmüller spaces where the base surface is equipped with a hyperbolic metric (and not only a conformal structure); Teichmüller space can be taken here to be the set of homotopy classes of hyperbolic metrics that are bi-Lipschitz equivalent to the base structure by a homeomorphism that is homotopic to the identity. In this setting one defines the mapping class group (or, what is more appropriately called in this setting, the bi-Lipschitz mapping class group) as the group of homotopy classes of bi-Lipschitz homeomorphisms. There are several other possibilities. 9 The fact that in the infinite-dimensional Teichmüller spaces the action of the mapping class group is not properly discontinuous was known since a long time (it is mentioned in Bers’ 1964 ETH lecture notes On moduli of Riemann surfaces).

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of points are bounded) and of divergent type (i.e., the intersection of the orbit with any bounded subset is finite). To such actions are associated limit points, limit sets, regions of discontinuity (the complements of limit sets) and regions of stability (the set of points where the Teichmüller modular group acts stably). Some of these concepts are imported into this setting from the dynamical theory of Kleinian groups. At the level of the surface itself, a useful notion of a surface of bounded geometry is introduced. Here, a surface equipped with a Riemannian metric is said to be of bounded geometry if the injectivity radius at each point is bounded from above and from below by uniform positive constants, except for neighborhoods of cusps. The region of stability of a group action is the set of points whose orbits are closed. In the present setting, this region is a dense open subset of Teichmüller space. This leads to a useful variation on the notion of moduli space which is proper to the case of surfaces of infinite type, namely the stable moduli space is defined as the metric completion of the quotient of the region of stability by the Teichmüller modular group. Matsuzaki also considers the asymptotic Teichmüller space. This is a parameter space for complex structures whose dilatation with respect to the base complex structure is arbitrarily small in the neighborhoods of the topological ends of the surface. The asymptotic Teichmüller space, like the Teichmüller space itself, is infinite-dimensional and non-separable. This space was introduced by Gardiner and Sullivan in the case where the underlying surface is the hyperbolic plane,10 and it was then studied by Earle, Gardiner and Lakic for more general surfaces. This space admits a complex structure, and its group of biholomorphic automorphisms is called the asymptotic Teichmüller modular group. The action of the asymptotic Teichmüller modular group on the asymptotic Teichmüller space was studied by Fujikawa. Teichmüller space can be regarded as a fiber space over the asymptotic Teichmüller space. The mapping class group acts on Teichmüller space preserving the fibers, and this action is studied separately on the asymptotic Teichmüller space and on the fibers. Notions of asymptotically elliptic modular transformations and of asymptotically elliptic subgroups of modular groups are introduced. Several particular subgroups of mapping class groups in this setting of surfaces of topological infinite type are highlighted. Examples include the following: • The stable mapping class group is the group of mapping classes that are supported on surfaces of finite type. This group acts naturally on the asymptotic Teichmüller space, and, in the case where the surface is of bounded geometry, the action is discontinuous. We note that the stable mapping class group had already appeared in other contexts; for instance, Tillman worked on the homotopy of the stable mapping class group, and Weiss studied the cohomology of that group. • The pure mapping class group is the group of mapping classes that fix all surface ends except the cuspidal ones. It is a closed normal subgroup of the mapping class group which contains the stable mapping class group. 10 Here,

one considers again the non-reduced Teichmüller theory of surfaces with nonempty ends.

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• A stationary group of the mapping class group is a subgroup G for which there exists a compact sub-surface V of S such that g.V / \ V 6D ; for every homeomorphism g representing an element of G. • The asymptotically trivial mapping class group is the kernel of the homomorphism from the mapping class group to the group of biholomorphic isometric automorphisms of the asymptotic Teichmüller space. The asymptotically trivial mapping class group contains the stable mapping class group. • Countable subgroups of the mapping class group have special dynamical properties. For instance, any countable and closed subgroup of the mapping class has a nonempty region of discontinuity. Matsuzaki points out in Chapter 15 a relation between quasiconformal mapping classes that act trivially on the asymptotic Teichmüller space and the asymptotic Nielsen realization problem, which concerns the asymptotic Teichmüller modular group. Another theme that is developed in this chapter is the fact that in the setting of surfaces of infinite type there are interesting intermediate moduli spaces, obtained by quotienting Teichmüller spaces by appropriate subgroups of the Teichmüller modular group. One reason for studying these intermediate spaces is that, as we already pointed out, the mapping class group does not act properly discontinuously on Teichmüller space, and consequently the quotient of this action does not inherit geometric structures in the usual sense from those of Teichmüller space. But some interesting subgroups of mapping class groups act properly discontinuously on Teichmüller space, and the quotients by these actions are examples of intermediate spaces. They lie between Teichmüller spaces and moduli spaces. The list of interesting intermediate moduli spaces that are considered in Chapter 15 includes the following: • The stable moduli space, which is a metric completion of the quotient of the region of stability by the Teichmüller modular group. • The enlarged moduli space, which is the quotient of Teichmüller space by the stable modular group. • The moduli space of stable points, which is the quotient of the region of stability by the Teichmüller modular group. • The moduli space of discontinuous points, that is, the quotient of the region of discontinuity by the Teichmüller modular group. • The geometric moduli space, that is, the quotient of Teichmüller space by the equivalence relation of closure equivalence, where two points are considered equivalent if one of them is contained in the closure of the orbit of the second one. The geometric moduli space is a quotient of the topological moduli space, and it is equipped with a complete metric. • The quotient of Teichmüller space by the subgroup consisting of mapping classes that act trivially on the asymptotic Teichmüller space.

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All these spaces as well as the relations between them are reviewed in Chapter 15.

3.2 Teichmüller theory and complex dynamics A rational map is a map of the Riemann sphere S 2 D C [ f1g which is of the form f .z/ D p.z/ where p and q are polynomial mappings. The theory of iteration of q.z/ rational maps was developed in the 1920s by Pierre Fatou (1878–1929) and Gaston Julia (1893–1978). In this theory, a rational map f is considered as a discrete dynamical system acting on S 2 . If f W S 2 ! S 2 is a rational map, then one is interested in the behavior of the orbits of points in S 2 under iteration by f . The sphere, under this action, is decomposed into two subsets: a closed subset called the Julia set, on which the dynamics of the iterates of a point is “chaotic”, and its complement, called the Fatou domain, on which the dynamics is “predictable”. More precisely, the Fatou domain of a rational map f is the set of points z on the Riemann sphere such that the family of iterates ff n g is a normal family in a neighborhood of z. (We recall that a family of analytic maps defined on an open subset U is said to be normal if every sequence has a convergent subsequence; convergence is in the sense of uniform convergence on compact sets of U ). The Fatou domain is also the largest subset of the Riemann sphere on which the family of iterates of f is a normal family. That the behavior of the iterates of f on the Julia set is “chaotic” is expressed by the fact that for any open set U meeting this set, the closure of the union of the forward orbits of U , S that is, the set n0 f n .U /, is equal to the whole Riemann sphere. There are several other features of chaotic behavior of points in the Julia set. For instance, the Julia set is the closure of the repelling periodic S points of f , that is, the periodic points whose multiplier is > 1. The backward orbit n0 f n .z/ of every point z in the Julia set is dense in this set. More generally, the backward orbit of any point of the Riemann sphere accumulates on the Julia set. Fatou and Julia, among other things, worked out a classification of the eventually periodic components of the Fatou domains. In the 1980s, the theory of iteration of rational maps became again fashionable, due in large part to the work of Sullivan, Douady, Hubbard and their collaborators. Sullivan proved in 1985 a conjecture stating that there are no “wandering domains” for rational maps, that is, every component of the Fatou domain of a rational map is eventually periodic and there are only finitely many periodic components. This question had been left open by Fatou and Julia in the 1920s. Perhaps more importantly, in the same paper, Sullivan developed a correspondence between the theory of iteration of rational maps and the theory of Kleinian groups, that is, discrete subgroups of PSL.2; C/ acting by conformal automorphisms on the Riemann sphere S 2 and by isometries on hyperbolic three-space H3 , S 2 being regarded as the boundary at infinity of H3 . He stated his results in the form of a dictionary translating notions, techniques, and results from one theory into the other one. The analogy between the two theories was not obvious from the beginning since the actions are a priori of different kinds: on the one hand, the dynamics is obtained by iterating a single rational map, and on the other hand it is

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induced on the sphere by the action of a group acting on hyperbolic three-space. The analogy immediately turned out to be most fruitful; it inspired several geometers, and it gave a huge impulse to research in both fields. Let us mention a few entries of the dictionary established by Sullivan. We recall: (1) Fatou domains of rational maps correspond to domains of discontinuity of Kleinian groups. (We recall that the domain of discontinuity of a Kleinian group  is the set of points z 2 S 2 which have a neighborhood U.z/ such that the set f 2 j.U / \ U 6D ;g is finite.) (2) Julia sets correspond to limit sets, that is, complements of domains of discontinuity (both sets being loci of chaotic behavior of the given dynamical system). (3) The Mandelbrot set, a parameter space for (degree two) polynomials, corresponds to Teichmüller space, and more precisely to a Bers slice representation of Teichmüller space in quasi-Fuchsian space. (4) Sullivan’s “no wandering domain” theorem for Fatou domains which we already mentioned corresponds to Ahlfors’ finiteness theorem (1964) saying that the quotient of the domain of discontinuity of a finitely-generated torsion-free Kleinian group by the group action has a finite number of components, and that each such component is a compact Riemann surface with a number of points removed. The measurable Riemann Mapping Theorem (1960), developed by Ahlfors and Bers as a fundamental tool in Teichmüller theory, was the essential ingredient used by Sullivan in the solution of the problem of wandering domains. There are other entries in the dictionary, and some of the most important ones have been formulated by McMullen. We shall further comment on this below. The proof of Ahlfors’ measure zero conjecture (stated in 1966) saying that the limit set of a finitely-generated Kleinian zero either is the whole Riemann sphere or it has measure zero was recently completed by Agol, Calegari and Gabai. This conjecture was put in parallel with a conjecture stating that the Julia set of a rational map either has full measure or has nonempty interior. The latter was recently shown to be false by Buff and Chéritat. Thus, in that particular case, the dictionary is not completely faithful. But it is probable that the formulation of the conjecture on the measure of Julia sets was motivated by Ahlfors’ measure zero conjecture, and this is also an instance of how the two fields influenced each other. Some of the open problems that were raised by Sullivan concern the local structure of limit sets (respectively Julia sets), for instance, local connectivity and fractal behavior. Other more general problems were expected to have several possible answers and to remain open for years. One such problem asks for associating a 3-manifold to a rational map of the sphere, in the same way a 3-manifold is associated to a Kleinian group (the quotient of the action of that group on hyperbolic three-space, seen as the interior of the Riemann sphere). McMullen proposed in 1994 a list of open problems on the dynamics of rational maps that are inspired from the theory of Kleinian groups, in a paper entitled Rational

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maps and Teichmüller space: Analogies and open problems.11 One of these problems asks for the description of a boundary structure for the space of polynomials of degree n with an attracting fixed point with all critical points in its immediate basin, in analogy with the construction of boundaries for Teichmüller space. Another problem concerns the analogy between the mating operation of proper holomorphic maps from the unit disk to itself and the mating of quasi-Fuchsian groups provided by Bers’ simultaneous uniformization theorem. At the 1990 ICM in Kyoto, McMullen gave a communication titled Rational maps and Kleinian groups. The whole subject of that communication was to highlight three items in the dictionary between iteration of rational maps and Kleinian groups. These items concern particularly the (then developing) theory of hyperbolization of 3-manifolds. More precisely, they ask the following: (1) Make a parallel between the combinatorics of critically finite rational maps and the geometrization of Haken 3-manifolds via iteration on Teichmüller theory. (2) Make a parallel between renormalization of quadratic polynomials and 3-manifolds which fiber over the circle. (3) Make parallels between the notions of boundaries and laminations in both theories, in particular between Teichmüller space in the Bers embedding and the Mandelbrot set. Several developments of the theory of iteration of rational maps took place during the last three decades. The aim of some of these works was to find new invariants of post-critically finite rational maps and to define new conformal dynamical systems by combining known ones. In particular, these works led to the notions of decomposition and of mating, and to other surgery techniques in complex dynamics. A cornerstone of the dictionary between the two theories is Thurston’s theorem on the topological characterization of (conjugacy classes of) post-critically finite rational maps among orientation-preserving branched covering maps of the Riemann sphere. The theorem and its proof made a profound link between iteration of rational maps and Teichmüller theory. Chapter 16 of this volume, by Xavier Buff, Guizhen Cui and Tan Lei concerns Thurston’s theorem. To state this theorem, we recall a few points of vocabulary. A critical point of an orientation-preserving branched covering F W S 2 ! S 2 is a point at which F is not locally injective; the critical set of F is the set of critical points of F ; the post-critical set of F is the closure of the set of forward images of the critical set. The post-critical set is also the smallest forward-invariant closed set containing the critical values of f . Thurston’s theorem concerns equivalence classes of orientation-preserving branched coverings F W S 2 ! S 2 whose post-critical set is finite. The theorem gives a characterization of post-critically finite branched coverings F W S 2 ! S 2 that are equivalent to rational maps, and it tells us to what extent such a rational map is unique 11 In: V. P. Havin and N. K. Nikolskii, editors, Linear and Complex Analysis Problem Book, volume 1574 of Lecture Notes in Math., p. 430-433. Springer, 1994.

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when it exists. The theorem is expressed in terms of a combinatorial obstruction for the existence of a rational map representing a post-critically finite branched covering. The obstruction consists in the existence of an invariant system of essential and pairwise non-homotopic curves (the system is called a “multicurve”) to which is associated a certain incidence matrix, with an eigenvalue  1 equal to its spectral radius. The theory uses the familiar Perron–Frobenius Theorem from linear algebra. As such, the obstruction has the flavor of a well-known obstruction for a mapping class to be of pseudo-Anosov type. In other words, Thurston’s result says that the post-critically finite map F is equivalent to a rational map if and only if every nonnegative eigenvalue of some induced action of this map on a vector space whose basis is the elements of the multi-curve is < 1. In the context of mapping classes, one has a much similar characterization of pseudo-Anosov maps: non-existence of stable curves for any iterate of the map, with several criteria formulated in terms of the action of an incidence matrix on some combinatorial data (train tracks) and the use of the Perron–Frobenius Theorem. McMullen established a relation between Thurston’s characterization and the geometrization of 3-manifolds. Although the relation with Teichmüller theory is not obvious from the statement, the proof of Thurston’s theorem uses an iteration on Teichmüller space, and the existence result provided by the theorem is obtained as a fixed point of an action on that space. Thurston obtained this theorem in 1982; he lectured on it on several occasions, and he circulated notes on the proof. A detailed proof was published in 1983 by Douady and Hubbard. After giving a proof of Thurston’s theorem, the authors of Chapter 16 present several recent applications of this theorem. The theory of iteration of rational maps combines hyperbolic geometry and complex analysis, as the classical Teichmüller theory does. The tools that are used involve quasiconformal maps, the Teichmüller metric and quadratic differentials. Associated to a rational map f of the Riemann sphere, there is a moduli space, a group QC.f / consisting of the quasiconformal automorphisms of the sphere that commute with f , a modular group Mod.f /, which is the quotient of QC.f / by the quasiconformal automorphisms that are isotopic to the identity, and there is a Teichmüller space T .f / of f , which is a space of rational maps equipped with a marking which is a quasiconformal conjugacy with f . The theory of Teichmüller spaces associated to holomorphic dynamical systems was developed by McMullen and Sullivan. A new ingredient in the theory, compared to the classical theory of surface homeomorphism actions on Teichmüller spaces, is the use of the action of the branched cover on the set of homotopy classes of essential multi-curves. The tools that are presented in Chapter 16 include actions on cotangent spaces to Teichmüller spaces (identified with spaces of integrable meromorphic quadratic differentials), estimates of norms of maps between cotangent spaces and between tangent spaces, and a search for contraction properties of such maps, which are in fact major tools in analytical Teichmüller theory.

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We should mention again that in Chapter 14 of this volume, the author highlights several analogies between the theory of iteration of rational maps of the sphere and the theory of iteration of quasiconformal maps in Rn

4 Part D. The quantum theory, 2 Chapter 17 of this volume concerns the quantization theory of Teichmüller space, and it is written by Ren Guo. This chapter can be considered as a sequel to the four chapters on quantization, written by Chekhov–Penner, Fock–Goncharov, Teschner and Kashaev that are contained in Volume I of this Handbook. We recall that the quantization theory of the Teichmüller space of a punctured surface is a theory of deformations of the C  -algebra of functions on that space. It was first developed independently by Chekhov–Fock and by Kashaev. The theory was motivated by theoretical physics, namely, by the physical interpretation of 2 C 1dimensional gravity as a Chern–Simons quantum field theory with noncompact gauge group. From the analytic point of view, both works of Chekhov–Fock and of Kashaev make use of self-adjoint operators on Hilbert spaces and of the quantum dilogarithm function. While the geometric setting of the work of Chekhov and Fock uses Thurston’s shear coordinates for Teichmüller spaces, the setting of Kashaev employs the notion of decorated ideal triangulation, that is, an ideal triangulation in which the set of triangles is equipped with a total order and where in each ideal triangle there is a mark at one of its corners. Kashaev’s theory also uses the lambda-length parameters that were introduced by Penner. More recently, a purely algebraic version of the Chekhov–Fock algebra was worked out by X. Liu, and a similar description was done for the Kashaev algebra in joint work by R. Guo and X. Liu. In this work, Guo and Liu established a natural link between the Chekhov–Fock algebra and an appropriate generalization of the Kashaev algebra by examining the relationship between the two kinds of coordinates for Teichmüller space that we mentioned: the shear coordinates and the Kashaev coordinates on decorated ideal triangles. In Chapter 17, Guo makes a survey of the algebraic aspect of this circle of ideas, and in particular on the relationship between the Chekhov–Fock algebra and the generalized Kashaev algebra. He also presents some recent work by Bonahon and Liu on the representation theory of the Chekhov–Fock algebra.

5 Part E. Sources Chapter 18 consists of a translation, by Annette A’Campo-Neuen, of Teichmüller’s paper Veränderliche Riemannsche Flächen (Variable Riemann Surfaces), published in 1944. This paper is the last one that Teichmüller wrote on the problem of mod-

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uli. He presents in it a construction of Teichmüller space which is completely new compared with the construction he introduces in his first seminal paper on the subject, Extremale quasikonforme Abbildungen und quadratische Differentiale (1939) and its sequel Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen (1943). In these two papers, Teichmüller laid down the foundations of the metric theory of Teichmüller space, showing that this space is homeomorphic to a ball of a certain dimension and proving in particular the existence and uniqueness of the extremal map in each homotopy class of homeomorphisms that was later on given the name Teichmüller extremal map. In the present paper, Teichmüller equips the space with a complex analytic structure, and he characterizes it by a certain universal property. At the same time, he introduces a fibre bundle which today is called the Teichmüller universal curve. The paper is rather sketchy, and it contains the following results: (1) The existence and uniqueness of the universal Teichmüller curve and the idea of a fine moduli space, that is, a fiber space where the isomorphism type of the fibre determines the point below it. (2) The proof of the fact that the automorphisms group of the universal Teichmüller curve is the extended mapping class group. (3) The idea that Teichmüller space represents a functor. (We are using a language that Grothendieck introduced a few years later.) (4) The idea of using the period map to define a complex structure on Teichmüller space. The paper is rather unknown in the mathematical community and it was read only by very few specialists. This is the reason why we decided to include it here in English translation, together with a commentary (Chapter 18 of this volume) written byAnnette A’Campo-Neuen, Norbert A’Campo, Lizhen Ji and the author of this introduction.

Part A

The metric and the analytic theory, 4

Chapter 1

Local and global aspects of Weil–Petersson geometry Sumio Yamada

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical and universal Teichmüller spaces . . . . . . . . . . . . . . 2.1 Classical Teichmüller spaces for closed surfaces . . . . . . . . 2.2 The universal Teichmüller space . . . . . . . . . . . . . . . . . Riemannian structures of L2 -pairing . . . . . . . . . . . . . . . . . . 3.1 L2 -pairing and its Levi-Civita connection . . . . . . . . . . . . 3.2 Tangential conditions and the Weil–Petersson metric . . . . . . 3.3 Weil–Petersson metric and Weil–Petersson cometric . . . . . . 3.4 L2 -decomposition theorem of Hodge-type . . . . . . . . . . . 3.5 L2 -decomposition theorem for the universal Teichmüller space 3.6 Weil–Petersson geodesic equation . . . . . . . . . . . . . . . . Harmonic map parameterizations . . . . . . . . . . . . . . . . . . . 4.1 General setting for harmonic maps . . . . . . . . . . . . . . . . 4.2 Harmonic maps between surfaces . . . . . . . . . . . . . . . . 4.3 The Teichmüller of the torus and its Weil–Petersson metric . . . 4.4 Teichmüller space of higher genus surface . . . . . . . . . . . . 4.5 Applications of Weil–Petersson convexity . . . . . . . . . . . . N 4.6 @-energy functional on the universal Teichmüller space . . . . . Metric completion and CAT(0) geometry . . . . . . . . . . . . . . . 5.1 Metric completion of Teichmüller space . . . . . . . . . . . . . 5.2 The Weil–Petersson metric tensor near the strata . . . . . . . . 5.3 The Weil–Petersson isometric action of the mapping class group and equivariant harmonic maps . . . . . . . . . . . . . . . . . Geodesic completion and CAT(0) geometry . . . . . . . . . . . . . . 6.1 Construction of the geodesic completion . . . . . . . . . . . . 6.2 Finite rank properties of Tx . . . . . . . . . . . . . . . . . . . . 6.3 Weil–Petersson geodesic completeness . . . . . . . . . . . . . 6.4 Weil–Petersson isometric action and symmetry of D.Tx ; / . . . 6.5 Embeddings of the Coxeter complex into UT . . . . . . . . . Teichmüller space as a Weil–Petersson convex body . . . . . . . . . 7.1 Tx as a convex subset in D.Tx ; / . . . . . . . . . . . . . . . . .

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7.2 7.3 7.4

Euclidean convex geometry and Funk metric . . . . . Weil–Petersson Funk metric . . . . . . . . . . . . . . The Teichmüller metric, Thurston’s asymmetric metric and the Weil–Petersson Funk metric . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . 103 . . . . . . . . 104 . . . . . . . . 105 . . . . . . . . 107

1 Introduction The study of moduli spaces of Riemann surfaces holds a central position in the history of modern mathematics, as the subject is a crossroad among several areas such as complex function theory, algebraic geometry, topology, number theory, partial differential equations and differential geometry. In those areas, the original motivations for the investigations differ from each other, which led to a seemingly distinct set of subfields all studying the moduli space of Riemann surfaces in essence. In this context, it seems that the field of differential geometry holds a curious position, as the very origin of surface theory was marked by Gauss in his study of embedded surfaces in R3 , leading to the Gauss curvature and the Theorema Egregium, which had become the starting point of modern differential geometry. Soon afterwards, however, the theory of algebraic curves initiated by Riemann and that of Fuchsian groups took over the classical differential geometry in the study of surfaces, which in a hindsight is a natural development considering the effectiveness of algebraic equations and the SL.2; R/ representation theory associated with the uniformaization theory of Poincaré and Koebe. In 1939, when Teichmüller investigated Teichmüller maps in order to relate a pair of conformal structures, he was most likely aware of the lack of differential geometric approaches in the theory of moduli space. Ahlfors followed Teichmüller in rewriting the theory of Fuchsian groups by regarding it as a deformation theory of conformal structures, using the theory of Beltrami differentials. André Weil was also instrumental in recognizing the geometric importance of Teichmüller’s work. Bers and Ahlfors pushed hard over the 1950s and 60s to make Teichmüller’s theory complete. At the same time, the theory of algebraic curves was developed independently by a set of algebraic geometers including Grothendieck, Serre, and Mumford. In the meantime, it is fair to say that among the differential geometers of the second half of the 20th century, the moduli space of Riemann surfaces remained an esoteric topic compared to their scientific interests. This chapter, especially the first half of it, is written with a second year graduate student in mind, who has taken a year long course in Riemannian geometry, but not necessarily well-versed in complex analysis, algebraic geometry, or Teichmüller theory. The author has ventured to write down the preceding paragraphs on the very abridged and very incomprehensive history of the subject, simply to make a point that much of so-called Teichmüller theory can be understood from a purely

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Riemannian geometric viewpoint, despite the non-differential-geometric development of the subject over time. We owe this approach to the following list of contributions: 1) Morrey’s proof of the existence of solutions to the Beltrami equation [53], 2) Earle– Eells’s attempt [18] to recognize Teichmüller space as a submanifold of the space of smooth metrics, 3) the theory of harmonic maps into non-positively curved manifolds by Eells–Sampson [21] allowing to track varying metrics by smooth harmonic maps, 4) the body of work by Thurston who took the hyperbolic geometry of surfaces to its full power to reinterpret the Ahlfors–Bers theory, and 5) Fischer–Tromba [23], [24], [70] who in the 70s and 80s rewrote Teichmüller theory, in particular the Weil–Petersson geometry from the deformation theory of hyperbolic metrics, partly initiated by the theory of traceless-transverse (TT) tensors by Fischer and Marsden [22] originally developed with applications to general relativity in mind. In the post-Ahlfors Weil–Petersson geometry, Wolpert’s contributions stand out for his singular pursuit wanting to understand the geometry of Teichmüller spaces. The chapter [79] written by Wolpert that has appeared in this Handbook series covers most of the ground in the development of Weil–Petersson geometry in the last few decades, particularly the last one, and the present article is meant to complement Wolpert’s. One should also consult the book of Wolpert [80] which offers a more comprehensive exposition. The reader will notice that many of the results stated in his chapter and the present chapter overlap, and the author uses several statements of Wolpert’s at crucial steps in developing the theory. However, also apparent should be that the languages used in describing the geometry and the techniques employed in the proofs of the theorems are distinct from each other in the two chapters. We now state several objectives of this exposition. The first is to write down the basics of the Riemannian approach to the Weil–Petersson geometry, complementing Wolpert’s expositions on the subject where much of the argument is made with Beltrami coefficients. This amounts to the local aspect of Weil–Petersson geometry of the title of the chapter. In doing so, we add some new material to the existing literature such as [23], [24], [32], [70]. Much of the exposition in this part of the chapter can be applied to settings of higher-dimensional moduli spaces, such as that of Calabi–Yau manifolds (for example [12].) Therefore insisting on the Riemannian geometric approach is meaningful in the sense that the theories of Beltrami equations and complex analysis are specific to dimension two, and most of it is not transferable to higher-dimensional situations. Some of the results in this exposition have been taken from the article [81] which appeared in the Journal of Differential Geometry in 1999. Unfortunately while in press numerous typographical mistakes were introduced in that article. All the results and proofs in the paper are valid, but over the years this situation has posed unnecessary challenges to the interested readers. We have tried to rectify the situation, and present here a comprehensive version of the content of the paper [81], including a proof of the Weil–Petersson convexity of energy of harmonic maps. Incidentally there has been some dispute (see for example [15]) whether the statement was first proved by Tromba [71] in 1996, where the domain of the harmonic maps is the surface itself. We point out in Section 3.6 of the present chapter, however reluctantly, a difference

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between the two proofs explicitly, in particular the difference in the Weil–Petersson geodesic equation, to let the matter rest. In utilizing harmonic maps for Teichmüller theory we have included a section on the Weil–Petersson geometry of the Teichmüller space of the torus, as this has not been explicitly written down in the literature although it is well known to the experts. Each conformal structure on the torus is uniformized by a flat metric, which can be identified with a harmonic map from a fixed reference torus. Then we show that the resulting Weil–Petersson geometry is isometric to the homogeneous space SL.2; R/=SO.2/ with its left invariant metric, or equivalently the Poincaré disc. As the harmonic map is an affine map, it is the Teichmüller map as well (see for example [6] for Teichmüller geometry). The second goal of this exposition is to present a series of developments the Weil– Petersson geometry has gone through over the last decade, after a paper by the author [82] (later as [83]) appeared where the CAT(0) geometry was introduced for augmented Teichmüller space for the first time. This part of the chapter corresponds to the global aspect in the title. We take steps in discussing 1) Teichmüller space, which is incomplete Weil–Petersson metrically and Weil–Petersson geodesically, 2) the Weil– Petersson metric completion of Teichmüller space, which is identified with the augmented Teichmüller space, and 3) the Weil–Petersson geodesic completion of Teichmüller space, which is realized as a Coxeter complex where the set which works as the simplex is the Weil–Petersson metric completion. The successive enlargement of the original Teichmüller space as traced through the author’s work [81], [83], [85], [87] is motivated from the point of view that there should be a genus-varying Teichmüller theory, a direction already actively pursued in the Teichmüller–Grothendieck theory from the algebro-geometric approach. With this in mind, we demonstrate that the Coxeter complex can be embedded in the universal Teichmüller space. The last objective of this chapter is to write down the Weil–Petersson geometry relevant to the universal Teichmüller space. The orthodox deformation theory in the universal Teichmüller space is written in terms of Beltrami differentials (see for example O. Lehto’s book [42]). The Nag–Verjovsky paper [56] has shed much light in clarifying how the Weil–Petersson metric can be considered as the L2 -pairing of the linearized Beltrami differentials induced by a smooth vector field on the unit circle. Takhtajan–Teo [67] then took over the idea and further developed it so that they succeeded in generalizing a collection of results by Wolpert [75] concerning the second derivatives of the Weil–Petersson metric tensor, in particular the Weil–Petersson curvature, by regarding the universal Teichmüller space as a Hilbert manifold. We look at the Nag–Vejovsky paper [56] again, and describe the tangent space of the universal Teichmüller space at the identity as the set of traceless transverse tensors constituting a component in an L2 -decomposition theorem of Hodge type. Additionally, we N demonstrate the Weil–Petersson metric tensor as a Hessian of the @-energy of harmonic maps. Those results have their counterparts in the compact surface cases, but were never investigated in the universal context before [81]. The relation between a negatively curved complete manifold and its geometric boundary has been an active area of

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investigation in the last decade, partly due to the excitement from so-called AdS-CFT correspondence, which in turn has triggered much incentive to study the conformally compact/Einstein–Poincaré manifolds. As the Poincaré disc is the simplest example of conformally compact manifold as well as Einstein–Poincaré manifold, we believe that the universal Teichmüller space offers a prototype of the moduli space of such manifolds. In doing so, it will become necessary to formulate the theory without the use of one-variable complex analysis, and the deformation theory of Riemannian metrics as explored in this chapter will be a basic model for higher-dimensional analogues. The author thanksAthanase Papadopoulos for his encouragement to write down this exposition. This work was partially supported by Grant-in-aid for Scientific Research 20540201, 23654061.

2 Classical and universal Teichmüller spaces 2.1 Classical Teichmüller spaces for closed surfaces Let † be a compact surface without boundary of genus g  1 (when g D 0 the situation is very simple.) By the existence theorem of an isothermal coordinate system by Korn and Lichtenstein, any Riemannian metric g can be identified with a Riemann surface, namely a Riemannian surface is a Riemann surface. The universal covering space of the surface is either the whole plane or the upper half space, and thus the surface can be uniquely equipped with a Euclidean metric when g D 1 or a hyperbolic metric when g > 1. This statement is the so-called Uniformization Theorem. Hence we can think of the space MK .K  0; 1/ of constant curvature metrics as a subset of the space of smooth metrics M on †, the latter being fibered by the elements of MK so that each fiber consists of the metrics conformal to a constant curvature/uniformized metric G 2 M. The Teichmüller space is then defined as the quotient space Tg D MK =Diff 0 †; where the equivalence relation is given as G1  G2 () G2 D '  G1 for some ' in Diff 0 †. Here Diff 0 † is the identity component of the orientationpreserving diffeomorphism group Diff†. Recall that the map ' W .†; '  G2 / ! .†; G2 / is an isometry. Note that in defining the identity element of Diff 0 † one requires a reference Riemann surface .†0 ; G0 / such that it acts as the domain of Id W †0 ! †. Namely .†0 ; G0 / gives homotopy markings on the target surface. By an important theorem of Earle–Eells [18], it is known that the identity component Diff 0 †  Diff† consists of diffeomorphisms homotopic to the identity map.

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The moduli space Mg is defined as Mg D MK =Diff†; where the equivalence relation is given as G1  G2 () G2 D '  G1 for some ' in Diff†. Thus the Teichmüller space projects down to the moduli space with the fibers identified with the discrete infinite group Diff†=Diff 0 †, called mapping class group, or Teichmüller modular group. We denote this group by Map.†/. We define now for a later use the full diffeomorphism group Diff† which, in addition to the elements of Diff†, also contains the orientation-reversing diffeomorphisms of †. Then the quotient group Diff†=Diff 0 † is called the extended mapping class group Map.†/.

2

b

b

2.2 The universal Teichmüller space The Uniformization Theorem by Poincaré and Koebe (see [42]) says that given a closed surface, all the smooth metrics on it can be uniquely uniformized by a constant curvature metric. When the surface is of genus greater than one, the constant curvature metrics are hyperbolic metrics, and the hyperbolic surface can then be written as H2 =  for some Fuchsian group   SL.2; R/. Namely, given a Riemannian surface .†; G/, the metric G is conformal to a hyperbolic metric G0 on †. The space of Fuchsian groups appearing in the Uniformization Theorem can be regarded as QC.0 /=SL.2; R/ D fw 2 QC.D/ W w0 w 1 is an element of SL.2; R/g for some fixed reference Fuchsian group 0 , where the set is identified with the set of all pulled-back metrics on the unit disc D WD fz 2 C W jzj < 1g, obtained by pulling back the Poincaré disc metric 4 G0 D jdzj2 .1  jzj2 /2 by 0 -equivariant quasi-conformal self-maps of D up to an equivalence via the Möbius transforms of the disc. Note that the original hyperbolic surface .†; G/ in this context appears as IdD 2 QC.D/. Note that this set is the set of all the hyperbolic metrics on the surface †: M1 D QC.0 /=SL.2; R/: On the other hand, recall that two metrics are deemed geometrically equivalent if one can be obtained from the other by a diffeomorphism on the manifold. Here the manifold is a hyperbolic surface realized as a quotient manifold of the open disc. As the hyperbolic metric G0 of the Poincaré disc model blows up near the geometric boundary S 1 D @D D fz W jzj D 1g, the diffeomorphisms of the surface regarded (by going to its universal covering space) as diffeomorphisms of the open disc are the

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elements of the set QC0 .0 / of quasi-conformal self-maps of the disc which extend to the identity map of the geometric boundary S 1 D @D. This can be understood from the picture that those diffeomorphisms can be lifted to the universal covering, which are periodic on the tessellation by a fundamental region of †, leaving the tessellation invariant, and that on the Poincaré disc the tessellation pattern gets increasingly dense as one approaches the geometric boundary @D. Hence the Teichmüller space in this context is regarded as T D M1 =Diff 0 †   D QC.0 /=SL.2; R/ =QC0 .0 / D QS.0 /=SL.2; R/; where QS.0 / WD QC.0 /=QC0 .0 / is the set of 0 -equivariant quasi-symmetric self-maps of S 1 D @D, a subset of the space of quasi-symmetric self-maps of the circle QS.S 1 /. The quotient space is defined under the following equivalence relation: two elements 1 and 2 in QC.0 /=SL.2; R/ are equivalent if they are related by 2 D 1 B for some in QC0 .0 /. Note that we have used the fact that SL.2; R/ \ QC0 .0 / D fIdg. Here we think of points in T as left cosets of the form SL.2; R/Bw D Œw where each w is a quasi-symmetric homeomorphism of the circle. We replace the closed surface † above by the hyperbolic plane H2 . This can be regarded as replacing the Fuchsian group   SL.2; R/ by the trivial group 0 D Id 2 SL.2; R/. Namely in the above construction, such a replacement results in a new space   UT D QC.fIdD g/=SL.2; R/ =QC0 .fIdD g/   D QC.D/=SL.2; R/ =QC0 .D/ D QS.fIdD g/=SL.2; R/ D QS.S 1 /=SL.2; R/: Note that we have used the fact that SL.2; R/ \ QC0 .D/ D fIdD g. As the resulting space contains all Teichmüller spaces of surfaces of the form H2 = , we call the space UT the universal Teichmüller space. For a comprehensive treatment of the subject, including the definitions of quasi-conformal and quasi-symmetric maps, we refer the reader to Lehto’s book [42]. The advantage in introducing quasi-conformal maps of the disc and quasi-symmetric maps of the unit circle is that one can then describe the deformations of hyperbolic metrics by the pull-back action of quasi-conformal maps w W D ! D each of which is a solution to the Beltrami equation wzN D .z/wz for some Beltrami coefficient , a C-valued measurable function on D with jj < 1, an element of the unit ball L1 .D/1 in the complex Banach space L1 .D/. In other words, we can identify each element w of QC.D/ with  uniquely, provided the solution w W D ! D fixes three points on the boundary. The existence and uniqueness of the solution to the Beltrami equation is due to Morrey [53]. The reader

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is referred to the books by Nag [54] and Lehto [42] for an exposition on the subject. We denote the solution to the Beltrami equation above by w . In particular, define the set of equivariant Beltrami differentials by n o g 0 .z/ D .z/ a.e. on D for all g in 0 L1 .0 / D  2 L1 .D/ j .gz/ 0 g .z/ which is a closed subspace in L1 .D/. The unit ball L1 .0 / \ L1 1 is denoted by 1 L .0 /1 . The Teichmüller space for a Fuchsian group 0 , including the case of the trivial group fIdg, is identified with T .0 / D L1 .0 /1 =  where    if and only if w D w on @D D S 1 . The identification between the space of Beltrami coefficients and the Fuchsian group is given by the following: If  2 L1 .0 /1 then w conjugates 0 to another Fuchsian group 1 :  D w 0 w

Hence the equivalence class Œ represents the hyperbolic surface H2 =  . Under the light of this identification, as presented in a paper by Nag–Verjovsky [56], the moduli of hyperbolic surfaces are determined by the quasi-symmetric homeomorphisms of the unit circle; a model mentioned but often insufficiently explained in the literature of the string theory. We remark that this viewpoint was established by Beurling and Ahlfors in the 50s [9]. In the linear theory to be developed in the following section, we will return to the viewpoint of the Beltrami equation.

3 Riemannian structures of L2 -pairing 3.1 L2 -pairing and its Levi-Civita connection 3.1.1 L2 -pairing of deformation tensors. The tangent space TG M of the space M at a metric G is the space of smooth symmetric .0; 2/-tensors on †. This linear space has a natural L2 -pairing defined as follows: Z hh1 ; h2 iL2 .G/ D hh1 .x/; h2 .x/iG.x/ dG .x/; †

where the hi ’s are symmetric .0; 2/-tensors indicating the directions of deformation of G along the path G C "hi C o."/. The integrand can be rewritten, using a local

Chapter 1. Local and global aspects of Weil–Petersson geometry

coordinate chart, as hh1 .x/; h2 .x/iG.x/ D

X

51

G ij G kl .h1 /ik .h2 /j l

1i;j;k;l2

  D Tr .G 1  h1 /  .G 1  h2 / ; where A  B denotes matrix multiplication and Tr A is the trace of the matrix A. This quantity is well defined, meaning it is invariant under change of coordinate charts. In particular it can be simplified by choosing a geodesic normal coordinate system where Gij .p/ D ıij at its center p as X .h1 /jk .p/.h2 /jk .p/ .D Tr.h1  h2 //; hh1 .p/; h2 .p/iG.p/ D j;k

the trace of the product of 2  2 matrices. From now on, we will use the Einstein notation of indices, omitting the summation symbols. 3.1.2 Levi-Civita connection of the L2 -pairing. The space of smooth metrics M defined on a manifold N has always the L2 -pairing defined above. Formally one can regard the L2 -pairing as a Riemannian metric on M and write down its Levi-Civita connection. We fix a coordinate chart around a point p in †. Let h1 , h2 and h3 be locally constant symmetric .0; 2/-tensors defined over the chart. Then the brackets among the hi ’s vanish, namely Œhi ; hj  D 0; where the bracket here is the Lie derivative of the tensor hj in the direction of hi , defined on M. Note that in what follows, as all the quantities appearing below are tensorial, it suffices to consider point-wise calculations, namely we may restrict to the locally constant tensors. The formula, which appears in the existence and uniqueness theorem of the LeviCivita connection in any standard differential geometry textbook, is 1 h1 hh2 ; h3 i C h2 hh1 ; h2 i  h3 hh1 ; h2 i 2  C hŒh1 ; h2 ; h3 i  hŒh1 ; h3 ; h2 i  hŒh2 ; h3 ; h1 i  1 D h1 hh2 ; h3 i C h2 hh1 ; h2 i  h3 hh1 ; h2 i : 2 On the other hand, the pairing is a function of G, and the hi ’s are deformation tensors of G. We write down the derivatives hi hhj ; hk i as Z ˇ   d Tr G t1  hi  G t1  hj dG t .x/ ˇ t D0 hk hhi ; hj iL2 .G/ D dt Z †   D Tr G 1  .hk /  G 1  hi  G 1  hj dG.x/ C hDh1 h2 ; h3 i D

†

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Z C

  Tr G 1  hi  G 1  .hk /  G 1  hj dG.x/

Z†

1  Tr G 1  hi  G 1  hj .tr G hk /dG.x/ 2 † D hh1  G 1  hj ; hk iL2 .G/  hhj  G 1  hi ; hk iL2 .G/ 1 1 C h.tr G hk /hi ; hj iL2 .G/ C h.tr G hk /hj ; hi iL2 .G/ ; 4 4 C

where G t D G C t hk . When the hi ’s are trace-free, the commutativity Tr.A  B/ D Tr.B  A/ is applied to the above integrands, and hk hhi ; hj iL2 .G/ is invariant under permutations of i , j and k: hk hhi ; hj iL2 .G/ D h.k/ hh.i/ ; h.j / iL2 .G/ for any element  of the symmetric group S3 . By substituting the above expression into the formula for the connection D, we obtain the following relatively simple expression: Lemma 3.1 (Levi-Civita connection for L2 -pairing). For trace-free symmetric .0; 2/ tensors hi and hj , the Levi-Civita connection for the L2 -pairing at the tangent space TG ¥calM of the space M of smooth metrics is written as  1 h1  G 1  h2 C h2  G 1  h1 2  1 C .Tr G h1 /h2 C .Tr G h2 /h1  hh1 ; h2 iG.x/ G : 4

Dh1 h2 D 

Note that the expression for the connection is symmetric in 1 and 2. The author thanks AkiraYoshizato for pointing out an error in the lemma that appeared previously.

3.2 Tangential conditions and the Weil–Petersson metric When G is a uniformizing metric of its conformal class, then the tangent space TG M decomposes into the deformation of G preserving the constant curvature condition, and its complement. This can be formally stated as follows. In dimension two, the Riemann curvature tensor is completely determined by one scalar function, the sectional curvature K. Then the Ricci curvature tensor is of the form Rij D KGij ; namely G is an Einstein metric. The well-known variational formula (see [8]) of the Ricci tensor under a deformation G C "h at " D 0 gives, after taking its trace: G ij RP ij D 4G Tr G h C ıG ıG h:

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Hence we have the following variational formula for the sectional curvature under the deformation of G in the direction of h: KP D G ij RP ij C GP ij Rij D G ij RP ij  hij KGij D 4 Tr G h C ıG ıG h  K Tr G h: We denote the quantity .4G C K/ Tr G h C ıG ıG h by LG h, where the differential operator LG is sometimes called Lichnerowicz operator. Hence if the deformation tensor h is tangential to MK , then h satisfies the following linear equation, which is the curvature-preserving condition LG h D 0: Having characterized the tangential condition to MK , we additionally require the deformation tensor h to be L2 -perpendicular to the diffeomorphism group Diff 0 † action. Consider a one-parameter family of diffeomorphisms ' t W † ! † with '0 D d Idj† and let dt ' t j tD0 D X be a vector field on †. Recall that the Lie derivative LX G of the tensor G in the direction X is defined by d  ˇˇ ' G : dt t t D0 Take a chart which gives a geodesic normal coordinate centered at p. Then LX G D

LX G.p/ D XiIj C Xj I i as Gij D ıij and Gij Ik D 0 at p. The condition that a symmetric .0; 2/-tensor h is L2 -perpendicular to the diffeomorphism group Diff 0 † action is described as 0 D hh; LX GiL2 .G/ for all X 2 X.†/. The right-hand side can be rewritten, with respect to a geodesic normal coordinate, as Z hh; LX GiL2 .G/ D hh.x/; LX G.x/iG.x/ dG .x/ Z† hij .XiIj C Xj I i / dG .x/ D † Z D2 hij XiIj dG .x/ † Z hij Ij Xi dG .x/ D 2 †

D 2hıG h; XiL2 .G/ ; where integration by parts has been used. There is no boundary contribution as the surface † is closed. Therefore, for the tensor h to be L2 -perpendicular to the diffeomorphism group Diff 0 † action, h is required to be divergence-free; ıG h D 0. Note

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that ıG h is regarded here as a tensor of .1; 0/-type, that is, a vector field. In the normal coordinate system, the divergence-free condition is the same as .ıG h/i D hij Ij D 0. Now let h be a deformation tensor tangential to M1 at a hyperbolic metric G. Then h satisfies the Lichnerowicz equation LG h D 0: .4G C K/ Tr G h C ıG ıG h D 0: In addition, we require h to be perpendicular to the diffeomorphism action, which implies ıG h D 0, which in turn says that h satisfies .4G C K/ Tr G h D 0. When K D 0; 1 which are the cases we are interested in, the linear partial differential equation .4G C K/ Tr G h D 0 has only the trivial solution on the closed surface, forcing an additional condition Tr G h D 0. Therefore, we have so far characterized the conditions that a tangential vector to the Teichmüller space Tg D MK =Diff 0 † needs to satisfy; namely the trace-free condition Tr G h D 0 which is hi i D 0 in a normal coordinate system, and the divergence-free condition, also called the transverse condition ıG h D 0: The so-called TT-tensors (for trace-free transverse) appear in the study of minimal surfaces where they are the second fundamental forms of minimally embedded surfaces (see [57] for details), as well as in the study of the Einstein equation where the tensors are a part of Cauchy initial values for the evolution problem associated to the so-called Einstein constraint equations (see [22] for references). We can now define the Weil–Petersson metric on Teichmüller space. Definition 3.2 (Weil–Petersson metric [23]). The L2 -pairing of TG M restricted to the trace-free, divergence-free tensors is called the Weil–Petersson metric on the Teichmüller space T D MK =Diff 0 †. As a 2  2 matrix, the tangential tensor h 2 TG T can be expressed as   h11 h12 h12 h11 with respect to a geodesic normal coordinate system centered at a point p in †. The integrand of the Weil–Petersson pairing evaluated at P becomes 2.h211 C h212 /. Then the divergence-free condition is equivalent to the Cauchy–Riemann equation for .h11  ih12 /.z/ at the origin. We next look into this situation more closely.

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3.3 Weil–Petersson metric and Weil–Petersson cometric First from the discussion in modeling the Teichmüller space as a homogeneous space of QS./ for the Fuchsian group , without loss of generality, by using a Möbius transformation we may assume any given point p to be the origin O of the Poincaré disc. Let z D x C iy be the standard Euclidean coordinate system at the origin. Note that this coordinate system matches with the geodesic normal coordinate system at O.D p/, namely G D .z/.dx 2 C dy 2 / with .O/ D 1 and @ jO D 0, as the first derivatives of 4=.1  jzj2 /2 at z D 0 all vanish, which in turn makes all the Christoffel symbols vanish. Then the function .h11  ih12 /.z/, where these indices denote the isothermal coordinates x and y, is holomorphic in z at the origin.  T at a conformal We recall that the cotangent space of Teichmüller space TŒG structure ŒG has been identified with the space QD.†/ of holomorphic quadratic differentials on the Riemann surface .†; ŒG/. Thus the correspondence between the tangent vectors and the cotangent vectors is h11 dx˝dxCh12 dx˝dyCh12 dy˝dxC.h11 / dy˝dy

! .h11 ih12 /.z/dz 2 ;

the former with respect to a geodesic normal coordinate chart, and the latter with an isothermal coordinate chart. The Weil–Petersson cometric defined for the elements of QD.†/ has the form Z jdzj2 .z/ N .z/ 2 ; hh1 ; h2 iL2 .G/ D

.z/ † where h1 .z/ D .z/dz 2 and h2 .z/ D .z/dz 2 locally, and the hyperbolic metric G with respect to the isothermal coordinate z is given as 2 .z/jdzj2 . It is clear from the preceding argument that the two L2 -parings coincide, when restricted to the respective deformations of trace-free divergence-free tensors, and of holomorphic quadratic differentials.

3.4 L2 -decomposition theorem of Hodge-type We consider the L2 -decomposition of the tangent space TG M. After having characterized the tangent vectors to the Teichmüller space M1 =Diff 0 †, it seems unnecessary to further investigate the linear structure. However, the precise formulation of the L2 -decomposition becomes crucial in formulating the nonlinear structure, namely the curvature of the spaces. The following statement is an adaptation to dimension two of the theorem by Fischer–Marsden [22] concerning the decomposition of the deformation space of a constant scalar curvature metric in higher (> 2) dimensions. It should be remarked that in the 1980s, Fischer and Tromba [23], [24], [70] undertook the task of rewriting Teichmüller theory from a Riemannian geometric viewpoint. In particular, they laid out the decomposition theory of the deformation tensors in TG M1 . Below, we develop a theory where the decomposition of the bigger linear space TG M D TG M1 ˚ .TG M1 /? is addressed.

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We have already identified the adjoint operator of the divergence operator ıG with the Lie derivative of G up to a constant, hh; LX GiL2 .G/ D 2hıG h; XiL2 .G/ which in turn can be stated as 1  W X 7!  LX G ıG 2 for X 2 X.†/, the space of smooth vector fields on †. We can also write down the adjoint operator of the Lichnerowicz operator LG by noting the following: hLG f; hiL2 .G/ D hf; LG hiL2 .G/ Z   D f .x/ .4G  K/ Tr G h C ıG ıG h .x/ dG .x/ Z† D hf.4G  K/f gG C HessG f; hiG.x/ dG .x/: †

Hence

LG f D .4G f  Kf /G C HessG f:

For the following decomposition theorem [81], we restrict ourselves to the case K  1, i.e. when the surfaces are uniformized by hyperbolic metrics. Theorem 3.3. Suppose that G is a hyperbolic metric on † and that h is a smooth symmetric .0; 2/-tensor defined over †. Then there is a unique L2 -orthogonal decomposition of h as a tangent vector in TG M, h D PG .h/ C LX G C L f; where PG .h/ is the projection of h onto TG T , LX G is a Lie derivative and LG f is a tensor perpendicular to M1 . Here the vector field X solves the following equation 1  X D  ıG h ıG ı G 2 uniquely and is smooth, the function f solves the following equation LG LG f D LG h uniquely and is smooth. Consequently PG .h/ is uniquely determined to be a smooth tensor given by PG .h/ D h  LX G  LG : Each of the three terms belongs to each of the mutually L2 -orthogonal components TG M D TG T ˚L2 .G/ TG Diff 0 † ˚L2 .G/ .TG M1 /? : We remark that this decomposition can be called of Hodge type for it identifies the tangential directions to Teichmüller space with the intersection of the kernel of the

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differential operator ıG and the kernel of LG ; for both there are associated elliptic  and LG LG . operators ıG ıG  and LG LG are both ellipProof of Theorem 3.3. The differential operators ıG ıG tic, self-adjoint, and with trivial kernel (and hence trivial co-kernel). The trivial  follows from first noting that 0 D hıG ıG X; X iL2 .G/ D ity of the kernel of ıG ıG    hıG X; ıG XiL2 .G/ implies ıG X D 0 and then from the non-existence of Killing vector fields on † due to the negative curvature. The triviality of the kernel of LG LG follows as 0 D hLG LG f; f iL2 .G/ D hLG f; LG f iL2 .G/ implies LG f D 0. By taking the trace of the equation LG f D 0; we obtain 4G f C 2f D 0 which implies f  0. This shows, by the standard theory of linear equations of elliptic type [27], that one can solve each of the two equations uniquely to specify the vector field X D X.h/ and the function f D f .h/, given the data h. In showing the L2 -orthogonality, we need the following two lemmas, which trigger a series of orthogonal relations.

Lemma 3.4. For any vector field Y on †, we have LG LY G D 0. This follows from the simple observation that LY G is a deformation tensor induced by a one-parameter family of isometries  t G with P 0 D Y , in particular preserving the curvature constraint, hence it is an element of TG M1 , which is the kernel of the differential operator LG . Lemma 3.5. For any smooth function  on †, we have ıG LG f D 0. Proof. First choose a geodesic normal coordinate chart centered at p, fx i g such that G D ıij and Gij Ik D 0 for all i , j and k where “I” stands for the covariant derivative. Then ıG LG f D ıG f.4G f C f /G C HessG f g D f4G f C f gj ıij C fij Ij D f4G f C f gj ıij C fjj Ii C Rij fj D 0; where the Ricci identity is used to interchange the order of the covariant derivatives for the second equality, and Rij D ıij on the hyperbolic surface †. We remark that an immediate consequence of the second lemma is that tensors of type LY G and type LG  are mutually L2 -perpendicular for an arbitrary vector field Y and an arbitrary function , due to the equality hıG LG ; Y iL2 .G/ D hLG ; LY GiL2 .G/ : Hence we get the first orthogonality: hLX G; LG f iL2 .G/ D 0:

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By projecting h to TG T and to .TG M1 /? respectively, we have hPG .h/; LG f iL2 .G/ D hh  LX G  LG f; LG f iL2 .G/ D hLG h  LG LX G  LG LG f; f iL2 .G/ D hLG h  LG LG f; f iL2 .G/ D 0: Finally the orthogonality between PG .h/ and LX G can be checked by hPG .h/; LX GiL2 .G/ D hh  LX G  LG f; LX GiL2 .G/ D hıG h  ıG LX G  ıG LG f; X iL2 .G/  D hıG h C 2ıG ıG X; X iL2 .G/

D 0: We have used above the fact that f and X solve the elliptic system LG LG f D LG h;

1  ıG ıG X D  ıG h 2

uniquely.

3.5 L2 -decomposition theorem for the universal Teichmüller space 3.5.1 L2 -decomposition theorem. In an attempt to introduce a Riemannian structure on the universal Teichmüller space, in particular the Weil–Petersson metric, we look at a subspace of the tangent space at the identity in UT consisting of L2 -integrable tensors. That subspace is a Hilbert space, and the quadratic form is the Weil–Petersson pairing. We generalize the L2 -decomposition theorem in the previous section in this context as follows [81]. Theorem 3.6. Suppose that G0 is the standard hyperbolic metric on the unit disc D, namely H2 D .D; G0 / and h is an L2 .H2 /-integrable symmetric .0; 2/-tensor defined over H2 . Then there is a unique L2 -orthogonal decomposition of h as a tangent vector belonging to TG0 UT as follows: h D PG0 .h/ C LX G0 C LG0 f; where LX G0 is a Lie derivative, X a vector field of finite L2 .H2 /-norm satisfying 1  X D  ıG0 h; ıG0 ıG 0 2  where LG0 f is a symmetric .0; 2/-tensor with the function f satisfying the equation LG0 LG0 f D LG0 h; and where PG0 .h/ is the projection of h onto the universal Teichmüller space specified as PG0 .h/ D h  LX G0  LG0 f .

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Proof. By a density argument, we can approximate h by a sequence fhi g of compactly supported smooth symmetric .0; 2/-tensors on H2 , such that limi!1 kh  hi k D 0. Hence we first consider the case where the tensor h is compactly supported. We treat the general case at the end of the proof. We first need to replace all the integration by parts argument in the L2 -decomposition theorem for the closed surface case by a strictly coercive property of the two  and LG0 LG0 . First we establish the statement elliptic differential operators ıG0 ıG 0 for the compactly supported cases.  2 and LG0 LG0 defined on X1 Lemma 3.7. The differential operators ıG0 ıG 0 .H / \ 0 H 1 .H2 / and C01 .H2 / \ H 2 .H2 /, respectively, satisfy the inequalities  2 hıG0 ıG X; X iL2  C kXkH 1; 0 2 hLG0 LG0 f; f iL2  C 0 kf kH 2

for some constants C; C 0 > 0. The proof of the lemma follows from integrations by parts inside the integral of the L2 -pairing, which are allowed since the functions and the tensors are compactly supported. This lemma, together with the fact that the two differential operators are selfadjoint and elliptic and a standard argument from linear PDE theory [27] give us  X D  12 ıG0 h and that there are unique solutions X and f to the equations ıG0 ıG 0 LG0 LG0 f D LG0 h for a compactly supported data h. This says that we have an L2 -decomposition of compactly supported tensors h’s. Now coming back to the general case where h is L2 -integrable, let fhi g be an approximating sequence, each compactly supported, convergent to h in L2 -norm. For  Xi D  12 ıG0 hi and LG0 LG0 fi D each i, we can solve the pair of equations ıG0 ıG 0 LG0 hi for Xi and fi . We need to show that the sequences fXi g and ffi g are both  convergent in the respective spaces, and that they solve the equations ıG0 ıG X D 0 1   2 ıG0 h and LG0 LG0 f D LG0 h. To see this, first note that Z kLXi Xj G0 k2 D hLXi Xj G0 ; LXi Xj G0 iG0 dG0 2 ZH D hhi  hj ; LXi Xj G0 iG0 dG0 H2

kh1  hj kL2 kLXi Xj G0 kL2 ; where the second equality is due to the L2 -decomposition for compactly supported  which says that C kXi Xj kL2 tensors. This, together with the coercivity of ıG0 ıG 0 kLXi G0  LXj G0 kL2 for some C > 0, gives that C kXi  Xj kL2 khi  hj kL2 . This shows that the sequence fXi g is Cauchy in the space of H 1 -integrable vector fields on H2 , and hence convergent to some X . The elliptic regularity says that for h smooth, so is X .

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By an analogous argument, one checks that ffi g is Cauchy in the Sobolev space H 2 on H2 , and it converges to some smooth f for a smooth data h. The mutual L2 -orthogonality of PG0 .hi /, LXi G0 and LG0 fi for each i then induces the orthogonality of PG0 .h/, LX G0 and LG0 f , proving the statement of the theorem. We now consider this L2 -decomposition theorem from the viewpoint of the universal Teichmüller space. Recall that the universal Teichmüller space is the quotient of the space of hyperbolic metrics consisting of the pulled-back metric of the standard Poincaré metric G0 by all the quasi-conformal (q.c.) diffeomorphisms of D where G1 and G2 are defined to be equivalent when G2 D   G1 for some quasi-conformal diffeomorphism  fixing the geometric boundary @D D S 1 . By linearizing this picture at G0 , namely considering a one-parameter family of quasi-conformal diffeomorphisms  t with 0 D IdD , and differentiating the pulledback metrics  t G0 at time t D 0, we can identify the tangent space of the universal Teichmüller space UT at G0 as TG0 M1 D fLZ G0 j Z vector fields generating q.c.-diffeomorphisms on Dg: Now let h be an L2 -integrable deformation tensor of G0 tangential to M1 , namely assume there is no third component of the type LG0 f in the L2 -decomposition of h; then h can be written as a Lie derivative LZ G0 of G0 for some vector field X. As the L2 -decomposition gives h D PG0 .h/ C LX G0 for some vector field X D X.h/, the tangential component PG0 .h/ of h to UT is of the form LZX G0 . Define ZzN WD , and  WD XzN . Recall from Section 3.5 that these equalities are the linearizations of the Beltrami equations at the identity map. Recall the Beltrami equation is of the form wzN D wz . Now take the Beltrami coefficients to be "0 for some fixed 0 and j"j sufficiently small such that "0 remains in the unit ball L1 .D/1 in the complex Banach space L1 .D/. Differentiate the equation w."/zN D "0 w."/z in " and evaluate at " D 0 to obtain wP zN .0/ D 0 as w.0/ D z. Denote the vector field w.0/ P by V .0 /. In general the equation VzN D  can be solved uniquely on the disc [2], [56], [54] using an integral kernel on the disc. The resulting vector field V .h/ WD Z  X defined on the disc is a particular type such that VzN DW P Œ is a so-called harmonic Beltrami differential. Note that P Œ D   . For detailed expositions on harmonic Beltrami differentials, see [2], [75], [56]. Let .z/ be the area density of the hyperbolic metric G0 with respect to the Euclidean area density of the unit disc. Recall that Z D Z.h/ is an L2 .H2 /-integrable 4 vector field. On the Poincaré disc model of H2 , as the area density .z/ D .1jzj 2 /2 blows up as jzj ! 1, Z.z/ has to decay as z approaches the geometric boundary fjzj D 1g: Namely the one-parameter families of quasi-conformal diffeomorphisms Z generates belong to QC0 .D/.

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The statement of the L2 -decomposition theorem corresponds to the so-called Ahlfors’s integral projection operator. Let B be the space L1 .0 /1 of Beltrami differentials with 0 the trivial group fIdg. Let B  B be the space of harmonic Beltrami differentials, where a Beltrami differential  is harmonic if .z/N is holomorphic, or  D 1 N for some holomorphic quadratic differential  on D. This correspondence can be better understood by looking at h d zN i D N dz 2 :

dzd zN  dz Ahlfors [2] introduced a bounded linear operator P W B ! B given by Z 3.z  zN /2 . / P Œ D d . /;  N 4 H2 .  z/ where d is the Euclidean area element, and z and are the Euclidean coordinates for the upper half space, which is a model of the hyperbolic plane H2 . This map is indeed a projection for one checks that P Œ D  when  2 B. The kernel of the projection map is denoted by N , and it is known that the space B=N is identified with the tangent space TG0 UT . Now recall our L2 -decomposition theorem says that an L2 -integrable tensor h tangential to the space of hyperbolic metrics M1 can be decomposed as h D PG0 .h/ C LX G0 while h D LZ G0 for some vector field Z. Each of the three Lie derivatives LZ G0 , LX G0 and LZX G0 is identified uniquely to Beltrami differentials ,  and   , respectively, via the N @-equations ZzN D ; XzN D  and .Z  X /zN D   : Now the correspondence between the two representations of the tangent space is given by PG0 Œ D    and  is an element of the kernel N of the projection operator PG0 W B ! B. 3.5.2 Weil–Petersson complex structure. We explain here that the paper of Nag– Verjovsky [56] identifies vector fields on S 1 with tangent vectors of the universal Teichmüller space at the identity, which is a natural thing to do as the universal Teichmüller space is defined as UT D QS.S 1 /=SL.2; R/; the space of quasi-symmetric self-maps of S 1 fixing three points (say, .1; 0/; .0; 1/ and .1; 0/ ) on the circle. Each tangent vector ‚ is obtained by linearizing the solution to the Beltrami equation near the identity as shown above and restricting the resulting vector field defined on the unit disc D onto the unit circle S 1 D @D; namely

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‚ D w"0 .z/ with jzj D 1 where P 0 .z/ C o."/; w"0 .z/ D z C "wŒ

" ! 0:

When the universal Teichmüller space UT is regarded as the space of Beltrami differentials L1 .D/1 = , there is a natural complex structure J W  7! i on the tangent space of UT at the identity. This induces a complex structure Jz on the other representation of the universal Teichmüller space as one can linearize the one-parameter family of quasi-conformal diffeomorphisms w"i0 at the origin " D 0 and restrict the resulting vector field to the unit circle. Nag–Verjovsky [56] showed that this Jz is the Hilbert transform, a statement attributed to S. Kerckhoff: Theorem Using as coordinate on S 1 , and z D e i , define ‚ D u. /.@=@ /, where wŒ P 0 .e i / D izu. /, namely u. / is the magnitude of the vector field ‚ at the point P 0 .e i / D izu . /, where u . / is given by z D e i . Then Jz‚ D u . / where wŒi u .z/ D Im.D.z// C .cz C cN zN C b/ on fjzj D 1g for a certain b 2 R and c 2 C, and D.z/ is an element of the disc x such that algebra A.D/ (namely functions holomorphic in D and continuous on D 1 ReD.z/ D u.z/ on z 2 S .) Note that the statement is for the universal Teichmüller space, but one can restrict to the Teichmüller space of a Fuchsian group, as the -equivariance can be incorporated into the proof of this theorem. 3.5.3 DiffS 1 =SL.2 ; R/ as a Hilbert manifold. One of the merits in looking at the L2 -integrable deformation tensors of the Poincaré metric G0 is that it provides a Hilbert space which acts as the tangent space equipped with the Weil–Petersson pairing. In the paper [56], the Lie algebra of DiffS 1 is identified as the algebra of C 1 -smooth vector fields on S 1 . The complexification of the Lie algebra is the Virasoro algebra generated by Ln D e i n

@ @ D iz nC1 ; @ @z

n 2 Z; z D e i :

A tangent vector to the homogeneous space DiffS 1 =SL.2; R/ at the identity ŒId is of the form X vm Lm ; vxm D vm : ‚D m¤1;0;1

Note that the omission of m D 1; 0; 1 is due to the fact that L1 ; L0 and L1 span the subspace that is the complexification of sl.2; R/ within the complexified Lie algebra of DiffS 1 . As the diffeomorphisms are C 1 -smooth, each vector field ‚ D u. /@=@ can be identified with a 2-periodic C 1 real-valued function u. /.

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There is a natural complex structure Jz at the identity, which is conjugation: X i sgn.m/vm Lm : Jz‚ D m¤1;0;1

Now Jz is an almost complex structure by definition, but it is also known that it is indeed integrable, and that the right multiplications/translations by the elements of DiffS 1 are biholomorphic automorphisms of the homogeneous space DiffS 1 =SL.2; R/ [56]. As each smooth diffeomorphism of S 1 extends to a smooth diffeomorphism of the closed disc S 1 [ D, which in turn is quasi-conformal, DiffS 1 is a subset of the space QS.S 1 / of quasi-symmetric maps. Thus one can think of the homogeneous space DiffS 1 =SL.2; R/ as a subset of the universal Teichmüller space UT D QS.S 1 /=SL.2; R/. We introduce the following theorem by Nag–Verjovsky [56]. Theorem The natural inclusion DiffS 1 =SL.2; R/ ,! UT is holomorphic. The proof of this statement follows, using the homogeneous structures of both DiffS 1 =SL.2; R/ and UT , from checking that the complex structure Jz and J coincide at the identity of UT . We omit the details and refer the reader to the paper [56]. In short, the conjugation Jz of ‚ D u. /@=@ can be shown to coincide with the Hilbert transform u . / of u. / so that by setting D.e i / D u. / C i u . /, D is identified with an element of the disc algebra A.D/, as appeared in the above theorem. In this situation Nag and Verjovsky show that the only possible homogeneous Kähler form ! on DiffS 1 =SL.2; R/ given at the identity is !.Lm ; Ln / D a.m3  m/ım;n ;

m; n 2 Znf˙1; 0g

for a a purely imaginary number. This form had been previously known as the Kirillov– Kostant symplectic form [67] on DiffS 1 =SL.2; R/. Having the complex structure Jz and the Kähler form ! at hand, there is a natural Kähler metric on DiffS 1 =SL.2; R/, specified as g.v; w/ D !.v; Jzw/. When the 1 tangentPvectors ‚1 and P ‚2 at the identity of DiffS =SL.2; R/ are written as Fourier series vm Lm and wm Lm , respectively, the metric has the form g.‚1 ; ‚2 / D 2ia Re

1 hX

i vm w Sm .m3  m/ :

mD2

This series converges absolutely when the vector fields ‚i are elements of the Sobolev space H 3=2 .S 1 /, when these vector fields ‚i are identified with ui . /@=@ . To keep the metric positive definite, we need a D ib for b > 0. Nag and Verjovsky proceed to show that this Kähler metric is indeed the Weil– Petersson metric defined on the L2 .G0 /-integrable tensors in TG0 UT . The proof is by finding an explicit correspondence between an L2 .G0 /-integrable Beltrami differential

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i N  and the vector field wŒ.e P / DW V .e i / on S 1 , by solving the @-equation VzN D  using the integral kernel. This series of results can be summarized by the fact that the Hilbert manifold structure imposed on the homogeneous space DiffS 1 =SL.2; R/ is affiliated to the space of H 3=2 -smooth vector fields on S 1 . In exponentiating those vector fields with respect to the Weil–Petersson pairing, we expect to understand the global structure of the homogeneous space. We will come back to this issue in Section 4.6.

3.6 Weil–Petersson geodesic equation Having the linear structure of the tangent space TG M at each hyperbolic metric G, and the covariant derivative D on M, we proceed to write down the Weil–Petersson geodesic equation. First note that the space M of smooth metrics on the surface † contains the space M1 of hyperbolic metrics as a smooth submanifold, as the function K which assigns to each metric its sectional curvature: K W M ! C 1 .†/ has the constant function 1 as a regular value. This follows from a standard argument (see [22], [8]), namely that the linearized operator, which is the Lichnerowicz operator LG , DK.G/ W TG M ! C 1 .†/ is surjective, as we have already seen in the proof of L2 -decomposition theorems. A consequence of the L2 -decomposition is that we can see that the quotient map Q W M1 ! M1 =Diff 0 † is a Riemannian submersion, a point of view initiated by Earle–Eells [18] in the 1960s, as the linearization of the map DQ.G/ W TG M1 ! TG T sends PG .h/ C LX G to PG .h/ where LX G is perpendicular to the tangent space TG T . Namely, the tangent space TG M1 is split into the horizontal space TG T and the vertical space TG Diff 0 †, and the latter is the kernel of the linear map DQ.G/. A standard result on Riemannain submersions then tells us that given a Weil– Petersson geodesic  W Œ0; T  ! T and a hyperbolic metric G0 with Q.G0 / D  .0/, there exists a unique path G t in M1 , itself a geodesic in M1 with Q.G t / D  .t / for t 2 Œ0; T , and its L2 -metric length is equal to the Weil–Petersson length of . The path G t is called the horizontal lift of  with initial point G0 . In what follows, we will identify each Weil–Petersson geodesic with its horizontal lift with a suitable initial metric. Let …G be the projection map …G W TG M ! .TG M1 /? defined by …G .h/ D LG f where f satisfies LG LG f D LG h.

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For (a horizontal lift of) a Weil–Petersson geodesic fG t g  M1  M, the tangent vector GP t WD dd G j Dt is an element of TG t M1 at each time t, namely …G t .GP t / D 0: Furthermore, as G t is a horizontal lift of a Weil–Petersson geodesic, we have GP t D PG t .GP t /. Since G t is a geodesic in M1 where the space M1 is a Riemannian manifold with its metric being the L2 -pairing, the geodesic curvature vector of the curve G t vanishes. Recall that we have specified the Levi-Civita connection D of the L2 -pairing defined on the tangent bundle T M in the above section. The connection D then induces the Levi-Civita connection r on its submanifold M1 by the Gauss formula rX Y D .DX Y /T M1 ; where the right-hand side is the tangential component of DX Y to TG M1 . Hence the fact that G t is a geodesic in M1 , is equivalent to rGP t GP t D 0 which in turn is equivalent to DGP t GP t D …G t .DGP t GP t / as .…G t .DGP t GP t //T M1 D 0 by definition. This last equation is the Weil–Petersson geodesic equation in the context of the L2 -geometry of the space of metrics M. Theorem 3.8. Given a horizontal lift G t of a Weil–Petersson geodesic, we have the following expression for the second t-derivative of G t in TG M: 



d 2 ˇˇ 1 P 2 kG0 k C ˛ G0 C LZ G0 ; G t tD0 D 2 dt 4 where ˛ D  12 .4G0  2/1 kGP 0 k2 which is nonnegative, and Z is a vector field on †. Remark. Tromba [71] has a similar calculation (Theorem 2.1) to obtain an expression of the second derivative of a horizontal lift of a Weil–Petersson geodesic, d 2 ˇˇ 1 G t tD0 D kGP 0 k2 G0 C LW G0 2 dt 2 with Tr G0 .LW G0 / D 0 which differs from the one above. The calculation is seemingly based on the assumption that the space M with respect to the L2 -metric is a linear space so that the geodesic curvature vector of an arc-length parameterized path G t is GR t . Thus the Weil–Petersson geodesic equation (Equation (2.1)) in [71] is GR t D …G t .GR t / instead of our DGP t GP t D …G t .DGP t GP t /. We also note here that it was claimed in the proof of the same theorem (Equation (2.3)) that the trace-free part

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of a symmetric .0; 2/-tensor which is divergence-free is again divergence-free, which does not hold in general. Proof. Recall that the geodesic curvature vector  of an arc-length parameterized curve u.t / has the expression ˛ .t / D ŒruP u.t/ P D uR ˛ .t / C ˇ uP ˇ .t /uP  .t /:

In our setting, this is equivalent to .t / D ŒDGPt GP t .t/ D GR t C DGPt GP t ; where D is the Levi-Civita connection for the L2 -metric defined on M. For a horizontal lift G t of a Weil–Petersson geodesic  .t /, the velocity vector is tangential to the Teichmüller space, …G t .GP t / D 0; while the geodesic curvature vector has no tangential component to the Teichmüller space, D P GP t D …G t .D P GP t /: Gt

Gt

Hence we have the expression GR 0 C DGP0 GP0 D …G0 ŒGR 0  C …G0 ŒDGP0 GP0  which is reorganized as GR 0 D …G0 ŒGR 0   ŒDGP0 GP0 TG0 M1 as DGP0 GP0  …G0 ŒDGP0 GP0  constitutes the tangential component to M1 in the L2 decomposition of TG0 M1 . On the other hand, differentiating the tangential condition …G t .GP t / D 0 in t yields ˇ ˇ d d …G t GP t ˇ tD0 D …G t GP 0 ˇ t D0 C …G0 GR 0 D 0: dt dt Combining these, we have an expression for GR 0 : ˇ d GR 0 D  …G t GP 0 ˇ tD0  ŒDGP0 GP0 TG0 M1 : dt The term ŒDGP0 GP0 TG0 M1 can be computed further by using the explicit expression for the Levi-Civita connection as follows. Recall the formula  1 Dh1 h2 D  h1  G 1  h2 C h2  G 1  h1 2  1 C .Tr G h1 /h2 C .Tr G h2 /h1  hh1 ; h2 iG.x/ G ; 4

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which becomes 1 1 1 Dh1 h2 D  h1  G 1  h2  h2  G 1  h1  hh1 ; h2 iG.x/ G 2 2 4 for the trace-free symmetric .0; 2/ tensors h1 and h2 . When G D G0 and h1 D h2 D GP 0 , we have 1 DGP0 GP0 D GP 0  G01  GP 0  kGP 0 k2G0 .x/ G0 : 4 By using the geodesic normal coordinates so that G0 D ıij and .GP 0 /11 D .GP 0 /22 , the matrix multiplication gives   1 .GP 0 /211 C .GP 0 /212 0 1 P P D  kGP 0 k2 G0 : G0  G0  G0 D  2 2 P P 0 .G0 /11 C .G0 /12 2 We recall that the explicit expression for the Levi-Civita connection is valid only for the locally constant symmetric .0; 2/ tensors. Here the tensor GP 0 is treated as such, as the quantities in the calculation are tensorial. This says, in the light of the L2 -decomposition theorem, that the deformation tensor DGP0 GP0 is purely conformal, hence point-wise (and thus L2 ) orthogonal to the trace-free tensors, which in turn implies that the tensor has no tangential component to the Teichmüller space. By taking the projection of DGP0 GP0 to the tangent space TG0 M1 , the resulting tensor is along the diffeomorphism fiber, hence ŒDGP0 GP0 TG0 M1 D LX G0 for some smooth vector field X. Hence we have so far established ˇ d GR 0 D  …G t GP 0 ˇ t D0  LX G0 : dt ˇ d …G t GP 0 ˇ t D0 . We proceed to calculate the term dt First of all as a consequence of the L2 -decomposition theorems, we have the following formula for the third component of the decomposition, which is in the orthogonal directions to the space of constant curvature metrics. Proposition 3.9. In the L2 -decomposition theorems, where an arbitrary smooth tensor h has the following decomposition h D PG .h/ C LX G C LG f;  where X and f are the unique solutions of the equation ıG ıG X D ıG h and LG LG f D LG h, respectively, we have

.4G  1/f D .4G  2/1 LG h:

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Proof. Note the following equalities LL f D .4  1/ Tr G .L f / C ıG ıG L f D .4  1/f2.4  1/f C 4f g C f.4  1/f ıij C fij gIij D .4  2/.4  1/f which is equal to Lh. As the differential operator 4  2 is invertible, we obtain the statement.

Having this statement at hand, we move on to write down the projection operator …G W h 7! LG h as …G t GP0 D f.4G t  1/f t gG t C HessG t f t D f.4G t  2/1 LG t GP0 gG t C HessG t f t ; where f t is the solution of LG t LG t f t D LG t GP 0 . We have at t D 0, LG0 GP 0 D 0 and thus f0 D 0. Using these equalities, the time-derivative of …G t GP0 at t D 0 can be written as    ˇ  d d ˇˇ d …G t GP0 D .4G0  2/1 LG t GP0 ˇ t D0 G0 C HessG0 f t t D0 : dt dt dt d ˇ  Note that the Hessian term HessG0 dt f t ˇ tD0 is a Lie derivative Lr fP0 G0 . ˇ d We now calculate the term dt LG t GP0 ˇ tD0 : ˇ ˇ d  d LG t GP0 ˇ tD0 D  .4G t  1/ Tr G t GP 0 C ıG t ıG t GP 0 ˇ t D0 dt dt ˇ ˇ d d D .4G0  1/ .Tr G t GP 0 /ˇ t D0 C ıG0 .ıG t GP 0 /ˇ t D0 dt dt 3 ij D .4G0  1/..GP 0 / .GP 0 /ij / C ıG0 . rkGP 0 k2 / 4 3 D .4G0  1/kGP 0 k2  4G0 kGP 0 k2 4   1 2 4G0  1 kGP 0 k : D 4 ˇ d .ıG t GP 0 /ˇ tD0 D  34 rkGP 0 k2 was used. This follows In the third equality, the fact dt from the following calculation; as p  hip jpk /; .ıG h/i D G j k hij Ik D G j k .hij;k  hpj ik

where the semi-colon is used to denote the covariant derivative, we have ˇ d .ıG t GP 0 /ˇ tD0 D .GP 0 /j k .GP 0 /ij;k dt

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  ˇ d 1 .G t /pq f.G t /iq;k C .G t /kq;i  .G t /ikIq g ˇ t D0  G0j k .GP 0 /pj dt  2  ˇ d 1 jk P pq .G t / f.G t /j q;k C .G t /kq;j  .G t /j kIq g ˇ t D0  G0 .G0 /ip dt 2 D .GP 0 /j k .GP 0 /ij;k   1 jk P pq P P P  .G0 / .G0 /pj .G0 / f.G0 /iq;k C .G0 /kq;i  .G0 /ikIq g 2   1  .G0 /j k .GP 0 /ip .G0 /pq f.GP 0 /j q;k C .GP 0 /kq;j  .GP 0 /j kIq g : 2

By using the fact that at t D 0, G0 can be expressed as ıij at each point, which in turn makes the traceless transverse deformation tensor .GP 0 /ij;k fully symmetric in i; j and k, we get a tensorial expression for our result: ˇ 3 d .ıG t GP 0 /ˇ tD0 D  kGP 0 k2Ii : dt 4 ˇ 1  d ˇ By inserting dt LG t GP0 tD0 D 4 4G0  1 kGP 0 k2 we have 



ˇ d d …G t GP 0 D .4G0  2/1 LG t GP0 ˇ t D0 G0 C Lr fP0 G0 dt dt    1 1 4G0  1 kGP 0 k2 G0 C Lr fP0 G0 D  .4G0  2/ 4  1 1 2 P .4G0  2  2/kG0 k G0 C Lr fP0 G0 D  .4G0  2/ 4  1 P 2 1 1 P 2 D  kG0 k G0 C .4G0  2/ kG0 k G0 C Lr fP0 G0 : 4 2 Lemma 3.10. For a non-negative function f on †, .4G0  2/1 f is non-positive. Proof. We will show that u WD .4G0  2/1 f 0. By supposing that u attains its maximum at a point p with u.p/ > 0, we have Œ4G0 u.p/ 0. As 2u.p/ < 0, f .p/ D Œ.4G0  2/u.p/ < 0, a contradiction to the hypothesis f  0. By setting ˛ D  12 .4G0  2/1 kGP 0 k2  0 and Y D r fP0 , we have an expression for GR 0 : ˇ d GR 0 D  …G t GP 0 ˇ tD0  LX G0 dt   1 P 2 D   kG0 k  ˛ G0 C LY G0  LX G0 4   1 P 2 D kG0 k C ˛ G0 C LZ G0 ; 4 where we denoted the vector field X  Y by Z.

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4 Harmonic map parameterizations 4.1 General setting for harmonic maps We recall the theory of harmonic maps. Our treatment of the subject is by no means complete, and interested readers are referred to several standard texts available (for example [32], [20], [63]). Let u W .M; g/ ! .N; G/ be a C 1 -map. Let du denote the section of the bundle E WD T  M ˝ u .T N / for which there is the induced metric hX ˝ Y; W ˝ ZiE WD gx .X; W /Gu.x/ .Y; Z/. When fx i g (and fy ˛ g) is a local coordinate system near a point p in M (and a point u.p/ in N respectively,) locally du W TM ! T N is expressed as du D

X @u˛ i;˛

@x i

dx i ˝

@ : @y ˛

Then one can define the energy of the map as Z 1 kduk2 dg ; E.u/ D M 2 where the integrand 12 kduk2E .x/ D denoted by e.u/, locally written as

1 hdu.x/; du.x/iE 2

is the energy density, also

1 @u˛ @uˇ 1 1 kduk2 .x/ D g ij .x/G˛ˇ .u.x// i D Tr g .u G/; 2 2 @x @x j 2 where g  .x/ D g ij .x/

g.x/ D gij .x/dx i ˝ dx j ;

@ @ ˝ j; @x i @x

G.y/ D G˛ˇ .y/dy ˛ ˝ dy ˇ ; u G is the pulled-back metric tensor of G by u, .u G/ij D G˛ˇ .u.x//

@u˛ @uˇ .x/ .x/ dx i ˝ dx j @x i @x j

and Tr g .u G/ D g ij .u G/ij .  This bundle E has an induced Levi-Civita connection from the connections r T M and r T N of g and G, respectively, rXE .Y ˝ Z/ D .rXT 

M

T N

Y / ˝ Z C Y ˝ rXu

Z

TN .u Z/.x/. where rXu T N Z.x/ WD rdu.X/ Consider the situation where u is stationary, that is, the first variation of the energy functional vanishes under arbitrary smooth variations of the map of the form u" .x/ with u0 D u and dd" u" j"D0 D W . Note that each W is a smooth section of the bundle

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E D T  M ˝ u T N . Writing down what this means pointwise, we obtain Z d d 1 ıE.u/.W / WD E.u" /j"D0 D hdu" ; du" iE dg j"D0 d" d" 2 Z M D hd W; du0 iE dg ZM D hW; d  du0 iE D 0: M

This holds for arbitrary W , which in turn implies that d  du0 D 0. The adjoint operator d  of the differential d acting on the smooth sections of T  M ˝ u T N is X 7! Tr g .r E X/. Locally, we have 



@ @y ˇ @ D uˇij dx j ˝ dx i ˝ ˇ @y  ˇ T M i  @ @  C ui .r@j dx / ˝ dx j ˝ ˇ C uˇk dx i ˝ r@uj T N ˇ @y @y   ˇ @ i ˇ D uij C uˇk kj .x/ C ˛ .u.x//u˛i uj dx i ˝ dx j ˝ ˇ @y   @ ˇ D ŒHess.u/ij C ˛ u˛i uj dx i ˝ dx j ˝ ˇ : @y

r E du D g ij r@Ej uˇi dx i ˝

By taking the g-trace of the above, d  du is written as ˇ d  du D Tr g .r E du/ D .4g uˇ C g ij ˛ u˛i uj /

@ : @y ˇ

The vanishing of d  du is called the harmonic map equation, and locally written as ˇ u˛i uj D 0 4g uˇ C g ij ˛

for 1 i; j dimM and 1 ˇ dimN. When dimM D 1, this is nothing but the geodesic equation. In what follows, the target manifolds of harmonic maps are of non-positive sectional curvature, and the following theorem covers all the situations we will be concerned with. We now quote the following existence and uniqueness statements of harmonic maps in situations we are interested in. This version comes from a collection of results by Eells–Sampson [21] who showed the existence and the regularity, and by Hartman [30] and Al’bers [3] independently who showed the uniqueness. Theorem (Existence and uniqueness of harmonic maps). Let .M n ; g/ be a closed manifold, and .†2 ; G/ a surface of non-positive sectional curvature. Suppose there is a continuous map  W .M n ; g/ ! .†2 ; G/. Then there exists a smooth harmonic map

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homotopic to . When the sectional curvature of G is strictly negative and the image of the map is not a point or a closed geodesic, then the harmonic map is unique. Furthermore by utilizing the inverse function theorem, Eells–Lemaire [19] and Koiso [39] showed: Theorem (Smooth dependence on target metric variations). Let .M n ; g/ be a closed manifold, and .†2 ; G/ a closed surface with a hyperbolic metric G. For a smooth deformation G t of the hyperbolic metric G DW G0 in the space of smooth metrics on †, the resulting harmonic maps u t W .M; g/ ! .†2 ; G t / are smoothly dependent in t . In Eells–Lemaire’s statement, there is a technical condition that the Hessian of the energy functional of the harmonic map u under variations of the map is positivedefinite. This is satisfied for the harmonic map u into the hyperbolic surface .†; G/, for the harmonic map in this case is the unique energy minimizing map in its homotopy class.

4.2 Harmonic maps between surfaces Recall that a two-dimensional Riemannian manifold is a Riemann surface, i.e. each Riemannian metric g is conformal to dz ˝ d zN D dx ˝ dx C dy ˝ dy for some local coordinate chart z D x C iy, so that g D .z/dz ˝ d zN for some function > 0. Such a z is called an isothermal coordinate. Now consider the situation when both the domain M and the target N are Riemannian surfaces, and for p and u.p/, choose isothermal coordinate z D x C iy and w D u1 C i u2 around p and u.p/, respectively. We then have g D .z/jdzj2

and

G D .w/jdwj2 ;

and with the differential operators 



@ @ 1 @ i ; D 2 @x @y @z



1 @ @ @ D Ci @Nz 2 @x @y

we have

ˇ



ˇ

ˇ

ˇ 

1

.u.z// ˇˇ @u ˇˇ2 ˇˇ @u ˇˇ2 1 Cˇ ˇ : e.u/ D kduk2E D g ij G˛ˇ u˛i ujˇ D 2 2 .z/ ˇ @z ˇ @Nz Introduce the following notation: ˇ

j@uj2 WD

ˇ

.u.z// ˇˇ @u ˇˇ2 .z/ ˇ @z ˇ

and

N 2 WD j@uj

Then we have N 2 e.u/ D j@uj2 C j@uj

ˇ

ˇ

.u.z// ˇˇ @u ˇˇ2 : .z/ ˇ @Nz ˇ

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while the Jacobian J.u/ of the map u is equal to   p

.u.z// @u1 @u2 @u1 @u2 N 2: J.u/ D det.du/ D  D j@uj2  j@uj .z/ @x @y @y @x The harmonic map equation expressed with those isothermal coordinates turns into uz zN C

.u/u uzN uz D 0:

.u/

Note that the conformal factor for g does not appear in the harmonic map equation above, which is explained by the fact that the energy is conformally invariant, namely 2 Q its value is unchanged by replacing g D .z/jdzj2 by gQ WD .z/jdzj , and hence if u is harmonic with respect to g, then it is harmonic with respect to g. Q Now we make a remark about linearizing the harmonic map equation using the isothermal coordinate z of the Poincaré disc .D; G0 D .z/jdzj2 / around the identity map. Let u t W .D; Gˇ0 / ! .D; G0 / be a one-parameter family of harmonic maps with d u t ˇ tD0 D V .z/. Then as the harmonic map equation is satisfied for u0 D IdD and dt each t, differentiating the equation in t at t D 0 when u0 .z/ D z, one obtains Vz zN C

.z/z VzN D 0

.z/

which in turn says that .z/VzN .z/ is anti-holomorphic. We have previously encountered the equation VzN D .z/ as the linearization of the Beltrami equations w."/zN D "w."/z with V .z/ WD dd" w."/j"D0 .z/. In particular,  was said to be harmonic when .z/.z/ N is locally holomorphic in the isothermal coordinate z. Combining these observations we conclude that the tangent space at the identity map to the space of harmonic diffeomorphisms fu t W H2 = 0 ! H2 =  t g where f t g is the set of deformations of the Fuchsian groups are represented by the space of harmonic Beltrami differentials. One further remark relevant to this observation is that instead of the harmonic diffeomorphisms u t , one can consider the family of Douady–Earle extensions [17] w t and its tangent space at the identity map, to get the same conclusion.

4.3 The Teichmüller of the torus and its Weil–Petersson metric We give an explicit description of the Teichmüller space of the torus. We furthermore specify its Weil–Petersson metric. The identification of the space is done through harmonic maps. This sets a model for higher genus surfaces in the next section. First choose a reference torus T02 which may as well be chosen to be R2 =Z2 where 2 Z is the standard integer lattice 0 ' Z2 in the x-y plane. We denote the resulting flat metric by g0 . Let M0 be the set of all flat metrics on T 2 of unit area. The space of all smooth metrics on the torus is thus uniformized by the elements of M0 . Each element .T 2 ; g/ of M0 can be parameterized by a harmonic map u W .T 2 ; g0 / ! .T 2 ; g/ in

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the same homotopy class as the identity map Id W T02 ! T 2 as shown above. Then a standard formula (see for example [20]) often referred to as the Bochner–Weitzenböck formula, applied to this situation says that 1 4g0 kduk2 D kr E duk2 : 2 Integrating this equality over the domain surface T0 we conclude that the map u is totally geodesic (r E .du/  0), namely u is an affine map. This in turn says that the pulled-back metric u g is locally constant; namely @u˛ @uˇ g˛ˇ dx i ˝ dx j @x i @x j is constant. Note that by looking at the pulled-back metric u g of the elements of g in M0 , we are actually looking at the point Œg in the Teichmüller space T1 D M0 =Diff 0 T 2 , as the pull-back actions of the diffeomorphisms are isometries so that Œg D Œ  g for  2 Diff 0 T 2 . By identifying T0 with the fundamental region Œ0; 1  Œ0; 1 in R2 , we can regard u g as an inner product structure on R2 with its determinant of the bilinear form equal to one, due to the unit volume normalization. By introducing an equivalence relation under the rotations around the origin parameterized by SO.2/ which preserve the standard inner product structure, we can identify the space of such inner product structures as n

g g  o 11 12 2 G D g21 g22 W g11 g22  g12 D 1 = D SL.2; R/=SO.2/ .u g/ij dx i ˝ dx j D

which is the hyperbolic space H2 as a set. To identify the metric structure, we now recall the characterization of the tangent vectors of Teichmüller spaces as trace-free, transverse symmetric .0; 2/ tensors: Tr G h D 0

and

ıG h D 0:

We also saw that such an h can be identified with a holomorphic quadratic differential h D .z/dz 2 . On a torus .T 2 ; g/, the (complex) dimension of the space of holomorphic quadratic differentials is one, namely differentials are locally constant c dz 2 for an isothermal complex coordinate z. This says that the trace-free transverse deformation tensor h 2 Tg T1 , with respect to the standard coordinate of R2 is of the form of a traceless matrix .hij / with constant components. By the above argument, a flat metric g with unit volume is represented as a point ŒG in SL.2; R/=SO.2/. Let h and k be two traceless transverse tensors in TŒG T , then the Weil–Petersson pairing is hh; kiŒG D G ij G kl hik hj l : Note that this is precisely the left-invariant Riemannian metric of the homogeneous space SL.2; R/=SO.2/, which makes the space isometric to the hyperbolic plane H2 . Theorem 4.1. The Teichmüller space of the torus with the Weil–Petersson metric is isometric to the hyperbolic disc.

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75

We remark that the affine harmonic map u W .T 2 ; g0 / ! .T 2 ; g/ is also the Teichmüller map in the sense that the map is extremal in minimizing the complex dilatation as a quasi-conformal map between the two surfaces, as explained in [42] and [6]. Hence in this instance, the Teichmüller geometry and the Weil–Petersson geometry coincide, and the Weil–Petersson geodesics are Teichmüller geodesics.

4.4 Teichmüller space of higher genus surface We saw the effectiveness of harmonic maps in parametrizing points in the Teichmüller space of the torus with respect to the Weil–Petersson metric. In higher genus surfaces .g > 1/, Weil–Petersson geometry involves much non-linearity, but curiously it has many geometric structures, due to the existence of many convex functionals. In what follows, we will be solely concerned with the cases where the target manifolds of harmonic maps are hyperbolic surfaces. In particular we are interested in the variational theory of the harmonic maps where the variable is the hyperbolic metric on the target surface, or rather, the equivalent classes of hyperbolic metrics, representing points in the relevant Teichmüller space. To be precise, we fix a domain manifold .M; g/ which is compact without boundary equipped with a Riemannian metric g, a topological surface † of higher genus g.†/ > 1, and a continuous map ˆ W M ! †. When the surface † is equipped with a hyperbolic metric G, by the above existence and uniqueness statements, there exists a smooth harmonic map u which is homotopic to , which is unique (up to rotations in case of M D S 1 ) in its homotopy class Œˆ. Naturally the harmonic map u depends on the hyperbolic z be a hyperbolic metric on † such that G z D '  G for some ' in metric G. Let G 1 z Diff 0 †. Then the map u B ' W .M; g/ ! .†; G/ is still a (unique) harmonic map z is an isometry by definition homotopic to ˆ, for the map ' 1 W .†; G/ ! .†; G/ of the pulled-back metric. We remark that when ' is an element in Diff† n Diff 0 †, z is an z is still harmonic as ' 1 W .†; G/ ! .†; G/ then ' 1 B u W .M; g/ ! .†; G/ isometry, but the composite map ' 1 B u is no longer homotopic to ˆ. This observation tells us that the correspondence G 7! u.G/ is well-defined when seen as ŒG 7! Œu.G/ where ŒG is a point in the Teichmüller space of †, and Œu.G/ is an equivalence class where u.G/  u.'  G/.D u B ' 1 / when ' is in Diff 0 †. In particular, the energy functional E.G/ WD E.u.G// of the map u.G/ E W M1 ! R can be seen as a functional defined on T : E W M1 =Diff 0 † ! R where E.ŒG/ WD E.u.G//. We now demonstrate the following theorem [81]. Theorem 4.2 (Weil–Petersson convexity of energy). The energy functional E W T ! R is strictly convex with respect to the Weil–Petersson metric.

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Remark 4.3. The proof closely follows the one given in Tromba [71] except for the value of the second time derivative of the path G t . Proof. We will show that for a horizontal lift G t 2 M1 of an arbitrary Weil–Petersson d2 geodesic  .t / 2 T , dt 2 E.G t / > 0. As seen above, for each G t , we have a unique harmonic map u t W .M; g/ ! .†; G t /. We denote the first variation tensor of G t d by GP t and the first variation vector field dt u t by W t , respectively. We note that the following calculation is done around t D 0 so that G t D G0 C GP 0 t C 12 GR 0 t 2 C o.t 2 / in M, but the proof works for any other value of t . The energy functional E.G t / D E.u t / is Z 1 Tr g .ut G t /dg E.u t / D 2 M where the dependence on t appears on the hyperbolic metric G t and the harmonic map u t . The first time-derivative is Z   ˇ 1 d  ˇˇ d E.G t /ˇ tD0 D u t G t t D0 dg Tr g dt 2 M dt Z     1 d  ˇˇ d  ˇˇ D Tr g u0 G t t D0 C Tr g u t G0 t D0 dg 2 M dt dt Z       1 D Tr g u0 GP 0 C Tr g u0 ŒLW0 G0  dg ; 2 M whereR u0 ŒLW0 G0  is the pulled-back tensor of the Lie derivative. Note that the second  term M Tr g u0 ŒLW0 G0  dg is the first variation of the energy in the direction of W0 .u0 .x//, which vanishes since the Rmap u0 is harmonic.Actually as u t is harmonic for all t , we have ı.E.u t //.W t / WD M Tr g ut ŒLW t G t  dg D 0 for all t . Hence the first time-derivative should be written as Z ˇ ˇ   1 d E.G t /ˇ tD0 D Tr g ut GP t dg ˇ t D0 : dt 2 M As for the second time-derivative, we get  Z  ˇ ˇ    d 1 d2 ˇ ˇ P E.G / D Tr u d G t tD0 g g t t t D0 2 dt dt 2 M Z     1 D Tr g u0 GR 0 C Tr g u0 ŒLW0 GP 0  dg : 2 M On the other hand, we differentiate the vanishing condition of the first variation of the energy Z   Tr g ut ŒLW t G t  dg D 0 M

in t and evaluate at t D 0, to obtain the following relation: Z     1 Tr g u0 ŒLW0 ŒLW0 G0  dg C Tr g u0 ŒLW0 GP 0  dg D 0: 2 M

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77

  R Note that the first term 12 M Tr g u0 ŒLW0 ŒLW0 G0  dg is the second variation ı 2 .E.u//.W0 ; W0 / of the energy in the directions of W0 and W0 . Hence for now, we have an expression of the second time-derivative of the energy functional E.G t /: ˇ d2 1 ˇ E.G / D t tD0 dt 2 2

Z M

  Tr g u0 GR 0 dg  ı 2 .E.u//.W0 ; W0 /:

We use the following estimate [71] for the second variation term: Lemma 4.4. We have ı 2 .E.u//.W0 ; W0 /

1 8

Z M

Tr g .u0 ŒkGP 0 k2 G0 /dg :

Proof. By using the equality obtained above, we have Z   1 2 ı .E.u//.W0 ; W0 / D  Tr g u0 ŒLW0 GP 0  dg 2 M Z  ˇ d 1 D  Tr g .ut GP 0 /dg ˇ t D0 dt 2 M Z  ˇ d 1 ˇ ij ˛ P D  g .x/.G0 /˛ˇ .u t .x//.u t /i .u t /j dg ˇ t D0 dt 2 M Z 1 D .GP 0 /˛ˇ; W0 g ij u˛i ujˇ dg 2 M  Z 1 1 C  .GP 0 /˛ˇ g ij Wi˛ ujˇ dg 2 2 M Z Z 1 1 ˛ ˇ P D .G0 /˛ˇ W 4g u dg  .GP 0 /˛ˇ g ij Wi˛ ujˇ dg 2 M 2 M Z 1 D .GP 0 /˛ˇ W ˛ g ij ˇı ui ujı dg 2 M Z 1  .GP 0 /˛ˇ g ij Wi˛ ujˇ dg 2 M Z 1  D .GP 0 /˛ˇ g ij .riu T † W /˛ ujˇ dg 2 M where the fifth equality comes from integration by parts, the sixth from the harmonic map equation, and the seventh uses the trace-free condition of .GP 0 / with respect to the geodesic normal coordinates.

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With respect to the geodesic normal coordinates we have g ij .x/ D ı ij and G˛ˇ .u.x// D ı˛ˇ and the Cauchy–Schwarz inequality says Z 1  2 ı .E.u//.W0 ; W0 / D  .GP 0 /˛ˇ g ij .riu T † W /˛ ujˇ dg 2 M n X 1 Z f.GP 0 /211 C .GP 0 /212 gf.u1i /2 C .u2i /2 gdg 8 M iD1 Z  1   C Œ.riu T N W /1 2 C Œ.riu T † W /2 2 dg 2 M Z Z 1 1   2 P D Tr g .u ŒkG0 k G0 /dg C kr u T † W k2 dg : 16 M 2 M Finally the second variation formula for the energy functional at a harmonic map u0 gives Z  ı 2 .E.u0 //.W0 ; W0 / D kr u T † W k2 dg  hR† .W0 ; du0 /du0 ; W0 iL2 .G/ ZM   kr u T N W k2 dg ; M

where the inequality is due to the negative sectional curvature of the surface †. Combining the pair of inequalities, we obtain Z 1 1 2 Tr g .u ŒkGP 0 k2 G0 /dg C ı 2 .E.u0 //.W0 ; W0 / ı .E.u0 //.W0 ; W0 / 16 M 2 which gives the statement of the lemma. Now we conclude the proof of the convexity by inserting the expression of GR 0 obtained above within the integrand Z ˇ   d2 1 ˇ E.G / D Tr g u0 GR 0 dg  ı 2 .E.u//.W0 ; W0 / t tD0 2 dt 2 M Z     1 1 P 2 kG0 k C ˛ G0 dg  Tr g u0 2 M 4 Z Z      1 Tr g u0 ŒLZ G0  dg  Tr g u0 ŒkGP 0 k2 G0  dg C 8 M Z M    1 D Tr g u0 Œ˛G0  dg  0: 2 M   R The term M Tr g u0 ŒLZ G0  dg vanishes as this is the first variation of the energy along a one-parameter family of isometries. Note that the inequality is an equality when ˛ D  12 .4G0  2/1 kGP 0 k2  0 is zero as well as the integral of the curvature hR† .W0 ; du0 /du0 ; W0 iL2 .G/ is zero. The former never occurs for nontrivial

Chapter 1. Local and global aspects of Weil–Petersson geometry

79

geodesics G t , which in turn implies that the energy functional is strictly Weil–Petersson convex.

4.5 Applications of Weil–Petersson convexity We now introduce some applications, by specifying the domain manifold .M; g/ and the homotopy classes of the harmonic maps. In [81], in addition to the convexity, a condition for the properness of the energy functional was obtained. Recall that a strictly convex functional, which is also proper, has a unique point in its domain where the value is minimized. Theorem 4.5. Suppose that we have a family of harmonic maps u W .M n ; g/ ! .†2 ; G/ with varying hyperbolic metrics G within a homotopy class so that the induced map u W 1 .M / ! 1 .†2 / has a finite-index image in 1 .†2 /. Then the energy functional E.G/ W T .†/ ! R is proper, and hence there exists a unique E-minimizing point ŒG in T . Schoen and Yau in [62] considered harmonic maps of Riemann surfaces into a three-dimensional manifold in order to find minimal immersions. There the energy functional of those maps as the conformal structures of the domain surfaces are varied is shown to be proper provided the induced maps on 1 ’s is injective. In a sense, our result is dual to theirs. The proof of this theorem is in [81]. Here we present a Sketch of proof. For each C 2 R, consider the sub-level set S.C / WD fG 2 M1 W E.G/ C < 1g of the energy functional. We show that M1 =Diff 0 N is sequentially compact in the Teichmüller space M1 =Diff 0 N . The Böchner–Weitzenböck formula combined with the standard elliptic estimate, often referred to as the De Giorgi–Nash– Moser iteration scheme, provide us with the following estimate: Z e.u/dg ; sup e.u/.x/ C x2M

M

where the constant C > 0 depends on the sectional curvature KG , which in our situation equals 1, but independent of the choice of the hyperbolic metric G. This says that on the sub-level set S.C /, the energy density e.u/ has an upper bound, uniform in the point x 2 M as well as in G 2 M1 . A geometric consequence of this fact is that the diameter of the image of the harmonic map u W .M n ; g/ ! .†2 ; G/ is uniformly bounded. This prohibits the hyperbolic surface .†2 ; G/ in S.C / to develop a pinching neck, a consequence of the Collar Lemma. This is because for all the closed geodesics (not necessarily simple), which are transverse to the pinching simple closed geodesic, their hyperbolic length blows up due to the finite-index condition of u 1 .M / in 1 .†/. This, namely the existence of a pinching neck, would contradict the upper bound condition for the diameter obtained above.

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Using the resulting lower bound of the length of the shortest simple closed geodesics, the Mumford–Mahler compactness theorem (see, for example, [70], [32]) says that the image of the sub-level set S.C /  M1 =Diff 0 † projected down to the moduli space M1 =Diff† is compact. Given a sequence of points Gi in Teichmüller space, this allows to find a convergent subsequence of the projected (from Teichmüller space) sequence ŒGi  in the moduli space. This can be rephrased as follows: there exists a sequence of diffeomorphisms ffk g of † and a subsequence fGk g of fGi g such that ffk Gk g is convergent in M1 . For the sequence of harmonic maps fuk W .M; g/ ! .†; Gk /g in the given homotopy class, each fk1 B uk is harmonic, though not necessarily in the same homotopy class. As fk Gk converges to a hyperbolic metric with the C 1 topology in M1 , for sufficiently large values of k, the homotopy type of the harmonic maps fk Gk stabilizes, and one checks that the fk ’s are homotopic to each other for the large k’s. This says that the subsequence fGk g converges to some hyperbolic metric G1 , proving the sequential compactness of the sub-level set S.C /. Now we make some specific choices of the domain manifold .M n ; g/ of the harmonic maps u W .M n ; g/ ! .†2 ; G/. 4.5.1 Harmonic maps from copies of S 1 to †. This situation has been first investigated by Wolpert [76], who showed that the hyperbolic length functional L W T ! R of each simple closed geodesic  is Weil–Petersson convex. He then chose a set fi gN 1 of  ’s which fill the surface N , namely, the complement of the loops are homeomorP phic to a set of open discs, to make the functional N iD1 Li W T ! R proper. This provides a proof of the so-called Nielsen Realization Problem demonstrated in [76], which was first proven by S. Kerckhoff [35] using the convexity of L along earthquake deformations. A comparison should be made between E and L when the harmonic map u W S 1 ! .†; G/ maps onto the simple closed geodesic . In this case, the harmonic map is not unique, and the lack of uniqueness corresponds to rotations of the domain S 1 .Š Œ0; 1= /, which induces an S 1 -action on the map u. However, the energy is invariant under the S 1 -action and it is equal to the square of the hyperbolic length E.G/ D L2 . Note that Wolpert’s Weil–Petersson convexity of L thus implies the Weil–Petersson convexity of E.G/, but not vice-versa. Another remark concerning convexity is that in our harmonic map setting, the closed geodesic need not be simple; for example the image of the harmonic map may be an immersed closed geodesic in the surface. In this regard, recently M. Wolf gave another proof of Weil–Petersson convexity [73] of hyperbolic length of geodesics, not necessarily simple nor closed. 4.5.2 Harmonic maps from the surface † to itself. When the domain is the surface † itself, and when the harmonic maps .†0 ; g/ ! .†; G/ with the hyperbolic metric G varying in M1 , are all homotopic to the identity map Id W †0 ! †, the energy functional is both convex and proper, as the map u W 1 .†/ ! 1 .†/ is the identity

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map, hence surjective. Therefore there exists a unique minimizer ŒG0  in T . These maps are known to be diffeomorphisms by the results of Jost–Schoen [33]. The minimizer is specified by the hyperbolic metric uniformizing the domain metric g, namely the hyperbolic metric G0 conformal to g. This can be seen by the following argument. Recall that the energy of the harmonic map u is written with respect to the isothermal coordinates z around x and w around u.x/ as Z Z N 2 /dg : e.u/dg D .j@uj2 C j@uj †

†

On the other hand, for the maps homotopic to the identity, the mapping degree is one. Hence, the integral of the pulled-back volume form u dG coincides with the integral of the pulled-back Kähler form which is equal to minus the Euler characteristic of the surface †: Z Z N 2 /dg D .†/: J.u/dg D .j@uj2  j@uj †

Note that

†

Z

Z .e.u/  J.u//dg D

E.u/ C .†/ D †

N 2 dg 2j@uj

†

R N N 2 dg implies that the energy functional E differs from the @-energy E@N WD † 2j@uj by a topological constant. Therefore the Weil–Petersson convexity of the energy N functional E is equivalent to the RWeil–Petersson convexity of the @-energy functional 2 N E@N W T ! R where E@N .G/ WD † 2j@uj dg for the harmonic map u W .†0 ; g/ ! .†; G/. N measures the quasi-conformality of the map u, and, We remark that the @-energy in particular, equals zero when the map u is conformal. Specifically, the usual DirichN let energy and the @-energy can be written down by using the Beltrami coefficient .z/ WD uzN =uz as Z Z 2 2 2jj2 j@uj2 dg .z/: E.G/ D .1 C jj /j@uj dg .z/; E@N .G/ D †

†

N The @-energy functional is strictly Weil–Petersson convex and proper, hence the collection of sublevel sets fS.C /gC 2R0 provide an exhaustion of the Teichmüller space, T N functional is uniquely minwith a point ŒG0  WD C >0 S.C / where the @-energy imized. This point ŒG0  is characterized by the conformal harmonic identity map, namely when g and G0 are conformal. The harmonic map is a canonical object in relating a pair of hyperbolic surfaces, where other canonical ways to relate them include the Teichmüller map (see [42] for references) and the minimal stretch map [68]. The Teichmüller map requires no metric structures, only the conformal structures of the domain and the target, while the harmonic map requires the hyperbolic metric of the target surface, but only the conformal structure of the domain surface. The minimal stretch map does require

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hyperbolic structures on both the domain and the target surfaces. In this context, we draw the reader’s attention to two papers on harmonic map theory between surfaces. One is Y. Minsky’s thesis [52], where the asymptotic behavior of harmonic maps are investigated in relation to measured foliations, and the other is C. Mese’s work [50] where a conjecture by M. Gerstenhaber and H. Rauch from 1954 was resolved, based on the preceding results by M. Leite [43] and E. Kuwert [41]. It says that the Teichmüller map is a harmonic map between a pair of Riemann surfaces where the target is equipped with a singular metric induced from the Teichmüller differential relating the two conformal structures. 4.5.3 Harmonic maps from Kähler manifolds to †. The next situation is when the domain is a closed Kähler manifold .M; g/. Consider a homotopy class of a map  W .M; g/ ! .†; G/ so that the condition for properness of the energy functional is met. The hyperbolic surfaces .†; G/ are Kähler, and by a result of Sampson (see [61]), any holomorphic map between Kähler manifolds is harmonic, and in particular, energy-minimizing. This holds for the homotopy class Œ. As was stated in [81], the point ŒG0  in T minimizes the energy functional E W T ! R.

N 4.6 @-energy functional on the universal Teichmüller space 4.6.1 Asymptotically conformal harmonic maps. In this section, we restrict our study to the situation in [81] where there is a Weil–Petersson geodesic G t in the universal Teichmüller space UT , and we parameterize the varying hyperbolic metrics by harmonic maps from the Poincaré disc .D; G0 /. Recall that we already looked at the linear structure of the universal Teichmüller space, when the deformation tensor is induced by sufficiently smooth vector fields. In what follows, we will look into some nonlinear structures of the Weil–Petersson geometry. To make the setting precise, first recall that the Uniformaization Theorem guarantees that each complete hyperbolic metric G t on the unit disc D can be represented by ' t G0 with a quasi-conformal diffeomorphism ' t W D ! D. We impose that for each t the map ' t fixes three points, say .1; 0/; .0; 1/ and .1; 0/ on the boundary @D D S 1 , to fix an SL.2; R/ gauge. We consider a harmonic map u t W .D; G0 / ! .D; G t / which, viewed as a map from the unit disc D2 to itself, has the trivial extension to the geometric boundary: u t jS 1 D IdS 1 . This condition is in place of specifying the homotopy type of the harmonic map into the compact surfaces. Recalling the definition of the pulled-back metric, we have ut

't

.D; G0 / ! .D; G t / ! .D; G0 /; where the map ' t is an isometry. Thus the composite map ' t B u t W .D; G0 / ! .D; G0 /

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is a harmonic map with the asymptotic boundary condition ' t B u t jS 1 D ' t jS 1 , which is a quasi-symmetric map from S 1 to itself. As defined above, given map from a surface to a surface, one can define R a harmonic N 2 dg . In our setting here, the usual energy is diverN the @-energy E@N .G/ WD † 2j@uj R gent as the area functional is divergent. Thus disregarding the @-energy † 2j@uj2 dg amounts to a renormalization of the energy functional. Indeed, in [81], this was made analytically rigorous. N of the harmonic map First we remark that for  2 QC.D/ the @-energy u W .D; G0 / ! .D; G D '  G0 / N is equal to the @-energy of the harmonic map ' B u W .D; G0 / ! .D; G0 /: N Hence our @-energy functional E@N .G/ formally defined on the universal Teichmüller N space UT is identified with the @-energy of the harmonic map ' t B u t W .D; G0 / ! .D; G0 / where the G dependence is replaced by the ' t jS 1 -dependence. This enables us to use harmonic maps to identify the points of the universal Teichmüller space QS.S 1 /=SL.2; R/, in analogy with the compact surface cases. Theorem 4.6. For a quasi-conformal map ' W D ! D with boundary restriction 'jS 1 W S 1 ! S 1 in QS.S 1 / \ C 2 .S 1 I S 1 /, let u be the harmonic map from .D; G0 / N E@N .'  G0 / to itself with asymptotic condition ujS 1 D 'jS 1 . Then the @-energy is finite. We note that for a C 1;˛ diffeomorphism f W S 1 ! S 1 , it is known by the work of Li–Tam [44] that there exists a unique proper harmonic map u W .D2 ; G0 / ! .D2 ; G0 / with ujS 1 D f . In particular the identity map is the unique harmonic map from the Poincaré disc to itself with the trivial asymptotic boundary condition. Therefore under the hypothesis of the above theorem, the correspondence between the hyperbolic metric '  G0 and the harmonic map u is justified. The proof of this statement [81] is by writing down the asymptotic expansion of the harmonic map near the geometric boundary. It turns out, from the analysis of Li–Tam, that the harmonic map is asymptotically conformal, namely the map u becomes conformal as the point approaches the geometric boundary measured with respect to its defining function. Then we impose the fact that the map has zero tension field  .u/, which is equivalent to the harmonic map equation. The deviation of u from a conformal map can be controlled by the fact that the consecutive terms in the asymptotic expansion decay in certain rates imposed by the vanishing of the tension field. The C 2 regularity condition was used here in the expansion. In fact, we obtain that in the upper half plane model G0 D .dx 2 C dy 2 /=y 2 , the energy density of u as well as the area density/Jacobian of u behave as e.u/ D 1 C O.y 2 /;

J.u/ D 1 C O.y 2 /;

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where y > 0 is the defining function of the geometric boundary fy D 0g. Thus 2 N N 2 the @-energy density e.u/ R  J.u/2 D 2j@uj is of the form O.y /, which in turn N implies that its integral H2 2j@uj dG0 is finite. Note that if the map is conformal, N we have e.u/ D J.u/ D 1 everywhere, and the @-energy vanishes in that case. We 2 do not expect C regularity of the asymptotic boundary map ' to be optimal. The optimal regularity should be closely linked to the regularity of the quasi-symmetric diffeomorphism obtained by integrating a family of H 3=2 -smooth vector fields on S 1 , where the integration is specified by the Weil–Petersson exponential map. However, little understanding of the exponential map in infinite-dimensional settings (cf. [29], [51]) exists, and such a direction of research would be highly nontrivial as well as important. N 4.6.2 @-energy as a Weil–Petersson potential. Having the R-valued functional E@N at hand, we look at its behavior near the hyperbolic metric G0 , or equivalently the identity map Id W .D2 ; G0 / ! .D2 ; G0 /. Theorem 4.7. Suppose that G t D ' t G0 is a Weil–Petersson geodesic so that GP 0 is a Lie derivative LZ G0 , where Z is a divergent (in G0 sense) vector field inducing an H 3=2 -smooth vector field on the geometric boundary S 1 . We further suppose that the one-parameter family of the quasi-symmetric maps ' t jS 1 are all C 2 smooth, N N E@N .G t / corresponding to the finite @-energy harmonic maps u t . Then the @-energy satisfies E@N .G0 / D 0;

ˇ d E@N .G t /ˇ tD0 D 0 and dt

ˇ d2 1 E@N .G t /ˇ t D0 D : 2 dt 2

N In particular, the @-energy E@N defined on DiffS 1 =SL.2; R/  UT is Weil–Petersson convex at ŒG0 . Proof. The first statement E@N .G0 / D 0 is clear, as the identity map u0 W .D; G0 / ! .D; G0 / is the unique harmonic map, which is conformal. d u t j t Dt0 . We first show We denote by W t0 the vector field dt Lemma 4.8. The vector field W0 is identically zero. Proof. Recall that for G t D ' t G0 , the map ' t Bu t W .D; G0 / ! .D; G0 / is a harmonic map with the asymptotic boundary condition ' t Bu t jS 1 D ' t jS 1 . Denote ' t Bu t by uQ t , d and its time derivative dt uQ t j tD0 by V .z/, where z is the standard complex coordinate for the unit disc D. The harmonic map equation for uQ t is then uQ tz zN C

.uQ t /uQ t t t uQ uQ D 0;

.uQ t / zN z

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4 where we have shifted the index t up for uQ t temporarily, and .w/ D .1jwj 2 /2 . By differentiating the harmonic map equation in t , and evaluating at t D 0, we have

Vz zN C

.z/z VzN D 0

which is equivalent to @z . .z/@zN V / D 0. Recall that this says that the deformation tensor LV G0 is a traceless divergence-free tensor. On the other hand, as ' t G0 is a Weil–Petersson geodesic, by the L2 -decomposition theorem, its time derivative a t D 0 is a traceless divergence free tensor, which can be d d G t j tD0 . If we set dt u t j t D0 D W , then by differentiwritten as LX G0 where X D dt ating ' t B u t at t D 0, we have V .z/ D X.z/ C W .z/. As the L2 .G0 / integrable space of traceless divergence-free tensors are linear, LW G0 is again a traceless divergencefree tensor, or equivalently @z . .z/@zN W / D 0. As u t is a one-parameter family of harmonic maps from .D; G0 / to .D; G t / fixing the geometric boundary @D2 D S 1 , we have W jS 1 D 0. As the only vector field satisfying @z . .z/@zN W / D 0 with @D D S 1 is the trivial vector field, we obtain that W0 D 0. This lemma should be contrasted with the observation by Ahlfors [2] that one can choose 3g  3 harmonic Beltrami differentials such that their complex linear combinations form a Weil–Petersson geodesic normal coordinate system. Also M. Wolf [72] shows that the Weil–Petersson geodesic G t is approximated by a path G0 C t GP0 up to order two where GP 0 is a deformation tensor induced by a harmonic Beltrami differential, which in this context, is equal to the linearized Hopf differential of the one-parameter family of harmonic maps whose target metrics are G t . Now we differentiate the energy density e.u t ; G t / D 12 Tr G0 .ut G t / and the area p det.u t G0 / density J.u t ; G t / D pdet G in t . 0

ˇ ˇ d 1 d e.u t ; G t /ˇ tDt D Tr G0 .ut G t /ˇ t Dt 0 0 dt dt 2 1 1 D Tr G0 .ut0 GP t0 / C Tr G0 .ut0 ŒLW t0 G t0 /: 2 2 ˇ d Note here that when t0 D 0, dt e.u t ; G t /ˇ t Dt D 0 as W0 D 0, u0 .z/ D z and 0 Tr G0 .GP 0 / D 0. Differentiate this expression one more time, and obtain iˇ iˇ ˇ d2 d h1 d h1  P  ˇ ˇ ˇ Tr Tr G e.u ; G / D .u / C .u ŒL G / t t tDt G0 G0 Wt t t t t t Dt0 t Dt0 0 dt 2 dt 2 dt 2 1 1 D Tr G0 .ut0 GR t0 / C Tr G0 .ut0 ŒLW t0 GP t0 / 2 2 1 1  C Tr G0 .u t0 ŒLWP t G t0 / C Tr G0 .ut0 fLW0 ŒLW t0 GP t0 g/ 0 2 2 1  C Tr G0 .u t0 ŒLW t0 GP t0 /: 2

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By evaluating the expression at t0 D 0, we get ˇ d2 1 1 e.u t ; G t /ˇ tD0 D Tr G0 .ut0 ŒLWP 0 G0 / C Tr G0 .u0 GR 0 /: 2 2 dt 2 Recall that in the Fuchsian setting the term 12 Tr G0 .ut0 ŒLW t0 G t0 / was identically zero, as this is the first variation of the energy and the maps u t were all harmonic as t varies. This would be the case here too if the variational vector field W were compactly supported. However in the current setting, the hyperbolic norm of the vector fields W t0 (t0 ¤ 0) is asymptotically divergent, and hence the term cannot be assumed to vanish. The derivatives of the area density are given by p ˇ d d det.ut G0 / ˇˇ ˇ J.u t ; G t / tDt D p 0 dt dt det G0 t Dt0 p det.ut0 G0 / 1  D Tr ut G t0 .u t0 ŒLW t0 / p 0 2 det G0 p  det.u 1 t0 G0 / C Tr ut G t0 .ut0 GP t0 / p 0 2 det G0 p  det.u t0 G0 / 1 : D Tr ut G t .ut0 ŒLW t0 / p 2 det G0 The second equality is due to the fact that 12 Tr ut G t0 .ut0 GP t0 / is zero for 12 Tr G t0 .GP t0 / is zero as GP t0 is traceless. When t0 is zero, the fact that W0 D 0 implies that d J.u t ; G t /j tDt0 dt

D 0. Combined with

d e.u t ; G t /j t Dt0 dt

D 0 as shown above, it

follows that ˇ d E@N .G t /ˇ t D0 D 0: dt Lastly we have p h1 ˇ det.ut G0 / iˇˇ d2 d  ˇ  G .u ŒLW G t / Tr J.u ; G / D p t t u t t t tDt0 t Dt0 t dt 2 dt 2 det G0 p  det.u t0 G0 / 1 D Tr ut G0 .ut0 ŒLWP t G t0 / p 0 0 2 det G0 p det.ut0 G0 /  hLW t0 G t0 ; LW t0 G t0 iut G t0 p 0 det G0 p  det.u t0 G0 /  hLW t0 G t0 ; GP t0 iut G t0 p 0 det G0 p h 1 d det.ut G0 / iˇˇ  C Tr ut G t .u t0 ŒLW t0 G t0 / : p t Dt0 2 dt det G0

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Setting t0 D 0 again, we obtain ˇ 1 d2 J.u t ; G t /ˇ tD0 D Tr G0 .LWP 0 G0 /: 2 2 dt N As the @-energy E@N .G t / is the integral of e.u t ; G t /  J.u t ; G t /, we have Z iˇ ˇ d2 h d2 ˇ ˇ dG e.u E .G / D ; G /  J.u ; G / N t t t t t 0 tD0 t D0 2 dt 2 @ H2 dt Z 1 Tr G0 .GR 0 /dG0 : D H2 2 Finally at t D 0, we have the following simple equality: ˇ d Tr G t .z/ GP t .z/ˇ tD0 D hGP 0 .z/; GP 0 .z/iG0 .z/ C Tr G0 .z/ GR 0 .z/ D 0 dt as the Weil–Petersson geodesic has traceless tangent vectors, Tr G t GP t D 0 for all t . Therefore we have Z ˇ 1 1 P P d2 ˇ E .G / D hG0 ; G0 iG0 dG0 D : N t tD0 2 dt 2 @ 2 2 H Variations of Weil–Petersson convexity have been obtained in several contexts previously. Wolf [72] showed that for the harmonic maps u t W .†; G0 / ! .†; G t / for a closed surface † homotopic to the identity map, the Hessian of the energy functional is equal to twice the Weil–Petersson pairing. Fischer–Tromba [25] showed that instead of varying the target hyperbolic metrics, by varying the domain metrics along Weil– Petersson geodesics, the Hessian of the resulting energy functional gives twice the Weil–Petersson pairing as well. Also one should mention the work of Takhtajan–Teo [67] where they looked at the universal Teichmüller space as a union of uncountable components each of which has a structure of Hilbert manifold. In particular, the connected component containing the identity is a topological group whose Hilbert structure is the space of H 3=2 -integrable vector fields we have encountered in the work of Nag–Verjovsky [56]. Also they formulated another Weil–Petersson Kähler potential at the identity, called universal Liouville action. One should note that this is not an exhaustive list of Weil–Petersson potentials, as there have been many different approaches to the universal Teichmüller space, some from theoretical physics.

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5 Metric completion and CAT(0) geometry 5.1 Metric completion of Teichmüller space We have shown the existence of many convex and proper functionals defined on T with respect to the Weil–Petersson metric. On the other hand, it has been known (Wolpert [74], Chu [13]) that the Weil–Petersson metric is not complete. Namely Proposition 5.1. Suppose that  W Œ0; T / ! T is a Weil–Petersson geodesic which cannot be extended beyond T < 1. Then for any sequence ftn g with limn!1 tn D T , the hyperbolic length of the shortest geodesic(s) on the surface .†;  .tn // goes to zero. The statement follows from the so-called Mumford–Mahler compactness of moduli space (see for example [32], [70]), namely if there exists a lower bound for the injectivity radius, then the Weil–Petersson geodesic  lies away from the nodal surfaces, namely in the interior of the Teichmüller space T , contradicting the inextendability of  beyond T . Hence along an inextendable Weil–Petersson geodesic  W Œ0; T / ! T , the convex and proper functional E..t// blows up as t ! T . We have already seen such an occurrence in the proof of the properness of the energy functional. Namely, when a simple closed geodesic is pinched, then any simple closed geodesic transverse to the pinched loop gets arbitrarily long, which would then induce the blow up of the energy of the harmonic map, due to the De Giorgi–Nash–Moser estimate. In other words, the pinching of necks are the only cause of the incompleteness. This observation is available since the 1970s when Bers [7] and Abikoff [1] formulated the so-called augmented Teichmüller space, and when H. Masur analyzed in 1976 the decay of the Weil–Petersson metric tensor as a neck is pinched. Masur in his paper even used the notation Tx to denote the augmented Teichmüller space. In 2000 (2001 arXiv paper [82], part of which appeared in [83] in 2004) the author proposed to look at the Weil–Petersson metric completion Tx of the Teichmüller space T as a CAT(0) space, that is, a non-positively curved geodesic space, which as a set is the augmented Teichmüller space. Recall that a metric space .X; d / is a CAT(0) space, (or an NPC space as called in [36] for non-positively curved space) when • .X; d / is a length space. That is, for any two points P and Q, in X , the distance d.P; Q/ is realized as the length of a rectifiable curve connecting P and Q. Such curves are called geodesics. • For any three points P , Q and R in X, and choices of geodesics PQ of length r, QR of length p, and RP of length q connecting the respective points, the following comparison property holds: For any 0 < < 1 write Q for the point on QR satisfying d.Q ; Q/ D p;

d.Q ; R/ D .1  /p:

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On the (possibly degenerate) Euclidean triangle of side length p; q and r, and x and R, x there is a corresponding point opposite vertices Px ; Q x D Q x C .R x  Q/: x Q The CAT(0) hypothesis is that the metric distance d.P; Q / from Q to its x j. This can opposing vertex is bounded above by the Euclidean distance jPx  Q be written as d 2 .P; Q / .1  /d 2 .P; Q/ C d 2 .P; R/  .1  /d 2 .Q; R/: When a metric space .X; d / is CAT(0), then it follows that a geodesic is unique given its end points, and that the space is simply connected and contractible [36], [11]. The most familiar examples of CAT(0) spaces are the simply connected complete Riemannian manifolds having non-positive sectional curvature. In particular, a Teichmüller space with Weil–Petersson metric is a CAT(0) space, although it is an incomplete metric space. We note that the space Tx is not a compactification but a metric completion, each point representing a Cauchy sequence in T . The author’s contribution in this regard is the simple observation that the triangle comparison property of the Weil–Petersson distance function (appearing in the definition of CAT(0) space) on T extends to the metric completion Tx as a point-wise convergence of a sequence of convex functions produces a convex function; an observation which took nearly 25 years to materialize since the paper of Masur [47] appeared, over which period the field of metric space geometry had sufficiently matured and begun to be widely studied. The space Tx is bigger than T by the set of all nodal surfaces resulting from degeneration of the neck-pinchings from the original surface †. We describe the setting of the metric completion Tx more carefully. Detailed descriptions can be found in [83] and also in [78]. We first let  be the set of free homotopy classes of homotopically nontrivial simple closed curves on the surface †0 . This set can be identified with the set of simple closed geodesics on the surface with a hyperbolic metric. Then define the complex of curves C./ as follows. The vertices/zero-simplices of C./ are the elements of . An edge/one-simplex of the complex consists of a pair of homotopy classes of disjoint simple closed curves. A k-complex consists of k C 1 homotopy classes of mutually disjoint simple closed curves. A maximal set of mutually disjoint simple closed curves, which produces a pants decomposition of †0 , has 3g 3 elements. We say a simplex  in C./ precedes a simplex  0 provided    0 , and we write    0 . We say that a simplex  in C./ strictly precedes a simplex  0 provided  ¨  0 , and write  >  0 . This defines a partial ordering by reverse inclusion in the complex of curves C.P /, and thus makes it a partially ordered set (poset.) We define the null set to be the .1/-simplex. Then there is a C./[;–valued function ƒ, called labeling, defined on Tx as follows. Recall a point p in T represents a marked Riemann surface .†; f / with an orientation-preserving homeomorphism f W †0 ! †. The Weil–Petersson completion Tx consists of bordification points of T so that † is allowed to have nodes, which are geometrically interpreted as simple closed geodesics of zero

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hyperbolic length. Thus a point p in Tx nT represents a marked noded Riemann surface .†; f / with f W †0 ! †. We now define ƒ.p/ to be the simplex of free homotopy classes on †0 mapped to the nodes on †. We denote the fiber of ƒ W Tx ! C./ [ ; at a point  2 C./ [ ; by T . We denote its Weil–Petersson completed space by Tx . The completed space Tx has a stratification in the sense of [11], S Tx D 2C./[; T ; where the original Teichmüller space T is expressed as T; . Each stratum T is the Teichmüller space of the nodal surface † . An important fact is that the set of nodal surfaces exactly corresponds to the set of admissible degenerations of conformal structures while the surfaces are uniformized by hyperbolic metrics. Properties of Tx . The following is a set of properties satisfied by the Weil–Petersson completion of Teichmüller space. 1) Tx is a union of strata Tx ;

2) if Tx D Tx then  D ; 3) if the intersection Tx \ Tx of two strata is non-empty, then it is a union of strata; 4) for each p 2 Tx there is a unique .p/ 2 C./ such that the intersection of the strata containing p is Tx.p/ .

The author showed in [83] that this stratification is very much compatible with the Weil–Petersson geometry. Namely, for each collection  2 C./, each boundary Teichmüller space T is a Weil–Petersson geodesically convex subset of Tx . Here geodesic convexity means that given a pair of points in T , there is a distance-realizing Weil–Petersson geodesic segment connecting them lying entirely in T . We do not reproduce the proof here, but we mention a key idea of the proof. Consider the sublevel set S.; "/ WD fx j L .x/ < "g of the geodesic length functional L of a simple closed geodesic indexed by  in C./, a subset in the CAT(0) space .Tx ; d /. As the length functional is Weil–Petersson convex, for each " > 0, S.; "/ is a convex subset of Tx . By varying the value of ", we have a collection of nested convex sets S.; "/: for 0 < "1 < "2 , S.; "1 /  S.; "2 / with S.; "1 / \ S.; "2 / D S.; "1 / convex. In light of this fact, the frontier set Tx D fxjL .x/ D 0g can be regarded as T ">0 S.; "/; a convex set, as an intersection of all the receding convex sets S.; "/ as " ! 0.

5.2 The Weil–Petersson metric tensor near the strata One can see what happens locally on the surface † when a neck pinches to become a node near the frontier sets T . The model case is the standard hyperbolic cylinder

Chapter 1. Local and global aspects of Weil–Petersson geometry

Ajt j D fz j jtj=c < jzj < cg dsjt2 j

ˇ



91

ˇ2

 log jzj ˇˇ dz ˇˇ  csc D log jt j log jt j ˇ z ˇ

:

Here the hyperbolic length L0 .t/ of the waist of the cylinder is equal to 2 2 = log.1=jtj/. 2 / converges pointwise to the hyperbolic As jt j ! 0, the hyperbolic annulus .Ajtj ; dwjtj metric on two copies of the punctured disc fz j 0 < jzj < cg 

ds02 D

jdzj jzj log jzj

2

which models the standard hyperbolic cusp. In [83] it was shown that the thin part of the surface can be approximated by j j 2 copies of the standard hyperbolic cylinder, and the Weil–Petersson metric tensor dsWP behaves as 2 dsWP

D

ds2

jj hX

C 4 .1 C O.kuk // 3

3

j D1

where



j D arg tj

and

uj D

log

i 1 duj2 C .uj /6 d j2 ; 4 1 jtj j

1=2

I

the constant 4 3 is due to Wolpert [75]. The proof of this asymptotic expansion follows the line of arguments in Masur’s 1976 paper [47], where the components of the Weil–Petersson cometric tensor was written as various pairings of holomorphic quadratic differentials. This expansion captures the singular behavior of the Weil–Petersson metric, namely as the necks pinch, the Fenchel–Nielsen twist parameters approximated by j become increasingly ineffective in terms of Weil–Petersson norm. Indeed, Wolpert (see his exposition [80]) demonstrated that the Weil–Petersson sectional curvature of the plane spanned by fr`j ; J r`j g blows down as O.`j1 /, where J is the Weil–Petersson complex structure. Geometrically the blow-down behavior of the Weil–Petersson sectional curvature describes that the transversal section to the frontier set T is modeled by an R-tree of uncountable degree, or the universal covering space of an incomplete cusp as pictured in [80], whose vertex/cuspidal point is represented by the nodal surface † . We will utilize this Weil–Petersson asymptotic structure to construct the Weil–Petersson geodesic completion in the next section.

5.3 The Weil–Petersson isometric action of the mapping class group and equivariant harmonic maps We consider harmonic maps into the Weil–Petersson completed Teichmüller space Tx . In particular, we are concerned with how the maps behave with respect to the

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stratification and isometric actions. It turns out that the harmonic map respects the Weil–Petersson stratification property as much as it can, in the sense that one can impose an equivariance condition on the harmonic map, and the push-forward 1 of the domain as a subgroup of the mapping class group forces the map to stay in certain strata. It should be remarked at this point that the full Weil–Petersson isometry group of Teichmüller space is known to be the extended mapping class group Map.†/ [48]. One can also refer to the exposition [15] in this Handbook. We first need to recall the Thurston classification theorem [69] of elements of the mapping class group Map.†/. An element  of Map.†/ is classified as having a representative, also called , which is of one of the following three types:

2

1) of finite order, also called periodic or elliptic; 2) reducible if it leaves a tubular neighborhood of a collection  of closed geodesics c1 ; : : : ; cn invariant; 3) pseudo-Anosov (also called irreducible) if there is r > 1 and transverse measured foliations FC ; F such that .FC / D rFC and .F / D r 1 F . In this case the fixed point set of the  action in P MF .†/ (the Thurston boundary of T ) is precisely FC ; F . As for the classification of subgroups, McCarthy and Papadapoulos [49] have shown that the subgroups of Map.†/ is classified into four classes: 1) subgroups containing a pair of independent pseudo-Anosov elements (called sufficiently large subgroups); 2) subgroups fixing the pair fFC ./; F . /g of fixed points in P MF .†/ for a certain pseudo-Anosov element  2 Map.†/ (such groups are virtually cyclic); 3) finite subgroups; 4) infinite subgroups leaving invariant a finite, nonempty system of disjoint, nonperipheral simple closed curves on † (such subgroups are called reducible.) Those group-theoretic classifications are relevant to the Weil–Petersson geometry in the sense that the pseudo-Anosov elements are loxodromic [77], [14], namely the infimum of the translation distance d.x; x/ is achieved on a unique pseudo-Anosov axis in T , and the reducible (by  ) elements are loxodromic in the respective stratum T . This implies that there are neither parabolic elements nor parabolic subgroups in the sense of symmetric spaces of noncompact type. This statement was first claimed by the author in [82], where a proof was presented by noting that for a one-dimensional harmonic map u W .a; b/ ! T and for the distance function d. ; Tx / to the stratum T , the pulled back distance function u d. ; T / is super-harmonic, namely concave, then using the minimum principle to show that either the image of the map u entirely lies in the stratum T or otherwise entirely in the interior T . The argument is suggestive, but the known differentiability of d near the stratum was not enough to make the proof complete. Wentworth–Daskalopoulos and Wolpert then came up with proofs

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by a length comparison argument, or no-refraction argument (see [77], [14] or [80], Chapter 5) that the harmonic image, or equivalently an open Weil–Petersson geodesic segment cannot lie partly in T and partly in T . Once this is established, the existence of the pseudo-Anosov axis follows readily. For harmonic maps with higher-dimensional domains, we define a functional on the Weil–Petersson completion Tx of the Teichmüller space T . Definition 5.2. Suppose  is a finitely generated subgroup of Map.†/, with fi g1il its generators. Then we define a functional ı on Tx by ı.x/ D max d.x; i x/: 1i l

Note here that ı W Tx ! R [ f1g is a convex functional, since each d.x; i x/ is convex on Tx due to the NPC curvature condition (see for example [11], II.2), and since the maximum of finitely many convex functionals is again convex. Definition 5.3. Given a subgroup  of Map.†/, the isometric action of  on Tx is said to be proper if the sublevel set S.M / D fx 2 Tx W ı.x/ < M < C1g is bounded in Tx for all M < 1. A simple yet important consequence of the lack of parabolic elements in the mapping class group, which follows from the results in [37], [14], is that a representation W 1 .M / ! Isom.Tx /, where .1 .M // is a sufficiently large subgroup of Map.†/ induces a proper isometric action on the Teichmüller space. Furthermore, when .1 .M // is reducible by  2 C./, then the isometric action of .1 .M // on T is proper. A general framework developed by Korevaar–Schoen [37] provides the existence of equivariant harmonic maps for representations which induces a proper isometric action on the Teichmüller space. Theorem (Existence). Suppose M is a compact manifold without boundary. Suppose that a representation W 1 .M / ! Isom.Tx / induces an isometric action on the Teichmüller space by a sufficiently large subgroup .1 .M //. Then there exists an energy z ! Tx which is -equivariant. (M z is the universal minimizing harmonic map u W M covering space of M .) Moreover the map u is Lipschitz continuous. When M is not compact but complete, under mild additional conditions which often are met for many applications, we still have an existence theorem. As for the uniqueness of the map, it follows from a standard argument using the so-called geodesic homotopy [30], with some extra attention needed when the harmonic map touches upon several distinct strata. A complete proof is provided in [86].

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Theorem (Uniqueness). The harmonic map is unique within the class of finite energy maps which are -equivariant, provided that the image of is not reducible by any element  of C./ and that the image of the map is not contained in a geodesic. Note that the condition for the theorem is satisfied when u 1 .M / is a sufficiently large subgroup of the mapping class group. We will discuss an application. A Kähler manifold M of real dimension four is said to have a structure of a holomorphic Lefschetz fibration if the following descriptions hold. There exists a holomorphic map … W M 4 ! B 2 where the base space B is a surface. The map … has finitely many critical points Ni , i D 1; : : : ; n, in disjoint fibers Fi D …1 .Pi /, i D 1; : : : ; n, each of which is a nodal surface, while away from those disjoint fibers, each fiber of the map … is a Riemann surface of varying conformal structures of a fixed genus g. The neighborhood of each critical point Ni can be described by local complex coordinates z, w on M and t on B such that … W .z; w/ 7! zw.D t / where Ni D .0; 0/ 2 C 2 and Pi D 0 2 C. The picture above can be transcribed as saying that there exists a -equivariant holomorphic map uQ W BnfPi g ! T , where W 1 .BnfPi g/ ! Map.†/ is the monodromy representation of the fibration. The uniqueness theorem of harmonic maps has an immediate application, which was first proved by Shiga [65] by a different method. See the exposition in [66].

B

Corollary. Given a holomorphic Lefschetz fibration whose non-singular fibers are closed surfaces of genus greater than one, its monodromy representation is sufficiently large. Proof. It is well known that a holomorphic map between two Kähler manifolds is energy-minimizing [61]. Hence the map uQ W BnfPi g ! Tx is the unique -equivariant harmonic map whose existence and uniqueness have been so far established. To see that  is sufficiently large, note that if it were not, then we have three other possible cases. The first being when  is a finite group can be excluded since each local monodromy is of infinite order. The second being the case when  is virtually cyclic, fixing a pair of points in the Thurston boundary. Then the image of the -equivariant harmonic map uQ W BnfPi g ! Tx is a  invariant Weil–Petersson geodesic, which lies entirely in the interior of the Teichmüller space T . The projection of the invariant geodesic down to the moduli space is a loop located away from any of the divisors, which in turn says that there is no sequence of points fqj g in BnfPi g over which a cycle on the Riemann surface represented by u.Pi / vanishes (or equivalently a neck is pinched), which contradicts the fact that M 4 has singular fibers/vanishing cycles. Lastly the third possible case to be excluded is when  can be reduced by a collection of C mutually disjoint simple closed curves. Then the harmonic image of uQ is entirely contained in TC , which implies that every fiber is a Riemann surface with nodes where the nodes are obtained by pinching each simple closed curves in C . This certainly is not the case when M is a Lefschetz fibration.

B

B

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95

6 Geodesic completion and CAT(0) geometry 6.1 Construction of the geodesic completion Having established the Weil–Petersson metric completion Tx , it is natural to seek for a geodesic completion of the Teichmüller space with respect to the Weil–Petersson distance function. Such a space was constructed in [85] where it is named Teichmüller Coxeter complex. We will briefly describe the construction of the space below. The theory of Coxeter group was developed in an attempt to understand combinatorial and geometric characterizations of tessellation of standard spaces such as Rn , S n and Hn by reflecting convex polygons across the sides of the polygons. the sides of those polygons are totally geodesic in the ambient space as they function as the interfaces between two open convex sets. Each reflection is of order two, and the group generated by all the reflections is called the Coxeter group. Naturally the geometry of the vertices (given as the cone angles) of the polygon comes into the structure of the Coxeter group. Around 1960, Jacques Tits introduced the notion of an abstract reflection group, which he named “Coxeter group” .W; S /. It is a group W generated by a set of reflections S and a collection of relations among the reflections 0 f.ss 0 /m.s;s / g. Here m.s; s 0 / denotes the order of ss 0 and the relations range over all unordered pairs s; s 0 2 S with m.s; s 0 / ¤ 1. In other words, m.s; s 0 / D 1 means no relation between s and s 0 . The data Œm.s; s 0 /.s;s 0 / can be regarded as a matrix and is said to constitute a Coxeter matrix. The following list presents a set of evidence that the Weil–Petersson geometry of Tx and the Coxeter theory are indeed compatible. • The Teichmüller space T is Weil–Petersson geodesically convex. • For every element  of the complex of curves C./ [ ;, the frontier set Tx is a complete convex subset of Tx , altogether forming a stratified space Tx . • Given a point p in T  Tx , the Alexandrov tangent cone with respect to the jj Weil–Petersson distance function is isometric to Rjj 0  Tp T , where R0 is the orthant in Rj j with the standard metric (Wolpert [78]). • Tx is the closure of the Weil–Petersson convex hull of the vertex set given by the zero-dimensional Teichmüller spaces of the maximally degenerate surfaces fT j j j D 3g  3g (Wolpert [77]). The Alexandrov tangent cone Cq X of a CAT(0) space (see for example [11]) is a space of equivalence classes of constant speed geodesics originating at q where two geodesics are deemed equivalent when they share the same speed and they form a zero Alexandrov angle. To be precise, the Alexandrov angle is defined by d.q; 0 .t//2 C d.q; 1 .t //2  d.0 .t /; 1 .t //2 : t!0 2d.q; 0 .t //d.q; 1 .t //

cos †.0 ; 1 / D lim

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The significance of the Weil–Petersson tangent cone structure in our context is that it describes the geometry around the vertices, given as the Weil–Petersson tangent cone angles, when Tx is seen as a convex polygon. This picture specifies a particular choice of the Coxeter matrix. Namely for each  with j j D 1, one can reflect Tx across the totally geodesic stratum Tx . Now for  D  [  0 with  and  0 representing a pair of disjoint simple closed geodesics, the relation m.s ; s 0 / D 2 has a geometric meaning where four copies of Tx can be glued together around a point q 2 T to form a space whose tangent cone at q is a union of four copies of R20  Tp T (each R20 is regarded as a quadrant in the plane) isometric to R2  Tp T on which the reflections s ; s 0 act as reversing of the orientations of the x; y axes for R2 . Hence we define a Coxeter group .W; S / by letting the generating set S be the elements of , and the relations among the elements of the generating set specified by the Coxeter matrix m.s; s 0 / whose components satisfy i) m.s; s/ D 1, ii) if s ¤ s 0 , and if there is some simplex  in C./ containing s and s 0 , then m.s; s 0 / D 2, and iii) if s ¤ s 0 , and if the geodesics representing s and s 0 intersect on †0 then m.s; s 0 / D 1. This group has a geometric realization acting on a collection of copies of Tx ’s. We remark that the Coxeter group with such a Coxeter matrix as above is said to form a cubical complex, for it has a canonical geometric realization where each generating element is represented as a linear orthogonal reflection across the face of the infinite-dimensional unit cube in RjSj . However we instead form a geometric realization D.Tx ; / as the set which is the quotient of W  Tx by the following equivalence relation: .g; y/  .g 0 ; y 0 / () y D y 0 and g 1 g 0 2 W.y/ ; where Tx .y/ denotes the smallest stratum containing y, and the subgroup W.y/ fixes the stratum Tx.y/ pointwise. We write Œg; y to denote the equivalence class of .g; y/. Furthermore  denotes a simple morphism of groups, which specifies a system of subgroups W  W , compatible with the poset structure of the complex of curves C./ (see [85] for details.) The remarkable phenomenon here is that despite the fact that the generating set is infinite, we have a geometric realization of the Coxeter group .W; S / action (which is very far from linear) on a space modeled on a finite-dimensional space T , albeit the partial bordification Tx encodes non-locally compact geometry due to the singular behavior of the Weil–Petersson metric tensor. Note that the singularity is also manifest in the fact that the reflecting wall Tx with j j D 1 is of complex codimension one, instead of being a real hypersurface as in the standard Coxeter theory, a situation reminiscent of the theory of complex reflection groups (see [64] for example.) We also make a remark that the Weil–Petersson metric defined on each stratum T is Kähler, hence the space D.Tx ; / providing a geometric realization of the Coxeter group W is a “simplicial” complex with each face equipped with a Kähler metric, a situation unattainable in a real simplicial complex, where the reflecting walls are real hypersurfaces. The space obtained by the action of the Coxeter group on Tx , which we will call development D.Tx ; /, is then shown to be CAT.0/ via the Cartan–Hadamard theorem

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[4], and also to be geodesically complete. The reason for the negative curvature is due to the following geometric construction: Theorem (Y. Reshetnyak [60], Bridson–Haefliger [11]). Let X1 and X2 be CAT.0/ spaces (not necessarily complete) and let A be a complete metric space. Suppose that for j D 1; 2, we are given isometries ij W A ! Aj where Aj  Xj is assumed to be convex. Then X1 tA X2 is a CAT.0/ space. ` Here the space X1 tA X2 is the quotient space of the disjoint union X1 X2 by the equivalence relation generated by i1 .a/  i2 .a/ for all a 2 A. The resulting space, called gluing space or amalgamation of the CAT(0) spaces X1 and X2 along A has a canonical distance between x 2 Xj and y 2 Xj 0 defined as follows: d.x; y/ D dj .x; y/ D dj 0 .x; y/ if j D j 0 ; d.x; y/ D inf fdj .x; ij .a// C dj 0 .x; ij 0 .a//g if j ¤ j 0 : a2A

This is used in [85] to glue the copies of Tx along the strata Tx , where the number of copies at each x 2 Tx is determined by the number of nodes jj.x/j; for example if j j.x/j D 2, the tangent cone at the point x is the product of a first quadrant in R2 and the tangent space of Tx T.x/ , and by putting together 4 D 2j.x/j copies of Tx , the tangent cone of the resulting enlarged space is isometric to R2 times the tangent space of T .x/ . This glueing construction provides a locally CAT(0) space around each point in Tx . Then knowing that the Coxeter complex D.Tx ; / is simply connected, one can apply the Cartan–Hadamard theorem to identify D.Tx ; / with a geodesically complete CAT(0) space.

6.2 Finite rank properties of Tx The Weil–Petersson metric defined on T is a smooth Riemannian metric whose sectional curvature is negative everywhere ([32], [70], [75].) Hence the only flats (i.e. isometric embeddings of Euclidean space Rn , n  1) are the Weil–Petersson geodesics. The fact that there is no strictly negative upper bound for the sectional curvature is explained by the fact that the Weil–Petersson completion Tx does have higher-dimensional flats (see the concluding remark in [83], as well as Proposition 16 in [77]). Those flats arise when the collection  of mutually disjoint simple closed geodesics separates the surface † into multiple components. Then the frontier set T is a product space of the Teichmüller spaces of the components separated by the nodes. The number  which is achieved when the of connected components is bounded by g C . g2  1/, g  surface † is a union of g once-punctured tori and 2  1 four-times-punctured spheres. One  construct isometric embeddings of Euclidean spaces of dimension  can up to g C . g2  1/ by considering a set of Weil–Petersson geodesic lines, whose existence is established in [14] and [77], in the different components of the product space T . In this sense the Weil–Petersson completion Tx is a space of finite rank

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  where the rank is bounded by g C . g2  1/. This rank has been known to coincide with Brock–Farb’s geometric rank of Map.†/, as studied by Behrstock–Minsky [5]. There is another definition of rank, which we call FR, and it first appeared in [38]. Definition 6.1. An NPC (CAT.0/) space .X; d / is said to be an FR space if there exist any subset of X with diameter D > D0 is contained in a "0 > 0 and D0 such thatp ball of radius .1  "0 /D= 2. We make several remarks about FR spaces. The definition can be interpreted as follows. If X is an FR space, then among all the closed bounded convex sets F in X with its diameter larger than D0 , there exists some positive integer k such that r D.F / p kC1  2 ; inf F X R.F / k where D.F / and R.F / are the diameter and the circum-radius of F , respectively. It is p well known that Rk is FR with the optimal/largest choice of "0 D 1  k=k C 1 > 0 which is realized by the standard k-simplex. An infinite-dimensional Hilbert space is p > 0. It was shown not an FR space, while a tree is an FR space with "0 D 1  1=2 p in [37] that a Euclidean building is an FR space with "0 D 1  k=k C 1 > 0 with k the dimension of chambers. A CAT(-1) space (e.g.pthe hyperbolic plane H2 ) is FR with "0 which can be made arbitrarily close to 1  1=2 by taking the value of D0 large. Heuristically the number "0 > 0 detects the maximal dimension of flats inside the space X, that is, the rank of the given NPC space. We show in [85] that Theorem 6.2. The Weil–Petersson geodesic completion D.Tx ; / of a Teichmüller space T is FR. Corollary 6.3. The Weil–Petersson completion Tx of a Teichmüller space T is FR. p The proof of the theorem determines a lower bound of "0 to be 1  k=k C 1 with k D 6g  6, which is larger (for g > 1) than the maximal dimension of the flats as described above, which was g C . g2  1/. The particular value of k here should be regarded as the maximal dimension of a flat. Namely if there is a flat, its dimension cannot exceed 6g 6. On the other hand, the existence of those flats arising from the product structure of the frontier sets of Tx does not necessarily imply that the space is FR, as there may be infinite-dimensional flats elsewhere. The definition of FR spaces utilizes only the convexity of the distance function to describe the finitedimensionality of possibly existent flats, without directly dealing with the singular behavior of the Weil–Petersson metric tensor near the frontier set @Tx . The statement of the theorem [85] says that despite the lack of local compactness near the frontier set @Tx , the Weil–Petersson completion Tx of the Teichmüller space T exhibits finite rank characteristics.

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99

6.3 Weil–Petersson geodesic completeness Given a Riemannian manifold M and a codimension-two submanifold S  M , the open manifold N WD M nS has M as metric completion as well as geodesic completion with respect to the Riemannian distance function. (Consider M D R2 , S D f0g and N the punctured plane, for example.) This of course is expected with the Hopf–Rinow theorem available in the manifold setting, which in effect demonstrates the equivalence of metric completeness and geodesic completeness. Here the analogous picture is given by taking N D T , S D Tx nT , and M being either Tx or D.Tx ; /, depending on whether the completion is taken to be metric or geodesic. The disparity, that Tx is metrically complete but not geodesically complete, is caused by the singular behavior of the Weil–Petersson metric near the strata, where the points in the frontier set can be modelled as vertex points of cusps [77]. A consequence of the geodesic extension property (namely that each geodesic can be extended to a geodesic line) of the development D.Tx ; / is that for each point p 2 D.Tx ; /, the inverse map expp1 of the “exponential map” from D.Tx ; / to the tangent cone Cp D.Tx ; /, which is isometric to Rj.p/j , is surjective, as every geodesic segment starting at p can be extended to a geodesic line so that the image by the inverse exponential map is an entire real line through the origin of Rj.p/j . It is precisely this point that will be needed in the proof of the finite rank theorem. Namely, the finite rank of the space D.Tx ; / is shown by using Carathéodory’s theorem about convex hulls in Rn , which then implies the inequality between the diameter D and the circumradius R of convex sets in D.Tx ; /. Without the surjectivity of the inverse exponential map expp1 , the correspondence between the space of geodesics and the space of directions breaks down. Also one notes that the geodesic completeness is understood in the sense that any geodesic segment can be extended to a geodesic line, and there may be more than one extension (in fact uncountably many extensions) making expp multi-valued. This is once again due to the singular behavior of the Weil–Petersson metric tensor near the frontier sets, where the sectional curvature can blow down to 1, which causes that the behavior of geodesics resembles that of geodesics in R-trees. One should also mention that the development D.Tx ; / can be seen from billiard theory. This point was raised in the proof of existence of pseudo-Anosov axes by Wolpert [77], where a conditional sequential compactness of Weil–Petersson geodesics is established. A convergence can be guaranteed up to Dehn twists at the strata the geodesics hit. Namely a billiard ball is bounced back at each stratum with equal incoming and outgoing angles at the tangent cone level, but not in the nonlinear level once Weil–Petersson exponentiated. In Section 5 of [85], the author has laid out a detailed comparison between Wolpert’s statement and the situation for the development D.Tx ; /: The billiard ball goes through the wall/stratum only to find on the other side not knowing which directions to go in the Fenchel–Nielsen twist directions. In short, the billiard ball trajectory is deterministic in the tangent cone at the origin of the Weil–Petersson geodesic, but highly non-deterministic due to the R-tree like structure at the strata.

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6.4 Weil–Petersson isometric action and symmetry of D.Tx ; / It was shown ([48]) that the full isometry group of Tx is the extended mapping class group. Recall that Royden showed that the extended mapping class group is the full isometry group with respect to the Teichmüller distance. Thus the isometry group of D.Tx ; / contains a group which is the semi-direct product of the extended mapping class group and the Coxeter group, in which the extended mapping class group is a normal subgroup. For a Coxeter complex, there is a natural construction of an isometry group of the Coxeter complex which contains the original Coxeter group as a normal subgroup. In the current context, the fundamental domain is the Weil– Petersson completion Tx of the Teichmüller space T , on which the extended mapping class group Map† acts isometrically. The extended mapping class group Map† is known ([31], [40], [46]) to be the full automorphism group of the complex of curves C./. Using this fact, it has been shown [48] that the extended mapping class group is indeed the full isometry group of the Weil–Petersson completed Teichmüller space. Note that each element  of the extended mapping class group Map† preserves the Coxeter matrix, namely m..s/; .t// D m.s; t /:

1

1

1

As the Coxeter group W is generated by , and the group W is completely determined by the Coxeter matrix Œm.s; t/s;t 2 , it follows that each element  in Map† induces an automorphism of W . Such an automorphism of W is called a diagram automorphism [16]. The formalism laid out in M. Davis’book gives us a natural action (Proposition 9.1.7 of [16]) of the semi-direct product G WD W Ì Map† on the development D.Tx ; / as follows: given u D .g; / 2 G and Œg 0 ; y 2 D.Tx ; /,

1

1

u  Œg 0 ; y WD Œg.g 0 /; y; where .g 0 / is the image of g 0 by the automorphism of W induced by  W C./ ! C./. Clearly the action G ,! D.Tx ; / is isometric. Thus G is a subgroup of the isometry group Isom.D.Tx ; //. It remains an open question whether this group is indeed the full isometry group, and if not, how much larger the isometry group is. By applying a result of Davis’ book (Lemma 5.1.7 [16]) we note also the following. Theorem 6.4. The action of the Coxeter group W  G on D.Tx ; / is properly discontinuous. For a proof, see [84].

6.5 Embeddings of the Coxeter complex into UT A loosely formulated guiding philosophy in studying Teichmüller spaces is that the geometry of the space is somehow inherited from the geometry of the Riemann surfaces

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it is parameterizing. In this section, we proceed along this line of thinking by considering the space of embeddings of the Coxeter complex into the universal Teichmüller space. We first note that each simple geodesic in the closed hyperbolic surface † can act as a mirror introducing a reflective Z2 symmetry to a doubled cover. The symmetry is defined by first providing another copy of the surface, then cutting across the simple closed geodesic c, in both the original surface and the new copy, and lastly identifying the four ends by pairs so that for each simple closed geodesic c 0 transverse to c, the union c 0 [ c 0 of the original c 0 and the new copy c 0 is either a simple closed geodesic in the new surface † [ †, or a pair of simple closed geodesics. We denote the resulting surface † tc †. The distinction here is caused by the nature of the simple closed geodesic c. If c is non-separating, namely the punctured surface †nfcg consist of a path-connected component, then † tc † is path-connected, and c 0 [ c 0 is a single simple closed geodesic. When c is separating, † tc † consists of two copies of †. In the former case, the genus of the surface † tc † is 2g  1, in the latter case 2g. ` In the case of a separating geodesic c, we can embed the surface † tc † Š † † in a surface of genus 2g1 by cutting each of the †’s at a simple closed non-separating geodesic c 00 disjoint from c and pasting the two surfaces along (the four copies of) c 00 . 00 We denote the resulting surface by † tcc †. Geometrically it is a surface of genus 2g  1 with a Z2 -symmetry across the pair of simple closed geodesics corresponding to c 00 . Recall the construction of the Coxeter–Teichmüller complex D.Tx ; /, a quotient space W  Tx =  whose points are written as Œg; y. We identify a point Œg; y with a point in Teichmüller space of a higher genus surface as follows. One characteristic of the Coxeter complex is that each element g in W is written as a product of generators …N i D1 si of the Coxeter group W . For s1 D sc1 in the product, we extend the surface † by introducing an unramified double cover † tc1 † which has a symmetry across two copies of c1 . Note the unramified cover † tc1 † needs to be decorated by c100 in case c1 is a separating geodesic, which we have suppressed for now. For s2 D sc2 in the product, we next extend the surface † tc1 † by introducing an unramified double cover .† tc1 †/ tc2 .† tc1 †/ which has a symmetry across four copies of c2 . Inductively we can define a tower of double covers over the original surface †, its largest cover has genus  B    B .g/ ( composed N times) where .g/ D 2g  1. We denote the resulting surface tw †. The fact that the correspondence w 7! tw † is well-defined modulo the choices of ci00 ’s follows from the properties of the Coxeter group [10]. The Weil–Petersson metric needs to be normalized by the volume j†g j, so that the Weil–Petersson metric on †g and Weil–Petersson metric † .g/ with a Z2 symmetry are compatible. Namely hh1 ; h2 iWP

1 D .†g /

Z hh1 .x/; h2 .x/iG.x/ dG .x/: †g

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This follows from the equalities j†..g//j D ..g// D 2.g/2 D 2.2g1/2 D 2.2g2/ D 2.g/ D 2j.g/j: This simply says that by the doubling procedure via a reflection across a simple closed geodesic, the volume is doubled, which can be normalized by the topological invariant  to have a well-defined Weil–Petersson metric. This construction of higher genus Riemann surfaces ftw †gw2W provides a way of embedding the Coxeter–Teichmüller complex D.Tx ; / Weil–Petersson isometrically in the universal Teichmüller space UT . We need to remind ourselves that the embedding is not unique for two reasons. First there are no canonical embeddings of the Teichmüller spaces Tg in UT for g  1. Secondly, we recall that when g 2 W is generated by sc with c a separating simple closed geodesic, we need to enlist an extra parameter c 00 to obtain a path-connected double cover. Understanding the space of embeddings is far from complete, and we hope to make things better organized in the near future.

7 Teichmüller space as a Weil–Petersson convex body 7.1 Tx as a convex subset in D.Tx ; / The Coxeter complex setting allows to view the Teichmüller space as a Weil–Petersson convex set in an ambient space D.Tx ; /, bounded by a set of complex-codimension one “supporting hyperplanes” fD.Tx ; / j j j D 1g of the frontier stratum fTx g with each  representing a single node. Every boundary point is contained in at least one of the set of the supporting hyperplanes fD.Tx ; / j jj D 1g. In this picture, each D.Tx ; / is a totally geodesic set, metrically and geodesically complete, and when D.Tx1 ; / and D.Tx2 ; / intersect along D.Tx1 [2 ; /, they meet at a right angle. One can also look at the translates of fD.Tx ; / j j j D 1g by the action of the Coxeter group W . They form a right-angled grid structure in D.Tx ; /, whose lattice points are the orbit image by the Coxeter group W of the set fTx j j j D 3g  3g with

indexing the maximal set of nodes on the surface. Under this setting, for each  with jj D 1, consider a half-space, namely the set H , containing Tx in the D.Tx ; /, and bounded by D.Tx ; /. We note here the fact obtained by Wolpert [78] that the Weil–Petersson metric completion Tx is the closure of the convex hull of the vertex set fTx j j j D 3g  3g, which suggests an interpretation of the Teichmüller space as a simplex. We can summarize the above discussion as T S Tx D 2 H with @Tx   D.Tx ; /; where every boundary point b 2 @Tx belongs to D.Tx ; / for some  in .

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7.2 Euclidean convex geometry and Funk metric Suppose that  is an open convex subset in a Euclidean space Rd . In what follows, we set the presentation by Papadopoulos and Troyanov [59] as our reference for Funk and Hilbert metrics. First we represent the convex set  as T  D .b/2P H.b/ ; where H.b/ is the half-space bounded by a supporting hyperplane .b/ of  at the boundary point b, containing the convex set . The index set P is the set of all supporting hyperplanes of . That for every boundary point p there exists a supporting hyperplane .b/ follows from the convexity of . In general, there can be more than one supporting hyperplane of  at p 2 @. The index set P is identified with the set of unit normal vectors to the supporting hyperplanes. It is also identified with a subset of S d 1 , equal to the entire sphere when the convex set is bounded. We denote by P .b/ the set of supporting hyperplanes at b 2 @. Definition 7.1. For a pair of points x and y in , the Funk asymmetric metric [26] is defined by d.x; b.x; y// ; F .x; y/ D log d.y; b.x; y// where the point b.x; y/ is the intersection of the boundary @ and the ray fx C t xy j t > 0g from x though y. Here xy is the unit vector along the ray. Remark. In this section only, we use the term metric on a set X for a function ı W X  X ! .RC [ f1g satisfying: (1) ı.x; x/ D 0 for all x in X , (2) ı.x; z/ ı.x; y/ C ı.y; z/ for all x, y and z in X . Now let 0 be a supporting hyperplane at b.x; y/, namely 0 2 P .b.x; y//. Then note the similarity of the triangle 4.x; …0 .x/; b.x; y// and 4.y; …0 .y/; b.x; y//, where …0 .p/ is the foot of the point p on the hyperplane 0 , or to put it differently, …0 W Rd ! 0 is the nearest point projection map. This says that log

d.x; 0 / d.x; b.x; y// : D log d.y; b.x; y// d.y; 0 /

Again by the similarity argument of triangles, the right-hand side of the equality is independent of the choice of 0 in P .b.x; y//. Using the convexity of , the quantity F .x; y/ can be characterized variationally as follows. Define T .x; ; / by  \ fx C t jt > 0g with  2 P where  is a unit vector. Consider the case  D xy where xy is the unit tangent vector at x to the ray from x through y. When the hyperplane supports  at p, we have T .x; xy ; / D b.x; y/. Otherwise the point T .x; xy ; / lies outside . When  … P .b.x; y//, by the

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similarity argument between the triangles 4.x; F .x/; T .x; xy ; //

and

4.y; F .y/; T .xy ; //

again we have d.x; T .x; xy ; b// d.x; / D : d.y; / d.y; T .x; xy ; b// Now the point b.x; y/ D T .x; xy ; / is actually the closest point to x along the ray fx C t xy W t > 0g. This in turn says that  which supports  at b.x; y/ maximizes d.x;T .x; xy ;// among all elements of P : the quantity d.y;T .x; xy ;// log

d.x; / d.x; .b.x; y/// D sup log : d.y; .b.x; y/// d.y; / 2P

Hence we have a new characterization of the Funk metric [87]: Theorem 7.2. The Funk metric defined as above over a convex subset   Rd has the following variational formulation: F .x; y/ D sup log 2P

d.x; / : d.y; /

We note that though variational formulations of Hilbert metric have been known (for example [45] for polygons), there had been none available for the Funk metric previous to [87].

7.3 Weil–Petersson Funk metric We now transcribe the Euclidean Funk geometry as well as its compatible Finsler structure in the previous section to the Weil–Petersson setting. First note that as each Tx lies in Tx as a complete convex set, for each point x 2 Tx , there exists a nearest point projection … .x/ 2 D.Tx ; /, and the Weil–Petersson geodesic x .x/ meets D.Tx ; / perpendicularly, its length uniquely realizing the distance inf y2Tx d.x; y/ D d.x; … .x//. We denote this number by d.x; Tx /. We also introduce the notation  .x/ for the unit vector at x along the Weil–Petersson geodesic between x and … .x/. In particular  .x/ is the Weil–Petersson gradient vector of the function d.x; Tx /. Definition 7.3. We define the Weil–Petersson Funk metric F on T as d.x; Tx / : F .x; y/ D sup log d.y; Tx / 2 In order to make the analogy with the Euclidean setting more obvious, and in order to make clearer the viewpoint that Teichmüller space is a convex body within

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an ambient space, we can instead define the metric, as F .x; y/ D sup log 2

d.x; D.Tx ; // : d.y; D.Tx ; //

We are allowed to replace Tx by D.Tx ; / in the above definition since for any z 2 T , … .z/ is in T  D.; Tx /, due to the fact that the frontier sets intersect perpendicularly. The equality follows from the discussion in the paragraph preceding the definition of Weil–Petersson Funk metric. Note that the triangle inequality for the Weil–Petersson Funk metric results from the following: F .x; y/ C F .y; z/ D sup log 2



d.x; D.Tx ; // d.y; D.Tx ; // C sup log d.y; D.Tx ; // 2 d.z; D.Tx ; //

 sup log 2

D sup log 2

 d.y; D.Tx ; // d.x; D.Tx ; // C log d.y; D.Tx ; // d.z; D.Tx ; //

d.x; D.Tx ; // D F .x; z/: d.z; D.Tx ; //

7.4 The Teichmüller metric, Thurston’s asymmetric metric and the Weil–Petersson Funk metric In this section, we make a comparison among three Funk-type metrics defined on Teichmüller spaces. The first is the Teichmüller metric, which is defined as follows: Definition 7.4. Let ŒG1  and ŒG2  be two conformal structures (uniformized by hyperbolic metrics Gi ) on †. The Teichmüller distance between ŒG1  and ŒG2  is given by 1 dT .ŒG1 ; ŒG2 / D inf log K.f /; 2 f where the infimum is taken over all quasi-conformal homeomorphisms f W .†; ŒG1 / ! .†; ŒG2 / that are isotopic to the identity. Kerckhoff [34] showed that the Teichmüller distance can be alternatively defined as dT .ŒG1 ; ŒG2 / D

Ext ŒG1  . / 1 sup log ; 2  2 Ext ŒG2  . /

where Ext ŒG ./ is the extremal length of the homotopy class of simple closed curves in †. Recall [58] that the extremal length of  is defined as 1=Mod† . /, where Mod† . / is the supremum of the moduli of the topological cylinders embedded in † with core curve in the class  .

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Secondly, Thurston’s asymmetric metric is defined as follows. Definition 7.5. Let G1 and G2 be the two hyperbolic metrics on †. A distance function can be defined as ` .G1 / T .G1 ; G2 / D sup log I ` .G2 / 2 it is called Thurston’s asymmetric metric. This quantity was shown by Thurston to be equal to the number L.G1 ; G2 / D inf Lip./;

Id†

where the infimum is taken over all diffeomorphisms  in the isotopy class of the identity, and Lip./ is the Lipschitz constant of the map : dG2 ..x/; .y// : dG1 .x; y/ x¤y2†

Lip./ D sup

For this reason, the quantity T .G1 ; G2 / is sometimes called Thurston’s Lipschitz metric. Thurston in his paper ([68], Chapter 4) emphasizes the underlying convex geometry for the metric. In particular, the space of projective measured laminations is embedded in the cotangent space TG T for a fixed point G in T as the boundary set of a convex body, where the embedding is given by the differential of the logarithm of the geodesic length function of geodesic laminations d log length W PL.†/ ! TG T ; where the map only registers the projective classes of geodesic laminations, since we take the logarithmic derivative of the length. Now recall the new Funk-type metric we have introduced above. Definition 7.6. The Weil–Petersson Funk metric F on T is defined as d.x; Tx / : F .x; y/ D sup log d.y; Tx / 2 Now the analogy among the Teichmüller metric, the Thurston metric and the Weil– Petersson Funk metric is clear; namely for each of them we have an embedding ‚ W T ! R ; where the target space has a weak metric d.x; y/ D sup log yx for x D .x /2 , and each of the three Finsler metrics is the pulled-back metric ‚ d defined on T  T . The author would like to thank H. Miyachi for pointing out this comparison among the three metric structures.

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Chapter 2

Simple closed geodesics and the study of Teichmüller spaces Hugo Parlier

Contents 1 2 3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . Simple closed geodesics versus closed geodesics . . . . . . 3.1 The non-density of simple closed geodesics . . . . . . 3.2 The growth of the number of simple closed geodesics . 3.3 Multiplicities of simple closed geodesics . . . . . . . 4 Short curves: systolic and Bers’ constants . . . . . . . . . . 4.1 Systolic constants . . . . . . . . . . . . . . . . . . . 4.2 Bers’ constants . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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113 114 116 116 117 118 121 121 128 130

1 Introduction The aim of this chapter is to present certain aspects of the relationship between the study of simple closed geodesics and Teichmüller spaces. The set of simple closed geodesics is more than a mere curiosity and has been central in the study of surfaces for quite some time: it was already known to Fricke that a carefully chosen finite subset of such curves could be used as local coordinates for the space of surfaces. Since then, the literature on the subject has been vast and varied. Recent results include generalizations of McShane’s Identity [39], [44], [45] and results on how to use series based on lengths of simple closed geodesics to find invariant functions over Teichmüller space to calculate volumes of moduli spaces. Questions surrounding multiplicity in the simple length spectrum are sometimes related to questions in number theory [29], [68]. In a somewhat different direction, and although this theme will not be treated here, a related subject is the combinatorics of simple closed curves. The geometry of the curve complex [24], [38], [51], [52], [69] and the pants complex [15] have played an important role in the study of the large scale geometry of Teichmüller spaces with its different metrics and the study of hyperbolic 3-manifolds.

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It should be noted that this chapter should not in any way be considered a survey of all relationships between Teichmüller space and simple closed geodesics, but just a presentation of certain aspects the author is familiar with. Specifically, this chapter will concentrate on two themes. The first theme is the study of the set of simple closed geodesics in contrast with the set of closed geodesics. There are a number of ways in which these sets differ and these illustrate the special nature of simple closed geodesics. Three subjects of contrast are exposed here. The first concerns results related to the non-density of simple closed geodesics and in particular the Birman–Series theorem [14]. The second subject is about the contrast in growth of the number of simple closed geodesics in comparison with closed geodesics, and in particular Mirzakhani’s theorem [46]. The third subject concerns how multiplicity differs in the full length spectrum in comparison with the simple length spectrum. The second theme is on systoles, their lengths, and other related quantities such as the lengths of pants decompositions. For systolic constants, many of the known results are due to Schmutz Schaller who wrote a survey article on the subject [64], and so the information provided is intended to somewhat complement his exposition. Bers’ constants, introduced by Bers [11], [12], are upper bounds on lengths of a shortest pants decomposition of a surface, and have been extensively studied by Buser, who proves a number of bounds on these [18], [19]. Here again, the information provided should be seen as a complement to [19]. There are many similarities between the problems of finding bounds on lengths of systoles and pants decompositions and one of the goals of this chapter is to illustrate this. Acknowledgements. Foremost, I would like to thank Athanase Papadopoulos for giving me the opportunity to present this piece of work. I would also like to take this opportunity to thank my co-authors for making working with them such a pleasure.

2 Generalities Consider Teichmüller space T D Tg;n , the set of marked finite area hyperbolic metrics on an orientable surface of genus g with n cusps. Surfaces with a non-empty set of cusps will be called punctured surfaces. On occasion, we will talk about non-complete surfaces with boundary geodesics or cusps instead of just cusps, and they will be called surfaces with holes. A free homotopy class of a closed curve is called non-trivial if it is not homotopic to a disk or to a puncture. It is called simple if it can be represented by a simple closed curve. We begin with the following essential property. Property 2.1. A non-trivial closed curve is freely homotopic to a unique closed geodesic. If the closed curve is simple then so is the freely homotopic closed geodesic. One way of seeing this is by considering the lifts of a non-trivial curve to the universal cover H. The lifts are all disjoint simple curves between boundary points

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of H. The geodesics between the same boundary points are invariant by the Fuchsian group, and the quotient is the desired simple closed geodesic. The same argument shows that the closed geodesic in the free homotopy class of a closed curve always minimizes its self-intersection number. Simple closed geodesics will be generally thought of as primitive and unoriented. The above property allows us to associate to a homotopy class Œ˛ a length function, which to a surface M 2 T associates the length `M .˛/ of the unique geodesic in the homotopy class of ˛. A fundamental property of these length functions is that they are analytic, with respect to the usual analytic structure on Teichmüller space [1]. For a given surface M , we shall denote .M / the marked set of lengths of closed geodesics. By the above property, this is a countable set of values. The simple length spectrum 0 .M / is the subset of .M / restricted to simple closed geodesics. Simple closed geodesics provide useful parameters for Teichmüller space via pants decompositions. A pants decomposition is a collection of disjoint simple closed geodesics of the surface such that the complementary region is a collection of threeholed spheres (pants). By a simple topological argument one sees that there are exactly 3g  3 C n curves in a pants decomposition, and this number is sometimes referred to as the complexity of the surface. This provides us with 3g  3 C n length functions, which although they do not determine the surface, they determine the geometry on the complement of the pants decomposition geodesics. To determine a marked surface, one uses a twist parameter to determine how the pants curves are glued together. Generally one measures twist by looking at perpendicular geodesic arcs on pants between distinct boundary curves. By cutting along these, one obtains a pair of symmetric hyperbolic right-angled hexagons. The twist is then measured by the displacement factor between two perpendicular arcs intersecting the boundary and coming from different sides. The collection of lengths and twists are called the Fenchel–Nielsen coordinates. A consequence of the decomposition of a hyperbolic surface into hexagons is the well-known collar lemma (see [17], [19], [36], [53] for different versions). Lemma 2.2 (Keen’s collar lemma). Around a simple closed geodesic  there is always an embedded hyperbolic cylinder (called a collar) of width w./ D arcsinh



1 / sinh `. 2



:

Furthermore, the collars around pants decomposition geodesics are all disjoint. The proof essentially follows from the above discussion and hyperbolic trigonometry (see [10], [19] for hyperbolic trigonometry formulas, and [26] for Fenchel and Nielsen’s original approach). Indeed, on each side of a simple closed geodesic  , one obtains two isometric right angled hexagons with one of their side lengths `. /=2. By hyperbolic  trigonometry, the subsequent edges cannot have length less 1 and the result follows. than arcsinh `. / sinh

2

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Although Fenchel–Nielsen coordinates are very useful, they do not provide a homogeneous set of parameters. However, if one allows more length functions of simple closed curves, one does obtain a complete local description of Teichmüller space. Theorem 2.3. There is a fixed finite set of simple closed geodesics 1 ; : : : ; m such that the map ' W M 7! .`M .1 /; : : : ; `M .m // is projectively injective on Tg;n . Recall that a map f W X ! V where V is a real vector space is projectively injective if f .x/ D tf .y/; for some t 2 R, implies x D y. Interestingly, this theorem fails to be correct in all generality if one allows surfaces with variable boundary length [40]. There are different versions of it and as stated has probably been known for some time. We refer to [30], [31], [59], [62] for different statements about the m and related questions if one allows the curves to not be simple and whether one wants a projectively injective map or just an injective map. A tool that will come up several times in this discussion is the following, which will be called the length expansion lemma. Lemma 2.4 (Length expansion lemma). Let S be a surface with n > 0 boundary curves 1 ; : : : ; n . For ."1 ; : : : ; "n / 2 .RC /n with at least one "i ¤ 0, there exists a surface Sz with boundary geodesics of length `.1 / C "1 ; : : : ; `.n / C "n such that all corresponding simple closed geodesics in Sz are of length strictly greater than those of S. There are different proofs and versions of this lemma [50], [71]. Recently, Papadopoulos and Théret [49] proved a stronger version where they show that not only can one increase the lengths of all simple closed geodesics but one can do so such that the infimum of the ratios of lengths between the long geodesics and the short geodesics is strictly greater than 1. They use this to show things about how one can and cannot generalize Thurston’s asymmetric metric defined in [70] to surfaces with boundary (see also [48] for an overview on this metric).

3 Simple closed geodesics versus closed geodesics One often studies the behavior of the set of simple closed geodesics in contrast with the set of closed geodesics. This section is devoted to showing that simple closed geodesics are rare in the set of closed geodesics in several ways.

3.1 The non-density of simple closed geodesics One of the first remarkable results in this direction is a theorem of Birman and Series. It is well known that on hyperbolic surfaces, points lying on the set of closed geodesics

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form a dense subset of the surface. In fact, they are even dense in the tangent bundle. Birman and Series [14] show that this is in sharp contrast with the set of simple closed geodesics (and more generally with the set of simple complete geodesics). Theorem 3.1 (Birman–Series). The set of points lying on a simple complete geodesic is nowhere dense and has Hausdorff measure one. More generally, they show this to be true for complete geodesics with bounded self-intersection number. One of the essential steps in their approach is he following. For a given L, the set of simple complete geodesics lie in an  neighborhood of some set of geodesic arcs of length less than L. They show that for any given surface, there are positive constants L, C , ˛ and a polynomial P such that the full set of simple complete geodesics lies in the " D C e ˛ n neighborhood of a set of at most P .n/ geodesic arcs of length at most L. This has to do with the fact that “long” simple complete geodesics spend great deals of time running parallel to themselves. Along similar lines, there have been different descriptions of algorithms which determine whether a word in the fundamental group corresponds to a simple closed geodesic, including one by Birman and Series [13]. In particular one can ask how “non-dense” the simple closed geodesics are on surfaces. In [21] the following theorem is shown. Theorem 3.2. There exists a constant cg > 0, depending only on g, such that any compact hyperbolic Riemann surface M of genus g contains a disk of radius cg into which the simple closed geodesics do not enter. In contrast, for any " > 0 there exists a surface S" on which the geodesics are "-dense. To show the existence of the constants, one uses Theorem 3.1 and one needs to show the continuity of the “gaps” in Teichmüller space, use the compactness of the thick part of Teichmüller space and a discussion of how the constant behaves in the thin part. The converse is essentially a consequence of a theorem of Scott’s [66], [67] which says that given any closed geodesic on a surface, there exists a finite cover of the surface where all of the primitive lifts of the original closed geodesic are simple. Rivin has recently asked [57] if one can quantify Scott’s result, i.e., compute or at least find bounds on the minimal degree cover which is necessary to “unravel” a closed geodesic with k self-intersections.

3.2 The growth of the number of simple closed geodesics A result of Mirzakhani’s computes the asymptotic growth of the number of simple closed geodesics of less than a given length, and as we shall outline, this provides another “simple closed geodesics are rare” analogy. If one counts the number of closed

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geodesics on hyperbolic surfaces of length less than L, the asymptotic result does not depend on the surface, or even on the topology of the surface. Quite remarkably, L this number always behaves asymptotically like eL (this is sometimes referred to as Huber’s asymptotic law [32]). In contrast, Mirzakhani has shown the following. Theorem 3.3 (Mirzakhani). Let NM .s; L/ be the number of simple closed geodesics of lengths  L on M . For L ! 1 this number has the asymptotic behavior NM .s; L/  cM L6g6 ;

where cM is a constant depending on the metric on M . Mirzakhani’s theorems show more than the above stated result, and in particular it is shown that the leading coefficient cM gives a continuous proper function over moduli space. One of the ideas of the proof is to notice that up to the action of the mapping class group, there are only a finite number of types of simple closed curves on a surface. For each type, one can count the growth of a simple closed geodesic under the action of the mapping class group (which she does) and then obtain the result by adding the finite number of types. There is quite a rich history to this problem which traces back to Dehn. Rivin [55] had previously obtained partial and related results. He also provided a simplified proof [56] and explained some of the history of the problem. Also prior to Mirzakhani’s work, was a theorem of McShane and Rivin [43], [42] in the particular case of a once-punctured torus. They obtained the same theorem, and showed [43] that the leading asymptotic coefficient has to do with the stable norm on the homology of the torus. Indeed, in the case of once-punctured tori, homology classes and oriented not necessarily primitive homotopy classes of simple closed curves coincide. Given a metric on a torus T , the length of the corresponding geodesics induces a norm on the integer homology of a torus H1 .T; Z/ which can be extended to a norm on H1 .T; R/. What they show is that the leading asymptotic coefficient is in fact the inverse of the area of the unit ball of this norm. They conjecture that the leading coefficient is maximal (i.e. the area of the unit ball is minimal) for the modular torus. The modular torus is the unique once-punctured torus with an isometry group of order 12, and it can be obtained for instance by a quotient of H by an index 3 subgroup of PSL2 .Z/. Similarly, one could ask the same questions about the constants that appear as the leading asymptotic coefficients of Mirzakhani’s formula.

3.3 Multiplicities of simple closed geodesics A well-known theorem of Randol [54], using a construction due to Horowitz, states that multiplicity is unbounded in the set of lengths of closed geodesics. The proof is by construction, and in fact one constructs arbitrarily large sets of homotopy classes of closed curves whose geodesic representatives all have the same length, regardless of

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the hyperbolic metric on the surface (the length of course does depend on the metric, but the lengths are always equal). There is a nice illustrated proof of Randol’s result in [19]. Leininger [37] has shown that these curves, sometimes called equivalent curves, have the property that they intersect all simple closed geodesics the same number of times. Essentially, this follows from the collar lemma. However, somewhat surprisingly, Leininger also shows that the converse fails, i.e., there are closed geodesics that are not equivalent but yet they intersect every simple closed geodesic the same number of times. Note that this is again in contrast with the set of simple closed geodesics. Indeed Thurston [25] uses the fact that homotopy classes of simple closed curves are determined by their intersection numbers with other simple closed curves to construct his compactification of Teichmüller space. In contrast one can study multiplicity in the simple length spectrum. The following is true [40]. Theorem 3.4. The set of surfaces with all simple closed geodesics of distinct length is dense in Teichmüller space and its complement is Baire meagre. As its statement suggests, one can show this by using the Baire category theorem. Denote by E.˛; ˇ/ the set of all surfaces where the distinct pair .˛; ˇ/ of homotopy classes of simple closed curves have geodesic representatives of the same length. These sets are zero sets of analytic functions (because length functions are analytic) and are thus closed. Also, these sets have no interior, because otherwise they would be equal over all Teichmüller space. Using the collar lemma, one can show this to be impossible. The set of surfaces with all simple closed geodesics of distinct length is the complement of the union E of the sets E.˛; ˇ/. The result then follows from the Baire category theorem as there are a countable number of homotopy classes. One can actually say more about the topology of the sets E.˛; ˇ/ and their union E [40]. Theorem 3.5. The sets E.˛; ˇ/ are connected sub-manifolds of Teichmüller space. The set E is connected. To show that these are indeed sub-manifolds and not just sub-varieties, one uses Thurston’s stretch maps [70]. The connectedness of E is a consequence of what follows. A natural question to ask is whether one can deform a surface within E c . In fact not [40]. Theorem 3.6. If A is a non-constant path in Teichmüller space then there is a surface on A which has at least two distinct simple closed geodesics of the same length. As a consequence, the set E is the complement of a totally disconnected set of Teichmüller space and as such is connected. The proof of this uses the projectively

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injective map to Teichmüller space described previously. Given two distinct surfaces M1 and M2 , one constructs pairs of curves ˛; ˇ such that `M1 .˛/ > `M1 .ˇ/ and `M2 .˛/ < `M2 .ˇ/ and the construction relies on the projectively injective map. Now by continuity of length functions, along any path between M1 and M2 there is a point on which the two simple closed geodesics have the same length. As a corollary of the above discussion one obtains the following. Corollary 3.7. The marked order in lengths of simple closed geodesics determines a unique point in Tg;n . One of the motivations of this study was to study the nature of the following conjecture, attributed to Rivin [64]. Conjecture (Rivin). Multiplicity in the simple length spectrum is bounded above by a constant which depends on the topology of the surface. It should be mentioned that lower bounds which depend on topology have been established in the work of Schmutz Schaller in his investigation of surfaces with a large number of systoles (see section 4.1). Also due to Schmutz Schaller is the following conjecture [64]. Conjecture (Schmutz Schaller). Multiplicity is bounded by 6 in the particular case of once-punctured tori. In fact, this is a geometric generalization of the well-known Markov uniqueness conjecture in number theory [27] which is in fact equivalent to whether Schmutz Schaller’s conjecture is satisfied by the modular torus. Conjecture (Frobenius). A solution .a; b; c/ of positive integers to the Markov cubic a2 C b 2 C c 2  3abc D 0

(3.1)

is uniquely determined by maxfa; b; cg. We refer the reader to [41], [64] and references therein on known results concerning this conjecture. Let us note however that the apparent difficulty of the Markov uniqueness conjecture suggests that Schmutz Schaller’s conjecture is very difficult. Furthermore, Schmutz Schaller [64] remarked that there were no known surfaces where one knows that multiplicity in the simple length spectrum was even bounded. Note that by Theorem 3.4 most surfaces have simple multiplicity bounded by 1, but whether or not a given surface has this property can be a difficult question. In [41], a family of examples of one-holed tori with multiplicity bounded by 6 are given and will be discussed in Proposition 4.3 and Remark 4.4. However, an example of either a punctured or a closed surface with bounded simple multiplicity remains to be found.

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4 Short curves: systolic and Bers’ constants The simple length spectrum has not been studied in as much detail as the full length spectrum. However, one particular length, namely the length of a shortest closed geodesic or the systole length, has been the object of numerous articles and investigations. The study of this function, largely developed by Schmutz Schaller (see for instance [64] and references therein), continues to be a subject of active study. Another length function which has proved useful in the study of Teichmüller space is the length of the shortest pants decomposition of a surface where the length is usually defined to be the length of the longest curve in the pants decomposition. Both of these quantities are bounded by constants that depend only on the topology of the surfaces (and not on the metrics themselves). Here we will try to compare some of the techniques and results used to study both problems.

4.1 Systolic constants Consider a hyperbolic surface of genus g with n cusps, or if one is ambitious, with n boundary geodesics. We define the systole to be the length of the shortest closed geodesic which does not belong to the boundary. Unless the surface is a pair of pants, such a curve is always simple. We define the systole function sys.S / to be the length of the systole of a surface S . This gives an interesting function over Teichmüller space as is portrayed by the following theorem of Akrout [3]. Theorem 4.1 (Akrout). The systole of Riemann surfaces is a topological Morse function on the Teichmüller space. Note that this is the best one could hope for as it could not be a regular Morse function because the curve realizing systole length changes as one moves around Teichmüller space. To show this, Akrout shows a more general result concerning generalized systole functions. Generalized systole functions are functions on manifolds defined locally as the minimum of a finite number of smooth functions. He shows that if the manifold admits a connexion such that the Hessians of length functions are positive definite, such a function is a topological Morse function such that a Morse point of index r is an eutatic point of rank r. (A point is eutactic if the zero vector of the tangent space lies in the interior of the convex hull of the gradients of the length functions of the generalized systoles.) Let us outline why this proves the above theorem. The manifold in our case is Teichmüller space endowed with the Weil–Petersson metric. In a neighborhood of a point, length functions are continuous and length spectra are discrete, there are only a finite number of simple closed curves that will realize the systole in the neighborhood, so the systole function is a generalized systole function. Furthermore, it is a result of Wolpert [72] that length functions have positive definite Hessians with respect to the Weil–Petersson metric, and thus the result follows.

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Although we begin the study of the systolic geometry of surfaces by Akrout’s theorem, it should be noted that this is not chronologically correct. Schmutz Schaller had previously obtained partial results in this direction [64], [65] and had largely initiated a systematic study of systoles on hyperbolic surfaces and established a parallel with n-dimensional sphere packings. Bavard [9], by generalizing the study of Hermite invariants to systolic problems on manifolds, provided a theoretical framework to study extremal points of systole type functions. Using these parallels, Akrout’s result is related to Ash’s results [4], [5]. As in the case of sphere packings or Hermite invariants, one is interested in extremal points, local and global maxima, and these turn out to be difficult to find. First of all, by Mumford’s compactness theorem, for each genus g  2, there is a global maximum for the systole, which we shall denote sys.g; n/ and sys.g/ for closed surfaces. Exact values of systolic constants In the non-compact case, Schmutz Schaller proved that surfaces corresponding to quotients of H by principal congruence subgroups of PSL2 .Z/ are in fact global maxima for the systole length of their corresponding signature [60]. There are a number of other known results about low complexity cases, even with (fixed length) boundary geodesics [58]. For closed surfaces there is only one known value, in genus 2, a result of Jenni [33]. Theorem 4.2 (Jenni). A systole  of a genus 2 surface satisfies `./ p  2C1 2 and equality occurs for a unique surface (up to isometry). cosh

The surface that attains the optimal bound in genus 2 is the so-called Bolza curve, and is also the genus 2 surface with the highest number of conformal self-isometries (it has 48). Since then there have been other proofs of Jenni’s result, namely [58] where among many results, one finds a complete list of all critical values of sys in genus 2. Another proof is given by Bavard [8] where the problem of the maximal systole among hyperelliptic surfaces is considered. He finds the maximal values in genus 2 and 5, and describes the surfaces which attain the bounds (and shows that they are unique up to isometry). As all genus 2 surfaces are hyperelliptic, the result coincides with Jenni’s. For higher genus hyperelliptic surfaces, consider the quotient of the surface by the hyperelliptic involution. One obtains a hyperbolic sphere with 2g C 2 cone points of angle  (corresponding to the Weierstrass points of the surface). A simple geodesic path of length ` between two distinct cone points lifts to a simple closed geodesic passing through two Weierstrass points on the closed surface above, of length 2`. Finding optimal bounds for this problem is a type of hyperbolic equivalent to the well-known Tammes problem on the Euclidean sphere whose solution is only

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Figure 1. The Bolza surface: on the upper left, one sees the 12 systoles, and on the lower right one sees the gaps from Theorem 3.2. See Peter Buser’s article [20] for a description of how this picture was made.

known for certain numbers of points. There is no reason to believe that this hyperbolic version is any easier. In genus 3, Klein’s quartic is the surface with the largest number of conformal self-isometries (it attains Hurwitz’s upper bound of 84.g  1/ D 168 but it fails to be maximum for the systole function). Schmutz Schaller has conjectured that another surface, also with a large number of symmetries, is maximal (the so-called M.3/ surface explicitly described in [58]). Conjecture (Schmutz Schaller). A systole  of a surface in genus 3 satisfies p `./ 2C 3 cosh 2 with equality occurring for a unique surface up to isometry. Although this is a “yes or no” type question, Schmutz Schaller has obtained partial results which include the fact that M.3/ is a local maximum [58] and that certain subsurfaces (with boundary) of this surface are optimal in their configuration. In order to give an idea about this last result, consider the following toy problem. One considers a one-holed torus with a boundary geodesic of length `. Among all such surfaces, which one has maximal systole length? The solution to this problem requires

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only cut and paste techniques and hyperbolic trigonometry. This same proposition, expressed differently and without proof, can be found in [58]. We give a full proof as an illustration of some of the difficulties of these techniques in higher surface complexity. Proposition 4.3. Let Q be a surface of signature .1; 1/ with boundary geodesic . Then Q contains a simple closed geodesic ı satisfying `.ı/ `. / 1  cosh C : 2 6 2 This bound is sharp and for a given length of  , there is a unique surface up to isometry which reaches this bound. cosh

Proof. The idea of the proof is to use hyperbolic polygons to obtain an equation from which we can deduce the sharp bound. Let ı be the shortest closed geodesic on Q with boundary geodesic  . Let c be the length of the perpendicular geodesic arc from ı to ı on the embedded pants .ı; ı;  / obtained by cutting Q along ı. By the formula for a right angled hexagon, c is given by the following formula: cosh c D

/ C cosh2 cosh `. 2

sinh2

`.ı/ 2

`.ı/ 2

:

Another way of expressing it is using one of the four isometric hyperbolic pentagons that form a symmetric pair of pants as in the following figure. c 2

ı

c 2

ı



Figure 2. A symmetric pair of pants.

The pentagon formula implies that cosh2

cosh2 `./ C cosh2 `.ı/ 1 c 4 2 : D `.ı/ 2 2 cosh 2  1

The smaller c is, the longer ı is. Thus to find an upper bound on ı, we need to find a minimal c. Consider ı 0 the second shortest simple closed geodesic. Its not too difficult

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to see that ı 0 intersects ı exactly once. For a given ı and , this ı 0 is of maximal length when Q is obtained by pasting ı with a half twist. The length of this maximal ı 0 can be calculated in the following quadrilateral.

`.ı 0 / 2

`.ı/ 4

c 2 c 2 `.ı/ 4

`.ı 0 / 2

From one of the two right-angled triangles that compose the quadrilateral we have cosh

`.ı 0 / c `.ı/ D cosh cosh : 2 2 4

Using the fact that ı  ı 0 we can deduce cosh2

`.ı/ c `.ı/  cosh2 cosh2 2 2 4 2 `.ı/ cosh2 `./ C cosh 1 `.ı/ 4 2 D cosh2 2 `.ı/ 4 cosh 2  1 D

cosh2

`./ 4

C cosh2

2.cosh

`.ı/ 2

`.ı/ 2

1

 1/

:

From this we obtain the following condition: 2 cosh3

`.ı/ `.ı/ `. /  3 cosh2 C 1  cosh2  0: 2 2 4

and C D cosh2 With x D cosh `.ı/ 2 polynomial

`./ 4

> 1 we can study the following degree 3

f .x/ D 2x 3  3x 2 C 1  C and find out when it is negative for x > 1. The function f satisfies f .1/ D C < 0 and f 0 .x/ > 0 for x > 1. The sharp condition we are looking for is given by the unique solution x3 to f .x/ D 0 with x > 0. Thus p 1 1 1 1 1 C : x30 D .1 C 2C C 2 C C C 2 / 3 C p 2 2 .1 C 2C C 2 C C C 2 / 13 2

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Now we replace x and C by their original values. Using hyperbolic trigonometry we can show 

`./ `./ 1 `.ı/ C sinh cosh  cosh 2 2 2 2 which in turn can be simplified to

1

3



`. / `. / C sinh C cosh 2 2

 1 3



C1

`.ı/ `. / 1  cosh C : 2 6 2 The bound is sharp and the length of ı satisfies this bound if and only if `.ı 0 / D `.ı/ and the twist parameter ı is pasted with a half twist. cosh

Remark 4.4. For fixed boundary length, the unique surface that reaches the upper bound on systole length has exactly 3 distinct systoles: the curves ı, ı 0 and a third ı 00 which is the mirror image of ı 0 reflected along ı. If the boundary is a cusp, then the surface obtained is in fact the modular torus mentioned previously, conformally equivalent to the torus obtained by taking a regular euclidean hexagon and identifying opposite sides. In general, if there is a boundary curve  , if one was to glue a euclidean hemisphere by gluing the equator to , one would once again obtain the same conformal structure. This family of tori have different characterizations: for instance, for each boundary length they are the unique one-holed tori with an isometry group of order 12. And one could ask whether they satisfy a generalization to one-holed tori of the Schmutz Schaller Conjecture 3.3 mentioned earlier. In [40] it is shown that there is a dense subset of these surfaces which fail to satisfy the generalization, and in [41] it is shown that if `./ is such that cosh `./ is a transcendental real, then it does satisfy 2 the Schmutz Schaller conjecture. We can now explain the partial result for closed surfaces of genus 3 obtained by Schmutz Schaller and mentioned above. He shows that among all closed genus 3 surfaces with a configuration of 3 distinct systoles lying inside an embedded one-holed tori (thus exactly like the ones explained above), the surface M.3/ has maximal systole. It should be said that Schmutz Schaller’s detailed analysis of systole configurations are extremely useful in studying some of the synthetic geometry of Teichmüller and moduli spaces for low complexity surfaces. In higher complexity surfaces, there are no significantly different known ways of attacking these problems and the combinatorics of the problem become quickly out of hand. To further underline the difficulty behind these problems, let us give another question which can already be found in [64], and which seems to wide open. Question 4.5. Is the maximal systole in genus g C 1 greater than the maximal systole in genus g? There is also a similar question for Riemannian surfaces which also seems to be wide open. The appropriate quantity that one studies for Riemannian surfaces is the

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systolic ratio and is given by sysg D sup

sys2 .S / ; area.S /

where sys.S/ is the length of the shortest non-trivial closed curve on the surface S, and the supremum is taken among all Riemannian surfaces of genus g (the notation is highly non-standard for Riemannian geometers but this is only to be able to relate the two subjects). The appropriate question here is whether sysgC1  sysg . Other questions and results about systolic geometry and topology can be found in [34]. Growth of the systolic constants Systolic constants also provide an interesting growth problem. For reasons outlined in [2], [60], the constants can only significantly grow if one increases genus (as opposed to adding cusps). The interesting question is on how the constants sys.g/ behave. Again, for reasons outlined above, the strategy of finding the optimal constants in each genus is at the very least hopeful, so one is interested in less precise results. The question of rough growth was solved by Buser and Sarnak [22] who obtained the following theorem. Theorem 4.6 (Buser–Sarnak). There exist constants A and B such that A log g < sys.g/ < B log g: Note that by an area argument the upper bound is trivial. To show the lower bound they constructed a family of surfaces of genus gk with gk ! 1 as k ! 1 and systole length with growth 43 log gk . Their construction relies on the use of congruence subgroups in a quaternion algebra. From their construction, you can extrapolate a full family of surfaces (i.e. a surface in every genus) with logarithmic systole growth. Since then, there have been other constructions of surfaces with large systole growth. For instance in [35], Katz, Schaps and Vishne consider constructions based on congruences in other matrix groups. In particular, they show that the family of surfaces which reaches Hurwitz’s bound of maximal number of automorphisms in a given genus also provides such a family (meaning a family of surfaces of genus gk with gk ! 1 as k ! 1 and systole length with growth 43 log gk ). Yet another construction is essentially due to Brooks: in [16] he considers the surfaces obtained by uniformizing the surfaces coming from principal congruence subgroups of PSL2 .Z/ mentioned above, all of them maximal in their respective signatures. Specifically, the quotients by the principal congruence subgroups give conformal structures on an underlying closed surface, and one considers the unique closed hyperbolic metric in the same conformal class. This gives a sequence of surfaces which Brooks calls the Platonic surfaces. Brooks explains how to compare surfaces with cusps with their compactifications, provided the cusps are sufficiently far apart. It seems that Brooks’ theorems have not been used for this purpose, as originally the goal was to find families of surfaces with “large” first eigenvalue of the laplacian (meaning

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uniformly bounded away from 0), but Brooks’ theorems imply that one only has to compute the systole in the non-compact surfaces, which is straightforward. And once again, this gives a sequence of surfaces of gk with gk ! 1 as k ! 1 and systole length with growth 43 log gk . Interestingly there seems to be a gap between the multiplicative constant in the upper and lower bounds as the best known multiplicative constant in the upper bound is 2 (and this is trivial as mentioned above). There are a number of conjectures about the exact bounds. The strongest conjecture related to this is the affirmative answer to the following question. Question 4.7. Does maximal systole length have asymptotic growth

4 3

log g?

Misha Katz has called this the Rodin problem. Also note that a positive answer to this question is an extremely strong statement and implies a number of partial results, as for instance the fact that these systole functions have asymptotic growth, i.e., limg!1 sys.g/ exists. It also implies the existence of an upper bound with 43 log g log g behavior conjectured by Schmutz Schaller [64]. Finally, note that Schmutz Schaller was also very interested in how many distinct systoles a surface could have. He showed a number of results including the exhibition of different families of surfaces, both closed and with punctures, with number of systoles growing more than linearly in the Euler characteristic . Specifically, the best result [63] is that an upper bound on the number of systoles cannot grow asymptotically 4 less than ./ 3 . In [61], he claims to show that for closed surfaces, there is an upper bound of order g 2 , but the proof is not very convincing. Finally, he conjectures [63] 4 that there should be an upper bound with growth ./ 3 . As systoles do not pairwise intersect more than once, one could be interested in the maximal number of simple closed geodesics with this property, but not much seems to be known.

4.2 Bers’ constants Another length function which has proved useful in the study of Teichmüller space is the length of the shortest pants decomposition of a surface. For a given surface S 2 Tg;n and a pants decomposition P of S , we define the length of P as `.P / D max `. /: 2P

We denote B.S/ the length of a shortest pants decomposition of the given surface S. The quantity Bg;n is defined as Bg;n D sup `.B.S //: S2Tg;n

This quantity is a finite quantity by a theorem of Bers [11], [12] and the constants Bg;n are generally called Bers’ constants. Depending on what one wants to use these

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results for, just the existence of the constants is good enough. For instance, it plays a crucial role in the proof of Brock’s theorem [15] that Teichmüller space endowed with the Weil–Petersson metric is quasi-isometric to the pants complex. Building on work of Wolpert (see for instance [73]), Brock covers Teichmüller space with regions corresponding to when given marked pants decompositions are short (called Bers regions in reference to Bers’ constants) and then sends these regions to the topological pants decompositions. However, if one wants to say something explicit (like for instance give bounds on the quasi-isometric constants of Brock’s theorem) then one needs bounds on Bers’ constants. Other uses include bounds on the number of non-isometric isospectral surfaces [19]. An explicit bound can be extracted from a proof of Bers’ theorem in [1]. Buser’s investigations led to a number of bounds [18], [23], where best lower and upper bound for closed surfaces of genus g can be found in [19], Theorems 5.1.3, 5.1.4. Theorem 4.8 (Buser). Bers’ constants satisfy

p

p 6g  2  Bg;0  6 3.g  1/.

Note that the first linear upper bounds were obtained in collaboration with Seppälä [23] where they also show that on a surface with a reflection, one can choose such a pants decomposition so that it remains globally invariant by the reflection. As in the case of the systole, one can ask about exact values and whether the “sup” in the definition above can be replaced by a “max”. This is not immediate but does follow nonetheless from Mumford’s compactness theorem as in the case of systoles [7]. As it is a short example of how some of the basic tools described previously are used, a proof is provided. Property 4.9. There exists a surface Smax 2 Tg;n such that B.Smax / D Bg;n . Furthermore sys.Smax /  sg;n where sg;n > 0 is a constant that only depends on g and n. Proof. Given a surface S 2 Tg;n , by the collar lemma, any simple closed geodesic that crosses a geodesic of length ` has length at least 2 arcsinh 1 ` : A surface S 2 Tg;n sinh

2

with a short enough systole  (shorter than a computable constant sg;n ) has the property that any simple closed geodesic that crosses a systole has length at least 2Bg;n . By the collar lemma again, because `./ < 2 arcsinh 1, all systoles of S , if there are several, are disjoint. Thus a shortest pants decomposition of S necessarily contains all the systoles of S. By using the length expansion lemma explained above, one can increase the length of all the systoles at least up until sg;n to obtain a new surface S 0 such that the lengths of all simple closed geodesics disjoint from the systoles increase (strictly). In particular B.S 0 / > B.S / and sys.S 0 / D sg;n . Thus we have moved to the thick part of moduli space while increasing the length of a shortest pants decomposition. The thick part of moduli space being compact [47], it suffices to find the sup for Bg;n on a compact set. As B is a continuous function over moduli space, this proves the existence of a Smax .

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Along similar lines, one can show that Bers’ constants satisfy certain inequalities [7]. Property 4.10. The following inequalities hold: a) Bg;nC1 > Bg;n , b) Bg;n > Bg1;nC2 , c) BgC1;n > Bg;n . Note that as a corollary one obtains that Bg;0 > Bg1;0 which one would be happy to show in the case of systoles. Thus one can ask which surfaces attain extrema, and if the corresponding surfaces have interesting geometry. Gendulphe [28] has recently identified the exact value of B2;0 , and as in the case of systoles, this is the only known value for closed surfaces. Theorem 4.11 (Gendulphe). The constant B2;0 is determined by B2;0 D x0 12 where x0 is the unique solution greater than 1 of the equation cosh

32x 5  32x 4  24x 3 C 24x 2  1 D 0: The surface that realizes this constant is unique up to isometry. Note that the surface that realizes this bound is not the Bolza curve. However, in contrast with the systolic constant case, even the rough asymptotic growth of Bg;n is not known. Many of the known results are due to Buser (see [19]). Buser has also conjectured what the rough growth should be. Conjecture (Buser). There exists a universal constant C such that p Bg;n  C g C n: With Florent Balacheff, we have obtained a positive answer to this question for punctured spheres [7] building on work of Balacheff and Sabourau [6], but the general case remains wide open.

References [1]

W. Abikoff, The real analytic theory of Teichmüller space. Lecture Notes in Math. 820, Springer, Berlin 1980.

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[3]

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F. Balacheff and H. Parlier, Bers’constant for punctured spheres and hyperelliptic surfaces. J. Topol. Anal. 4 (2012), no. 3, 271–296.

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C. Bavard, Systole et invariant d’Hermite. J. Reine Angew. Math. 482, (1997), 93–120.

[10] A. F. Beardon, The geometry of discrete groups. Corrected reprint of the 1983 original, Grad. Texts in Math. 91, Springer, New York 1995. [11] L. Bers, Spaces of degenerating Riemann surfaces. In Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Ann. of Math. Stud. 79, Princeton University Press, Princeton, N.J., 1974, 43–55. [12] L. Bers, An inequality for Riemann surfaces. In Differential geometry and complex analysis, Springer, Berlin 1985, 87–93. [13] J. S. Birman and C. Series, An algorithm for simple curves on surfaces. J. London Math. Soc. (2) 29 (1984), no. 2, 331–342. [14] J. S. Birman and C. Series, Geodesics with bounded intersection number on surfaces are sparsely distributed. Topology 24 (1985), no. 2, 217–225. [15] J. F. Brock, The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores. J. Amer. Math. Soc. 16 (2003), no. 3, 495–535. [16] R. Brooks, Platonic surfaces. Comment. Math. Helv. 74 (1999), no. 1, 156–170. [17] P. Buser, The collar theorem and examples. Manuscripta Math. 25 (1978), no. 4, 349–357. [18] P. Buser, Riemannsche Flächen und Längenspektrum vom trigonometrischen Standpunkt. Habilitation Thesis, University of Bonn, 1981. [19] P. Buser, Geometry and spectra of compact Riemann surfaces. Progr. Math. 106, Birkhäuser, Boston, MA, 1992. [20] P. Buser, An algorithm for simple closed geodesics. In Geometry of Riemann surfaces, London Math. Soc. Lecture Note Ser. 368, Cambridge University Press, Cambridge 2010, 38–87. [21] P. Buser and H. Parlier, The distribution of simple closed geodesics. In Complex analysis and its applications (Proc. of the 15th ICFIDCAA (Osaka 2007)), OCAMI Stud. 2, Osaka Munic. Univeraity Press, Osaka 2007, 3–10. [22] P. Buser and P. Sarnak, On the period matrix of a Riemann surface of large genus. With an appendix by J. H. Conway and N. J. A. Sloane. Invent. Math. 117 (1994), no. 1, 27–56. [23] P. Buser and M. Seppälä, Symmetric pants decompositions of Riemann surfaces. Duke Math. J. 67 (1992), no. 1, 39–55.

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[24] Y. Choi, K. Rafi, and C. Series, Lines of minima and Teichmüller geodesics. Geom. Funct. Anal. 18 (2008), no. 3, 698–754. [25] A. Fathi, F. Laudenbach, and V. Poénaru (eds.), Travaux de Thurston sur les surfaces. Astérisque 66—67 (1979). [26] W. Fenchel and J. Nielsen, Discontinuous groups of isometries in the hyperbolic plane. Edited and with a preface by Asmus L. Schmidt, De Gruyter Stud. Math. 29, Walter de Gruyter & Co., Berlin 2003. [27] G. Frobenius, Über die Markoffschen Zahlen. Preuss. Akad. Wiss. Sitzungsberichte 1913, 458–487. [28] M. Gendulphe, La constante de Bers en genre 2. Math. Ann. 350 (2011), no. 4, 919–951. [29] A. Haas, Diophantine approximation on hyperbolic Riemann surfaces. Acta Math. 156 (1986), no. 1–2, 33–82. [30] U. Hamenstädt, Length functions and parameterizations of Teichmüller space for surfaces with cusps. Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 1, 75–88. [31] U. Hamenstädt, Parametrizations of Teichmüller space and its Thurston boundary. In Geometric analysis and nonlinear partial differential equations, Springer, Berlin 2003, 81–88. [32] H. Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. Math. Ann. 138 (1959), 1–26. [33] F. Jenni, Über den ersten Eigenwert des Laplace-Operators auf ausgewählten Beispielen kompakter Riemannscher Flächen. Comment. Math. Helv. 59 (1984), no. 2, 193–203. [34] M. Kapovich, Hyperbolic manifolds and discrete groups, Progr. Math. 183, Birkhäuser, Boston, MA, 2001. [35] M. G. Katz, M. Schaps, and U. Vishne, Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups. J. Differential Geom. 76 (2007), no. 3, 399–422. [36] L. Keen, Collars on Riemann surfaces. In Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Ann. of Math. Stud. 79. Princeton University Press, Princeton, N.J., 1974, 263–268. [37] C. J. Leininger, Equivalent curves in surfaces. Geom. Dedicata 102 (2003), 151–177. [38] H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. I. Hyperbolicity. Invent. Math. 138 (1999), no. 1, 103–149. [39] G. McShane, Simple geodesics and a series constant over Teichmuller space. Invent. Math. 132 (1998), no. 3, 607–632. [40] G. McShane and H. Parlier, Multiplicities of simple closed geodesics and hypersurfaces in Teichmüller space. Geom. Topol. 12 (2008), no. 4, 1883–1919. [41] G. McShane and H. Parlier, Simple closed geodesics of equal length on a torus. In Geometry of Riemann surfaces, London Math. Soc. Lecture Note Ser. 368, Cambridge University Press, Cambridge 2010, 268–282. [42] G. McShane and I. Rivin, A norm on homology of surfaces and counting simple geodesics. Internat. Math. Res. Notices 1995 (1995), no. 2, 61–69.

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[43] G. McShane and I. Rivin, Simple curves on hyperbolic tori. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 12, 1523–1528. [44] M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. Math. 167 (2007), no. 1, 179–222. [45] M. Mirzakhani, Weil-Petersson volumes and intersection theory on the moduli space of curves. J. Amer. Math. Soc. 20 (2007), no. 1, 1–23. [46] M. Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces. Ann. of Math. (2) 168 (2008), no. 1, 97–125. [47] D. Mumford, A remark on Mahler’s compactness theorem. Proc. Amer. Math. Soc. 28 (1971), 289–294. [48] A. Papadopoulos and G. Théret, On Teichmüller’s metric and Thurston’s asymmetric metric on Teichmüller space. In Handbook of Teichmüller (A. Papadopoulos, ed.), Vol. I, EMS Publishing House, Zürich 2007, 111–204. [49] A. Papadopoulos and G. Théret, Shortening all the simple closed geodesics on surfaces with boundary. Proc. Amer. Math. Soc. 138 (2010), no. 5, 1775–1784. [50] H. Parlier, Lengths of geodesics on Riemann surfaces with boundary. Ann. Acad. Sci. Fenn. Math. 30 (2005), no. 2, 227–236. [51] K. Rafi, A characterization of short curves of a Teichmüller geodesic. Geom. Topol. 9 (2005), 179–202 (electronic). [52] K. Rafi, A combinatorial model for the Teichmüller metric. Geom. Funct. Anal. 17 (2007), no. 3, 936–959. [53] B. Randol, Cylinders in Riemann surfaces. Comment. Math. Helv. 54 (1979), no. 1, 1–5. [54] B. Randol, The length spectrum of a Riemann surface is always of unbounded multiplicity. Proc. Amer. Math. Soc. 78 (1980), no. 3, 455–456. [55] I. Rivin, Simple curves on surfaces. Geom. Dedicata 87 (2001), no. 1–3, 345–360. [56] I. Rivin, A simpler proof of Mirzakhani’s simple curve asymptotics. Geom. Dedicata 114 (2005), 229–235. [57] I. Rivin, Geodesics with one self-intersection, and other stories. Adv. Math. 231 (2012), no. 5, 2391–2412. [58] P. Schmutz, Riemann surfaces with shortest geodesic of maximal length. Geom. Funct. Anal. 3 (1993), no. 6, 564–631. [59] P. Schmutz, Die Parametrisierung des Teichmüllerraumes durch geodätische Längenfunktionen. Comment. Math. Helv. 68 (1993), no. 2, 278–288. [60] P. Schmutz, Congruence subgroups and maximal Riemann surfaces. J. Geom. Anal. 4 (1994), no. 2, 207–218. [61] P. Schmutz, Systoles on Riemann surfaces. Manuscripta Math. 85 (1994), no. 3–4, 429– 447. [62] P. Schmutz Schaller, A cell decomposition of Teichmüller space based on geodesic length functions. Geom. Funct. Anal. 11 (2001), no. 1, 142–174. [63] P. Schmutz Schaller, Extremal Riemann surfaces with a large number of systoles. In Extremal Riemann surfaces (San Francisco, CA, 1995), Contemp. Math. 201, Amer. Math. Soc., Providence, RI, 1997, 9–19.

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[64] P. Schmutz Schaller, Geometry of Riemann surfaces based on closed geodesics. Bull. Amer. Math. Soc. (N.S.) 35 (1998), no. 3, 193–214. [65] P. Schmutz Schaller, Systoles and topological Morse functions for Riemann surfaces. J. Differential Geom. 52 (1999), no. 3, 407–452. [66] P. Scott, Subgroups of surface groups are almost geometric. J. London Math. Soc. (2) 17 (1978), no. 3, 555–565. [67] P. Scott, Correction to: “Subgroups of surface groups are almost geometric” [J. London Math. Soc. (2) 17 (1978), no. 3, 555–565], J. London Math. Soc. (2) 32 (1985), no. 2, 217–220. [68] C. Series, The geometry of Markoff numbers. Math. Intelligencer 7 (1985), no. 3, 20–29. [69] C. Series, Kerckhoff’s lines of minima in Teichmüller space. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Vol. III, EMS Publishing House, Zürich 2012, 123–153. [70] W. Thurston, Minimal stretch maps between surfaces. Preprint, arXiv:math.GT/9801039. [71] W. Thurston, A spine for Teichmüller space. Preprint, 1985. [72] S. A. Wolpert, Geodesic length functions and the Nielsen problem. J. Differential Geom. 25 (1987), no. 2, 275–296. [73] S. A. Wolpert, The Weil-Petersson metric geometry. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Vol. II, EMS Publishing House, Zürich 2009, 47–64.

Chapter 3

Curve complexes versus Tits buildings: structures and applications Lizhen Ji

Contents 1

2

3 4 5 6

7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The origin of Tits buildings . . . . . . . . . . . . . . . . . . . . . 1.3 The origin of curve complexes . . . . . . . . . . . . . . . . . . . . Definition of buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 A geometric definition of Tits buildings via symmetric spaces . . . 2.2 Axioms for spherical buildings via apartments . . . . . . . . . . . 2.3 Euclidean and hyperbolic buildings . . . . . . . . . . . . . . . . . 2.4 Rational Tits buildings of linear algebraic groups . . . . . . . . . . Definition of curve complexes . . . . . . . . . . . . . . . . . . . . . . . Geometric and topological properties of buildings . . . . . . . . . . . . Geometric and topological properties of curve complexes . . . . . . . . Selected applications of buildings . . . . . . . . . . . . . . . . . . . . . 6.1 Automorphism groups of buildings . . . . . . . . . . . . . . . . . 6.2 Mostow strong rigidity and generalizations . . . . . . . . . . . . . 6.3 Compactifications of locally symmetric spaces . . . . . . . . . . . 6.4 Cohomological dimension and duality properties of arithmetic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Simplicial volumes of locally symmetric spaces . . . . . . . . . . . 6.6 Asymptotic cones of symmetric spaces and locally symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Applications of Euclidean buildings . . . . . . . . . . . . . . . . . 6.8 Compactifications of Euclidean buildings . . . . . . . . . . . . . . Applications of curve complexes . . . . . . . . . . . . . . . . . . . . . . 7.1 Automorphism groups of curve complexes . . . . . . . . . . . . . 7.2 Isometry groups of Teichmüller spaces . . . . . . . . . . . . . . . 7.3 The ending lamination conjecture of Thurston . . . . . . . . . . . . 7.4 Quasi-isometric rigidity of mapping class groups . . . . . . . . . . 7.5 Finite asymptotic dimension of mapping class groups and the Novikov conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

136 136 136 137 139 139 141 142 144 144 147 150 153 153 153 155

. 159 . 160 . . . . . . . .

163 163 165 168 168 169 170 173

. 174

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7.6 7.7 7.8 7.9

Non-hyperbolicity of Weil–Petersson metric of Teichmüller space . Heegaard splittings and Hempel distance of 3-manifolds . . . . . . Partial compactifications of Teichmüller spaces and their boundaries Cohomological dimension and duality properties of mapping class groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Tangent cones at infinity of Teichmüller spaces, moduli spaces, and mapping class groups . . . . . . . . . . . . . . . . . . . . . . 7.11 Simplicial volumes of moduli spaces . . . . . . . . . . . . . . . . 7.12 The action of Modg;n on C .Sg;n / and applications . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 178 . 178 . 179 . 182 . . . .

183 185 186 187

1 Introduction 1.1 Summary Tits buildings Q .G/ of linear algebraic groups G defined over Q have played an important role in understanding partial compactifications of symmetric spaces and compactifications of locally symmetric spaces, cohomological properties of arithmetic subgroups and S-arithmetic subgroups of G.Q/. On the other hand, curve complexes C .Sg;n / of surfaces Sg;n were introduced to parametrize boundary components of partial compactifications of Teichmüller spaces and were later applied to understand properties of mapping class groups of surfaces and the geometry and topology of 3-dimensional manifolds. Tits buildings are spherical buildings. Another important class of buildings consists of Euclidean buildings, for example, the Bruhat–Tits buildings of linear algebraic groups defined over local fields. In this chapter, we summarize and compare some properties and applications of buildings and curve complexes. We try to emphasize their similarities but also point out differences. In some sense, curve complexes are combinations of spherical, Euclidean and hyperbolic buildings. We hope that such a comparison might motivate more questions and at the same time suggest methods to solve them. Furthermore it might introduce buildings to people who study curve complexes and curve complexes to people who study buildings.

1.2 The origin of Tits buildings Buildings were originally introduced by Tits [166], [167] in order to realize exceptional Lie groups as the symmetry groups of spaces (or geometry)1 so that one can construct 1 Classical simple Lie groups over C are the symmetry groups of quadratic forms or sesquilinear forms of finite-dimensional vector spaces over C. The same construction works for vector spaces over finite fields and produces classical finite groups of Lie type, but this method does not extend to exceptional simple Lie groups.

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geometrical analogues of exceptional simple Lie groups over arbitrary fields.2 See [152] for an overview of motivations and the history of Tits buildings. Since then, Tits buildings have been applied to many different situations with great success. There are several types of buildings: (1) spherical buildings, for example, Tits buildings, (2) Euclidean buildings, for example, Bruhat–Tits buildings, (3) hyperbolic buildings, (4) R-buildings, (5) topological buildings, (6) twin buildings. We will recall briefly some of these buildings below. See the book [2] for detailed definitions and structures of buildings, and the survey [87] for references on many different applications of buildings. There are several basic ways in which Tits buildings are used: (1) Buildings describe the large-scale geometry or the geometry at infinity of Lie groups, symmetric spaces and locally symmetric spaces. (2) Buildings describe the topology at infinity of partial compactifications of symmetric spaces and of the boundary of compactifications of locally symmetric spaces, and also the topology of the ends of locally symmetric spaces. (3) Buildings provide natural combinatorial and metric spaces on which groups such as Lie groups, arithmetic subgroups and p-adic Lie groups act. (4) Buildings can be used to study cohomological properties of arithmetic groups. In this chapter we are mainly interested in infinite buildings. For finite buildings and their applications in finite groups and their representation theory, see [44].

1.3 The origin of curve complexes Motivated by the Borel–Serre compactification of locally symmetric spaces [20], Harvey [69], [70], [73] introduced the curve complex C .Sg / of a compact oriented surface Sg to parametrize the boundary components of partial compactifications of the Teichmüller space Tg , the space of marked compact Riemann surfaces of genus g, which induce compactifications of the moduli space Mg of Riemann surfaces of genus g. In some sense, it was an exact analogue of the spherical Tits building Q .G/ of a linear semisimple algebraic group G defined over Q which serves as a parameter space for the boundary components of the Borel–Serre partial compactification of the symmetric space X D G=K. The same definition works for a more general oriented surface Sg;n of genus g with n punctures and gives a curve complex C.Sg;n /. Motivated by the analogy between arithmetic groups and mapping class groups Modg;n D DiffC .Sg;n /=Diff0 .Sg;n /, the curve complex C .Sg;n / was used to study cohomological properties of Modg;n [66], [80], [69]. It turns out that the curve complexes C.Sg;n / can also be used to study many problems in (lower-dimensional) topology and geometry, in particular the ending lamination conjecture of Thurston for 3-dimensional hyperbolic manifolds, quasi-isometric 2According to [153], p. 292, “... it is perhaps worth remarking that one of the initial motivations for the theory of buildings, at a time when Chevalley’s fundamental “Tohoku paper” had not yet appeared, was the search for a geometric way of obtaining algebraic analogues of the exceptional Lie groups.”

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rigidity of mapping class groups Modg;n and the finiteness of the asymptotic dimension of Modg;n . Some applications of curve complexes are motivated by results for Tits buildings, and others are quite different. There are several basic ways in which the curve complexes are used. (1) Curve complexes describe the large-scale geometry or the geometry at infinity of Teichmüller spaces and of the moduli spaces of Riemann surfaces. (2) Curve complexes parametrize boundary components of partial compactifications of Teichmüller spaces, they describe the topology at infinity of these partial compactifications, and they also describe the boundary of compactifications of moduli spaces, and the topology of the ends of moduli spaces. (3) Curve complexes provide natural combinatorial and metric spaces on which mapping class groups act, and structures and representations of the mapping class groups can be understood and constructed from these actions. (4) Curve complexes can be used to study cohomological properties of mapping class groups. (5) Surfaces and simple closed curves occur naturally in the study of 3-dimensional manifolds and hence curve complexes can be used to understand 3-dimensional geometry and topology. In this chapter, we will present some applications of buildings and curve complexes to support these general points. The above two lists suggest many similarities between these two classes of simplicial complexes. On the other hand, there are also dramatic differences between buildings and curve complexes. Besides the curve complex C.Sg;n /, there are also several related complexes for surfaces Sg;n , which are important to understand the geometry and topology of Teichmüller spaces Tg;n and mapping class groups Modg;n . We will address these issues as well. This chapter can be seen as a sequel of the survey papers [87] and [88] in some sense. Since it emphasizes curve complexes and their applications, it can complement the other two papers. Acknowledgments. This chapter is an expanded version of lecture notes of an invited talk at the International Conference on Buildings, Finite geometries and Groups in Bangalore, India, August 29–31, 2010. I would like to thank the organizers for their invitation. This chapter was written for the book with the same title in the series “Springer Proceedings in Mathematics”. I would like to thank Springer and the editor of the book, N. Sastry, for their permission to reprint this chapter in the Handbook of Teichmüller theory. I would also like to thank Dick Canary for many helpful suggestions and references, Bill Harvey for sending valuable reprints, Feng Luo, Juan Souto and Ralf Spatzier for helpful correspondences, Misha Kapovich and B. Sury for helpful comments, and Athanase Papadopoulos for reading several versions of the chapter very carefully and

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for many helpful comments. I would also like to thank an anonymous referee for helpful suggestions and comments. A part of the work in this chapter was done during a visit to MSC, Tshinghua University, Beijing, in 2010 and I would like to thank the people at this center for providing a stimulating environment.

2 Definition of buildings In this section, we first introduce in §2.1 the spherical Tits building .G/ of a semisimple Lie group G through a classification of geodesics in the associated symmetric space X D G=K . This justifies the point of view that .G/ describes the geometry at infinity of the symmetric space X . Then in §2.2, we discuss a more common definition of buildings through a system of apartments. In §2.3, we introduce Euclidean and hyperbolic buildings and we mention other buildings. In §2.4, we define the rational Tits building Q .G/ of a linear algebraic group G defined over Q.

2.1 A geometric definition of Tits buildings via symmetric spaces Let G be a semisimple noncompact Lie group with finitely many connected components, and let K  G be a maximal compact subgroup. Then X D G=K with an invariant metric is a symmetric space of noncompact type, and X is a simply connected nonpositively curved Riemannian manifold. Let X.1/ be the set of equivalence classes of geodesics in X. Specifically, we assume that all geodesics in X are of unit speed and directed. Two geodesics 1 .t / and 2 .t / in X are defined to be equivalent if lim sup d.1 .t/; 2 .t // < C1; t!C1

where d.; / is the distance function of X. It can be shown that for any point x0 2 X, X.1/ can be canonically identified with the unit sphere in the tangent space Tx0 X . It is known that there is a natural topology on X [X.1/ such that it is a compactification of X , called the geodesic compactification, or visual compactification, and X.1/ is hence called the sphere at infinity (or visual sphere) of X . See [4]. When X D SL.2; R/=SO.2/ is identified with the upper half place H2 , then X.1/ D R [ fi 1g. When X D SL.2; R/=SO.2/ is identified with the unit disc D D fz 2 C j jzj < 1g, then X.1/ is equal to the unit circle S 1 D @D. Clearly the isometry action of G on X preserves the equivalence relation between geodesics and hence acts on X.1/. A natural question is whether these points in X.1/, i.e, equivalence classes of geodesics, are the same, i.e., belong to one common G-orbit. If the answer is negative, a natural problem is to parametrize G-orbits in X.1/. It turns out that both questions can be answered using parabolic subgroups P of G, i.e., closed subgroups P such that G=P is compact. If P D G, it is a parabolic

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subgroup by definition. Any parabolic subgroup P with P ¤ G is called a proper parabolic subgroup of G. The set of proper parabolic subgroups of G is a partially ordered set by inclusion, and there are infinitely many maximal and minimal elements in this partially ordered set, which are called maximal proper parabolic subgroups and minimal parabolic subgroups of G. Proposition 2.1. The G-action on X extends to a continuous action on X [ X.1/. For any point z 2 X.1/, its stabilizer Gz in G is a proper parabolic subgroup, and every proper parabolic subgroup P of G arises as the stabilizer of a point z 2 X.1/. For each proper parabolic subgroup P of G, let P be the set of points in X.1/ that are fixed by P , equivalently the set of points z 2 X.1/ whose stabilizer Gz in G contains P . Proposition 2.2. For every proper parabolic subgroup P , P is a spherical simplex, and its interior Po , i.e., the open simplex when P is considered as a simplex, is equal to the set of points z whose stabilizer is equal to P . The simplex P consists of one point if and only if P is a maximal proper parabolic subgroup. If P is a minimal parabolic subgroup of G, then the dimension of P is equal to r  1, where r is the rank of X, i.e., the maximal dimension of totally geodesic flat submanifolds of X. The decomposition of X.1/ into P , a X.1/ D [P p D P0 ; P

gives X.1/ the structure of an infinite simplicial complex, denoted by .X/. The simplicial complex .X/ is called the Tits building associated with the symmetric space X of noncompact type. If the rank r of X is equal to 1, then .X/ is a 0-simplicial complex, i.e., a disjoint union of points. If r > 0, it can be shown that .X/ is connected. This can be seen from the axioms of buildings in the next subsection. By definition, .X/ classifies geodesics of X into different types. Proposition 2.3. Let P0 be a minimal parabolic subgroup of G. Then every G-orbit in X.1/ meets P0 in exactly one point. G acts on the Tits building .X/ by simplicial maps, and every simplex P is contained in the G-orbit of a face of P0 , i.e., Gn.X/ can be identified with P0 . Corollary 2.4. G acts transitively on X.1/ if and only if a minimal parabolic subgroup of G is also a maximal parabolic subgroup G, i.e., the rank r of X is equal to 1. For more discussion and proofs of the above results, see [60].

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2.2 Axioms for spherical buildings via apartments The Tits building .X/ defined in the previous subsection can be described directly in terms of proper parabolic subgroups of G. Let P be the partially ordered set of proper parabolic subgroups of G, where the partial order is given by containment, i.e., the opposite of the inclusion mentioned in the previous subsection. The structure theory of parabolic subgroups of G shows that this poset can be realized by an infinite simplicial complex .G/ satisfying the following conditions: (1) Every parabolic subgroup P 2 P corresponds to a unique simplex P in .G/, and every simplex of .G/ is of this form. (2) For any two parabolic subgroups P1 ; P2 2 P , P1  P2 if and only if P1 contains P2 as a face. (3) A simplex P is a point if and only if P is a maximal parabolic subgroup of G. For a non-maximal parabolic subgroup P , the vertices of P correspond to maximal parabolic subgroups that contain P . This simplicial complex .G/ is called the Tits building of G. By Proposition 2.2, .G/ is isomorphic to .X/. These are important examples of spherical Tits buildings. Definition 2.5. A simplicial complex  is called a spherical Tits building if it contains a family of subsets called apartments and satisfies the following conditions: (1) Every apartment is a finite Coxeter complex. (2) Any two simplices are contained in some apartment. (3) Given two apartments † and †0 and simplices ;  0 2 † \ †0 , there exists an isomorphism of † onto †0 which keeps ,  0 pointwise fixed. In the above definition, for any finite Coxeter group W , i.e., a finite group generated by reflections with respect to hyperplanes in a fixed Euclidean space, there is a Coxeter complex, which is a finite simplicial complex constructed as follows. Every reflection ˛ 2 W fixes a hyperplane H˛ . The collection of such hyperplanes H˛ is invariant under W . Connected components of the complement of the union of fH˛ g in V are called chambers, which are simplicial cones. The chambers and their faces together give a partition of V into simplicial cones. Let S be the unit sphere in V . Then the intersection of S with these simplicial cones gives a finite simplicial complex, called the Coxeter complex of W , whose underlying topological space is S , i.e., the Coxeter complex gives a finite triangulation of the unit sphere. To see that .X/ is a spherical Tits buildings, we start with the construction of apartments. For any flat totally geodesic submanifold F of X , which is isometric to Rr , its closure in X [ X.1/ is equal to F [ F .1/, where F .1/ is homeomorphic to the sphere S r1 . The simplicial complex structure of X.1/ given by .X/ induces a simplicial complex structure on F .1/. In fact, F .1/ is equal to the union of P

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for some parabolic subgroups P . Denote this finite simplicial complex by †F . Then it is a Coxeter complex associated with the Weyl group of the Lie group G. Proposition 2.6. With respect to the collection of finite subcomplexes †F associated with flat subspaces F , the infinite simplicial complex .X/ is a spherical Tits building. To understand the spherical building structure of .G/, we need to define apartments in terms of group structure. For any maximal compact subgroup K of G, let g D k ˚ p be the Cartan decomposition. Let a  p be a maximal abelian subalgebra. Let A D exp a be the corresponding Lie subgroup of G. Then there are only finitely many parabolic subgroups P that contain A, and the union of their simplices P gives a triangulation of the unit sphere a.C1/ of a and is a Coxeter complex for the Weyl group of G. Denote it by †a . Proposition 2.7. With respect to the collection of finite subcomplexes †a associated with maximal abelian subspaces, the infinite simplicial complex .X/ is a spherical Tits building. Proposition 2.7 is proved using the Bruhat decomposition. To derive Proposition 2.6 from Proposition 2.7, we need to identify their apartments. The finite simplicial complexes †F and †a can be identified as follows. For any maximal flat totally geodesic submanifold F of X, let x0 2 F be any point, and K D Gx0 be the stabilizer of x0 in G. Then there exists a maximal abelian subalgebra a  p with respect to the Cartan decomposition of g induced by K such that F D exp a x0 , and †F is identified with †a under the identification between .X/ and .G/. For the proof of the above propositions and more discussion, see the book [60] and references therein.

2.3 Euclidean and hyperbolic buildings It will be shown below that the curve complex C .Sg;n / shares some properties with spherical buildings, and also with Euclidean buildings and hyperbolic buildings. For convenience, we briefly introduce their definitions here. First we define Euclidean reflection groups. Let V be a Euclidean space. An affine reflection group W on V is a group of affine isometries generated by reflections with respect to affine hyperplanes such that the set H of affine hyperplanes fixed by reflections in W is locally finite. Clearly, a finite reflection group is an affine reflection group. Assume that W is an infinite irreducible reflection group. Then the hyperplanes in H divide V into simplices, and W acts simply transitively on the set of simplices. These simplices and their faces form a Euclidean Coxeter complex. Definition 2.8. A simplicial complex  is called a Euclidean building if it contains a family of subsets called apartments and satisfies the following conditions:

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(1) Every apartment is an infinite Euclidean Coxeter complex. (2) Any two simplices are contained in some apartment. (3) Given two apartments † and †0 and simplices ;  0 2 † \ †0 , there exists an isomorphism from † onto †0 which keeps  ,  0 pointwise fixed. An important source of Euclidean buildings comes from the Bruhat–Tits building BT .G/ associated with a linear semisimple algebraic group G defined over a nonArchimedean local field k. In this case, the simplices of BT .G/ are parametrized by parahoric subgroups of G.k/. There is no simple definition or characterization of parahoric subgroups of G.k/ as in the case of parabolic subgroups. When G is simply connected, any maximal open compact subgroup of G.k/ is a maximal parahoric subgroup, and the converse is also true. See §3 in [87] for more details and references. Now we introduce hyperbolic buildings. Let Hn be the real hyperbolic space of dimension n. Let P be a compact convex hyperbolic polyhedron such that the reflections with respect to its codimension 1 faces generate a group W that acts properly on Hn with P as a fundamental domain. Then the totally geodesic hypersurfaces of Hn that are fixed by reflections in W are locally finite, and the connected components of the complement of the union of these hyperplanes are hyperbolic polyhedra. These polyhedra and their faces form a polyhedral complex, which is called a hyperbolic Coxeter complex associated with P . Hyperbolic Coxeter complexes can only exist for n  29. Definition 2.9. A polyhedral complex  is called a hyperbolic building of type P if it contains a family of subcomplexes called apartments and satisfies the following conditions: (1) Every apartment is a hyperbolic Coxeter complex determined by P . (2) Any two polyhedra are contained in some apartment. (3) Given two apartments † and †0 and polyhedra ;  0 2 † \ †0 , there exists an isomorphism of † onto †0 which keeps ,  0 pointwise fixed. See [55] for more detail and examples. When n D 2, P is given by a compact polygon in the hyperbolic plane H2 such that the angle at each vertex is equal to =m for some integer m, and we get a hyperbolic building of Fuchsian type [22]. A natural generalization of spherical buildings is the notion of twin buildings. An important example is a pair of spherical Tits buildings associated to one linear algebraic group, for example, SL.n; Fp Œt; t 1 /. See [1] for the definition and applications to S -arithmetic subgroups of linear algebraic groups defined over function fields. See [45], [46] for a more geometric group theoretic point of view of buildings and Coxeter complexes.

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2.4 Rational Tits buildings of linear algebraic groups As explained before, for any semisimple Lie group G, there is a spherical Tits building .G/ that can be related to the geodesic compactification of its symmetric space X D G=K. If G is the real locus of a linear algebraic group G  GL.n; C/ defined over Q with positive Q-rank, then there are several other spherical Tits buildings associated with G. They are important for compactifications of locally symmetric spaces and in the study of arithmetic subgroups of G.Q/ and more generally S arithmetic subgroups of G.Q/. Let PQ be the set of Q-parabolic subgroups of G, i.e., parabolic subgroups defined over Q. Then there is an infinite simplicial complex Q .G/ whose simplices P are parametrized by parabolic subgroups P in PQ and which satisfies the following conditions: (1) When P is maximal Q-parabolic subgroup, P is a point. (2) For two Q-parabolic subgroups P1 , P2 , the inclusion relation P1  P2 holds if and only if P1 contains P2 as a face. The simplicial complex Q .G/ is called the Q-Tits building of G. It cannot be realized as the boundary of a compactification of X as the Tits building .G/, where G D G.R/ is the real locus of G, or as a subset of the boundary of a compactification of X if the R-rank of G is strictly greater than the Q-rank of G. But Q .G/ can be realized as the boundary of a partial compactification of X, which gives the Tits compactification of nX in [94]. The main application of Q .G/ is that Q .G/ naturally parametrizes boundary components of partial compactifications of X whose quotients by arithmetic subgroups   G.Q/ are compactifications of locally symmetric spaces nX. See §6.3. For every prime p, let Qp be the field of p-adic numbers. When G is considered as a linear algebraic group defined over Qp , the set of parabolic subgroups of G defined over Qk gives a spherical Tits building Qp .G/. As mentioned before, there is a Euclidean building, the Bruhat–Tits building, associated with the p-adic Lie group G.Qp /. Denote this building by BT .G.Qp //. We will see below that BT .G.Qp // can be compactified by adding the spherical building Qp .G/ (Proposition 6.25).

3 Definition of curve complexes Let Sg;n be an oriented surface of genus g with n punctures (or n boundary components). In the rest of this paper, we assume that the Euler characteristic .Sg;n / is negative so that Sg;n admits complete hyperbolic metrics of finite area. A simple closed curve c in Sg;n is called essential if it is not homotopic to a point or a loop around a puncture or a boundary component. The curve complex C .Sg;n / is a simplicial complex such that:

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(1) Its vertices are parametrized by homotopy classes of essential simple closed curves Œc in Sg;n . (2) Homotopy classes Œc0 ; Œc1 ; : : : ; Œck  form the vertices of a k-simplex if and only if they are pairwise distinct and admit disjoint representatives. If we put a complete hyperbolic metric of finite area and with geodesic boundary on Sg;n , then the homotopy class of each essential simple closed curves in Sg;n contains a unique simple closed geodesic, and hence the vertices of C .Sg;n / correspond to simple closed geodesics of the hyperbolic metric. The simplest example is C.S1;1 /. Let †1;1 be a hyperbolic surface of genus 1 with one puncture. In this case, for every simple closed geodesic of †1;1 , there is no other simple closed geodesic that is disjoint from it. Therefore, C .S1;1 / is a countable collection of points and can be identified with the rational boundary points Q [ f1g of the upper half-plane H2 . For the surface S1;1 , a slight modification gives an interesting complex C1 .S1;1 /. The vertices of C1 .S1;1 / still correspond to simple closed geodesics  in †1;1 , and k C 1 simple closed geodesics 1 ; : : : ; kC1 form the vertices of a k-simplex if and only if for every pair of distinct geodesics i and j , the intersection number .i ; j / D ˙1. Note that C1 .S1;1 / is 2-dimensional. Then the 1-skeleton of C1 .S1;1 / can be identified with the Farey graph, whose vertices are numbers in Q [ f1g, the vertex 1 is joined to every integer n, and two points rs and yx in reduced form (i.e., the numerators and denominators do not contain common primes) are connected by an edge if and only if ry  sx D ˙1. The simplicial complex C1 .S1;1 / gives an ideal triangulation (or tessellation) of the upper half plane H2 , which is equal to the Farey tessellation, and hence C1 .S1;1 / is contractible. (It should be emphasized that a general curve complex C1 .Sg;n / is not contractible, and its nontrivial homotopy is used crucially to prove that the mapping class group Modg;n is not a virtual Poincaré duality group. See §7.9 below.) In the general case, C.Sg;n / is fairly complicated and there is no simple geometric model. For example, it is not locally finite in general. It is not locally finite because the complement of a simple closed curve on a surface contains infinitely many homotopy classes of essential simple closed curves except in some special cases (the torus with at most one hole or the sphere with at most four holes). Note that C .S1;1 / is a disjoint union of countably many points and hence is locally finite. But the nonlocally finiteness phenomenon can be seen in the above example C1 .S1;1 / already. One way to see this is to note that SL.2; Z/ acts on the upper half plane, the Farey tessellation and the Farey graph. Each vertex of C1 .S1;1 / has an infinite stabilizer in SL.2; Z/, which permutes the 2-simplices that contain the vertex, and hence there are infinitely many edges of C1 .S1;1 / out of each vertex, and C1 .S1;1 / is not locally finite. The curve complex is not “homogeneous” in general as in the case of buildings. We will discuss some of its properties later. Compared with the definition of the spherical Tits building .G/, this suggests that a homotopy class Œc of essential simple closed curves plays a role similar to the one of a maximal parabolic subgroup, and a simplex of C .Sg;n / plays the role of a parabolic subgroup. More specifically, we have the following similarities:

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(1) The boundary components of the Borel–Serre partial compactification of a symmetric space X are parametrized by proper Q-parabolic subgroups of G, and maximal boundary components correspond to maximal Q-parabolic subgroups. On the other hand, the boundary components of an analogous Borel–Serre partial compactification of the Teichmüller space Tg;n or of the completion of Tg;n in the Weil–Petersson metric are parametrized by simplices of C .Sg;n /, and the maximal boundary components correspond to the vertices. (See §6 and §7 below for more details). (2) For every Q-parabolic subgroup P of G, there is an associated Langlands decomposition of X. When P is a minimal Q-parabolic subgroup, it is reduced to the Iwasawa decomposition of X (or rather of the Lie group G). On the other hand, for every top-dimensional simplex of C .Sg;n /, there is a Fenchel–Nielsen coordinate system (or decomposition) of Tg;n . It is easy to see that the maximal number of homotopy classes of essential simple closed curves with disjoint representatives is equal to 3g  3 C n. Therefore C .Sg;n / has dimension 3g  4 C n. Simplices of C .Sg;n / with dimension equal to 3g  4 C n play a similar role as minimal parabolic subgroups. There are also several other related complexes. (1) The arc complex for surfaces with at least one puncture or boundary component in [66], [67], [106]. The topology of the arc complex at infinity is closely related to the topology of the curve complex C.Sg;n / [67], Theorem 3.4. The arc complex is used crucially in obtaining a Modg;n -equivariant cell decomposition and a spine of optimal dimension of Tg;n when n > 0 [67], Theorem 1.3 and Theorem 2.1. (2) The Torelli curve complex in [51] (see also [101]), its vertices are homotopy classes of separating simple closed curves of Sg and bounding pairs of curves. It plays a similar role for the Torelli group as the curve complex C .Sg;n / for the mapping class group Modg;n , which is equal to Diff C .Sg;n /=Diff 0 .Sg;n / (see §7.5 below). (3) The pants complex in [74] whose vertices are pants decompositions of Sg;n . It was originally used to find generators and relations for the mapping class group Modg;n . It was also used to describe the coarse geometry of the Weil–Petersson metric of Teichmüller space [33]. (Note that the pants complex is a CW-complex but not a simplicial complex.) (4) The train-track complex in [65], [143]. It was used in [65] to prove quasiisometric rigidity of the mapping class group Modg;n (see Proposition 7.15 below). (5) Sub-complexes of C.Sg;n / such as the complex of separating curves [145], [115], [101] (see also [72]). (6) The complex of domains and its various subcomplexes [129]. ([129] contains a comprehensive list of complexes associated to surfaces.) (7) A family of complexes related to the curve complex [136].

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Remark 3.1. A class of groups closely related to arithmetic groups and mapping class groups consists of outer automorphism groups Out.Fn / of the free group Fn on n generators. There are several candidates for the analogue of the spherical Tits building Q .G/ and the curve complex C.Sg;n /. They are infinite simplicial complexes on which Out.Fn / acts simplicially, and they are also homotopy equivalent to a bouquet of spheres. See [75], [99], [88].

4 Geometric and topological properties of buildings As mentioned in the introduction, the original motivation of Tits buildings was to give a geometric interpretation of exceptional Lie groups and hence to construct their analogues over finite fields. Their geometric and topological structures have been used for various applications, some of which will be explained in this chapter. First we define canonical metrics on buildings, their diameters, and curvature properties. Then we state the Solomon–Tits Theorem which determines the topology of buildings. Finally we explain a relation between Euclidean buildings and spherical buildings. Recall that a geodesic segment in a metric space .M; d / is an isometric embedding  W Œa; b ! M . .M; d / is called a geodesic metric space if every two points in M are joined by a geodesic segment. M is called geodesically complete if every geodesic segment in M ,  W Œa; b ! M , can be extended infinitely in both directions to a map  W .1; C1/ ! M such that for any t 2 .1; C1/, when jt  t 0 j is sufficiently small, d..t 0 /; .t// D jt  t 0 j, i.e., the map is only locally distance minimizing. Note that this is different from the notion of a complete geodesic metric space. Clearly any complete Riemannian manifold is geodesically complete. On the other hand, any Riemannian manifold with nonempty boundary is not geodesically complete, since a geodesic segment perpendicular to the boundary cannot be extended. Proposition 4.1. Every spherical Tits building  admits a metric such that (1) its restriction to each apartment is isometric to the unit sphere S r1 in Rr , where r is the rank of the building, i.e., the number of vertices of the top-dimensional simplices; (2) it is a complete geodesic metric space; (3) it is geodesically complete. This metric is called the Tits metric. If  D .G/, then G acts simplicially and isometrically on .G/. The idea of the proof is as follows. By definition, each apartment of  is a finite Coxeter complex, which is a triangulation of the unit sphere S r1 . The metric on S r1 induces a geodesic metric on each apartment. The axioms for buildings show that the

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metrics on all the apartments are compatible, for example they agree on intersection, and they can be patched together to form a metric on . Since all apartments are isometric and can be mapped isometrically from one to another, it can be shown that  is a geodesic metric space. Since each apartment is geodesically complete, it can be shown similarly that  is also geodesically complete. Proposition 4.2. For every spherical Tits building , its diameter with respect to the Tits metric is equal to 2. Proof. It is clear that the diameter of every apartment is equal to 2. Since every two points of  are contained in an apartment, the diameter of  is less than or equal to 2. Since any apartment can be retracted to a fixed apartment, it can be shown that each apartment is a totally geodesic subspace and hence the diameter of  is equal to 2. Proposition 4.3. Every Euclidean building  admits a metric such that (1) its restriction to each apartment is isometric to the Euclidean space Rr , where r is the rank of the building, i.e., r is equal to the dimension of the top-dimensional simplices of , or equivalently, r C 1 is the number of vertices of the topdimensional simplices of the Euclidean building ;3 (2) it is a complete geodesic metric space; (3) it is geodesically complete; (4) with respect to this metric,  is a CAT.0/-space in the sense that every two distinct points are connected by a unique geodesic, and every triangle in  is thinner than the corresponding geodesic in Rr of the same side lengths. This metric is called the Tits metric. If  D BT .G/, the Bruhat–Tits building of a linear semisimple algebraic group defined over a non-Archimedean local field k, then G.k/ acts isometrically on . See [2] for more details and also [30] for a general discussion on CAT.0/-spaces. Corollary 4.4. The diameter of any Euclidean building with respect to the Tits metric is infinite. Proof. This follows from the fact that every apartment is a totally geodesic subspace and hence has infinite diameter. 3 It might be worthwhile to emphasize that the rank of a spherical Tits building in Proposition 4.1 is equal to 1 plus the dimension of the top-dimensional simplices. This convention is different from the convention of rank of Euclidean buildings here. Roughly speaking, the reason for these conventions is that the rank of buildings should be equal to the rank of the algebraic groups which define them. For example, a simple algebraic group over R of rank 1 gives a 0-dimensional Tits building, and a simple algebraic group defined over a p-adic number field of rank 1 gives a 1-dimensional Euclidean building. Both buildings are defined to have rank 1. Another reason for such conventions is that for each Euclidean building of rank r, there is a spherical building of the same rank r which can be added to the infinity of the Euclidean building. See §6.8 below.

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Proposition 4.5. Every hyperbolic building  admits a metric such that (1) its restriction to each apartment is isometric to the hyperbolic space Hn ; (2) it is a complete geodesic metric space; (3) it is geodesically complete; (4) with respect to this metric,  is a CAT.1/-space in the sense that every two distinct points are connected by a unique geodesic, and every triangle in  is thinner than the corresponding geodesic in Hn of the same side lengths; (5) it has infinite diameter. This metric is called the Tits metric on . Recall that a geodesic metric space is called a ı-hyperbolic space if for every triangle in the space, any one side is contained in the ı-neighborhood of the union of the other two sides. It is known that the real hyperbolic space Hn is ı-hyperbolic. More generally, every simply connected Riemannian manifold with sectional curvature bounded from above by a negative constant is also ı-hyperbolic, and every CAT.1/ geodesic metric space is ı-hyperbolic. Every metric tree is 0-hyperbolic (i.e., ı-hyperbolic, where ı D 0 if we use closed ı-neighborhoods.) Note that every metric space with a finite diameter is automatically hyperbolic for trivial reasons. On the other hand, when n  2, Rn is not ı-hyperbolic. Corollary 4.6. Every hyperbolic building is ı-hyperbolic. Proposition 4.7. For a Euclidean building , if its rank is equal to 1, then it is a tree and is a ı-hyperbolic space, otherwise it is not a ı-hyperbolic space. Proof. The first statement was mentioned earlier, and the second statement follows from the fact that for any ı-hyperbolic space, any totally geodesic subspace is also ıhyperbolic. (Recall that a subspace Y of a geodesic space X is called a totally geodesic subspace if for any two points x; y 2 Y , any geodesic segment in X connecting x and y is contained in Y .) The topology of spherical buildings is given by the Solomon–Tits Theorem [2]. Proposition 4.8. Let  be a spherical Tits building of rank r. Then  is homotopy equivalent to a bouquet of spheres S r1 . If  D Q .G/, then the bouquet contains infinitely many spheres. Basically, fix one simplex  in ; then  is the union of apartments containing . Each apartment is homotopy equivalent to a sphere, and this union gives the bouquet. When  D Q .G/, there are infinitely many Q-parabolic subgroups and hence there are infinitely many spheres in the bouquet. On the other hand, the topology of Euclidean and hyperbolic buildings is trivial, i.e., they are contractible.

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Proposition 4.9. Euclidean buildings and hyperbolic buildings are contractible. Proof. By Proposition 4.3, every Euclidean building  is a CAT.0/-space. Fix a base point x0 2 . Then any other point x 2  can be connected to x0 by a unique geodesic, and deformation retraction along such rays from x0 shows that  is contractible. Since a CAT.1/-space is also a CAT.0/-space (or by Proposition 4.5), the same proof works for hyperbolic buildings.

5 Geometric and topological properties of curve complexes The curve complex C.Sg;n / has some properties that are similar to all three types of buildings in the previous subsection. We can put a metric on C .Sg;n / by making each simplex a standard Euclidean one with side length 1. Proposition 5.1. C.Sg;n / is a complete geodesic space, i.e., it is a geodesic space and is complete as a metric space. In order to prove this proposition, we need mapping class groups and their actions on C .Sg;n /. Let Diff C .Sg;n / be the group of orientation-preserving diffeomorphisms of Sg;n , and Diff 0 .Sg;n / its identity component, which is a normal subgroup. Diff C .Sg;n /=Diff 0 .Sg;n / is called the mapping class group of Sg;n and denoted by Modg;n . A closely related group is the extended mapping class group 0 Mod˙ g;n D Diff.Sg;n /=Diff .Sg;n /, where Diff.Sg;n / is the group of all diffeomorphisms of Sg;n including both orientation-preserving and orientation-reversing diffeomorphisms. When 3g  3 C n  2, for any essential simple closed curve c on Sg;n , there are infinitely many essential simple closed curves that are disjoint from c and not homotopy equivalent to c. This implies that the curve complex C .Sg;n / is not locally finite. Gluing up the metrics on the simplices certainly gives a length metric on the space, but it is not obvious why it is a geodesic metric, i.e., every two points are connected by a geodesic segment. This follows from a theorem on metric simplicial complexes in [29], Theorem 1.1. Proposition 5.2. If K is a metric simplicial complex with only finitely many isometry types of simplices, then the metric on K glued up from the metrics of the simplices is a complete geodesic metric space. To apply Proposition 5.2 to prove Proposition 5.1, we need to show that C .Sg;n / has a large symmetry group.

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Proposition 5.3. The mapping class group Modg;n and the extended mapping class group Mod˙ g;n act simplicially and isometrically on C .Sg;n / with respect to the metric which is glued up from the simplices, and the quotients Modg;n nC .Sg;n / and Mod˙ g;n nC .Sg;n / are finite CW-complexes, i.e., unions of finitely many cells. Proof. Since diffeomorphisms of Sg;n map simple closed curves to simple closed curves, preserve the homotopy classes, and preserve existence of disjoint representatives, it is clear that Modg;n acts simplicially on C .Sg;n /. Since a top-dimensional simplex of C .Sg;n / corresponds to a pants decomposition of Sg;n , the statement that Modg;n nC .Sg;n / is a finite complex is equivalent to the statement that there are only finitely many nonisomorphic pants decompositions of Sg;n . The latter is reduced to finiteness of the set of homeomorphism classes of trivalent graphs on the surface. See §3.5 of [40] for detail. The proof for Mod˙ g;n is the same. Proof of Proposition 5.1. By combining Propositions 5.2 and 5.3, we can see that C .Sg;n / is a complete geodesic space. (Proposition 5.1 was known and stated in [132], p. 1007.) Remark 5.4. Since the curve complex C.Sg;n / is a thick chamber complex in case g  2 [69], Proposition on page 266, in the sense that every co-dimension 1 simplex is contained in at least three top-dimensional simplices, it might be reasonable to conjecture that C .Sg;n / is a geodesically complete metric space, i.e., every geodesic segment in it can be extended infinitely in both directions. An important metric property of C.Sg;n / is the next result ([127], Theorem 1.1, Proposition 4.6). Proposition 5.5. The curve complex C .Sg;n / is a ı-hyperbolic space of infinite diameter. For simplified proofs, see [23], [63], [64]. As a ı-hyperbolic space, C .Sg;n / has a boundary @C .Sg;n /, which is similar to the geodesic boundary X.1/ of a symmetric space of noncompact type mentioned earlier and consists of equivalence classes of quasi-geodesics. (Note that for a proper ı-hyperbolic space, the boundary can also be defined as the set of equivalence classes of geodesics [98], Proposition 2.10.) Proposition 5.6. The boundary @C .Sg;n / is naturally homeomorphic to the space EL.Sg;n / of laminations of Sg;n which are filling and minimal (every leaf is dense in the support). Proposition 5.6 is due to Klarreich [103]. See also [63] for a proof. For every complete hyperbolic metric of finite volume on Sg;n , a geodesic lamination is a closed subset which is a disjoint union of complete simple geodesics. A geodesic lamination is called filling if it intersects every simple closed geodesic. The set of filling geodesic

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laminations has a natural topology, which is the induced topology from the quotient of Thurston’s topology. It is known that different choices of hyperbolic metrics on Sg;n give rise to homeomorphic spaces of filling geodesic laminations. Therefore, for the surface Sg;n of negative Euler characteristic, there is a well-defined space of filling and minimal laminations up to homeomorphism, denoted by EL.Sg;n / in the proposition. A large-scale invariant of a noncompact metric space is the asymptotic dimension introduced by Gromov. Another large-scale invariant (or rather property) is PropertyA, which is a weak amenability-type condition. See [140] for a definition. Proposition 5.7. The asymptotic dimension of C .Sg;n / is finite and has Property A. The former was proved in [8] and the latter in [100]. For some results on the quasi-isometry type of C.Sg;n /, see [148]. The homotopy type of C.Sg;n / was determined in [67] (see also [66] and [80]). Proposition 5.8. The curve complex C .Sg;n / is homotopy equivalent to a bouquet of spheres S d , where d D 2g  2 D .Sg / if n D 0, d D .Sg;n /  1 D 2g  3 C n if g  1 and n > 0, and d D .S0;n /  2 D n  4 if g D 0. The natural question on how many spheres are contained in the bouquet was answered in [86]. Proposition 5.9. The curve complex C.Sg;n / has infinite topology, i.e., the bouquet of spheres in Proposition 5.8 contains infinitely many spheres. Propositions 5.8 and 5.9 are an analogue of the Solomon–Tits theorem for spherical buildings (Proposition 4.8). Remark 5.10. Apartments are special and important subcomplexes of buildings and they explain easily the Solomon–Tits theorem. Though the curve complex C .Sg;n / shares many properties with buildings, one important difference or rather a mystery is that inside C.Sg;n /, in general, there are no known corresponding distinguished finite subcomplexes whose underlying spaces are spheres or are homotopy equivalent to spheres. In the special case g D 2, n D 1, spheres in C .Sg;n / that generate the topdimensional homology group have been constructed in [31], §4.3. There are also some candidates of spheres in C.Sg;n / when n D 1 in [31], §4.3. It might be helpful to note that C.Sg;n / contains many infinite subcomplexes corresponding to the curve complexes of sub-surfaces of Sg;n . These subcomplexes have played an important role in understanding the large-scale geometry of C .Sg;n / and Tg;n in [125], [126].

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6 Selected applications of buildings There are many applications of buildings. We select some which are similar to the applications of curve complexes discussed below, for the purpose of comparison. For more applications of buildings, see [87].

6.1 Automorphism groups of buildings One reason why buildings are useful in geometry and topology is that they are rigid and also admit a large symmetry group. We state briefly several general results for spherical Tits buildings. For the precise statements, see theorems and propositions numbered by 5.9, 6.3, 6.13, 8.4.5, 9.1, 10.2 in [166], 5.8. Recall that a building is called thick if every simplex of codimension 1 is contained in at least three top-dimensional simplices. Proposition 6.1. Every thick irreducible spherical Tits building of rank at least 3 is the spherical building of a linear semisimple algebraic group. Proposition 6.2. If k .G/ and k 0 .G0 / are irreducible thick spherical Tits buildings of rank at least 2 associated with linear semisimple algebraic groups G and G0 defined over fields k and k 0 respectively, then any isomorphism between k .G/ and k 0 .G0 / is essentially determined by an isomorphism between the algebraic groups G and G0 and an isomorphism between k and k 0 . See also Theorem 16.1 and Corollary 16.2 of [138]. Corollary 6.3. If k .G/ is an irreducible thick spherical Tits building of rank at least 2 associated with a semisimple linear algebraic group defined over a field k, then any automorphism of k .G/ is essentially induced by automorphisms of the group G and the field k.

6.2 Mostow strong rigidity and generalizations One major application of Tits buildings is Mostow strong rigidity (also called Mostow– Prasad strong rigidity) for locally symmetric spaces [138], [144]. Let X D G=K be a symmetric space of noncompact type. Any discrete subgroup   G acts properly on X and the quotient nX is called a locally symmetric space. The invariant metric on X defines an invariant measure on X which induces a measure on nX.  is called a lattice if the volume of nX with respect to the induced measure is finite. If G D G.R/ is the real locus of a linear semisimple algebraic group and   G.R/ is an arithmetic subgroup (see the next subsection for a precise definition), then  is a lattice in G, by the reduction theory of arithmetic subgroups.

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A locally symmetric space nX is called irreducible if no finite cover splits isometrically as a product. Theorem 6.4 (Mostow strong rigidity). Let nX and  0 nX 0 be two irreducible locally symmetric spaces of finite volume. Assume that one of the symmetric spaces X and X 0 is not isometric to the hyperbolic plane H2 , and  and  0 are isomorphic as abstract groups. Then nX and  0 nX 0 are isometric after suitable scaling of the metrics on the irreducible factors of X and X 0 . We note that if  is torsion-free, then  acts on X fixed-point freely, and nX is a manifold and its fundamental group 1 .nX / is equal to . Otherwise, nX is an orbifold, and its fundamental group as an orbifold is equal to . Mostow strong rigidity says roughly that if a locally symmetric space of finite volume is not a hyperbolic surface, then its isometry type is determined by its fundamental group. The proof in [138] depends on the rank of X and works under the assumption that nX and  0 nX 0 are compact. The basic idea is that an isomorphism between  and  0 induces an equivariant quasi-isometry between X and X 0 . When the rank of X is at least 2, it induces an isomorphism between the Tits buildings .X/ and .X 0 /, which are of rank at least 2, and the rigidity of Tits buildings in §6.1 implies the desired rigidity. The rank 1 case requires a different proof that involves quasi-conformal maps and ergodic actions of lattices on the sphere X.1/, which is a real-analytic manifold in this case. The remaining cases where nX and  0 nX 0 are noncompact were proved in [144] when the rank of X is equal to 1, and in [123] when the rank of X is at least 2. Remark 6.5. Since the original proofs in [138], there have been several different proofs of Mostow strong rigidity. When the locally symmetric spaces are irreducible and the covering symmetric spaces are of rank at least 2, the result also follows from the stronger super-rigidity of Margulis [124]. When nX and  0 nX 0 are compact, there are also proofs of Mostow strong rigidity in some cases by the method of harmonic maps (see [59] for a summary and references). When the rank of X is equal to 1, in particular when X is the real hyperbolic space, there are at least two completely new proofs: (1) A proof by Gromov using the notion of simplicial volume and the fact that the simplicial volume of a finite-volume hyperbolic manifold is proportional to the volume of the hyperbolic metric. See the books [10] and [149] for detailed descriptions. (2) A proof in [13] using the notion of barycenter map that characterizes locally symmetric spaces among compact negatively curved Riemannian manifolds in terms of minimal entropy. There are also proofs in [165], [84] of Mostow strong rigidity for hyperbolic spaces using quasi-conformal maps that generalize and simplify the original proof in [138].

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Mostow strong rigidity has been generalized in several different ways. One generalization in [4] replaces one locally symmetric space, say  0 nX 0 , by a compact Riemannian manifold M 0 of nonpositive curvature, and still concludes that if nX and M 0 are homotopy equivalent, nX is irreducible and X is of rank at least 2, then M 0 is also a locally symmetric space isometric to nX after a scaling of the metric. A further generalization in [109] replaces the above Riemannian manifold M 0 by a geodesically complete metric space whose universal covering space is a CAT.0/-space without changing the above conclusion. For this generalization, the following characterization of symmetric spaces of noncompact type and Euclidean buildings was proved in [109], Main Theorem 1.2. Proposition 6.6. Let X be a locally compact geodesically complete CAT.0/-space. Let dTits be the Tits metric on X.1/. If .X.1/; dTits / is isomorphic to a connected thick irreducible spherical building of rank at least 2 with the Tits metric, then X is either a Riemannian symmetric space of noncompact type or a Euclidean building. Another type of rigidity problems concerns the characterization of locally symmetric spaces in terms of intrinsic geometric properties. For example, the rank rigidity of nonpositively curved Riemannian manifolds in [3], [38] says that any irreducible nonpositively curved Riemannian manifold M of finite volume with rank at least 2 is a locally symmetric space. The proof of [38] consists of two steps: z .1/, where M z is (1) The construction of a topological Tits building structure on M the universal covering space of M , (2) The use of rigidity and classification of topological buildings in [39]. All these rigidity results have one thing in common: the asymptotic geometry of the spaces at infinity is described by Tits buildings, and rigidity of Tits buildings implies the desired rigidity of the spaces.

6.3 Compactifications of locally symmetric spaces Let G  GL.n; C/ be a linear semisimple algebraic group defined over Q, and G D G.R/ D G \ GL.n; R/ the real locus, a real Lie group with finitely many connected components. Let K  G be a maximal compact subgroup. Then X D G=K with an invariant metric is a symmetric space of noncompact type. Let G.Q/ D G\GL.n; Q/ be the rational locus of G. A subgroup   G.Q/ is called an arithmetic subgroup if it is commensurable with G.Z/ D G.Q/ \ GL.n; Z/, i.e.,  \ G.Z/ has finite index in both  and G.Z/. By the reduction theory of arithmetic subgroups, it is known that nX has finite volume and nX is compact if and only if the Q-rank rQ .G/ of G is equal to 0, which is equivalent to the condition that there is no proper Q-parabolic subgroup of G. For the rest of this subsection, we assume that nX is noncompact unless otherwise indicated and let PQ be the collection of proper Q-parabolic subgroups of G.

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For various applications, we need to compactify nX . We will discuss one compactification here. See [19] for other motivations and applications. For any discrete group , there is a classifying space B which is a CW-complex satisfying the conditions: 1 .B/ D  and i .B/ D f1g when i  2. Such a space  is B is unique up to homotopy equivalence. Its universal covering space E D B a universal space for proper and fixed point free actions of . E is a -CW complex characterized by the conditions: (1)  acts properly and fixed point freely on E, and (2) E is contractible. It is known that if  contains nontrivial torsion elements, then there does not exist finite-dimensional B or E spaces. In this case, another important space is the universal space for proper actions of  which is usually denoted by E and is characterized by the conditions: (1) E is a -CW complex and  acts properly on it, (2) for any finite subgroup F  , the fixed point set .E/F is nonempty and contractible. In particular, E is contractible. If  is torsion-free, then the only finite subgroup of  is the trivial one, and an E-space is an E-space. Definition 6.7. A classifying space B is called finite if it is a finite CW-complex. Equivalently, E is called cofinite if the quotient nE Š B is a finite CW-complex. Similarly, E is called a cofinite universal space for proper actions of  if nE is a finite CW-complex. For topological problems involving , in particular cohomological properties and invariants of , it is important to find explicit and cofinite models of E and E [116]. Proposition 6.8. Let   G.Q/ be an arithmetic subgroup as above. Then X is an E-space. It is a cofinite E-space if and only if nX is compact. Proof. It is known that X is a simply connected and nonpositively curved Riemannian manifold. For any finite subgroup F  , by the Cartan fixed point theorem, the set of fixed points X F is nonempty. In fact, for any point x 2 X , the so-called center of gravity of the orbit F x, which is finite, is fixed by G. Since X F is a totally geodesic submanifold, it is also contractible. If X is a cofinite E space, then nX is compact. Conversely, if nX is compact, then the existence of an equivariant triangulation implies that X has the structure of a -CW-complex such that nX is a finite CW-complex. When  is torsion-free, then nX is a B-space. Assume that nX is noncompact. One approach to obtain a compact B-space is to construct a compactification nX such that the inclusion nX ! nX is a homotopy equivalence. The Borel–Serre compactification nX BS in [20] satisfies this property. For every Q-parabolic subgroup P 2 PQ , let P D P.R/ be its real locus, NP be its unipotent radical, and NP D NP .R/ its real locus. Let AP be a Q-split component of

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P that is stable under the Cartan involution of G with respect to the maximal compact subgroup K of G, and AP D AP .R/ its real locus. Then P admits a Q-Langlands decomposition P D NP AP MP Š NP  AP  MP : The subgroup MP is a reductive subgroup and MP \K is a maximal compact subgroup of MP . The quotient XP D MP =MP \ K with an invariant metric induced from X is a symmetric space of nonpositive sectional curvature (it might contain a flat factor and may not be of noncompact type), and is called the boundary symmetric space of P. The Q-Langlands decomposition of P induces a horospherical decomposition of X with respect to the Q-parabolic subgroup P: X Š N P  AP  XP : One example to illustrate this is G D SL.2; C/, and  D SL.2; Z/. Then  ² ³ ˇ a b  ˇ ; b 2 R a 2 R PD 0 a1 is a Q-parabolic subgroup, and   ² ³ ² ³ ˇ 1 b ˇˇ a 0 ˇ b 2 R ; AP D a>0 ; NP D 0 1 0 a1  ² ³ ˇ a 0 ˇ a D ˙1 : MP D 0 a1 The boundary symmetric space XP is one point, and the horospherical decomposition of X D H2 D SL.2; R/=SO.2/ with respect to P is the .x; y/-horospherical coordinates of the upper half plane H2 . See [19] for more detail. For every Q-parabolic subgroup P, define e.P/ D NP  XP and call it the boundary component of P. According to the slightly modified procedure in [19], the Borel–Serre compactification nX BS is constructed in the following steps. (1) For every P 2 PQ , attach the boundary symmetric space e.P/ at the infinity of X using the horospherical decomposition of X with respect to P to obtain a Borel–Serre partial compactification a BS DX[ e.P/: QX P2PQ BS

(2) Show that the Borel–Serre partial compactification Q X is a real-analytic manifold with corners. BS (3) Show that the -action on X extends to a proper real-analytic action on Q X with a compact quotient, which is the Borel–Serre compactification of nX and denoted by nX BS .

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From the above description, it is clear that the simplices of the Tits building Q .G/ parametrize the boundary components of the Borel–Serre compactification nX BS . Corollary 6.9. When  is torsion-free, nX BS is a finite B-space. BS

Proof. When  is torsion-free, it acts fixed point freely on Q X and hence nX BS is a real-analytic compact manifold with corners whose interior is equal to nX. Then it is clear that the inclusion nX ! nX BS is a homotopy equivalence by retracting from the boundary faces into the interior. Hence 1 .nX BS / D  and i .nX BS / D f1g for i  2, and nX BS is a B-space. Since a compact manifold with corners admits a finite triangulation, nX BS is a finite B-space. When  contains torsion elements, the following result was proved in [89]. Proposition 6.10. The Borel–Serre partial compactification Q X space.

BS

is a cofinite E-

The only non-obvious condition to check is that for any finite subgroup F  , BS the fixed point set .Q X /F is contractible. The point is that if F fixes a point in the boundary component NP  XP , then F is contained in P . For applications in the next subsection, we note BS

Proposition 6.11. The boundary @Q X is homotopy equivalent to the spherical Tits building Q .G/, and hence to a bouquet of infinitely many spheres S rQ .G/1 , where rQ .G/ is the Q-rank of G. Proof. Since each boundary space XP is contractible, and since for any two Qparabolic subgroups P1 , P2 , the inclusion relation P2  P2 holds if and only if XP1 is contained in the closure of XP2 , the first statement follows. The second statement follows from Proposition 4.8 and the fact that Q .G/ is a spherical building of rank rQ .G/. Remark 6.12. To compare with the two partial compactifications of Teichmüller space in §7.5, we mention that in the above construction of the Borel–Serre compactification, if we replace the boundary component e.P/ by the boundary symmetric space XP , then we obtain the reductive Borel–Serre partial compactification a RBS DX[ XP : QX P2PQ

Its boundary decomposes into contractible components XP , and these boundary comRBS ponents are also parametrized by the Tits building Q .G/ and its boundary @Q X is also homotopy equivalent to a bouquet of spheres S rQ .G/1 . The -action on X also extends to a continuous action on

QX

RBS

with a compact quotient nQ X

RBS

.

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All these results on the boundary of reductive Borel–Serre partial compactification are BS

similar to the Borel–Serre partial compactification Q X . But the difference is that this RBS RBS extended action on Q X is not proper any more. The inclusion nX ! nQ X is not a homotopy equivalence. In some sense, nX BS is a blow-up of the reductive RBS Borel–Serre compactification nQ X .

6.4 Cohomological dimension and duality properties of arithmetic groups Let   G.Q/ be an arithmetic subgroup as in the previous subsection. Once good models of classifying spaces for  such as B and E spaces are found, they can be used to study cohomological properties of . Recall that for any discrete group , the cohomological dimension cd./ is defined by cd./ D supfi j H i .; M / ¤ 0; for some Z-module M g: It is known that if  contains nontrivial torsion elements, then cd./ D C1. For any discrete group  that is virtually torsion-free, i.e., if  contains a finiteindex torsion-free subgroup  0 , then cd. 0 / is independent of the choice of  0 and is called the virtual cohomological dimension of , denoted by vcd./. It is also known that for any classifying space B 0 , cd. 0 /  dim B 0 . See [36] for more details. It is also known that every arithmetic subgroup   G.Q/ is virtually torsionfree. Since X is an E-space for a torsion-free arithmetic subgroup, an immediate corollary is the following bound on .vcd/: vcd./  dim X: The precise value of vcd./ was determined in [20] using the Borel–Serre partial BS compactification Q X and the topology of the spherical Tits building Q .G/. Proposition 6.13. The vcd./ D dim X  rQ .G/, where rQ .G/ is the Q-rank of G, i.e., the maximal dimension of Q-split tori in G. This was proved together with a stronger result on duality properties of  in [20]. A group  is called a Poincaré duality group of dimension n if for every Z-module M and every i , there is an isomorphism H i .; M / Š Hni .; M /: If  admits a B-space given by a closed orientable manifold of dimension n, then  is a Poincaré duality group. But the converse is not true in general. An important question is under what further conditions a Poincaré duality group  admits a closed manifold as a B-space. More generally,  is called a duality group of dimension n if there exists a Zmodule D, called the dualizing module, such that for every Z-module M and every

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integer i, there is an isomorphism H i .; M / Š Hni .; D ˝ M /: In this case, it is known that cd./ D n. Since groups  containing nontrivial torsion elements have cd./ D C1, they cannot be duality groups. On the other hand, if  admits a finite-index torsion-free subgroup that is a duality group, it is called a virtual duality group. Similarly, the notion of virtual Poincaré duality group can be defined. The stronger result proved in [20] is the following: Proposition 6.14. Every arithmetic subgroup   G.Q/ as above is a virtual duality group of dimension dim X  rQ , where rQ D rQ .G/. It is a Poincaré duality group if and only if rQ D 0, i.e., nX is compact. When rQ > 0, the dualizing module of  is x is the reduced homology group. xr 1 .Q .G//, where H equal to the Z-module H Q The conclusion that  is a virtual Poincaré duality group if nX is compact is clear since it is a closed orientable manifold if  is torsion-free. Since G.Q/ and hence  acts on the set of Q-parabolic subgroups of G and hence on the spherical Tits building Q .G/, for every i , Hi .Q .G// is a Z-module. By Proposition 4.8, the reduced homology of Q .G/ is non-zero in only degree rQ .G/1. The basic idea of the proof of Proposition 6.14 is as follows. Assume that  is BS torsion-free and nX is noncompact. Then nX BS is a finite B-space and Q X is a cofinite E-space. By general results on cohomology of groups [36], Proposition 7.5, BS Proposition 8.2, Theorem 10.1, it suffices to show that H i .; Z/ Š Hci .Q X ; Z/ is BS not zero only when i D dim X rQ .G/, where Hci .Q X ; Z/ denotes the cohomology group with compact support. By Poincaré–Lefschetz duality, Hci .Q X

BS

where d D dim X. Since Q X

BS

Hd i .Q X

; Z/ Š Hd i .Q X

BS

; @Q X

BS

/;

is contractible,

BS

; @Q X

BS

xd i1 .@Q X BS /; /ŠH

x is the reduced homology. By Proposition 6.11, the latter is not zero if and where H only if d i 1 D rQ .G/1, i.e., i D d rQ .G/. This proves that  is a duality group xd r .G/ .Q .G/; Z/. of dimension d  rQ .G/ and the dualizing module is equal to H Q

6.5 Simplicial volumes of locally symmetric spaces An important homotopy invariant of manifolds is the simplicial volume introduced by Gromov [58]. Suppose that M is a connected oriented compact manifold of dimension n, let ŒM  be the fundamental class in Hn .M; R/, or rather the image of the fundamental

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class in HnP .M; Z/ under the natural map Hn .M; Z/ ! Hn .M; R/. For each nchain c D  a  with R-coefficients, where  are n-singular simplices, define the simplicial `1 -norm X kck1 D j j: 

Then the simplicial volume of M , denoted by kM k, is defined by kM k D inffkck1 j c is an n-chain with R-coefficients in the fundamental class ŒM g: z be its double cover and If M is a connected non-orientable manifold, then let M z k: If M is an oriented orbifold, then it has a fundamental class define kM k D 12 kM and hence the usual notion of simplicial volume. If M admits a finite smooth cover N , then it also has a orbifold simplicial volume kM korb D kN k=d , where d is the degree of the covering N ! M . It follows from the multiplicative property of simplicial volumes of manifolds that kM korb is independent of the choice of a finite smooth cover N . It is known that kM korb  kM k; and the strict inequality can occur. See [91] for details. Assume that M is a connected orientable noncompact manifold of dimension n. Let Hlf .M; R/ be the locally finite homology group of M .P Let ŒM lf be the fundamental lf class in H .M; PR/. For any locally finite n-chain c D  a  , define the simplicial norm kck1 D  j j as above. Recall that to say that a chain is locally finite means that every compact subset of M meets the images of only finitely many singular simplices  in the chain. Hence, kck1 could be equal to infinity. The simplicial volume kM k of a noncompact manifold M is defined by kM k D inffkck1 j c is a locally finite n-chain in the fundamental class ŒM lf g: One of the motivations of Gromov for introducing the simplicial volume was to give a lower bound on the minimal volume of a manifold M . Consider all complete Riemannian metrics g on M whose sectional curvature K.g/ satisfies the bound jK.g/j  1 at all points. Let Vol.M; g/ denote the volume of M with respect to the metric g. Then the minimal volume of M is defined by Min-Vol.M / D finf.Vol.M; g/ j gis a complete metric; jK.g/j  1g: Another major application of simplicial volume is a different proof by Gromov of Mostow strong rigidity for compact hyperbolic spaces of dimension at least 3 as mentioned above. See [10], [149] for detailed discussions. A basic result in §0.5 of [58] states that there exists a universal constant Cn only depending on the dimension n such that Min-Vol.M /  Cn kM k: Therefore, a natural problem is to understand when the simplicial volume kM k is equal to zero.

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It is known that if a compact manifold M admits a self-map of degree greater than or equal to 2, then its simplicial volume kM k D 0. If a noncompact manifold M admits a proper self-map of degree greater than or equal to 2, then kM k D 0 or kM k D C1. As a consequence, the simplicial volumes of spheres and tori are equal to zero. It is also known that the simplicial volume of Rn is equal to 0. For spaces related to locally symmetric spaces, the following results on simplicial volume are known: Proposition 6.15. If M is a complete hyperbolic manifold of finite volume, then kM k > 0. More generally, if M admits a complete metric such that its sectional curvature K is bounded uniformly between two negative constants, then kM k > 0. This is a result due to Thurston ([58], §0.3). Proposition 6.16. If M D nX is a compact locally symmetric space of noncompact type, then kM k > 0. This was conjectured by Gromov [58], p. 11, and proved in [107] and [37]. Proposition 6.17. If M D nX is an arithmetic locally symmetric space whose Q-rank, denoted by rQ .G/, is greater than or equal to 3, then kM k D 0. This vanishing result was proved in [114]. In the proof, Proposition 6.9 and Proposition 6.11 are used crucially. Briefly, a vanishing criterion [58], p. 58, was applied. In order to apply this, a suitable covering of nX with multiplicity at most dim nX is needed. For this purpose, the conditions that the map 11 .nX / D 1 .nX BS / ! 1 .nX/ D 1 .nX BS / is injective and that the virtual cohomological dimension of  is at most dim nX  2 are needed. Since the virtual cohomological dimension of  is equal to dim nX  rQ .G/, the condition that rQ .G/  3 is more than enough. BS The assumption that rQ .G/  3 is needed to show that the boundary @Q X , which is homotopy equivalent to a bouquet of spheres S rQ .G/1 , is simply connected. If rQ .G/ D 1, this vanishing result does not hold in general. For example, it was proved in [113] that if M is a Hilbert modular variety, then kM k > 0. Note that Hilbert modular varieties are important examples of locally symmetric spaces of Q-rank 1. If the rank of X is equal to 1 and nX is noncompact, which implies that rQ .G/ D 1, then the sectional curvature of nX is bounded by two negative constants and hence the simplicial volume of nX is positive [58], §0.3. Remark 6.18. For any topological space M of dimension n that admits a suitable fundamental class in Hn .M; Z/ or Hnlf .M; Z/, we can define its simplicial volume. We can show that for any arithmetic locally symmetric space nX of Q-rank at least 3, and any irreducible arithmetic locally symmetric space nX of Q-rank at least 1 and the rank of X at least 2, the simplicial volumes of the reductive Borel– Serre compactification and the Baily–Borel compactification (if nX is Hermitian) also vanish. See [91] for details.

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6.6 Asymptotic cones of symmetric spaces and locally symmetric spaces Buildings also occur naturally in the large-scale geometry of symmetric spaces X and locally symmetric spaces nX . For any metric space .M; d /, choose a basepoint x0 2 X. For any " > 0, consider the family of pointed metric spaces .M; "d; x0 /. The limit lim"!0 .M; "d; x0 /, if it exists, is called the asymptotic cone (or tangent cone at infinity) of M and denoted by Cone1 .M /. Though the ordinary limits many not exist, there are always ultra limits, which may not be unique. Any Cone1 .M / is a metric cone and does not depend on the choice of the basepoint x0 . If .M; d / has finite diameter, then Cone1 .M / consists of one point. If M D Rn with the standard Euclidean metric, then Cone1 .Rn / is isomorphic to Rn due to the scaling invariance of the Euclidean metric. On the other hand, if M D Hn , n  2, Cone1 .M / is an R-tree that branches at every point. Recall that a usual simplicial tree branches only at a discrete set of points on every geodesic. The asymptotic cone of a general symmetric space of noncompact type was determined in [104]. Proposition 6.19. If X is a symmetric space of noncompact type, then Cone1 .X / is an R-Euclidean building. One application of this result is the quasi-isometric rigidity of symmetric spaces [104], [48], which is a generalization of Mostow strong rigidity. Proposition 6.20. Let X and X 0 be two symmetric spaces of noncompact type that have no irreducible factors of rank 1. If X and X 0 are quasi-isometric, then they are isometric up to suitable scaling of the metrics on irreducible factors. If   G.Q/ is an arithmetic subgroup and nX is noncompact, then Cone1 .nX / was determined in [76], [110], [94]. Proposition 6.21. If nX is a noncompact arithmetic locally symmetric space, then the asymptotic cone at infinity Cone1 .nX / is a metric cone over the finite complex nQ .G/. Recall that  acts on Q .G/ via the action on the set PQ of parabolic subgroups of G. By the reduction theory of arithmetic subgroups, there are only finitely many -conjugacy classes of proper Q-parabolic subgroups and hence nQ .G/ is a finite simplicial complex.

6.7 Applications of Euclidean buildings In the previous subsections, we have mainly concentrated on applications of spherical Tits buildings. Now we briefly consider some applications of Euclidean buildings.

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A natural generalization of arithmetic subgroups is the class of S-arithmetic subgroups. Let S D fp1 ; : : : ; pl ; 1g, where p1 ; : : : ; pl are prime distinct numbers, and ZŒ p11 ; : : : ; p1l  be the ring of S -integers, denoted by ZS . For any linear semisimple algebraic group G  GL.n; C/ defined over Q, let G.ZS / D G.Q/ \ GL.n; ZS /. Then a subgroup  of G.Q/ is called an S-arithmetic subgroup if it is commensurable with G.ZS /. It is known that under the diagonal embedding,  is a discrete subgroup of G  G.Qp1 /      G.Qpk /, where G D G.R/ is the real locus of G. For each pi , let pi be the Bruhat–Tits building of the group G.Qpi /. Let X D G=K be the symmetric space of noncompact type associated with G as above. Define XS D X  p1      pk : Then  acts properly on XS , since each G.Qpi / acts properly on pi . Proposition 6.22. Any S-arithmetic subgroup  acts properly on XS and XS is an E-space. The reason is that each pi is a CAT.0/-space and hence the product XS is also a CAT.0/-space. Since the Cartan fixed point theorem holds for CAT.0/-spaces, for any finite subgroup F  , the fixed point set .XS /F is nonempty. It is also a totally geodesic subspace. If the quotient nXS is compact, then XS is a cofinite E-space by [93]. Otherwise, we need an analogue of the Borel–Serre partial compactification Q XS

BS

D QX

BS

 p1      pk ;

on which  acts properly with a compact quotient. See [89]. Remark 6.23. Bruhat–Tits buildings, or rather the Bruhat–Tits theory, have played an important role in the representation theory of reductive p-adic Lie groups. Due to the lack of knowledge of the author, we only mention one application which is connected with the fact that Bruhat–Tits buildings are CAT.0/-spaces. In [139], it was proved that for any Bruhat–Tits building BT .G.k//, and any g 2 Aut.BT .G.k///, the displacement function dg W BT .G.k// ! R0 ; x 7! dist.x; gx/; is a convex function. This has important applications in representation theory. See [137] for detail. Remark 6.24. On the set of vertices of a Bruhat–Tits building, there is also a combinatorial distance, which is induced from the distance on the 1-skeleton of the building. This has applications to arithmetic algebraic geometry. In [170], non-Archimedean intersection indices on projective spaces over a non-Archimedean local field of characteristic zero are expressed in terms of the combinatorial distance of the Bruhat–Tits building for G D PGL.n/.

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6.8 Compactifications of Euclidean buildings It is clear from the definition that Euclidean buildings are noncompact, since each apartment is a closed noncompact subspace. For many purposes, Euclidean buildings play a similar role for p-adic Lie groups as symmetric spaces of noncompact type for noncompact real Lie groups. There has been a lot of work on compactifications of symmetric spaces motivated by applications in group theory, geometry, topology and analysis. See [19] for a history, more details and references. By Proposition 4.3, every Euclidean building  is a CAT.0/-space. It is also known that a symmetric space of noncompact type with an invariant Riemannian metric is a complete nonpositively curved simply connected Riemannian manifold and is hence also a CAT.0/-space. This is one important common property shared by both Euclidean buildings and symmetric spaces. It is known that for any CAT.0)-space , its set of equivalence classes of geodesics X.1/ forms the geodesic (or visual) boundary of  as in the case of a symmetric space that we saw in §2.1. If  is a proper metric space, then there is a natural compact topology on  [ .1/ [30], [21]. This compactification X [ X.1/ is called the geodesic compactification. The following result was also proved in [21]. Proposition 6.25. (1) If  is a Euclidean building, then its geodesic boundary .1/ has a natural structure of a spherical Tits building. If the rank of  is equal to r, then the rank of .1/ is also equal to r. (2) If  D BT .G/ is the Bruhat–Tits building of an algebraic group G over a locally compact local field k, then  D BT .G/ is locally compact, and .1/ is the spherical Tits building k .G/. Furthermore, the geodesic compactification BT .G/ [ BT .G/.1/ is contractible. This geodesic compactification explains a close relation between spherical and Euclidean buildings. Suppose 1 , 2 are two locally compact Euclidean buildings. Then there are at least two natural compactifications of the product 1  2 . The first one is obtained by taking the product of the geodesic compactifications 1 [ 1 .1/ and 2 [ 2 .1/, prod

1   2

D .1 [ 1 .1//  .2 [ 2 .1//:

Note that the product 1  2 is also a CAT.0/-space. Then the geodesic compactification gives another compactification of 1  2 , 1  2 [ .1  2 /.1/: It is clear that these two compactifications are not isomorphic, and each of them is natural in its own way. It turns out that the second type of compactification has an application to the Novikov conjecture in geometric topology for certain classes of S -arithmetic subgroups of algebraic groups. Let  be an S -arithmetic subgroup of a linear algebraic group as in §6.7. Assume that  acts cocompactly on XS D

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X  p1      pk . By Proposition 6.22, XS is a cocompact universal space for proper actions of , i.e., a cocompact model of E. Since X is also a CAT.0/-space as mentioned above, the product XS is a CAT.0/-space and hence admits a geodesic compactification. The -action on XS extends continuously to the compactification. An important point of the geodesic compactification of XS is that the extended action of  on the compactification is small at the boundary (or infinity). Then it follows from a general criterion for the validity of Novikov conjectures that the integral Novikov conjecture holds for . See [92], [93] for precise statements of the Novikov conjecture, for the criterion and for other details. On the other hand, if we take the product compactification of XS induced from compactifications of the factors, i.e., the product of X [X.1/, p1 [p1 .1/; : : : ; pk [ pk .1/, the action of  might not be small at infinity, and it might not be used for proving the integral Novikov conjecture for . Motivated by many different compactifications of symmetric spaces of noncompact type (see [19]), there are also corresponding compactifications of Euclidean buildings. As discussed in §2.2, one important feature of rich structure of a symmetric space is reflected by the flat subspaces F of maximal dimension in X , which determine the Tits building of X. There are several ways to understand a compactification Xx of X, and they can be generalized to construct compactifications of Euclidean buildings: (1) Decompose the boundary Xx X into smaller subspaces which enjoy some natural structures. (2) Determine the closure of maximal flat subspaces F in the compactification Xx and the intersection pattern of these compactified flat subspaces, and recover the compactification of X from the compactified flats. (3) Interpret X as a moduli space of certain objects and the boundary points in Xx X as degenerate objects. For the geodesic compactification X [ X.1/ in §2.1, the boundary X.1/ is naturally decomposed into simplices and can be viewed as the underlying space of the Tits building of X. As pointed out earlier, a geodesic compactification can be constructed for every locally compact Euclidean building. Another important compactification of X is the maximal Satake compactification S , which is the maximal element in the partially ordered finite set of Satake comXxmax pactifications of X . See [60]. Each maximal flat F of X has a natural decomposition S is canonically by Weyl chambers and Weyl chamber faces. The closure of F in Xxmax homeomorphic to a polyhedron, its boundary is a finite cell complex dual to the Weyl S can be constructed by gluing chamber decomposition, and the compactification Xxmax these compactified flats [60]. It turns out that a similar construction works for any Euclidean building  by gluing polyhedral compactifications of apartments of . See [108]. To explain Satake compactifications of symmetric spaces, we start with a special Satake compactification of the symmetric space SL.n; C/=SU.n/. Let Hn be the real

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vector space of .nn/-Hermitian matrices, and P .Hn / the associated projective space. Then the map A 7! AAxt defines an embedding of SL.n; C/=SU.n/ into P .Hn /, and the closure of SL.n; C/=SU.n/ under this embedding gives a Satake compactification. In this case, every point in SL.n; C/=SU.n/ corresponds to a positive definite Hermitian matrix of determinant 1 (or equivalently a positive definite Hermitian quadratic form of determinant 1), and the boundary points correspond to projective classes of degenerate positive semi-definite Hermitian matrices. For a general symmetric space X D G=K of noncompact type, any finite-dimensional projectively faithful representation of G gives an isometric embedding of X into SL.n; C/=SU.n/ for some n, and the closure of X in the above special Satake compactification of SL.n; C/=SU.n/ gives a Satake compactification of X. Though there are infinitely many representations of G, it turns out that they only give rise to finitely many non-isomorphic Satake compactifications of X . For the Bruhat–Tits building BT .G/ of a reductive algebraic group over a local field, the corresponding Satake compactifications have been constructed in [151], [150], [171], [172], [173]. The paper [150] constructs the compactification by using Berkovich analytic geometry over complete non-Archimedean fields, and the paper [151] uses irreducible representations of the algebraic group and is more similar to the Satake compactifications of symmetric spaces. The construction in [171] is also similar to the Satake compactifications of symmetric spaces. Compactifications of some special buildings were treated in [172] and [173]. S of the maximal Satake compactification is naturally decomThe boundary @Xxmax posed into symmetric spaces of noncompact type of smaller dimension, which are naturally parametrized by proper parabolic subgroups of G, where X D G=K, or by simplices of the Tits building .X/. Since each boundary symmetric space is a cell, S has a natural cell structure which is dual to the this shows that the boundary @Xxmax S Tits building. See [60]. The boundary @Xxmax can also be constructed in this way by adding these boundary components using the idea of the Borel–Serre compactification of locally symmetric spaces. See [19]. For Bruhat–Tits buildings, a similar construction should also work. For the Bruhat–Tits building of PGL.n/ over a local compact field, such a construction has been carried out in [171]. S can also be constructed by embedding The maximal Satake compactification Xxmax X into the compact space of closed subgroups of G [60]. A similar compactification of Bruhat–Tits buildings has been constructed in [61]. An important application of compactifications of symmetric spaces is to harmonic analysis on symmetric spaces, for example, the determination of the Poisson boundary. The analogue of the Poisson boundary of certain Bruhat–Tits buildings has been determined in [57]. Remark 6.26. For a hyperbolic building , its geodesic boundary .1/ has the structure of a trivial (or rank 1) spherical building structure, i.e., the top-dimensional simplices of the building are points. The reason is that the boundary Hn .1/ of the real hyperbolic space Hn has the structure of a rank-1 spherical building.

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7 Applications of curve complexes In this section, we briefly discuss some applications of curve complexes. In §7.1, we discuss the identification of the automorphism group of C .Sg;n / with Modg;n , which is responsible for several rigidity results on the geometry of Tg;n in §7.2. In §7.3, we briefly describe the ending lamination conjecture of Thurston on the rigidity of 3-dimensional hyperbolic manifolds with finitely generated fundamental group and its formulation in terms of the boundary @C.Sg;n / of C .Sg;n / as a ı-hyperbolic space. This is probably one of the most striking applications of the curve complex C .Sg;n /. In §7.4, we discuss an application to quasi-isometric rigidity of Modg;n . In §7.5, we describe several different notions of the Novikov conjectures and an application of the curve complex to finiteness of the asymptotic dimension of Modg;n . In §7.6, we mention an application to non-Gromov-hyperbolicity of the Weil–Petersson metric of Tg;n . In §7.7, we define the Hempel distance of a Heegaard splitting via the image of Heegaard diagrams in C.Sg /. After these applications to 3-dimensional manifolds, we describe the Borel–Serre partial compactification of Tg;n via C .Sg;n /, the original motivation of introducing C .Sg;n /. Cohomological properties of Modg;n are studied in §7.9. The asymptotic cone at infinity of Mg;n is described in §7.10 and the simplicial volume of Mg;n is discussed in §7.11. In the last subsection, §7.12, we mention applications of C .Sg;n / to the classification of elements of Modg;n , presentations and unitary representations of this group.

7.1 Automorphism groups of curve complexes As mentioned in §5, the extended mapping class group Mod˙ g;n acts on C .Sg;n / sim˙ plicially. This defines a map Modg;n ! Aut.C .Sg;n //, where Aut.C .Sg;n // is the simplicial automorphism group of C .Sg;n /. It turns out that the image is essentially the whole automorphism group [81], [117] (Theorem on p. 7), [105]. Proposition 7.1. (1) If the dimension of C .Sg;n / is at least 1 and .g; n/ ¤ .1; 2/, then the map Mod˙ g;n ! Aut.C.Sg;n // is surjective. (2) When .g; n/ D .1; 2/, the image of the map Modg;n ! Aut.C .Sg;n // contains all automorphisms of C .S1;2 / that preserve vertices represented by separating curves. Furthermore, this map is injective and the image has index 2 in Aut.C .Sg;n //. (3) When Sg;n is not a torus with 1 or 2 punctures, or a sphere with at most 4 punctures, or a closed surface of genus 2, then the map Mod˙ g;n ! Aut.C .Sg;n // is an isomorphism. This is an analogue of Corollary 6.3 for Tits buildings and it will be used in the next subsection to prove several rigidity results. Remark 7.2. There are several related results on rigidity of simplicial maps of C .Sg;n / which imply co-Hopfian property of the mapping class group Modg;n [79], [162], [9].

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Some quasi-isometric embeddings of curve complexes are constructed in [147], and they lead to some new quasi-isometric embeddings between mapping class groups. Remark 7.3. A closely related result on the simplicial automorphism group of the arc complex of a surface is proved in [106]. The automorphism group of the pants complex of a surface is determined in [122]. Remark 7.4. The paper [17] studies the automorphism group Aut.Cy .Sg;n // of the profinite completion of the complex of curves C .Sg;n / and compares it with the Grothendieck–Teichmüller group.

7.2 Isometry groups of Teichmüller spaces Motivated by the proof of Mostow strong rigidity for irreducible compact locally symmetric spaces of rank at least 2 via rigidity of Tits buildings, Ivanov [82] gave a new proof of the following theorem of Royden [154] by using the rigidity of C .Sg;n / in Proposition 7.1. Proposition 7.5. The Teichmüller metric dTei of the Teichmüller space Tg;n is invariant under the natural action of the extended mapping class group Mod˙ g;n . If g  2, or g D 1 and n  3, or g D 0 and n  5, then Mod˙ is equal to the full isometry g;n group Iso.Tg;n ; dTei /. The Teichmüller metric is historically the first metric defined on Tg;n . It is a complete Finsler metric and it gives the natural topology on Tg;n . The Teichmüller space Tg;n also admits another metric, the Weil–Petersson metric dWP . This is an incomplete Kähler metric. Its isometry group was identified in [128]. Proposition 7.6. For any Teichmüller space Tg;n , the extended mapping class group Mod˙ g;n is canonically mapped into the isometry group Iso.Tg;n ; dWP /. If g  2, or g D 1 and n  3, or g D 0 and n  5, then Mod˙ g;n is equal to the full isometry group Iso.Tg;n ; dWP /. The idea of the proof, simplified in [174], is similar to the previous proposition and can be explained as follows. The completion of Tg;n in the Weil–Petersson metric is the augmented Teichmüller space Tyg;n . The boundary components of Tyg;n are Teichmüller spaces of stable Riemann surfaces of smaller genus with more punctures (the same Euler characteristic). These boundary components are parametrized by simplices of C.Sg;n /. Since any isometry of .Tg;n ; dWP / extends to its completion, it induces an automorphism of C.Sg;n /. Proposition 7.1 implies that the isometry is induced by an element of Modg;n . Remark 7.7. Teichmüller spaces Tg;n are the counterpart of symmetric spaces in the analogy between Tits buildings and arithmetic groups on the one hand, and curve

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complexes and mapping class groups on the other hand. The above proposition shows one difference. Any symmetric space is a homogeneous space and has continuous symmetry, but Tg;n has only discrete symmetry, which implies that there is a uniform lower bound on volumes of nTg;n . Maybe one result to repair this difference is that for any symmetric space X, there is a uniform lower bound on volumes of all locally symmetric spaces nX which are quotients of X .

7.3 The ending lamination conjecture of Thurston Another generalization of Mostow strong rigidity is the ending lamination conjecture of Thurston. A special case of Mostow strong rigidity concerns 3-dimensional hyperbolic manifolds of finite volume. Let 1 nH3 and 2 nH3 be two hyperbolic manifolds of finite volume. If 1 Š 2 , then 1 nH3 and 2 nH3 are isometric. On the other hand, the conclusion does not hold if the volumes of 1 nH3 and 2 nH3 are not finite. For example, let   SL.2; R/ be a torsion-free cocompact Fuchsian group, i.e., a discrete subgroup such that nH2 is a compact hyperbolic surface †. Consider  as a discrete subgroup of SL.2; C/. Then nH3 is a 3dimensional hyperbolic manifold of infinite volume and is diffeomorphic to †  R. There are injective morphisms W  ! SL.2; C/ such that ./nH3 is not isometric to nH3 , but is still diffeomorphic to †  R. A deformation ./ of  is called a quasi-Fuchsian deformation if ./ is a quasi-Fuchsian group in the sense that the limit set of ./ in H3 .1/ is a Jordan curve. The complement of the limit set consists of two simply connected domains 1 and 2 . Many quasi-Fuchsian deformations of  have the property that ./nH3 is not isometric to nH3 . In this case, clearly  and ./ are isomorphic. Therefore, a direct generalization of Mostow strong rigidity does not hold for infinite-volume hyperbolic manifolds of dimension 3, i.e., if 1 nH3 and 2 nH3 are both of infinite volume, then 1 Š 2 does not imply that 1 nH3 and 2 nH3 are isometric. (Note that by Theorem 8.3 of [119], if one of the manifolds 1 nH3 and 2 nH3 has finite volume and 1 Š 2 , then the other manifold also has finite volume and hence 1 nH3 and 2 nH3 are isometric.) For each quasi-Fuchsian deformation ./, the noncompact infinite-volume 3dimensional hyperbolic manifold ./nH3 has two ends4 and admits a natural compactification ./nH3 by adding the two Riemann surfaces ./n1 and ./n2 as the boundary. Therefore, the Riemann surfaces ./n1 and ./n2 are called the Riemann surfaces at infinity of the ends of ./nH3 . The following fact establishes a connection between the Riemann surfaces at infinity and the geometry of the interior. 4 By an end of a 3-dimensional manifold, we mean a connected unbounded component of the complement of a sufficiently large compact subset. Strictly speaking, an end is a limit of such unbounded connected components over an exhausting family of compact subspaces. For a precise definition, see [159].

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Proposition 7.8. Two quasi-Fuchsian deformations 1 ./ and 2 ./ of  give rise to isometric 3-dimensional hyperbolic manifolds 1 ./nH3 and 2 ./nH3 if and only if they have the same Riemann surfaces at infinity of the ends. Furthermore, all possible conformal structures on the Riemann surfaces at infinity can arise. This is a proper generalization of Mostow strong rigidity for the class of quasiFuchsian hyperbolic manifolds, and it gives a complete classification of quasi-Fuchsian hyperbolic manifolds of dimension 3. See [12], [121], Chapter 5, for more details. Quasi-Fuchsian deformations are important examples of geometrically finite hyperbolic manifolds. Recall that a three-dimensional hyperbolic manifold nH3 is called geometrically finite if  admits a finite-sided convex fundamental domain (see [26] for discussion of several equivalent definitions), which is also equivalent to the fact that nH3 [ ./ is homeomorphic to M  P where M is a compact 3-manifold with boundary and P is a finite collection of disjoint annuli and tori in @M , and ./ is the largest open subset of H3 .1/ where  acts properly [43], p. 139. A similar rigidity result and classification of hyperbolic metrics holds for geometrically finite hyperbolic manifolds M of dimension 3. Roughly speaking, for each end of M , there is a Riemann surface at infinity. Such a collection of Riemann surfaces at infinity together with the topology of the manifold uniquely determines the hyperbolic manifold [119], [120] (Theorem 8.1), [121] (Chapter 5), [12], [132] (p. 188), [43] (§7.3). We note that the topology of a quasi-Fuchsian hyperbolic manifold is determined by any of the two Riemann surfaces at infinity, and hence the condition on the topology of the manifold is contained in the Riemann surfaces at infinity. For a finitely generated torsion-free subgroup   SL.2; C/, the hyperbolic manifold nH3 has only finitely many ends. This case is more complicated than the geometrically finite case. For simplicity, in the following discussion, we assume that  does not contain any nontrivial parabolic element (see §5.5 of [121] for the complication caused by the presence of parabolic elements.) A three-dimensional hyperbolic manifold M D nH3 is called geometrically tame if each of its ends is either geometrically finite or simply degenerate (defined below). It is called topologically tame if M is homeomorphic to the interior of a compact manifold with boundary. It is known that if  is finitely generated, then nH3 is topologically tame if and only if it is geometrically tame [41], [18]. Therefore, the solution of Marden’s tameness conjecture implies that nH3 is geometrically tame. See [42] for a precise description of these two notions of tameness and a history of Marden’s tameness conjecture. Thurston associated an invariant to an incompressible end of nH3 , and an end invariant for a general end was defined in [41] (see also [18]). If an end is geometrically finite and of infinite volume, then there is a Riemann surface at infinity which gives the end invariant. For a geometrically finite end of finite volume, there is no invariant. For a simply degenerate end, there is a filling lamination on an associated surface, which is an invariant of the end. For simplicity, we assume that nH3 has no cuspidal

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ends. Then for each end, there is a surface Sg;n such that the end is homeomorphic to Sg;n  .0; C1/. For every simple closed curve ˛ in Sg;n , there is a well-defined homotopy class of simple closed curves in the end of nH3 and hence in nH3 . Let ˛  be the unique geodesic representative in nH3 . Then the end is called simply degenerate if there is a sequence of simple closed curves ˛i on Sg;n such that their geodesic representatives ˛i exit the end, i.e., they are eventually contained in any small neighborhood of the end. The family of curves ˛i in Sg;n converges to a lamination, called the ending lamination of the end. It can also be interpreted through the curve complex of a surface in the end. The sequence of simple closed curves ˛i gives a sequence of vertices in the curve complex C.Sg;n /. It is known that they converge to a boundary point @C .Sg;n /, which is a filling lamination on Sg;n by Proposition 5.6. Thurston made the following ending lamination conjecture. Conjecture 7.9. A three-dimensional hyperbolic manifold nH3 with finitely generated fundamental group  is determined by its topological type and its end invariants. If nH3 has finite volume, then each end has finite volume and is a cusp, the end invariant is trivial, and the space nH3 is determined by its topology, i.e.,  as the fundamental group, up to isometry by Mostow strong rigidity. This ending lamination conjecture has been proved by Brock–Canary–Minsky (see [34], [130], [131]). Slightly different proofs have also been given. See [27], [28]. A crucial step in the proof is to build a bi-Lipschitz model of each of the ends of nH3 . (In [27], [28], a weaker Lipschitz model of each end is needed and constructed.) The large-scale geometry of the curve complex C .Sg;n / and its connection with the Teichmüller metric of Tg;n , in particular results in [126], [127], were used crucially for this purpose. (It seems that the papers [126], [127] were motivated by the ending lamination conjecture.) One result in [126] (see [64], Theorem 4.1) says that when Tg;n is given the Teichmüller metric dTei , there is a Modg;n -coarsely-equivariant map W .Tg;n ; dTei / ! C.Sg;n / which is quasi-Lipschitz in the following sense: (1) there exists a constant c > 1 such that for every point x 2 Tg;n and  2 Modg;n , d. .x/;  .x//  cI (2) for any two points x; x 0 2 Tg;n , d. .x/; .x 0 //  c dTei .x; x 0 / C c: This is an important instance of the philosophy that C .Sg;n / describes the large-scale geometry of Tg;n . For detailed descriptions of the ending lamination conjecture and methods to prove it, see [132], [133], [134], [27]. Remark 7.10. Besides giving a complete classification of 3-dimensional hyperbolic manifolds, the ending lamination conjecture has applications to the local connectivity of limit sets of Kleinian groups and the Cannon–Thurston map. See [135].

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Remark 7.11. See [146] for related results on a model for the Teichmüller metric in terms of combinatorial information on short curves on the hyperbolic surfaces that correspond to points on a Teichmüller geodesic. The paper [102] compares the Thurston boundary of Tg;n and the boundary @C .Sg;n /.

7.4 Quasi-isometric rigidity of mapping class groups In this subsection, we discuss another important application of curve complexes to quasi-isometric rigidity of Modg;n . As mentioned before, the notion of quasi-isometry was used in the proof of Mostow strong rigidity. Recall that two metric spaces .M1 ; d1 / and .M2 ; d2 / are called quasi-isometric if there are positive constants C; D and a map f W M1 ! M2 such that for every two points x1 ; x2 2 M1 , C 1 d2 .f .x1 /; f .x2 //  D  d1 .x1 ; x2 /  Cd2 .f .x1 /; f .x2 // C D; and every point of M2 lies in the D-neighborhood of f .M1 /. It is clear that if a finitely generated group  acts properly and isometrically on a metric space X with a compact quotient, then  with any word metric is quasiisometric to X. The quasi-isometry type of the group  does not depend on the choice of a word metric. A finitely generated group  is called quasi-isometrically rigid if the following property holds: for any finitely generated group  0 , if  0 is quasi-isometric to , then there exists a finite-index subgroup  00   0 and a homomorphism  00 !  with finite kernel and finite cokernel, i.e.,  00 and  are virtually isomorphic. The three propositions below are combinations of results in several papers including [49], [48], [54], [160], [161], [104]. See [50] for the history, references and more detailed statements of these propositions. There are also other rigidity results on mapping class groups. See [80]. Proposition 7.12. If   G is an irreducible non-uniform lattice of a semisimple Lie group G, then  is quasi-isometrically rigid. The assumption that  is not uniform is necessary. All uniform lattices of one semisimple Lie group G are quasi-isometric, but they are not necessarily commensurable up to conjugation by elements of G, and hence they are not necessarily isomorphic up to finite-index subgroups and quotients. On the other hand, this class of uniform lattices is quasi-isometrically rigid. Proposition 7.13. If a finitely generated group  0 is quasi-isometric to a uniform lattice in a semisimple Lie group G, then there exists a finite-index subgroup  00 and a homomorphism W  00 ! G with a finite kernel such that its image is a uniform lattice in G.

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Proposition 7.14. Non-uniform lattices of SL.2; R/ form one quasi-isometry class of groups. Quasi-isometric rigidity and the quasi-isometric classification of lattices of semisimple Lie groups were obtained by joint efforts of many people. See [50] for the history and more detailed statements of the above three propositions. In the analogy between the mapping class groups Modg;n and lattices of semisimple Lie groups, Modg;n correspond to non-uniform lattices. Therefore, the next proposition on the quasi-isometrical rigidity of the mapping class group Modg;n , proved in [65], [6], is as expected. Proposition 7.15. Assume that 3g  3 C n  2. Then Modg;n is quasi-isometrically rigid. One ingredient used in the proof in [6] is the hyperbolicity of the curve complex C .Sg;n / in the sense of Gromov, and one ingredient used in the proof of [65] is the train track complex. Results on the asymptotic cones at infinity Cone1 .Modg;n / in [7] (see Proposition 7.42 below) were used crucially. The papers [65], [6] also prove the following result on quasi-isometries of Modg;n . Proposition 7.16. Assume that 3g  3 C n  2 and .g; n/ ¤ .1; 2/, then quasiisometries of Modg;n are uniformly close to isometries induced by left-multiplication of elements of Modg;n .

7.5 Finite asymptotic dimension of mapping class groups and the Novikov conjectures In this subsection, we briefly describe another important application of the hyperbolicity of the curve complex C .Sg;n / to prove that the asymptotic dimension of the mapping class group Modg;n is finite. We also explain applications of this result to several different versions of the Novikov conjecture. For any noncompact metric space .M; d /, an important large-scale geometry invariant is the asymptotic dimension of M , denoted by asd.M /. It is defined to be the smallest integer n, which could be 1, such that for every r > 0, there exists a cover C D fUi g, i 2 I , of M by uniformly bounded sets Ui with r-multiplicity less than or equal to n C 1, i.e., every ball in M of radius r intersects at most n C 1 sets in C . For any finitely generated group , its asymptotic dimension asd./ is defined to the asymptotic dimension of  endowed with any word metric. Finiteness of asd./ has applications to the Novikov conjectures for , including various versions of the integral Novikov conjectures. For the convenience of the reader, we briefly recall several versions of the Novikov conjectures, both the integral and rational versions. See [92] for more discussions, details and proper references for the conjectures and results stated below.

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To motivate the Novikov conjectures, we first recall the Hirzebruch index theorem. Let M 4k be a compact oriented manifold (without boundary) of dimension 4k. The cup product defines a non-degenerate quadratic form on the middle dimension cohomology group: Q W H 2k .M; Q/  H 2k .M; Q/ ! H 4k .M; Q/ D Q:

(7.5.1)

This quadratic form can be diagonalized over R to the form Diag.1; : : : ; 1I 1; : : : ; 1/, and the number of C1’s minus the number of 1’s is called the signature of M and denoted by Sgn.M /. Since the identification H 4k .M; Q/ D Q depends on the orientation of M , the signature Sgn.M / depends on the orientation and is an oriented homotopy invariant of M . The Hirzebruch class L.M / is a power series in Pontrjagin classes P1 ; P2 ; : : : ; with rational coefficients, L.M / D 1 C L1 C L2 C    ; where Li are polynomials of Pontrjagin classes, for example, L1 D 1 .7P2  P12 /. 45 Then the Hirzebruch index theorem is the following equality: Sgn.M / D hL.M /; ŒM i;

1 P , 3 1

L2 D

(7.5.2)

where the right-hand side is the evaluation of L.M / on the fundamental class ŒM . The Hirzebruch class L.M / depends on the characteristic classes of the tangent bundle of M and a priori it also depends on the differentiable structure of M . (In fact, these rational classes in H  .M; Q/ are homeomorphism invariants of M ). As pointed out earlier, the left-hand side in the above formula is an oriented homotopy invariant, and hence the above equality shows that hL.M /; ŒM i only depends on the oriented homotopy type of M . To get more homotopy invariants, Novikov introduced the higher signatures. Let  D 1 .M /. Let B be a classifying space of the discrete group , i.e., a K.; 1/space, 1 .B/ D ; i .B/ D f1g; i  2: The universal covering space E of B is contractible and admits a free -action. Equivalently, we can reverse this process and define first E as a contractible space with a free -action, and then define B as the quotient nE. For example, when  D Z, it acts freely by translation on R and hence E D R and B D R=Z D S 1 . For each group , the spaces E and B are unique up to homotopy. The universal z ! M determines a classifying map f W M ! B, which is unique covering map M up to homotopy. For any ˛ 2 H  .B; Q/, f  ˛ 2 H  .M; Q/, and define a higher signature Sgn˛ .M / D hf  ˛ [ L.M /; ŒM i: The original Novikov conjecture is stated as follows:

(7.5.3)

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Conjecture 7.17 (Novikov conjecture). For any ˛ 2 H  .B; Q/, the higher signature Sgn˛ .M / is an oriented homotopy invariant of M , i.e., if N is another oriented manifold and g W N ! M is an orientation-preserving homotopy equivalence, then h.g B f / ˛ [ L.N /; ŒN i D hf  ˛ [ L.M /; ŒM i: The Novikov conjecture can be reformulated in terms of the rational injectivity of the assembly map in surgery theory, or L-theory. The surgery obstruction groups L .ZŒ/, or L-groups of ZŒ, are briefly defined as follows. For m D 2k, Lm .ZŒ/ is the Witt group of stable isomorphism classes of .1/k -quadratic forms on finitely generated free modules over the group ring ZŒ, and L2kC1 .ZŒ/ is a stable automorphism group of hyperbolic .1/k -quadratic forms on finitely generated free modules over ZŒ. Since .1/k is 4-periodic in m, the groups Lm .ZŒ/ are 4-periodic in m. Let L.Z/ be the surgery spectrum: m .L.Z// D Lm .Z/;

m 2 Z:

The spectrum L.Z/ defines a general homology theory with coefficient in L.Z/. For any topological space X, there are general homology groups H .X I L.Z// D  .XC ^ L.Z//, where XC is the disjoint union of X and a point. There is an important notion of assembly map: A W H .X I L.Z// ! L .ZŒ1 .X //:

(7.5.4)

Proposition 7.18. The Novikov conjecture, i.e., the oriented homotopy invariance of the higher signatures in Conjecture 7.17, is equivalent to the rational injectivity of the assembly map in Equation (7.5.4), i.e., the following rational assembly map is injective: A ˝ Q W H .BI L.Z// ˝ Q ! L .ZŒ/ ˝ Q: The injectivity of the map A ˝ Q is called the rational Novikov conjecture. Conjecture 7.19 (Integral Novikov conjecture). If  is torsion-free, then the assembly map A W H .BI L.Z// ! L .ZŒ/ is injective. This conjecture is also called the L-theory (or surgery theory) integral Novikov conjecture. In the integral Novikov conjecture, the torsion-free assumption on  is important. In fact, it is known that the conjecture is often false for finite groups. Clearly, the integral Novikov conjecture implies the rational Novikov conjecture and gives an integral version of homotopy invariance of higher signature. There are also several other reasons to consider the integral, rather than the original (rational) Novikov conjecture: (1) The relation to the rigidity of manifolds, in particular, the Borel conjecture for rigidity of aspherical manifolds, which says that two closed aspherical manifolds with the same fundamental group are homeomorphic.

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(2) The computation of the L-groups L .ZŒ/ in terms of a generalized homology theory, i.e., the injectivity of the assembly map in Equation (7.5.4) shows that the left-hand side is a summand of the groups L .ZŒ/. Once formulated in terms of the assembly map, there are also other versions of the Novikov conjecture. For any associative ring with unit R, there is a family of algebraic K-groups Ki .R/, i 2 Z. For example, K0 .R/ is defined as the stable equivalence classes of finitely generated projective modules, and K1 .R/ D GL.R/=ŒGL.R/; GL.R/. The higher K-groups Ki .R/, i  2, are defined to be the homotopy groups of the space BGL.R/C , where BGL.R/ is the classifying space of GL.R/ considered as a discrete group, and BGL.R/C is the space obtained by applying the Quillen +-construction to the perfect subgroup E.R/ D ŒGL.R/; GL.R/, in particular, the homology groups of BGL.R/C and BGL.R/ are equal to each other under inclusion. The K-theory spectrum K.R/ with i .K.R// D Ki .R/, i 2 Z, is given by the delooping of the infinite loop space BGL.R/C  K0 .R/. Let  be a group as above, and H .BI K.Z// the generalized homology of B with coefficients in K.R/. There is also an assembly map A W H .BI K.Z// ! K .ZŒ/:

(7.5.5)

Conjecture 7.20 (Integral Novikov conjecture in algebraic K-theory). Assume that  is torsion free. Then the assembly map A W H .BI K.Z// ! K .ZŒ/

(7.5.6)

is injective. There is also a rational version of the Novikov conjecture in algebraic K-theory, i.e., the induced map A ˝ Q W H .BI K.Z// ˝ Q ! K .ZŒ/ ˝ Q is injective. One fruitful approach to prove the Novikov conjecture for a group  is to show that the asymptotic dimension of  is finite. Theorem 7.21. If a finitely generated group  has finite asymptotic dimension, asdim  < 1, and has finite B, i.e., if its classifying space B can be realized as a finite CW-complex, then the integral Novikov conjectures in K-theory and L-theory hold for . We note that the existence of a finite model of B implies that  is torsion-free. For groups containing torsion elements, there is also a modified version of the integral Novikov conjecture. See [92]. In [14], the following result was proved. Proposition 7.22. The asymptotic dimension of Modg;n is finite. For any finite-index torsion-free subgroup  of Modg;n , the quotient by  of the thick part of the Teichmüller space Tg;n ."/ gives a cofinite model of B-spaces by [95]. One corollary of the above proposition is the following conclusion.

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Corollary 7.23. For any finite-index subgroup  of Modg;n , the rational Novikov conjecture in L-theory holds for . This result was proved earlier in [62], [100]. Even though it was not stated in [14], the above discussion shows that the following stronger corollary also holds. Proposition 7.24. For any finite-index torsion-free subgroup  of Modg;n , the integral Novikov conjecture in L- and K-theories holds for . This shows that the results of [14] imply a stronger version of the Novikov conjecture than what was known previously.

7.6 Non-hyperbolicity of Weil–Petersson metric of Teichmüller space As mentioned before, the Teichmüller space Tg;n admits a Modg;n -equivariant Kähler metric, the Weil–Petersson metric. This is an incomplete metric with strictly negative sectional curvature. It is known that if any simply connected Riemannian manifold with strictly negative sectional curvature admits a compact quotient, then it is a hyperbolic space in the sense of Gromov. The quotient Modg;n nTg;n is non-compact, and one question raised by Bowditch and others was whether the Weil–Petersson metric is hyperbolic in the sense of Gromov. This problem was solved in [32], and the hyperbolicity of the curve complex C.Sg;n / was used in the proof. Proposition 7.25. The Weil–Petersson metric on Tg;n is Gromov-hyperbolic if and only if 3g  3 C n  2. Besides the curve complex C.Sg;n /, the pants complex and its quasi-isometry with Tg;n endowed with the Weil-Peterson metric was also used crucially in the proof. See [32] for details.

7.7 Heegaard splittings and Hempel distance of 3-manifolds Another important application of the curve complex C .Sg / in 3-dimensional topology is the Hempel distance for Heegaard splittings of 3-dimensional manifolds. A handlebody is a 3-manifold homeomorphic to a submanifold with boundary of R3 with only one connected boundary component, which is a compact orientable surface Sg . Any closed oriented three-dimensional manifold M 3 can be written as the union of two handlebodies V1 ; V2 of the same genus glued along their boundary, M D V1 [Sg V2 ;

@V1 Š Sg ;

@V2 Š Sg :

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Such a decomposition of M 3 is called a Heegaard splitting, and Sg is called the splitting surface. It is not unique and there are infinitely many different isotopy classes of Heegaard splittings of M 3 . A simple closed curve in Sg is called essential if it does not bound a disc. The identification of Sg with the boundary of a handlebody V1 is determined by a maximal set 1 of disjoint non-homotopic essential simple closed curves in Sg that bound essential discs in the handlebody V1 (these curves are called meridians). Let 2 be a corresponding set for the identification of Sg as the boundary of V2 . The pair .1 ; 2 / is called a Heegaard diagram and corresponds to a pair of simplices in C .Sg /. For each Heegaard splitting V1 [Sg V2 with Sg as the splitting surface, the set of all possible simplices 1 and their faces form a subcomplex K1 of C .Sg / (the subcomplex spanned by the vertices corresponding to all essential simple closed curves of Sg that bound some meridians of the handlebody V1 ). Similarly, there is a subcomplex K2 of C.Sg / for the other handlebody V2 . The Hempel distance for the Heegaard splitting V1 [Sg V2 of M 3 is the distance d.K1 ; K2 / between K1 and K2 with respect to the geodesic length function of C .Sg /. This defines a numerical invariant of the Heegaard splitting of M and is closely related to topological properties of the manifold M and of the Heegaard splitting. For details, see [77]. The paper [137] discusses the behavior of the Hempel distance under stabilization of splitting. Remark 7.26. In [96], the curve complex C .Sg;n / is replaced by the pants complex of Sg;n and a similar distance function is introduced. The paper [97] contains some results relating the bridge number of hyperbolic knots and an invariant defined in terms of the distance function on C .Sg;n /. The paper [155] applies the idea of Hempel distance to knots and defines a distance for certain knots in lens spaces. Remark 7.27. The paper [68] shows that for a closed orientable 3-manifold containing an incompressible surface of genus g, any Heegaard splitting has Hempel distance at most 2g. See also [158], [157], [112], [168] for related results on the Hempel distance. Remark 7.28. Based on the Hempel distance, the paper [156] defines a translation distance on open book decompositions of three-dimensional manifolds, and relates it to the topological properties of the manifolds.

7.8 Partial compactifications of Teichmüller spaces and their boundaries Recall that Tg;n is the Teichmüller space of marked complex structures on Sg;n , where a marking is a choice of a set of generators of Sg;n . Then the mapping class group Modg;n acts on Tg;n by changing the markings, and the quotient Modg;n nTg;n is the moduli space Mg;n of Riemann surfaces of genus g with n punctures.

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It is known that Tg;n is a complex manifold of complex dimension 3g  3 C n and diffeomorphic to R6g6C2n , and that Modg;n acts holomorphically and properly on Tg;n . In particular, Mg;n has a natural structure of a complex orbifold. Proposition 7.29. For any torsion-free subgroup   Modg;n , nTg;n is a B-space. Proof. Since  acts properly on Tg;n and is torsion-free,  acts fixed-point freely on Tg;n . Since Tg;n is contractible, it is an E-space, and nTg;n is a B-space. A more general result is true for subgroups of Modg;n containing torsion elements. Proposition 7.30. For any subgroup   Modg;n , Tg;n is an E-space. We only need to check that for any finite subgroup F  , the set of fixed points F is nonempty and contractible. The former follows from the solution of the Nielsen Tg;n realization problem, and the latter from either the existence of a unique left (or right) earthquake between any two points or the convexity of the Weil–Petersson metric. See [95] for more details. It is known that Mg;n D Modg;n nTg;n is noncompact. Therefore, Tg;n is not a cofinite EModg;n -space. To see this, we assume for simplicity that 2g  2 C n > 0. Then every Riemann surface in Mg;n admits a unique hyperbolic metric of finite area which is conformal to the complex structure. By pinching a simple closed geodesic on a hyperbolic surface †g;n , we obtain a family of Riemann surfaces in Mg;n that has no limit. For a non-uniform torsion-free arithmetic subgroup of a linear semisimple algebraic group G, nX is a non-cofinite B-space. To obtain a cofinite B-space, this problem was solved by the Borel–Serre compactification of nX , which is a quotient of the Borel–Serre partial compactification of the associated symmetric space X D G=K. In order to construct an analogue of the Borel–Serre partial compactification of Tg;n , Harvey [69], [70], [73] introduced the curve complex C .Sg;n /. To explain the relation between partial compactifications of Tg;n and simplices of C.Sg;n /, we note the following compactness criterion of Mumford. Proposition 7.31. A sequence of Riemann surfaces †j in Mg;n has no accumulation point in Mg;n if and only if there is a disjoint collection of simple closed geodesics j;1 ; : : : ; j;rj with respect to the hyperbolic metric of †j whose lengths go to 0 as j ! C1, i.e., that can be simultaneously pinched. Note that in the above proposition, the geodesics must be disjoint since the collar theorem for hyperbolic surfaces implies that two sufficiently short geodesics are disjoint and hence simultaneously pinched geodesics must be disjoint. Conversely, a collection of disjoint simple closed geodesics in a hyperbolic surface †g;n can be pinched simultaneously if and only if they are disjoint. This can be seen from pants decompositions of hyperbolic surfaces.

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Given the above discussion, a natural compactification of Mg;n is obtained by adding degenerate Riemann surfaces. This is the Deligne–Mumford compactification DM

Mg;n of Mg;n . DM As a topological space, Mg;n can be constructed as a quotient of the augmented Teichmüller space Tyg;n . For every simplex  of C .Sg;n /, there is a boundary Teichmüller space Tg;nI , which is the Teichmüller space of the surface obtained from Sg;n by cutting along the curves in . In terms of the hyperbolic metric, every homotopy class of curves in  determines a unique simple closed geodesic in every marked hyperbolic surface in Tg;n , and degenerated hyperbolic surfaces in the boundary Tg;nI are obtained by pinching exactly the geodesics in  . Then the augmented Teichmüller space Tyg;n is a Tyg;n D Tg;n [ Tg;nI : 2C.Sg;n /

The action of Modg;n on Tg;n extends to a continuous action on Tyg;n and the DM quotient Modg;n nTyg;n is homeomorphic to Mg;n . An important point here is that the curve complex C .Sg;n / is a parameter space for the boundary components of the augmented Teichmüller space Tyg;n [69], [174]. Since each boundary Teichmüller space Tg;nI is contractible, the boundary @Tyg;n is homotopy equivalent to C .Sg;n / and hence is homotopy equivalent to a bouquet of spheres. Proposition 7.32. The action of Modg;n on Tyg;n is not proper. For every point in a boundary Teichmüller space Tg;nI , its stabilizer in Modg;n contains an infinite subgroup, which is generated by the Dehn twists along the curves in  and is isomorphic to Z , as a subgroup of finite index. This result implies that for any finite-index subgroup   Modg;n , the inclusion nTg;n ! nTyg;n is not a homotopy equivalence. This is similar to the situation with the reductive Borel–Serre compactification of nX in §6.3. To overcome the problem of infinite stabilizers Z for points in Tg;nI , we need to enlarge the boundary component to R  Tg;nI . The resulting space is a Borel–Serre BS partial compactification Tg;n of Tg;n constructed in [70], [73], [82]: a BS Tg;n D Tg;n [ R  Tg;nI : 2C .Sg;n /

Proposition 7.33. The action of Modg;n on Tg;n extends to a continuous and proper BS BS action on Tg;n with a compact quotient Modg;n nTg;n . The boundary of the Borel– BS Serre partial compactification Tg;n is homotopy equivariant to C .Sg;n / and hence to a bouquet of spheres S d , where d D 2g 2 D .Sg / if n D 0, d D .Sg;n /1 D 2g  3 C n if g  1 and n > 0, and d D .S0;n /  2 D n  4 if g D 0.

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Corollary 7.34. For any finite-index torsion-free subgroup   Modg;n , Tg;n cofinite E-space and the quotient nTg;n

BS

BS

is a

is a finite B-space.

BS

Proof. Since Tg;n is a manifold with corners whose interior is equal to Tg;n , it BS is contractible. Since  is torsion-free, it acts fixed-point freely on Tg;n , and the quotient   Modg;n is a compact manifold with corners. Therefore, the quotient BS nTg;n is a finite B-space. On the other hand, for Modg;n or its finite-index subgroups  which contain torsion BS elements, it is not clear whether Tg;n is an E-space. To solve this problem, we can take suitable subspaces of Tg;n instead of a compactification. For any sufficiently small " > 0, define the "-thick part Tg;n ."/ D f† 2 Tg;n j for any simple closed geodesic   †; `. /  "g; where `. / is the length of the geodesic . Proposition 7.35. The thick part Tg;n ."/ is a submanifold with corners and invariant under the action of Modg;n with a compact quotient. The boundary faces of Tg;n ."/ are also parametrized by the simplices of C .Sg;n / and contractible, and the boundary @Tg;n ."/ is homotopy equivalent to a wedge of spheres S d , where d D 2g  2 D .Sg / if n D 0, d D .Sg;n /  1 D 2g  3 C n if g  1 and n > 0, and d D .S0;n /  2 D n  4 if g D 0. In [95], we proved Proposition 7.36. There is a Modg;n -equivariant deformation retraction of Tg;n to Tg;n ."/, and hence Tg;n ."/ is a cofinite EModg;n -space. If n > 0, there is another equivariant deformation of Tg;n which is of the smallest possible dimension, i.e., the virtual cohomological dimension of Modg;n in [66]. Problem 7.37. When n D 0, construct a Modg;0 -deformation retract of Tg;0 of dimension equal to the virtual cohomological dimension of Modg , which is equal to 4g  5. This problem seems to be completely open. It is not clear what subspaces of Tg;n are possible candidates for such a retract. It is not obvious if such a retract exists.

7.9 Cohomological dimension and duality properties of mapping class groups As discussed earlier, one important application of the Borel–Serre partial compactification of a symmetric space concerns the virtual cohomological dimension and duality

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properties of arithmetic subgroups. The analogous partial compactifications of Tg;n in the previous subsection have similar applications. Proposition 7.38. When the Euler characteristic .Sg;n / is positive, Modg;n is a virtual duality group of dimension d , where d D 4g  5 D dim Tg C .Sg /  1 if n D 0, and d D 4g  4 C n D dim Tg;n C .Sg;n / if g  1 and n > 0, and d D n  3 D dim T0;n C .S0;n / C 1. The dualizing module is equal to the only nonzero reduced homology group of C.Sg;n /, called the Steinberg module of Modg;n . The proof is similar to the proof for arithmetic groups using results on the Borel– Serre partial compactification of Tg;n in the previous subsection. See [66], [67], [80], [69] for details. In [86], using Proposition 5.9 on the topology of C .Sg;n /, we proved Proposition 7.39. Modg;n is not a virtual Poincaré duality group. This proposition is equivalent to the fact that when dim Tg;n > 0, the Steinberg module of Modg;n is of infinite rank. See [31] for more information on this module. Since Modg;n nTg;n is noncompact, Proposition 7.39 is consistent with the result that an arithmetic subgroup  is not a virtual Poincaré duality group if and only if nX is noncompact (Proposition 6.14). The Euler characteristic of Modg;n is given by special values of the Riemann zeta function, and the homology groups of Modg;n in low degrees are also known. See [66] and references there. Problem 7.40. Compute Hi .Modg;n ; Z/ in all degrees. Maybe compute them for special values of g and n, and also for special values of i first.

7.10 Tangent cones at infinity of Teichmüller spaces, moduli spaces, and mapping class groups As discussed earlier, the asymptotic cones at infinity of symmetric spaces and locally symmetric spaces are given by buildings. A natural problem is to consider the asymptotic cones of the Teichmüller space Tg;n and its quotient Mg;n D Modg;n nTg;n . Proposition 7.41. With respect to the Teichmüller metric, for any finite-index subgroup   Modg;n , the asymptotic cone at infinity Cone1 .nTg;n / is equal to the metric cone over nC .Sg;n /. This result was proved in [111], [53]. There are some results on Cone1 .Modg;n / and Cone1 .Tg;n ; dWP /, where dWP is the Weil–Petersson metric. By [5], for any asymptotic cone Cone1 .Modg;n /, every point is a global cut-point, i.e., Cone1 .Modg;n / is tree-graded.

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Since the quotient Modg;n nTg;n is noncompact, it is reasonable to expect that Modg;n is not quasi-isometric to Tg;n , and that Cone1 .Tg;n / and Cone1 .Modg;n / will be different. The following bounds on the dimension of Cone1 .Modg;n / and Cone1 .Tg;n ; dWP / were obtained in [7]. Proposition 7.42. The maximal dimension of locally compact subsets of the asymptotic cone Cone1 .Modg;n / is equal to the maximal rank of free abelian subgroups of Modg;n , which is equal to 3g  3 C n. Consequently, the geometric rank of Modg;n is equal to the maximal rank of free abelian subgroups of Modg;n . The geometric rank of a group  is defined as the largest integer n for which there exists a quasi-isometric embedding Zn ! . This result was used to prove the quasi-isometric rigidity of Modg;n in [6], [65]. See §7.4. Proposition 7.43. The maximal dimension of locally compact subsets of the asymptotic   cone Cone1 .Tg;n ; dWP / is equal to 3gCp2 , , that is, the integral part of 3gCp2 2 2   and hence the geometric rank of .Tg;n ; dWP / is equal to 3gCp2 . 2 The geometric rank of a metric space is the maximal dimension of quasi-flats in the metric space. The geometric ranks of some complexes related to the curve complex were determined in [136]. Problem 7.44. Understand the global structure of the asymptotic cones at infinity, Cone1 .Modg;n / and Cone1 .Tg;n ; dWP /, and determine the maximal dimension of compact subsets of the asymptotic cone at infinity Cone1 .Tg;n ; dTei /, where dTei is the Teichmüller metric. There are some other results that connect the geometry and topology of Mg;n at infinity with the curve complex C .Sg;n /. Proposition 7.45. Assume that dim Tg;n  2. Then for any finite-index subgroup   Modg;n , the quotient nTg;n has only one end. Briefly, the reason is that under the assumption that dim Tg;n  2, Sg;n contains at least two simple closed essential curves, and C .Sg;n / is connected. Now a suitable neighborhood of the infinity of nTg;n is homotopy equivalent to nC .Sg;n / and hence is connected. Proposition 7.46. For any finite-index subgroup   Modg;n , the quotient nTg;n has a compactification nTg;n [ nC .Sg;n / whose boundary is equal to the finite complex nC.Sg;n /. This is similar to the Tits compactification of locally symmetric spaces in [94]. See [90], Proposition 12.6.

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7.11 Simplicial volumes of moduli spaces As recalled in §6.5, for an arithmetic locally symmetric space nX , whether its simplicial volume vanishes or not depends on the size of its Q-rank, which is related to the dimension of the spheres that determine the homotopy type of the Tits building Q .G/. It turns out that similar results hold for Mg;n [91]. Proposition 7.47. The orbifold simplicial volume of Mg is equal to zero if and only if g  2. The simplicial volume of Mg vanishes for all g  1. It is known that Modg;n admits finite-index torsion-free subgroups and hence Mg;n admits finite smooth covers. Since the orbifold simplicial volume is greater than or equal to the simplicial volume [91], it suffices to show that the orbifold simplicial volume of Mg is equal to zero if and only if g  2, and the simplicial volume of M1 is equal to zero. For the first statement, the proof is similar to the proof of Proposition 6.17 by replacing the Borel–Serre partial compactification of the symmetric space by the thick part Tg ."/ of the Teichmüller space Tg , and the rational Tits building Q .G/ by the curve complex C.Sg /. Since C.Sg / is homotopy equivalent to a bouquet of spheres of dimension 2g  2 for g  2, it is simply connected. This is a crucial ingredient of the proof of the above proposition. The nonvanishing of the orbifold simplicial volume of M1 Š SL.2; Z/nH2 follows from the result of Thurston for finite-volume hyperbolic manifolds (Proposition 6.16). To prove that the simplicial volume of SL.2; Z/nH2 vanishes, we note that SL.2; Z/nH2 is homeomorphic to R2 . Since the orbifold simplicial volume and hence the simplicial volume of SL.2; Z/nH2 is finite, this implies that the simplicial volume of R2 is finite. Since R2 admits proper self-maps of degree greater than 2, its simplicial volume is 0. Hence the simplicial volume of SL.2; Z/nH2 is 0. This also gives a new proof of the fact that the simplicial volume of R2n vanishes. See [91] for details. More generally, for the general moduli space Mg;n , we have Proposition 7.48. When g  1 and n > 0, both the simplicial volume and orbifold simplicial volume of Mg;n are equal to zero if .Sg;n /  1  2, i.e., 2g C n  5; and when g D 0 and n  4, the simplicial volume of M0;n if .S0;n /  2  2, i.e., n  6. It is known that M1;1 Š M1;0 D SL.2; Z/nH2 , which is homeomorphic to R2 . Since the simplicial volume of R2 is equal to 0, the simplicial volume of M1;1 and M1;0 is also equal to 0. On the other hand, by Proposition 6.16, any finite smooth cover of M1;1 and M1;0 have positive simplicial volume. Therefore, the orbifold simplicial volume of M1;1 and M1;0 is positive. Since T0;4 Š T1;1 Š H2 [142] and Mod0;4 is commensurable with SL.2; Z/, it follows from that for .g; n/ D .1; 0/; .1; 1/; .0; 4/, the orbifold simplicial volume of Mg;n is positive.

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Problem 7.49. Decide whether the simplicial volume and orbifold simplicial volume of Mg;n vanish or not for .g; n/ D .1; 2/; .0; 5/:

7.12 The action of Modg;n on C .Sg;n / and applications As mentioned before, Modg;n acts on C .Sg;n /, and this action can be used to understand structures and representations of Modg;n . One of the early applications of the curve complex by Harvey [71] concerns a simplified proof of the classification of elements of Modg;n by Thurston [164]. Recall that the group SL.2; R/ acts on H2 by Möbius transformations and that any element  2 SL.2; R/ belongs to one of the following types: (1)  is elliptic and is characterized by the property that it fixes a point in H2 ; (2)  is parabolic and is characterized by the property that it fixes exactly one point in the boundary H2 .1/ Š S 1 ; (3)  is hyperbolic and is characterized by the property that it fixes exactly two points in the boundary H2 .1/ Š S 1 . Since H2 [ H2 .1/ is a closed ball and  acts on it continuously, it must fix some point in H2 [ H2 .1/, and the three possibilities above cover all cases. For any discrete subgroup   SL.2; R/, any element  2  belongs to one of the following types: (1)  is periodic, i.e., there exists some integer n such that  n D Id ; (2)  is parabolic and fixes a boundary point in H2 .1/; (3)  is hyperbolic and fixes exactly two points in the boundary H2 .1/ Š S 1 . Thurston [164] defined a compactification Tg;n sured laminations such that (1) Tg;n

Th

Th

of Tg;n through projective mea-

is homeomorphic to a closed ball of dimension 6g  6 C n; Th

(2) the action of Modg;n extends to a continuous action on Tg;n . Using this action, he classified elements  2 Modg;n into corresponding three types: (1)  is periodic, i.e., there exists some integer n such that  n D Id , and is characterized by the condition that  fixes a point in Tg;n ; (2)  is reducible, i.e.,  leaves invariant a collection of disjoint simple closed essential curves of Sg;n ; (3)  is pseudo-Anosov and is characterized by the condition that  fixes exactly Th two points in the boundary @Tg;n .

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Remark 7.50. The action of Modg;n on C .Sg;n / can be used to define a translation length of an element  of Modg;n : for any simple closed curve ˛ in Sg;n , the translation length of  is defined by dC .˛; f j .˛// lim inf ; j !C1 j where dC .˛; f j .˛// is the distance between the two vertices ˛ and f j .˛/ of C .Sg;n /. It was proved in [125] that an element  is pseudo-Anosov if and only if its translation length is positive. See [52], [56] for some estimates on the translation length. Remark 7.51. An important application of the action of Modg;n on the curve complex C.Sg;n / is to give an explicit presentation of Modg;n in [78], [169] (see also [72]). Another presentation was constructed via the action of Modg;n on a variant of the curve complex in [11]. Remark 7.52. The action of Modg;n on the curve complex C .Sg;n / is also used essentially in the proof of the Tits alternative for Modg;n [16], [85]. Remark 7.53. The action of Modg;n on the pants complex [74] was also used to construct a presentation of Modg;n . Remark 7.54. Another application of the action of Modg;n on C .Sg;n / is to construct irreducible unitary representations of Modg;n . It was shown in [141] that the action of Modg;n on C.Sg;n / has non-commensurable stabilizers and a general method was used to produce irreducible unitary representations. Remark 7.55. By [125], there is a relative metric on Modg;n , which is quasi-isometric to C .Sg;n /. (A relative metric on a group is a word metric on the group with respect to an infinite generating set, consisting of a finite generating set and a finite collection of subgroups.) The paper [118] shows that a random walk on Modg;n makes linear progress in this relative metric.

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[163] B. Szepietowski, A presentation for the mapping class group of a non-orientable surface from the action on the complex of curves. Osaka J. Math. 45 (2008), no. 2, 283–326. [164] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. [165] P. Tukia, A rigidity theorem for Möbius groups. Invent. Math. 97 (1989), no. 2, 405–431. [166] J. Tits, Buildings of spherical type and finite BN-pairs. Lecture Notes in Math. 386, Springer, Berlin 1974. [167] J. Tits, On buildings and their applications. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., 1975, 209–220. [168] M. Tomova, Distance of Heegaard splittings of knot complements. Pacific J. Math. 236 (2008), no. 1, 119–138. [169] B. Wajnryb, A simple presentation for the mapping class group of an orientable surface. Israel J. Math. 45 (1983), no. 2–3, 157–174. [170] A. Werner, Non-Archimedean intersection indices on projective spaces and the Bruhat– Tits building for PGL. Ann. Inst. Fourier (Grenoble) 51 (2001), no. 6, 1483–1505. [171] A. Werner, Compactifications of Bruhat–Tits buildings associated to linear representations. Proc. Lond. Math. Soc. (3) 95 (2007), no. 2, 497–518. [172] A. Werner, Compactification of the Bruhat–Tits building of PGL by lattices of smaller rank. Doc. Math. 6 (2001) 315–341. [173] A. Werner, Compactification of the Bruhat–Tits building of PGL by seminorms. Math. Z. 248 (2004) 511–526. [174] S. Wolpert, Geometry of the Weil–Petersson completion of Teichmüller space. In Surveys in differential geometry, Vol. VIII (Boston, MA, 2002), Surv. Differ. Geom. VIII, Internat. Press, Somerville, MA, 2003, 357–393.

Chapter 4

Extremal length geometry Hideki Miyachi

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . 1.2 Our extremal length geometry . . . . . . . . . . . 1.3 Plan of this chapter . . . . . . . . . . . . . . . . . 2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Teichmüller space . . . . . . . . . . . . . . . . . 2.2 Measured foliations . . . . . . . . . . . . . . . . 2.3 Extremal length . . . . . . . . . . . . . . . . . . 2.4 Teichmüller geodesics . . . . . . . . . . . . . . . 3 The Gardiner–Masur compactification . . . . . . . . . . 3.1 Definition . . . . . . . . . . . . . . . . . . . . . . 3.2 The functions Ep . . . . . . . . . . . . . . . . . . 3.3 Simple closed curves and uniquely ergodic points . 3.4 Horofunction compactification . . . . . . . . . . . 3.5 Unification via intersection number . . . . . . . . 4 Appendix on geodesic currents . . . . . . . . . . . . . . 4.1 Geodesic currents . . . . . . . . . . . . . . . . . 4.2 Bonahon’s embedding . . . . . . . . . . . . . . . 4.3 Hyperboloid model in hyperbolic geometry . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction The aim of this chapter is to survey the development of the extremal length geometry of Teichmüller space, after S. Kerckhoff, F. Gardiner and H. Masur’s pioneering works (cf. [22] and [15]). Here, we mainly treat the author’s results given in [37], [38] and [41]. In general, these are stated without proofs, but some of the proofs are sketched. Notice that our Teichmüller space here is always finite-dimensional.

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1.1 Background 1.1.1 Teichmüller space. Teichmüller space is a parameter space of equivalence classes of marked Riemann surfaces. The space is topologized with a canonical distance, the so-called Teichmüller distance, which is defined using a Grötzch’s type extremal problem for quasiconformal mappings (cf. [50]). Teichmüller space with the Teichmüller distance is complete and uniquely geodesic. H. Masur [30] showed that Teichmüller space is not negatively curved in the sense of Busemann. Indeed, he found two geodesic rays emanating from a common initial point which are asymptotically bounded distance apart. The Teichmüller distance is also related to the complex structure of Teichmüller space. Indeed, H. Royden gave a remarkable observation that the Teichmüller distance coincides with the Kobayashi hyperbolic distance (cf. [48]). It was also shown that Teichmüller space is a Stein manifold (cf. [3] and [24]). Furthermore, every geodesic is contained in a complex geodesic, which is called a Teichmüller disk in Teichmüller theory. 1.1.2 Extremal length geometry. Extremal length is a conformal invariant introduced by L. Ahlfors and A. Beurling for families of curves on Riemann surfaces (cf. [1]). Since extremal length has the quasiconformal distortion property, extremal length not only distinguishes between two Riemann surfaces, but also measures how two Riemann surfaces are different geometrically. Thus, extremal length is one of the important geometric quantities in geometric function theory and in Teichmüller theory (e.g. [1], [2], [18] and [26]). In a beautiful work [22], S. Kerckhoff applied the asymptotic behavior of extremal length at infinity in Teichmüller space to investigate the global structure of Teichmüller space. In fact, he gave an important geometric formula for the Teichmüller distance, called Kerckhoff’s formula, and he showed that the Teichmüller compactification and the Thurston compactification are different by studying the behavior of the extremal lengths of simple closed curves along Teichmüller geodesic rays (cf. §2.4.3). Thanks to Kerckhoff’s formula, extremal length is recognized as a standard invariant for studying the geometry of the Teichmüller distance. Thus, the geometry of the Teichmüller distance is nothing but the extremal length geometry of Teichmüller space. In the paper [15], F. Gardiner and H. Masur gave a further development of the extremal length geometry of Teichmüller space. Indeed, they characterized the Teichmüller geodesics as the lines of minima of the products of the extremal lengths of two measured foliations (compare with [23]). By applying a procedure similar to the Thurston compactification, they defined a canonical compactification in the extremal length geometry of Teichmüller space that encodes the asymptotic behaviors of the extremal lengths of simple closed curves. Recently, this compactification was called the Gardiner–Masur compactification (cf. §3). They observed that the Gardiner–Masur boundary strictly contains the space of the projective measured foliations when the complex dimension of Teichmüller space is at least two. When the complex dimension

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of Teichmüller space is one, the Gardiner–Masur compactification coincides with the Thurston compactification (cf. [15], see also [36]). 1.1.3 Coarse geometry. Recently, the extremal length geometry of Teichmüller space was well studied in a coarse sense. In a pioneering work [44], Y. Minsky gave his product region theorem for characterizing the Teichmüller geometry at the thin parts of Teichmüller space in a coarse sense. In [33], Masur and Minsky observed that the electric Teichmüller space defined by making all thin parts of bounded diameter is quasi-isometric to the complex of curves. Thus, though the Teichmüller space is not negatively curved, it behaves like a negatively curved space at the thick part in a coarse sense. In [45], K. Rafi gave a combinatorial formula for the Teichmüller distance. His formula is a powerful tool for investigating not only the global structure of Teichmüller space, but also its topological aspect (e.g. the mapping class group action, etc.). For details, see Masur’s survey [32].

1.2 Our extremal length geometry Our extremal length geometry in this chapter is based on Kerckhoff, Gardiner and Masur’s extremal length geometry. Unlike Masur, Minsky and Rafi’s coarse geometry, our geometry is sensitive in the sense that we often do care about being “bounded distance apart” and for “bounded distortion (in the Lipschitz sense)”. For instance, we distinguish two geodesic rays emanating from a common initial point (cf. Remark 3.3). On the other hand, we sometimes admit these coarse properties (cf. Remark 3.10). Thus, our extremal length geometry may be categorized as a (kind of) mildly coarse geometry. 1.2.1 Geometry via intersection number. Gardiner and Masur’s compactification is defined in a way similar to Thurston’s compactification of Teichmüller space in the setting of hyperbolic geometry. Therefore, the Gardiner–Masur compactification has some flavor of the Thurston compactification. In this chapter, we will try to give an affirmative evidence for this impression. In fact, we will see that Kerckhoff, Gardiner and Masur’s (and hence our) extremal length geometry is unified using the intersection number, in analogy with the hyperbolic geometry in Thurston’s theory (cf. [9]). For instance, the extremal length of a measured foliation is regarded as the intersection number between a marked Riemann surface and the given measured foliation, and the intersection number between two marked Riemann surfaces is represented by the exponential of the Teichmüller distance between them. Furthermore, the Gromov product with respect to the Teichmüller distance is also represented explicitly by intersection number. As a corollary, we see that the Gromov product extends continuously to the Gardiner–Masur closure. Thus, like in the Thurston compactification, the Gardiner–Masur compactification links the analytic aspect of Teichmüller theory (the geometry of the Teichmüller distance) and

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the topological aspect of Teichmüller theory (the geometry of the intersection number) via our intersection number. See §3.5 for details. 1.2.2 Horofunction compactification. The Gardiner–Masur compactification is also a canonical compactification with respect to the Teichmüller distance from the standpoint of the theory of metric spaces. In fact, L. Liu and W. Su [27] observed that the Gardiner–Masur compactification coincides with the horofunction compactification defined by M. Gromov in [16] (cf. §3.4). This observation is comparable to the one made by C. Walsh, saying that the Thurston compactification coincides with the horofunction compactification with respect to Thurston’s asymmetric Lipschitz metric (cf. [54]). Several comparable properties between the Thurston compactification and the Gardiner–Masur compactification can be seen via the horofunction compactification in Table 1. Table 1. Comparison in view of the horofunction compactification.

Hyperbolic length

Extremal length

Distance

Thurston’s metric

Teichmüller distance

Horofunction compactification

Thurston compactification ([54])

Gardiner–Masur compactification ([27]).

Topology

Closed ball B6g6Cn ([9])

Boundary point

Intersection number function associated to a point in MF ([9])

Intersection number function associated to a point in @CGM (cf. §3.5)

Geometric object for boundary point

Singular foliation with a transverse measure ([9])

?

Busemann point

Every boundary point is a Busemann point ([54])

Some boundary points are not Busemann points ([40])

Gromov product

?

Extends continuously to the closure (cf. §3.5)

?

Question marks in the table means that the author does not know answers or facts for the blocks. About the topology, we can easily see that the Gardiner–Masur boundary is not homeomorphic to the .6g  7 C 2n/-dimensional manifold when 3g  3 C n  2, but the author does not know any other topological properties. About the geometric objects, in hyperbolic geometry, every boundary point is the projective class of a measured foliation, which is a pair of a singular foliation and a transverse measure. This geometric interpretation is very useful and applied for yielding rich

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results. In extremal length geometry, we actually have the intersection number, but we do not have geometric objects corresponding to boundary points. About the Gromov product, the author does not know any properties of the Gromov product in the theory of horofunction compactifications. However, since the Gromov product gives a relation between horofunctions and the distance, the extendability of the Gromov product to the horofunction compactification seems to be an interesting problem in view of the theory of horofunction compactifications. 1.2.3 Hyperboloid models. In the unification via the intersection number, we shall realize Teichmüller space in a cone CGM . The embedding is defined by a canonical lift of the Gardiner–Masur embedding of Teichmüller space (cf. (3.1)). Actually, this realization gives a hyperboloid model of Teichmüller space with respect to the Teichmüller distance in terms of the “quadratic form” i.  ;  /. This is in analogy with Bonahon’s observation that Teichmüller space is realized as a hyperboloid in the space of geodesic currents (cf. [7]). From these realizations, we can consider the distance induced from the intersection number in each model by imitating the hyperboloid model of hyperbolic space (cf. Chapter I.2 in [8]). In hyperbolic geometry, the hyperbolic distance admits two representations, which we call the hyperbolic distance defined in the infinitesimal and global sense. In the infinitesimal sense, the hyperbolic distance is given as the length metric defined by summation of the lengths (by the quadratic form) of subdivision of the paths. In the global sense, the cosine hyperbolic of the hyperbolic distance is equal to the quadratic form. See (3.27) and (3.28) in §3.5.4. p In Bonahon’s theory, the induced distance in the infinitesimal sense is =3 times the Weil–Petersson distance (cf. [7]). The author does not know what the induced distance is in the global sense. In the case of extremal length geometry, the induced distance is unfortunately different from the Teichmüller distance. In fact, in the infinitesimal sense, the induced distance seems to be non-existent, and in the global sense, it coincides with a distance defined by modifying the Teichmüller distance. See Table 2 and §3.5.4 for details. Table 2. Comparison in terms of hyperboloid models. See [7] and §3.5.4.

Bonahon’s theory

Extremal length geometry

Hyperboloid

fi.; / D  j.X/jg

fi.; / D 1g

Light cone

fi.; / D 0g D MF p =3 times the Weil– Petersson distance

fi.; / D 0g D @CGM

Induced distance (Infinitesimal) Induced distance (Global)

2

?

? (possibly being nonexistent) Modified Teichmüller distance

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1.3 Plan of this chapter This chapter is a survey on our extremal length geometry of Teichmüller space. We tried to make the chapter self-contained and we start with basic definitions and elementary facts. In §2, we recall basic objects in Teichmüller theory around extremal length geometry. We first give definitions and properties of holomorphic quadratic differentials, Teichmüller space and measured foliations (cf. §§2.1 and 2.2). Then we treat extremal length. Usually, extremal length is defined in two equivalent ways, the so-called analytic definition and the geometric definition. We shall state these two definitions and give a brief confirmation of the equivalence between them (cf. §2.3). We also recall the notions of Teichmüller geodesics and Kerckhoff’s formula of the Teichmüller distance in §2.4. In §3, we recall the definition of the Gardiner–Masur compactification of Teichmüller space. In §3.2, we deal with continuous functions associated to boundary points, which are prototypes of our intersection number. We treat simple closed curves and uniquely ergodic points in the boundary in §3.3. We recall Liu and Su’s theorem on the horofunction compactification in §3.4. In §3.5, we discuss the unification of the geometry of Teichmüller space with respect to the Teichmüller distance via intersection number. We also discuss the hyperboloid model and the characterization of the isometry group of Teichmüller space in terms of the Teichmüller distance. In the appendix (§4), we recall Bonahon’s theory on the realization of Teichmüller space via geodesic currents. Acknowledgement. The author would like to express his heartfelt gratitude to Professor Athanase Papadopoulos for giving an opportunity for writing this chapter and for useful comments on the first version. This work was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (C), 2154017.

2 Notation 2.1 Teichmüller space 2.1.1 Riemann surfaces and quadratic differentials. Throughout this chapter, X denotes a Riemann surface of analytically finite type .g; m/. Namely, X is a closed Riemann surface of genus g with m points removed. We also always assume that 2g  2 C m > 0. A holomorphic quadratic differential q D q.z/dz 2 on the Riemann surface X is a section of the line bundle OX˝2 over X, where OX is the structure sheaf on X . Any q 2 H 2 .X; OX˝2 / determines a singular flat metric jqj D jq.z/jjdzj2 on X. Hence,

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we have a well-defined integral Z jq.z/jdxdy:

kqk D X

We define QX D fq 2 H 2 .X; OX˝2 / j kqk < 1g: Then QX is a Banach space with norm k  k. One can see that any q 2 QX is a meromorphic quadratic differential on the completion Xx of X such that any pole is at a puncture of X and the order at any point in Xx is at least 1. Let q 2 QX and I  R an interval. A differential path  W I ! X is said to be a trajectory of q if arg q..t//.t P /2 is constant on I . A trajectory  is horizontal 2 if q..t //P .t/ > 0, and vertical if q..t //P .t /2 < 0. We always assume that a trajectory is complete in the sense that it is not contained in any trajectory as a proper sub-trajectory. We also call the image of a trajectory a trajectory, for simplicity. A trajectory is called critical if one of its ends is at a critical point of q. Otherwise it is called regular. If two ends of a trajectory are at critical points, we call the trajectory a saddle connection. A trajectory is closed if it is regular and its image is a closed curve on X. Any closed trajectory ˛ is a simple closed curve on X , and admits an annular neighborhood foliated by closed trajectories which are homotopic to ˛. The maximal such foliated annular neighborhood is said to be a characteristic annulus of ˛ for a quadratic differential q. The characteristic annulus is unique since X is not a torus. Trajectories of q are geodesics with respect to the jqj-metric in their homotopy classes rel endpoints. In general, any geodesic in terms of the jqj-metric is a broken line consisting of (critical) trajectories or saddle connections with a certain angular condition at critical points. We denote by `q .˛/ the infimum of the jqj-length which runs over all curves homotopic to a given curve ˛ (rel endpoints when ˛ is not closed). The jqj-length is always attained at a jqj-geodesic in Xx . See, for instance [14] and [49]. Example 2.1 (Holomorphic quadratic differential on annulus). Though an annulus is not of analytically finite type, it gives a canonical model for discussing Jenkins–Strebel differentials. Let Ann.r/ D fr 1=2 < jwj < r 1=2 g be a round annulus in C. Consider a quadratic differential q D q.w/dw 2 D

`2 dw 2 4 2 w 2

on Ann.r/. Fix 0 2 R and 1 .t/ D e tCi0 for jtj < .log r/=2. Then q.1 .t//.P1 .t//2 D

`2 > 0: 4 2

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Hence 1 .t / is a horizontal trajectory of q. Fix t0 with jt0 j < .log r/=2. Consider 2 . / D e t0 Ci . Then `2 < 0; 4 2 and hence 2 is a vertical (closed) trajectory of q. One can see that the length of a closed trajectory is equal to ` (cf. Figure 1). q.2 .t//.P2 .t//2 D 

horizontal trajectory

vertical trajectory

Figure 1. Quadratic differential with closed trajectories on an annulus.

2.1.2 Teichmüller space. The readers can refer to [1], [14] [18]. The Teichmüller space Tg;m of Riemann surfaces of analytically finite type .g; m/ is the set of equivalence classes of marked Riemann surfaces .Y; f /, where Y is a Riemann surface and f W X ! Y is a quasiconformal mapping. Two marked Riemann surfaces .Y1 ; f1 / and .Y2 ; f2 / are Teichmüller equivalent if there is a conformal mapping h W Y1 ! Y2 such that h B f1 is homotopic to f2 . If necessary, we consider x0 D .X; id/ as the basepoint of Tg;m . Teichmüller space Tg;m admits a canonical distance, which we call the Teichmüller distance dT , defined by dT .y1 ; y2 / D

1 log inf K.h/; 2 h

where yi D .Yi ; fi / (i D 1; 2) and h runs over all quasiconformal mappings from Y1 to Y2 such that h B f1 is homotopic to f2 . We topologize Tg;m with the Teichmüller distance. From the compactness of a family of quasiconformal mappings with uniformly bounded dilatations, we can check that the metric space .Tg;m ; dT / is complete. Furthermore, .Tg;m ; dT / is a uniquely geodesic space. We will discuss geodesics in Tg;m in §2.4. It is known that Teichmüller space Tg;m admits a natural complex structure inherited from the complex Banach space of Beltrami differentials of quasiconformal mappings on X . Under this complex structure, the Teichmüller distance coincides with the Kobayashi hyperbolic distance, and for y D .Y; f / 2 Tg;m , the space Qy WD QY is seen as the holomorphic cotangent space at y. Therefore, the complex vector bundle Qg;m D [y2Tg;m Qy is the holomorphic cotangent bundle over Tg;m .

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2.2 Measured foliations 2.2.1 Measured foliations. Let X be a Riemann surface as above. We denote by  D X the set of free homotopy classes of essential simple closed curves on X, where we call a simple closed curve essential if it is non-trivial and non-peripheral. Consider the set of formal products R0 ˝  D ft ˛ j t  0; ˛ 2 g. For any t ˛ 2 R0 ˝ , we define a function i .t ˛/ on  by i .t ˛/.ˇ/ D t i.˛; ˇ/; where i.  ;  / is the geometric intersection number function. This i defines an embedding i W R0 ˝  3 t ˛ 7! i .t ˛/ 2 R0 ; where R0 is the space of non-negative functions on  topologized by pointwise convergence. The closure MF  R0 of the image i .R0 ˝ / in R0 is called the space of measured foliations on X. We identify any ˛ 2  with 1  ˛ 2 R0 , and we consider  as a subset of MF . For G 2 MF , we denote by i.G; ˛/ the value G.˛/ at ˛ 2 , and call it the intersection number between G and ˛. By definition, when G 2 , i.G; ˛/ coincides with the original geometric intersection number between simple closed curves. We call a measured foliation G rational if there are collections of positive numbers fwj gjND1 and f˛j gN iD1 in  with i.˛j1 ; ˛j2 / D 0 and ˛j1 ¤ ˛j2 for j1 ¤ j2 such that i.ˇ; G/ D

N X

wj i.ˇ; ˛j /

j D1

P for all ˇ 2 . In this case, we write G D jND1 wj ˛j . The existence of rational measured foliations is not trivial from our definition. Practically, in our setting, a rational measured foliation is constructed as an accumulation point of the orbit of a simple closed curve under actions of Dehn-twists (see also Appendice of Exposé 4 in [9]). We can easily see the existence of such measured foliations via the geometric description of measured foliations given in §2.2.2 below. The set RC of positive numbers acts on R0 by the multiplication RC  R0 3 .s; f / 7! sf 2 R0 . Hence we get the projective space PR0 D .R0  f0g/=RC : Let pr W R0  f0g ! PR0 denote the projection. The image i .R0 ˝ / is invariant under the above action and hence we obtain the quotient space P MF D .MF  f0g/=RC  PR0 . The space P MF is called the space of projective measured foliations on X. For simplicity, we denote by ŒG the projective class of G 2 MF . By definition, the projection pr is injective on   MF  f0g. Hence, we may also think of  as a dense subset of P MF . It is known that MF and P MF are homeomorphic to R6g6C2m and S6g7C2n , respectively.

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The geometric intersection number i W    ! Z extends homogeneously on .R0 ˝ /  .R0 ˝ / by i.t ˛; sˇ/ D t s i.˛; ˇ/:

(2.1)

Furthermore, the intersection number admits a continuous extension on MF  MF (cf. [6], [7] and [46]). 2.2.2 Foliations with transverse measures. A measured foliation is a singular foliation with transverse measure. Indeed, for F 2 MF , there exist points fp1 ; : : : ; pm g, an open covering fUj gj of Xx  fp1 ; : : : ; pm g, and C 1 real valued function j on Uj such that the following holds: (1) jdj1 j D jdj2 j on Uj1 \ Uj2 . (2) At each point p of X , there is a neighborhood V of p in Xx and a local C 1 chart .; / W V ! R2 such that, for D  C i, we have ˇ ˇ jdj j D ˇIm. k=2 d /ˇ on Uj \ V for some k 2 Z. (3) The integer k in (2) depends on the location of the point p; If p 2 Xx  fp1 ; : : : ; pm g, k D 0. If p D pl for some l but p is not a puncture of X, k  0. Otherwise, k  1. (4) For any ˛ 2 ,

Z F .˛/ D i.F; ˛/ D inf 0

˛ ˛

˛0

jdj;

where jdj D jdj j on Uj . We have a singular foliation on Xx with transverse measure jdvj by gluing foliations on Uj defined by the kernels of jdj j (cf. [9], [14] and [17]). For instance, in Example 2.1, a differential jdj D jRe

p

ˇ  ˇ ˇ ` judu C vdvj ` dw ˇˇ ˇ qj D ˇRe ˇD 2 2

2 w

2

u Cv

(2.2)

on an annulus Ann.r/ defines a measured foliation with transverse measure (2.2), where w D u C iv D exp..2=`/ / D exp..2=`/. C i//. The leaves of the foliation are round circles, and Ann.r/ is fully foliated by these closed leaves. As we see in §2.3.4, this measured foliation is nothing but the vertical foliation of q.

2.3 Extremal length 2.3.1 Definition of the extremal length. We start with the definition of extremal length in a general setting (cf. [1] and [2]). Let D be a domain in a Riemann surface

Chapter 4. Extremal length geometry

207

R. Let be a family of rectifiable curves in D. For a measurable conformal metric  D .z/jdzj, we set Z Z .z/jdzj and Area./ D .z/2 dxdy: ` . / D inf 2



D

The extremal length of is defined by D . / D sup 

` . /2 ; Area./

(2.3)

where  runs over all measurable conformal metrics  D .z/jdzj on D with Area./ < 1. If a metric 0 attains the supremum (2.3), we call 0 an extremal metric (cf. §4-7 in [2]). The extremal length satisfies the quasiconformal distortion property 1 D . /  D 0 .f . //  K D . /; (2.4) K where f W D ! D 0 is a K-quasiconformal mapping. For simplicity, we shall check (2.4) briefly in the case when f is differentiable. A similar argument is available for a general quasiconformal mapping (cf. Remark at the end of Chapter II of [1]; see also [14].) Indeed, for any conformal metric  D .z/jdzj on D, we consider a conformal metric on D 0 defined by    B f 1 0 D jfz j  jfzN j in local parameters of D and D 0 . Notice that jfz j > jfzN j since f is quasiconformal (and orientation preserving). Then, for any  2 , Z Z 0 .w/jdwj D 0 .f .z//jfz dz C fzN d zj N f ./  Z Z .z/  .jfz j  jfzN j/jdzj D .z/jdzj  ` . /;  jfz j  jfzN j  Z  2  B f 1 .w/ dudv Area.0 / D 0 jf j  jf j z zN D Z  2 .z/ .jfz j2  jfzN j2 /dxdy D D jfz j  jfzN j Z jfz j C jfzN j D dxdy  KArea./: .z/2 jfz j  jfzN j D Therefore,

`0 .f . //2 ` . /2 K  K D 0 .f . //: Area./ Area.0 /

Since  is arbitrary, we get the left-hand side of (2.4). The right-hand side is obtained by the same argument applied to f 1 .

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2.3.2 Extremal lengths of simple closed curves. In this section, we shall define one of the main objects in this chapter, extremal length of simple closed curves. In a general setting, our extremal length is nothing but the extremal length of the family of simple closed curves homotopic to a given simple closed curve (cf. [15] and [22]). Let ˛ 2  and y D .Y; f / 2 Tg;m . The extremal length Exty .˛/ of ˛ on y is defined by (2.5) Exty .˛/ D Y .f .˛//; where in (2.5) ˛ is considered as a family of rectifiable simple closed curves homotopic to ˛ on X. More precisely, Exty .˛/ is defined as Exty .˛/ D sup 

where

` .˛/2 ; Area./

Z ` .˛/ D

inf

(2.6) Z

˛ 0 f .˛/ ˛ 0

.z/jdzj

and

Area./ D

.z/2 dxdy; Y

and  runs over all measurable conformal metrics on Y with Area./ < 1. Extremal length Exty .˛/ has another description: Exty .˛/ D inf f1=Mod.A/g; A

(2.7)

where A runs over all annulus on Y whose core is homotopic to ˛. The modulus 1 Mod.A/ of an annulus A (as a Riemann surface) is defined as 2 log r when A is conformally equivalent to a flat annulus fz 2 C j 1 < jzj < rg. The definitions (2.6) and (2.7) called the analytic definition and the geometric definition of extremal length, respectively. 2.3.3 Coincidence of two definitions of extremal length and Jenkins–Strebel differentials. The coincidence of the two quantities (2.6) and (2.7) follows from solutions to extremal problems on moduli of disjoint corrections of annuli on Riemann surfaces (e.g. Chapter VI in [49]). P For any rational measured foliation G D kiD1 wi ˛i and y D .Y; f / 2 Tg;m , there is a unique holomorphic quadratic differential JG;y 2 Qy with the following properties. (1) Any non-critical vertical trajectory is closed, and any closed trajectory is homotopic to some f .˛i /. Conversely, for any i , there is a closed trajectory homotopic to f .˛i /. (2) The characteristic annuli Ai of ˛i fills Y in the sense that Y n [kiD1 Ai consists of finitely many saddle connections. (3) The height of Ai in terms of JG;y is equal to wi . Equivalently, Mod.Ai / D wi =`i . where `i D `JG;y .˛i / denotes the JG;y -length of the closed trajectory homotopic to f .˛i /.

Chapter 4. Extremal length geometry

209

We call JG;y the Jenkins–Strebel differential for G on y (see [49]). It follows from theorems by Jenkins and Strebel that the Jenkins–Strebel differential P for G D kiD1 wi ˛i on y has the following three extremal properties: (a) For any locally square integrable metric  with Z .z/jdzj  `i ˛i

(2.8)

and for almost all closed trajectories ˛i of JG;y homotopic to f .˛i / for i D 1; : : : ; k, Z k X .z/2 dxdy; (2.9) wi `i D kJG;y k  Y

iD1

where by almost all we mean that when the characteristic annulus Ai of f .˛i / for JG;y is identified with a round annulus fR1 < jzj < Rg, (2.8) holds for closed trajectories corresponding to round circles fjzj D rg for almost all r in an open interval .R1 ; R/ with respect to the one-dimensional Lebesgue measure. Equality holds in (2.9) if and only if  D jJG;y j1=2 almost everywhere on Y (cf. Corollary 20.3 in [49]). (b) Let fA0i gkiD1 be mutually disjoint annuli on Y such that the core curve of Ai is homotopic to f .˛i / for each i. Then k X

`q .˛i /2 Mod.A0i /  kqk

(2.10)

iD1

for q 2 Qy . Equality holds in (2.10) if and only if there is a rational measured foliation G 2 MF with support f˛i gkiD1 with q D cJG;y for some c ¤ 0 and A0i D Ai for each i (cf. Theorem 20.4 in [49]). (c) For fA0i gkiD1 as in (b), k X iD1

X wi2 wi2  D kJG;y k: Mod.A0i / Mod.Ai / k

(2.11)

iD1

Equality holds in (2.11) if and only if A0i D Ai for each i (cf. Theorem 20.5 in [49]). Suppose G D ˛ 2 , that is, k D 1 and w1 D 1. Let ` D `J˛;y .˛/. For any conformal metric , we consider the normalized metric .`=` .˛//   in (a). Then we 2 have Area.jJ˛;y j/ D kJ˛;y k  ` `.˛/2 Area./ and 

2

` .˛/ `2 `2  D : Area./ Area.jJ˛;y j/ kJ˛;y k

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Therefore, from (2.11), the right-hand side of (2.6) in the analytic definition of extremal length is equal to `2 D ` D kJ˛;y k: kJ˛;y k Furthermore, from (2.11), the right-hand side of (2.7) in the geometric definition of the extremal length is equal to kJ˛;y k D

1 ; Mod.A˛ /

where A˛ is the characteristic annulus of J˛;y . Therefore, we get the equality between the analytic definition and the geometric definition of the extremal length. We summarize this equality as follows: Exty .˛/ D `J˛;y .˛/ D kJ˛;y k D

1 Mod.A˛ /

(2.12)

for ˛ 2 . Example 2.2 (Jenkins–Strebel differential). We consider a decagon R0 D fz D x C iy 2 C j 0  x  1; jyj  2g with vertices .i; j / where i 2 f˙1g and j 2 f0; ˙1; ˙2g. We label the edges of R0 by e˙ D Œ0; 1  f˙2g, aj;˙ D fj g  Œ1 ˙ 1; ˙1 and bj;˙ D fj g  Œ˙1; 1 ˙ 1 (j D 0; 1). See Figure 2. We identify aj C with aj  and bj C with bj  by translations with -rotation, and eC with e by translation. Then we get a Riemann surface X0 . A holomorphic quadratic differential dz 2 on R0 descends to the Jenkins–Strebel differential for ˛. Vertices define critical points, and edges aj ˙ and bj ˙ of R0 correspond to critical trajectories aj and bj on X0 , respectively. In the same way, we can also construct a Riemann surface X0 and a Jenkins–Strebel differential from the annulus f1 < jzj < e =2 g by gluing the circular arcs appropriately. (Cf. the upper-left in Figure 2. See also Example 2.1.) Conversely, any Jenkins–Strebel differential of a simple closed curve gives a procedure of (re-)construction of Riemann surface. For instance, see [30] and [35]. 2.3.4 Horizontal and vertical foliations and Hubbard–Masur differentials. Let y D .Y; f / 2 Tg;m . Any q 2 Qy determines two measured foliations Hq and Vq satisfying Z ˇ p ˇ ˇIm q ˇ ; (2.13) i.Hq ; ˛/ D inf 0 ˛ f .˛/ ˛ 0 Z ˇ p ˇ ˇRe q ˇ : i.Vq ; ˛/ D inf (2.14) ˛ 0 f .˛/ ˛ 0

We call Hq and Vq the horizontal and vertical foliations of q. Notice that Vq D Hq . For instance, the vertical foliation of J˛;y is equal to ˛. In [17], Hubbard and Masur proved that (2.15) Qy 3 q 7! Vq 2 MF

211

Chapter 4. Extremal length geometry b1C

a1C b0C

a0C

a0

b0

2i

˛ b 1

a1

eC

b0C

b1C

a0C

a1C ˛

0 b0

a0

˛

b1

a1

1 C 2i

1

b0

b1

a0

a1

2i

e

1

2i

Figure 2. Jenkins–Strebel differential.

is a homeomorphism for all y 2 Tg;m (see also [14], [22] and [55]). We denote by JG;y the holomorphic quadratic differential on Y whose vertical foliation is f .G/. We call JG;y the Hubbard–Masur differential for G on y. By definition, when G D ˛ 2   MF , the Hubbard–Masur differential for ˛ 2   MF is nothing but the Jenkins–Strebel differential for ˛. 2.3.5 Extremal length for measured foliation. Notice that from the uniqueness of Jenkins–Strebel differentials we have J t˛;y D t 2 J˛;y for all t ˛ 2 R0 ˝. Therefore, from (2.12), it is natural to define Exty .t˛/ D t 2 Exty .˛/

(2.16)

for t ˛ 2 R0 ˝ . Actually, in [22], Kerckhoff showed that when we define the extremal length of t ˛ 2 R0 ˝  as in (2.16), the extremal length function extends continuously to MF . Furthermore, we have Z Exty .G/ D kJG;y k D jJG;y .z/jdxdy (2.17) Y

for all G 2 MF . Actually, Kerckhoff observed that fJ tn ˛n ;y gn is a Cauchy sequence in Qy when tn ˛n ! G in MF with tn ˛n 2 R0 ˝  by a direct calculation (cf. §3 in [22]). The continuity of intersection number and the height theorem imply that J tn ˛n ;y converges to JG;y (see also [14]). The convergence J tn ˛n ;y ! JG;y in norm also follows from the fact that the mapping (2.15) is a homeomorphism. From the quasiconformal distortion property (2.4) of extremal length, we have e 2dT .y1 ;y2 / Exty1 .G/  Exty2 .G/  e 2dT .y1 ;y2 / Exty1 .G/

(2.18)

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for all y1 ; y2 2 Tg;m and G 2 MF (cf. (2.5)). Hence, we see that Tg;m  MF 3 .y; G/ 7! Exty .G/

(2.19)

is continuous. Recently, the author observed that (2.19) is totally differentiable in an appropriate sense. See [42]. 2.3.6 Minsky’s inequality. Let ˛; ˇ 2  and y 2 Tg;m . It follows from (2.6), (2.12) and (2.14) that ² ³2 Z p 2 2 i.˛; ˇ/ D i.˛; VJˇ;y / D inf jRe Jˇ;y j 0 ²

 D

˛ ˛

³2

Z

inf

˛ 0 ˛

˛0

jJˇ;y j1=2

˛0

D `Jˇ;y .˛/2

(2.20)

`Jˇ;y .˛/2 kJˇ;y k  Exty .˛/  Exty .ˇ/: kJˇ;y k

Since intersection number and extremal length are continuous on MF , from (2.1) and (2.16), we have (2.21) i.F; G/2  Exty .F /  Exty .G/ for all F; G 2 MF and y 2 Tg;m . The inequality (2.21) is called Minsky’s inequality (cf. Lemma 5.1 of [43]). This inequality is sharp in the sense that for any G 2 MF and y 2 Tg;m , there is a unique F 2 MF up to multiplication by a positive constant such that (2.22) i.F; G/2 D Exty .F /  Exty .G/: Actually, F in (2.22) is projectively equivalent to the horizontal foliation of JG;y . Moreover, it is known that for any F and G, the locus of y satisfying (2.22) in Tg;m consists of the Teichmüller geodesic defined by the holomorphic quadratic differential q with Hq D H and Vq D G, if the locus is not empty (cf. Theorem 5.1 in [15]; see also §2.4.) Minsky’s inequality leads to the following formula i.F; G/2 : F ¤0 Exty .F /

Exty .G/ D sup

(2.23)

Since the extremal length in the left-hand side is represented by the extremal lengths in the right-hand side, the formula (2.23) cannot be adopted as the definition of the extremal lengths of measured foliations. However, the formula plays a key rule in unifying the extremal length geometry via intersection number (cf. (3.20)). Example 2.3 (Equality in Minsky’s inequality). We shall check briefly that in the case where F D ˛ and G D ˇ with ˛; ˇ 2 , the equality (2.22) implies that F is projectively equivalent to the horizontal foliation of JG;y .

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Chapter 4. Extremal length geometry

Indeed, let A˛ be the characteristic annulus of J˛;y . Since Exty .˛/ D 1=Mod.A˛ /, from the inequality in the third line of (2.20), we have `Jˇ;y .˛/2 Mod.A˛ /  kJˇ;y k: Therefore, from the extremal property (b) in §2.3.3, if the equality (2.22) holds for F D ˛ and G D ˇ, we have J˛;y D cJˇ;y for some c ¤ 0. Namely, A˛ is foliated by closed trajectories of Jˇ;y homotopic to ˛. Notice that the intersection number i.˛; VJˇ;y / D i.˛; ˇ/ is attained by the geodesic representative ˛  of ˛ with respect to the singular flat metric jJˇ;y j (cf. Theorem 24.1 of [49]). In our case, ˛  is a closed trajectory of Jˇ;y homotopic to ˛. From the inequality of the second line of (2.20), we obtain Z Z p jRe Jˇ;y j D jJˇ;y j1=2 ˛

˛

and hence ˛  is a leaf of a horizontal foliation of Jˇ;y . It follows that the horizontal foliation of J˛;y is projectively equivalent to F D ˛. 2.3.7 Properties of extremal length. Fix y 2 Tg;m . As discussed before, the extremal length function Exty is continuous on MF . The extremal length function is non-degenerate in the sense that Exty .F / D 0 implies F D 0. Indeed, suppose F ¤ 0. By definition, there is an ˛ 2  with 0 ¤ i.˛; F / D i.˛; VJF;y /. From (2.14) and (2.17), JF ;y ¤ 0 and hence Exty .F / D kJF ;y k ¤ 0. The extremal length function is proper in the sense that for any M > 0, MF M D fF 2 MF j Exty .F /  M g is compact. Indeed, fix a family fˇj gjND1 of simple closed curves filling up X . By Minsky’s inequality, we have N X j D1

i.F; ˇj /2  Exty .F /  Exty .ˇj /  M 

N X

Exty .ˇj /

j D1

for all F 2 MF M . Hence MF M is compact (cf. [9]; see also [19]). Unfortunately, extremal length does not satisfy any additivity, but its square root satisfies the following sub-additive property: Exty .F1 C F2 /1=2  Exty .F1 /1=2 C Exty .F2 /1=2

(2.24)

for F1 C F2 2 MF (the readers can refer to [20] for the summation of measured foliations). From the continuity of the extremal length, it suffices to show that Exty .G/1=2  w1 Exty .˛1 /1=2 C w2 Exty .˛2 /1=2 for a rational measured foliation G D w1 ˛1 C w2 ˛2 . This is true since `JG;y .G/ D kJG;y k D w1 `JG;y .˛1 / C w2 `JG;y .˛2 /

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Hideki Miyachi

from (2.9) and (2.11), and we have we have .w1 `JG;y .˛1 / C w2 `JG;y .˛2 //2 kJG;y k   `JG;y .˛1 / `JG;y .˛2 / 2 D w1 C w2 kJG;y k1=2 kJG;y k1=2

Exty .G/ D kJG;y k D

 .w1 Exty .˛1 /1=2 C w2 Exty .˛2 /1=2 /2 from the analytic definition (2.6) of extremal length.

2.4 Teichmüller geodesics 2.4.1 Teichmüller geodesics. Let y D .Y; f / 2 Tg;m and q 2 Qy . For any t 2 R, we consider the solution f t of the Beltrami equation jqj @f @f D tanh.t / @zN q @z

(2.25)

on Y . We have a mapping Gq W R 3 t 7! .Y t ; f t B f / 2 Tg;m ; where Y t D f t .Y /. If is known that Gq is a geodesic in .Tg;m ; dT /. Namely, dT .Gq .t1 /; Gq .t2 // D jt1  t2 j holds for any t1 ; t2 2 R. We call Gq the Teichmüller geodesic associated to q. Let G 2 MF  f0g and y 2 Tg;m . Notice that the Beltrami differential jJG;y j JG;y depends only on the projective class of G. We call RG;y D GJG;y jŒ0;1/ W Œ0; 1/ ! Tg;m the Teichmüller geodesic ray emanating from y with respect to ŒG 2 P MF . Teichmüller’s fundamental theorem tells us that when we fix x 2 Tg;m , P MF  Œ0; 1/=P MF  f0g 3 .ŒG; t / 7! RG;x .t / 2 Tg;m

(2.26)

is a homeomorphism. In particular, any two points in Tg;m lie on a unique Teichmüller geodesic in Tg;m . 2.4.2 Distortion of extremal lengths along Teichmüller rays. For G 2 MF  f0g, the extremal length of G satisfies ExtRG;y .t/ .G/ D e 2t Exty .G/ D e 2dT .y;RG;y .t // Exty .G/

(2.27)

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Chapter 4. Extremal length geometry

for y 2 Tg;m and t  0. Indeed, from the continuity of the extremal length function on Tg;m MF , it suffices to check (2.27) in the case where G is a simple closed curve. For simplicity, we let G D ˛ 2  and y D .Y; f /. Let A˛ be a characteristic annulus of J˛;y on Y . Notice that `0 WD Exty .˛/ is the jJ˛;y j-length of ˛. Fix p0 2 A˛ , and consider the mapping   Z 2 p p z W A˛ 3 p 7! exp J˛;y 2 C: `0 p0 We can see that z is univalent and the image is a round annulus in C with center the origin. In fact, closed trajectories in A˛ are mapped to round circles with center the origin. When p0 is suitably chosen, the image of A˛ under z is a flat annulus Ann.r/ D fr 1=2 < jzj < r 1=2 g with Exty .˛/ D

1 2 D : Mod.A˛ / log r

In this coordinate .A˛ ; z/, the Beltrami differential jJ˛;y j=J˛;y is represented by zN d zN ; z dz and therefore the solution f t of the Beltrami equation (2.25) with q D J˛;y is A˛ 3 z 7! zjzje

2t 1

D z .e

2t C1/=2

zN .e

2t 1/=2

:

The image of Ann.r/ under this quasiconformal mapping is 2t

Ann.r e / D fr e

2t =2

< jzj < r e

2t =2

g

2t

whose modulus is equal to e Mod.A˛ /. On the other hand, Y is obtained by gluing circular arcs in the boundary of Ann.r/ appropriately (cf. Example 2.2). Moreover, since f t is an affine stretch with respect to the jJ˛;y j-metric, the Riemann surface f t .Y / admits a similar (re-)construction using the Jenkins–Strebel differential J˛;R˛;y .t / from the gluing procedure of Y via J˛;y , and the image f t .A˛ / becomes the characteristic annulus of J˛;R˛;y .t/ (cf. [30] and [35]). Therefore we have Ext R˛;y .t/ .˛/ D

1 1 D 2t D e 2t Exty .˛/; Mod.f t .A˛ // e Mod.A˛ /

which is what we wanted. 2.4.3 Kerckhoff’s formula. In [22], Kerckhoff gave the following formula of the Teichmüller distance: 1 Exty1 .˛/ dT .y1 ; y2 / D log sup : (2.28) 2 ˛2 Exty2 .˛/ This formula is now called Kerckhoff’s formula of the Teichmüller distance. For convenience, we shall sketch the proof of this formula. Let y1 ; y2 2 Tg;m . From the

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Hideki Miyachi

distortion property (2.18), we have Exty1 .˛/ 1 log sup  dT .y1 ; y2 /: 2 ˛2 Exty2 .˛/ On the other hand, from (2.27), for any y1 ; y2 2 Tg;m with y1 ¤ y2 , there exists G 2 MF  f0g such that Exty2 .G/ D e 2dT .y1 ;y2 / Exty1 .G/ and dT .y1 ; y2 / 

Exty1 .G/ 1 log : 2 Exty2 .G/

Since the extremal length function is continuous on Tg;m  MF , we get dT .y1 ; y2 / 

1 Exty1 .˛/ log sup ; 2 ˛2 Exty2 .˛/

which implies the equality in (2.28).

3 The Gardiner–Masur compactification 3.1 Definition z GM from Tg;m to R defined by Consider the mapping ˆ 0 z GM W Tg;m 3 y 7! Œ 3 ˛ 7! Exty .˛/1=2  2 R0 : ˆ

(3.1)

Since Exty .˛/ > 0 for all ˛ 2  and y 2 Tg;m , we have a well-defined mapping z GM W Tg;m ! PR0 : ˆGM WD pr B ˆ

(3.2)

Gardiner and Masur observed that ˆGM is an embedding and that its closure is relatively compact (cf. [15]). The relative compactness is deduced from the following observation: When we fix a family fˇj gjND1 of simple closed curves on X which fill up X, for any ˛ 2  there is a constant C.˛/ > 0 such that Exty .˛/  C.˛/  max Extx0 .ˇj / 1j N

for all y 2 Tg;m . Then the relative compactness follows from Tychonoff’s theorem (cf. Theorem 6.1 in [15]). The injectivity of ˆGM follows from Kerckhoff’s formula (2.28). We shall check it briefly. Let y1 ; y2 2 Tg;m with ˆGM .y1 / D ˆGM .y2 /. By definition, there is c > 0

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Chapter 4. Extremal length geometry

such that Exty2 .˛/ D cExty1 .˛/ for all ˛ 2 . Therefore, Exty1 .˛/ 1 1  log c D log sup 2 2 ˛2 Exty2 .˛/ D dT .y1 ; y2 / D dT .y2 ; y1 / 1 Exty2 .˛/ 1 D log c; D log sup 2 2 ˛2 Exty1 .˛/ and hence c D 1 and y1 D y2 (cf. Lemma 6.1 in [15]). The closure cl.Tg;m / of the image of ˆGM is called the Gardiner–Masur compactification of Tg;m and the boundary @GM Tg;m D cl.Tg;m / n ˆGM .Tg;m / is called the Gardiner–Masur boundary of Tg;m . Gardiner and Masur gave the following property P MF  @GM Tg;m

(3.3)

(cf. Theorem 7.1 in [15]). To see this, they checked that for any ˛ 2 , the projective class Œ˛ is in @GM Tg;m by showing that the Teichmüller geodesic ray R˛;x0 terminates at Œ˛ (see also [22]). We will explain (3.3) with an alternative approach in §3.3. First countability. Though this might be well known in general, we provide a simple observation that the Gardiner–Masur closure satisfies the first countability axiom. Let p0 2 cl.Tg;m / and fix a lift pQ0 W  ! R0 of p0 . Take ˛ 2  with pQ0 .˛/ ¤ 0. Notice from the definition that a subset of PR0 consisting of p 2 PR0 with p.˛/ ¤ 0 is open in PR0 . (The condition p.˛/ ¤ 0 makes sense because it depends only on the projective class.) Let Pc ./ be the set of finite subsets of . For > 0 and S 2 Pc ./, we set ˇ

²

ˇ

³

ˇ p.ˇ/ pQ0 .ˇ/ ˇˇ Q z < ; p.˛/ ¤ 0; ˇ 2 S ;  U.SI

/ D p 2 cl.Tg;m / j ˇˇ p.˛/ Q pQ0 .˛/ ˇ

where pQ is a lift of p. Then z fU.SI 1=n/ j S 2 Pc ./; n 2 Ng is a countable basis of neighborhoods of p0 , since Pc ./ is countable.

3.2 The functions Ep Fix a basepoint x0 D .X; id/ 2 Tg;m . For y 2 Tg;m , we define a continuous function on MF by ² ³ Exty .F / 1=2 Ey .F / D D e dT .x0 ;y/ Exty .F /1=2 (3.4) Ky where Ky D exp.2dT .x0 ; y//. We devote attention to the family fEy gy2Tg;m of continuous functions on MF .

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3.2.1 The family is normal. In [37], the author observed that the family fEy gy2Tg;m is normal. For the convenience of the reader we sketch the proof. From the distortion property (2.18) of extremal length, we have Ey .F /  Extx0 .F /1=2 for all F 2 MF . This means that the family fEy gy2Tg;m is uniformly bounded on any compact set of MF (cf. §2.3.7). We check the equicontinuity of the family. For G0 ¤ 0 and ı, we define Uı .G0 I H / D fF 2 MF j ji.F; H /  i.G0 ; H /j2 < Extx0 .H /Extx0 .G0 /ıg; Uı .0I H / D fF 2 MF j i.F; H /2 < Extx0 .H /ıg and we set Uı .G0 / D

T H ¤0

Uı .G0 I H /:

T Observe that Uı .G0 / is an open neighborhood of G0 which satisfies ı>0 Uı .G0 / D fG0 g (cf. Lemma 4.1 in [37]; see also §3 in [22]). We can see the equicontinuity by using the system fUı .G0 /gı>0 of neighborhoods of G0 . The proof is rather technical. We sketch the idea: Let G0 2 MF with G0 ¤ 0. For G 2 Uı .G0 / and y 2 Tg;m , we have Exty .G/1=2 Exty .HJG;y /1=2 D i.G; HJG;y / < i.G0 ; HJG;y / C Ext x0 .G0 /1=2 Ext x0 .HJG;y /1=2 ı  Exty .G0 /1=2 Exty .HJG;y /1=2 C Ext x0 .G0 /1=2 Extx0 .HJG;y /1=2 ı: By the quasiconformal distortion property of extremal length, we have Exty .G/1=2  Exty .G0 /1=2 

Extx0 .HJG;y /1=2 Exty .HJG;y /1=2

Extx0 .G0 /1=2 ı

 Ky1=2 Extx0 .G0 /1=2 ı; and hence Ey .G/  Ey .G0 /  Extx0 .G0 /1=2 ı for all G 2 Uı .G0 /. The case where G0 D 0 is deduced in the same way. The reverse direction follows from a similar calculation together with the fact that G0 2 Uı.1ı/ .G/ for G 2 Uı .G0 / with ı < 1 (cf. Lemma 4.2 of [37]). 3.2.2 Functions Ep . From the above observation, when a sequence fyn gn in Tg;m converges to p 2 @GM Tg;m , there is a subsequence fynj gj such that Eynj converges to a continuous function Ep on compact subsets of MF . One can easily see that Ep is non-trivial. Namely,  3 ˛ 7! Ep

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represents p (cf. Lemma 5.1 in [37]; see also (3.8) below). Therefore Ep is determined independently of the choice of fyn gn , up to multiplication by a positive constant. On the other hand, from Minsky’s inequality, max Ey .F / D 1

F 2MF 1

where MF 1 D fF j Ext x0 .F / D 1g. This means that the function Ep above also satisfies (3.5) max Ep .F / D 1 F 2MF 1

since MF 1 is compact. This means that there is no ambiguity in multiplying Ep by a positive constant. Namely, Ep is independent of the choice of sequences fyn gn , it depends only on p. We summarize the following (cf. [37] and [39]). Theorem 3.1 (The functions Ep ). For any p 2 cl.Tg;m /, there is a unique continuous function Ep with the following properties: (a) The projective class of the assignment  3 ˛ 7! Ep .˛/ is equal to p. (b) For any fyn gn  Tg;m converging to p 2 cl.Tg;m /, Eyn converges to Ep uniformly on compact subsets of MF . Applying the diagonal argument, we deduce that the mapping cl.Tg;m /  MF 3 .p; F / 7! Ep .F /

(3.6)

is continuous in two variables. From (2.24), Ep satisfies Ep .F1 C F2 /  Ep .F1 / C Ep .F2 /

(3.7)

for F1 C F2 2 MF . As in the case of the Thurston compactification, we can see the following. (Cf. [39]. Compare with Proposition 1.1 of Exposé 8 in [9].) Proposition 3.2. For p 2 cl.Tg;m /, the following are equivalent. • p 2 @GM Tg;m . • There is G 2 MF  f0g with Ep .G/ D 0. 3.2.3 Geodesics have limits. As we shall discuss in §3.4, Liu and Su [27] observed that the Gardiner–Masur compactification coincides with the horofunction compactification with respect to the Teichmüller distance. As a corollary, they deduced that any Teichmüller geodesic ray converges to a point in the Gardiner–Masur compactification (cf. [47]). In this section, we shall give a simple proof of this convergence. It suffices to check that the limit exists for any Teichmüller geodesic ray emanating from x0 . Let ŒG 2 P MF . We set y t D RG;x0 .t / for t  0. Let t; s  0 with s  t . For any F 2 MF with F ¤ 0, from Kerckhoff’s formula, we have Exty t .˛/ Exty t .F /  sup D e 2dT .y t ;ys / D e 2.t s/ ; Extys .F / ˛ Extys .˛/

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and hence Ey t .F /  Eys .F /: This means that for all ˛ 2 , Œ0; 1/ 3 t 7! Ey t .˛/ is a non-increasing function and has limit E1 .˛/ as t ! 1 for any ˛ 2 . Let ˛ 2  with i.˛; G/ ¤ 0. Then, from (2.27), 0 ¤ i.˛; G/  Exty t .˛/1=2 Exty t .G/1=2 D Ey t .˛/Ext x0 .G/1=2 :

(3.8)

Hence we have E1 .˛/  i.˛; G/=Extx0 .G/1=2 > 0: E1 is a non-trivial function on  (see also Lemma 5.1 in [37]). This means that ˆGM B RG;x0 .t/ tends to the projective class of E1 as t ! 1. Remark 3.3. By a similar idea, we can see that any two Teichmüller geodesic rays emanating from x0 have different limit points. Namely, for any ŒG1 ; ŒG2  2 P MF with ŒG1  ¤ ŒG2 , lim RG1 ;x0 .t/ ¤ lim RG2 ;x0 .t /: t!1

t !1

See [38].

3.3 Simple closed curves and uniquely ergodic points In this section, we shall give a characterization of functions Ep via their zeros. We also deduce (3.3) from our characterization. A measured foliation G(¤ 0) is said to be uniquely ergodic if i.F; G/ D 0 implies that F and G are projectively equivalent. For G 2 MF , we define the null space of G by N .G/ D fF 2 MF j i.F; G/ D 0g: Notice that when G is either a simple closed curve or is uniquely ergodic, if a measured foliation F satisfies i.F; H / D 0 for all H 2 N .G/, then F and G are projectively equivalent. Since  is dense in P MF in PR0 , (3.3) follows from the following. Theorem 3.4 (Characterization via zero [38]). Let G be either a simple closed curve or a uniquely ergodic measured foliation. Let p 2 cl.Tg;m /. If Ep .H / D 0 for all H 2 N.G/, then i.F; G/ Ep .F / D Extx0 .G/1=2 for F 2 MF .

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3.3.1 Background of the proof. The proof of Theorem 3.4 is accomplished by using Duchin–Leininger–Rafi’s compactification of the space Flatg;m of singular flat structures on a surface of type .g; m/ (cf. §§5 and 6 of [10]). The space Flatg;m of singular flat structures is identified with the quotient bundle 1 1 Qg;m =S1 over Teichmüller space where Qg;m is the unit sphere bundle in the vector 1 i bundle Qg;m ! Tg;m and S D fe j  2 Rg acts fiberwise on Qg;m by multiplication. Duchin, Leininger and Rafi realized the space of singular flat structures in the space of projective classes of geodesic currents, and the compactification is taken in the ambient space (cf. §4.1). They also observed that the boundary of the compactification is realized as the space of mixed structures, and it contains the space P MF of the projective measured foliations. 1 , since the space of geodesic currents is a convex cone, we can define For q 2 Qg;m a geodesic current Lq by the Riemann integral Z 1  V i d: (3.9) Lq D 2 0 e q Namely, for any ˛ 2 , we have 1 i.Lq ; ˛/ D 2

Z 0



i.Vei q ; ˛/d:

Notice that the current (3.9) is well defined for q 2 Flatg;m . In general, for q 2 Qg;m with q ¤ 0, we define Lq D kqk1=2  Lq=kqk : Duchin, Leininger and Rafi showed that `q .˛/ D i.Lq ; ˛/:

(3.10)

This implies that the jqj-length function has a continuous homogeneous extension on Flatg;m  MF (cf. Corollary 28 in [10]). In particular, when we set QG;y D JG;y =kJG;y k for G 2 MF and y 2 Tg;m , from (2.12), we obtain Exty .G/ D `QG;y .G/2 D `JG;y .G/ D i.LJG;y ; G/:

(3.11)

i.LJG;y ; F / D `QG;y .F /2  Exty .F /

(3.12)

In general, since kQG;y k D 1. In (3.12), the equality holds if F is projectively equivalent to G. 3.3.2 Sketch of the proof. Take p 2 cl.Tg;m / and G 2 MF satisfying the assumption in Theorem (3.4). The idea of the proof is to study the asymptotic behavior of geodesic currents LQF;yn =Ky1=2 n for F 2 MF and fyn gn  Tg;m with yn ! p. A key observation is the following: From (3.12) if LQF;yn =Ky1=2 n converges to a geodesic current L1 , we obtain  lim Eyn .H / D Ep .H / i.L1 ; H / D lim `QF;y .H /=Ky1=2 n n!1

n!1

(3.13)

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for F; H 2 MF . From (3.11), the equality in (3.13) holds if H is projectively equivalent to F . Let F 2 MF with Ep .F / ¤ 0. Then, by taking a subsequence if necessary, we can see that LQF;yn =Ky1=2 n converges to a non-zero measured foliation F1 as geodesic currents (cf. Proposition 5 of [38]). From (3.13) and the assumption, we obtain i.F1 ; H /  Ep .H /

(3.14)

for any H 2 MF . Moreover the equality holds if H is projectively equivalent to F . In particular, i.F1 ; F / D Ep .F / ¤ 0 and i.F1 ; H /  Ep .H / D 0 for H 2 N .G/. Therefore, F1 D t1 G for some t1 > 0. Take another F 0 2 MF with Ep .F 0 / ¤ 0. Then, by the same argument as above, one can find t2 > 0 such that (a subsequence of) fLQF;yn =Ky1=2 n gn converges to t2  G as geodesic currents such that i.t2 G; H /  Ep .H /

(3.15)

for any H 2 MF and i.t2 G; F 0 / D Ep .F 0 / ¤ 0. By substituting H D F 0 and F1 D t1 G in (3.14), we get t2  i.G; F 0 / D i.t2 G; F 0 / D Ep .F 0 /  i.t1 G; F 0 / D t1  i.G; F 0 / and hence t2  t1 since i.G; F 0 / ¤ 0. By applying the same argument to (3.15), we conclude t1 D t2 . This means that there is t0 > 0 such that Ep .F / D t0  i.F; G/

(3.16)

for F 2 MF with Ep .F / ¤ 0. We can easily see that (3.16) also holds for all F 2 MF (cf. the proof of Theorem 3 in [38]). By (2.23) and (3.5), we obtain t0 Extx0 .G/1=2 D t0

sup F 2MF 1

i.F; G/ D

sup F 2MF 1

Ep .G/ D 1;

which implies what we wanted.

3.4 Horofunction compactification 3.4.1 Horofunction compactifiation. In this section, we recall the horofunction compactification of a proper metric space defined by Gromov in [16]. A point in the boundary is a kind of a generalization of a Busemann function for a geodesic ray. Let .M; dM / be a proper metric space. Let C.M / be the space of continuous functions on M topologized by the uniform convergence on compact sets. On C.M /, R acts by translations. Let C .M / be the quotient space C.M /=R. We topologize C .M / with the quotient topology. We define a mapping M 3 y 7! ŒM 3 x 7! dM .x; y/ 2 C .M /:

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We can see that the mapping is an embedding and the image is relatively compact. We call the closure the horofunction compactification of M . When we fix a basepoint m0 2 M , the compactification coincides with the closure of the embedding M 3 y 7! ŒM 3 x 7! dM .x; y/  dM .m0 ; y/ 2 C.M /

(3.17)

(cf. [47]). 3.4.2 Liu and Su’s result. In [27], Liu and Su showed that the horofunction compactification of .Tg;m ; dT / canonically coincides with the Gardiner–Masur compactification. Their observation indicates that the Gardiner–Masur compactification is a natural compactification with respect to the Teichmüller distance. Indeed, from their observation, we see that any p 2 cl.Tg;m / corresponds to a horofunction Ep .F / : (3.18) p .x/ D log sup 1=2 F ¤0 Ext x .F / (Notice that our normalization in (3.5) coincides with theirs in [27].) For instance, when y 2 Tg;m , y .x/

Ey .F / 1=2 F ¤0 Ext x .F /

D log sup

e dT .x0 ;y/ Exty .F / Extx .F /1=2 F ¤0

D log sup

D dT .x; y/  dT .x0 ; y/ for x 2 Tg;m . This follows from Kerckhoff’s formula (2.28). Hence, y coincides with the function (3.17). Due to this result, we deduce that any isometry with respect to the Teichmüller distance extends homeomorphically to the Gardiner–Masur compactification (cf. [47]). We will discuss the horofunctions with the (extended) extremal length given in §3.5.2 (cf. Remark 3.8).

3.5 Unification via intersection number 3.5.1 Embedding into a cone. Recall that the Gardiner–Masur closure cl.Tg;m / is in the projective space PR0 . We define cones CGM and TGM in R0 by CGM D pr 1 .cl.Tg;m // [ f0g; TGM D pr 1 .ˆGM .Tg;m // [ f0g: z GM in (3.1) By definition, TGM is a dense subset of CGM . The image of the mapping ˆ z is contained in TGM because ˆGM .y/ D pr B ˆGM .y/ 2 cl.Tg;m / for y 2 Tg;m . Notice that MF  CGM from (3.3). The following is given in [41].

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Theorem 3.5 (Unification). There is a unique continuous function i.; / W CGM  CGM ! R0 with the following properties. (i) For any y 2 Tg;m , the projective class of the function z GM .y/; ˛/  3 ˛ 7! i.ˆ is exactly the image of y under the Gardiner–Masur embedding. Actually, we have z GM .y/; ˛/ D Exty .˛/1=2 i.ˆ for all ˛ 2 . (ii) For a; b 2 CGM , i.a; b/ D i.b; a/. (iii) For a; b 2 CGM and t; s  0, i.t a; sb/ D t s i.a; b/. (iv) For any y; z 2 Tg;m , z GM .y/; ˆ z GM .z// D exp.dT .y; z//: i.ˆ z GM .y// D 1 for y 2 Tg;m . z GM .y/; ˆ In particular, we have i.ˆ (v) For F; G 2 MF  CGM , the value i.F; G/ is equal to the usual geometric intersection number between F and G on MF  MF . Recall that x0 D .X; id/ is the basepoint of Tg;m . We define ‰x0 W Tg;m 3 y 7! Œ 3 ˛ 7! Ey .˛/ 2 R0 :

(3.19)

From the definition, we notice that z GM .y/ D e dT .x0 ;y/  ˆ z GM .y/ ‰x0 .y/ D Ky1=2  ˆ for y 2 Tg;m . From the continuity of (3.6) on cl.Tg;m /, ‰x0 extends continuously on cl.Tg;m /. Our unification is rephrased as follows. Theorem 3.6 (Unification with basepoint). There is a unique continuous function i.; / W CGM  CGM ! R0 independent of the choice of basepoint with the following properties. (i) For any y 2 Tg;m , the projective class of the function  3 ˛ 7! i.‰x0 .y/; ˛/ is exactly the image of y under the Gardiner–Masur embedding. Actually, we have z GM .y/; ˛/ D Ey .˛/ i.ˆ for all ˛ 2 .

Chapter 4. Extremal length geometry

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(ii) For a; b 2 CGM , i.a; b/ D i.b; a/. (iii) For a; b 2 CGM and t; s  0, i.t a; sb/ D t s i.a; b/. (iv) For any y; z 2 Tg;m , i.‰x0 .y/; ‰x0 .z// D exp.2hy j zix0 /; where hy j zix0 is the Gromov product of y and z with basepoint x0 with respect to the Teichmüller distance dT , that is: 1 .dT .x0 ; y/ C dT .x0 ; z/  dT .y; z//: 2 In particular, we have i.‰x0 .y/; ‰x0 .y// D exp.2dT .x0 ; y// for y 2 Tg;m . hy j zix0 D

(v) For F; G 2 MF  CGM , the value i.F; G/ is equal to the usual geometric intersection number between F and G on MF  MF . Theorem 3.6 links an analytic aspect (the Teichmüller distance) and a geometric aspect (the intersection number) of Teichmüller space. Actually we have the following (see also §3.5.5). Corollary 3.7 (Extension of the Gromov product for dT ). For any x0 2 Tg;m , there is a unique continuous function h j ix0 W cl.Tg;m /  cl.Tg;m / ! Œ0; C1 such that (1) for y; z 2 Tg;m , hy j zix0 D

1 .dT .x0 ; y/ C dT .x0 ; z/  dT .y; z//I 2

(2) for ŒF ; ŒG 2 P MF  @GM Tg;m , exp.2hŒF  j ŒGix0 / D

Extx0

i.F; G/ :  Ext x0 .G/1=2

.F /1=2

3.5.2 Background of the proof of unification. The idea of the proof is to recognize our function Ep as an intersection number. From (3.6) and (3.19), we can identify cl.Tg;m / as a subset of CGM . Namely, for any a 2 CGM , there are t  0 and p 2 cl.Tg;m / such that a D t ‰x0 .p/. In this case, we write a D tp for simplicity. Intersection number and extremal length We define the intersection number between tp 2 CGM and F 2 MF  CGM  R0 by ix0 .tp; F / D ix0 .t‰x0 .p/; F / D t Ep .F /:

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Notice that ix0 depends on the choice of basepoint x0 . Then we define the extremal length of a D tp 2 CGM on y 2 Tg;m by ix0 .a; F /2 Ep .F /2 D t 2 sup F ¤0 Exty .F / F ¤0 Exty .F /

Exty .a/ D sup

(3.20)

by imitating the formula (2.23). From the continuity of (3.6), Exty is continuous on CGM . Furthermore, from the distortion property (2.18), we observe that e 2dT .y1 ;y2 / Exty1 .a/  Exty2 .a/  e 2dT .y1 ;y2 / Exty1 .a/

(3.21)

for a 2 CGM and y1 ; y2 2 Tg;m . We can also see that our extremal length coincides with the original intersection number on MF from (2.23) (cf. Theorem 5.1 in [41]). Notice that Ez .F /2 F ¤0 Exty .F /

Exty .tz/ D Exty .t‰x0 .z// D t 2 sup

Ext z .F / F ¤0 Kz Exty .F /

D t 2 sup

(3.22)

D t 2 exp.dT .y; z/  dT .x0 ; z// for t z D t ‰x0 .z/ 2 TGM . In particular, one can see that Extx0 .a/ D 1 if and only if a is on the image of Tg;m under ‰x0 (cf. Proposition 6.1 in [41]). As a corollary, we obtain that for any R > 0, fa 2 CGM j Extx0 .a/  Rg is compact. This means that CGM is  -compact. Though ix0 depends on the basepoint x0 , the extremal length Exty is independent of the choice of x0 . Indeed, take x00 2 Tg;m . Suppose that t ‰x0 .z/ D t 0 ‰x00 .z/ for z 2 Tg;m . Then 0

t e dT .x0 ;z/ Extz .˛/ D t 0 e dT .x0 ;z/ Extz .˛/ for all ˛ 2 . Therefore, t 2 exp.dT .y; z/  dT .x0 ; z// D t 2 exp.dT .x0 ; z// exp.dT .y; z// D .t 0 /2 exp.dT .x00 ; z// exp.dT .y; z// D .t 0 /2 exp.dT .y; z/  dT .x00 ; z//: Since TGM is dense in CGM , from (3.22), we deduce that the extremal length (3.20) is independent of the choice of basepoints. Remark 3.8. From (3.18) and (3.22), we obtain a geometric representation of the horofunction y : y .x/

for z 2 Tg;m .

D log Exty .‰x0 .z//

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227

Functions Ey and the Gromov product From (3.21), we have a continuous function on TGM  CGM by ² ³ Exty .a/ 1=2 D se dT .x0 ;y/ Exty .a/1=2 : Esy .a/ D s Ky Notice that Ey .z/ D s 2 t 2 e dT .x0 ;y/ exp.dT .y; z/  dT .x0 ; z// D s 2 t 2 exp.2hy j zix0 /

(3.23)

for y D t ‰x0 .y/; z D s‰x0 .z/ 2 TGM , s; t > 0 and y; z 2 Tg;m . In particular, for y; z 2 TGM , we have the symmetry Ey .z/ D Ez .y/:

(3.24)

3.5.3 Sketch of the proof of unification. By a construction similar to that in (3.2.1), we can define “nice” neighborhoods of a 2 CGM , and conclude that fEy gy2TGM is a normal family (cf. §§6.2 and 8.1 of [41]). Let z 2 CGM . Take sequences fyn gn and fy0n gn in TGM which converge to z. 0 Suppose that Eyn converges to E1 and Ey0n converges to E1 uniformly on compact 0 sets of CGM . Notice that E1 and E1 are continuous on Tg;m . Then, for w 2 TGM , we have E1 .w/ D lim Eyn .w/ D lim Ew .yn / D Ew .z/: n!1

n!1

Since the right-hand side is independent of the choice of sequences converging to z 0 on CGM . Thus, our function Ez is a and TGM is dense in CGM , we obtain E1 D E1 well-defined extension for z 2 CGM and, we obtain the intersection number i.a; b/ D Ea .b/ for a; b 2 CGM as the extension. Since CGM is  -compact, from the above argument, we see that the intersection number is continuous in two variables. From (3.24), the above intersection number is symmetric since TGM is dense in CGM . Finally, we check that the intersection number is independent of the choice of basepoint x0 by an argument similar to that of extremal length that we gave above (cf. §8.2 of [41]). 3.5.4 Hyperboloid model in extremal length geometry. Theorem 3.5 gives a realization of Teichmüller space in the cone CGM . Indeed, from (iv) of Theorem 3.6, the boundary of CGM is the “light cone” fa 2 CGM j i.a; a/ D 0g

(3.25)

z GM coincides with the “hyperboloid” and the image of ˆ fa 2 CGM j i.a; a/ D 1g:

(3.26)

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Namely, Teichmüller space .Tg;m ; dT / is realized as a hyperboloid model with respect to the quadratic form i.; / in extremal length geometry (cf. Figure 3; see also [41].) z GM the image of ˆ CGM z GM .y/ ˆ

‰x0 .y/ the image of ‰x0 z GM .x0 / D ‰x0 .x0 / ˆ z GM and ‰x0 . Figure 3. Cone CGM and the images of ˆ

This observation makes an analogy with Bonahon’s theory of (re-)producing the Thurston compactification in the space of geodesic currents (cf. [7]; see also §4.1). We now discuss the analogy with Bonahon’s theory more detailed. The hyperbolic metric of Hn is the Riemannian metric defined by restricting the Lorentz form ..u1 ; : : : ; unC1 /; .v1 ; : : : ; vnC1 // D unC1 vnC1 C

n X

ui vi

iD1

to the tangent space of Hn  En;1 . Thus, the length of a path  in Hn is the superior limit of n X p .tk  tk1 ; tk  tk1 / (3.27) iD1

as the mesh of the subdivision ft0 ; t1 ; : : : ; tn g of  tends to 0. The hyperbolic distance between two points in Hn is the infimum of the length of all paths connecting the two points (Notice that the sign of our Lorentz form is different from that given in Chapter I.2 of [8].) In [7], Bonahon gave an embedding of Teichmüller space into the space of geodesic currents, and observed that the image looks like a hyperboloid in terms of the intersection number function on the space of geodesic currents. He also observed that the induced distance (in an infinitesimal sense (3.27)) from the intersection number, which is a quadratic form on the space of geodesic currents, coincides with the Weil– Petersson distance up to multiplying by a positive constant. In our case, unfortunately, we do not re-produce the Teichmüller distance by the above infinitesimal method. In fact, from (iv) of Theorem 3.5, the intersection number is not tangent at the second order with respect to the Teichmüller distance in a usual

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sense. Hence, the superior limit of the sum (3.27) for any rectifiable path in terms of the Teichmüller distance diverges. To make a connection between hyperbolic space and Teichmüller space via our intersection number, we recall the following global expression cosh.dHn .t1 ; t2 // D .t1 ; t2 /

(3.28)

of the hyperbolic distance where t1 ; t2 2 Hn  En;1 (cf. Proposition 2.6 of Chapter I.2 in [8]). We now introduce the modified Teichmüller distance ıT defined by p (3.29) ıT .x; y/ D log.e dT .x;y/ C e 2dT .x;y/  1/: The modified Teichmüller distance satisfies e dT .x;y/ D cosh.ıT .x; y// for all x; y 2 Tg;m . Then we have z GM .y/; ˆ z GM .z// cosh.ıT .x; y// D i.ˆ for y; z 2 Tg;m by (iv) of Theorem 3.5. Thus, we may think of the modified Teichmüller distance as induced (in a global sense) from our intersection number function. The modified Teichmüller distance is not comparable with the Teichmüller distance in an infinitesimal sense. However, we can easily see dT .x; y/ < ıT .x; y/ < dT .x; y/ C log 2 for all x; y 2 Tg;m . Remark 3.9. The functionp(3.29) is actually a distance function on Tg;m . Indeed, the function f .x/ D log.x C x 2  1/ is an increasing function on fx  1g satisfying f .xy/  f .x/ C f .y/ for x; y  1. 3.5.5 Isometry group of Teichmüller space. It is known that with few exceptions, the isometry group of Teichmüller space is canonically isomorphic to the extended mapping class group (cf. Royden [48], Earle–Kra [11], Ivanov [20], and Earle– Markovic [12]). From our unification, we can also deduce a characterization of the isometry group of Teichmüller space. We give here the idea. The complete proof is given in [41]. The idea is the following. Let ! be an isometry on Tg;m . Liu and Su’s result tells us that ! extends homeomorphically to cl.Tg;m / (cf. §3.4 and [47]). From Theorem 3.6, we have e dT .x0 ;!.x0 // i.‰x0 .p/; ‰x0 .q//  i.‰x0 .!.p//; ‰x0 .!.q///  e dT .x0 ;!.x0 // i.‰x0 .p/; ‰x0 .q//

(3.30)

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for p; q 2 cl.Tg;m /. In particular, we obtain that i.‰x0 .p/; ‰x0 .q// D 0 if and only if i.‰x0 .!.p//; ‰x0 .!.q/// D 0. Notice that simple closed curves and uniquely ergodic measured foliations are characterized by their null spaces (cf. §3.3). By applying Theorem 3.4 and (3.30), we see that ! preserves the set of uniquely ergodic boundary points in @GM Tg;m , and hence !.P MF / D P MF (§9.3.2 in [41]). By applying (3.30) again, we can see that !./ D  by analyzing the correspondence of null spaces. This implies that ! induces an automorphism of the complex of curves (cf. §9.4 in [41]). By results by Ivanov, Korkmaz and Luo, if .g; m/ ¤ .1; 2/, the above automorphism is induced by a mapping class (cf. [19], [25] and [28]). Finally, we check that ! coincides with the isometric action induced from the mapping class on Tg;m (cf. §9.5.2 in [41]). Remark 3.10. The argument allows us to treat coarse mappings on Teichmüller space via intersection number. In fact, we consider the following mapping ! W Tg;m ! Tg;m : • There are D1 ; D2 > 0 such that 1 hy j zix  D2  h!.y/ j !.z/i!.x/  D1 hy j zix C D2 D1 for any x; y; z 2 Tg;m . • ! admits a quasi-inverse ! 0 W Tg;m ! Tg;m . Namely, there is D3 > 0 such that sup fdT .x; ! B ! 0 .x//; dT .x; ! 0 B !.x//g < D3 : x2Tg;m

• ! and ! 0 extend continuously to the Gardiner–Masur compactification. A mapping ! satisfying the above three properties canonically induces an isomorphism of the complex of curves. In fact, the extension of ! to P MF  @GM Tg;m coincides with that of an isometry of Tg;m . See [41] for details.

4 Appendix on geodesic currents The reader may refer to Bonahon’s papers [6] and [7].

4.1 Geodesic currents We consider a purely hyperbolic Fuchsian group acting on H such that H= is homeomorphic to X. We fix an identification between X and H= . A geodesic current X is a -equivariant Radon measure on ƒ  ƒ  diag. We denote by G C g;m the set of geodesic currents on X. The set G C g;m is topologized by the weak* topology. The space MF of measured foliations is embedded in G C g;m . Indeed, any

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simple closed curve is identified with the infinite sum of Dirac measures with support at the endpoints of all lifts of the geodesic representative. The intersection number function on MF  MF extends continuously on the product G C g;m  G C g;m (cf. §IV in [6]). Moreover, ˛ 2 G C g;m is in MF if and only if i.˛; ˛/ D 0 (cf. Proposition 4.8 in [6]).

4.2 Bonahon’s embedding We now suppose that g  2 and m D 0. Bonahon observed that Teichmüller space Tg D Tg;0 is embedded in G C g D G C g;0 by assigning to a hyperbolic structure the projective class of its Liouville current, where the Liouville current for y 2 Tg;m is defined as the pull-back measure of the Isom.H2 /-invariant measure on S1 S1 diag by the diagonal action of the equivariant homeomorphism on S1 induced from the marking of y (cf. §2 and Corollary 16 in [7]). We denote by Ly the Liouville current for y 2 Tg . Then the intersection number function satisfies i.Ly ; Ly / D 2 2 .g  1/ D  2 j.X /j; i.Ly ; ˛/ D hly .˛/ for ˛ 2  and y 2 Tg (cf. Propositions 14 and 15 in [7]). In particular, the closure of the image of Bonahon’s embedding Tg 3 y 7! Ly 2 G C g  f0g ! P G C g D .G C g  f0g/=RC

(4.1)

is canonically identified with the Thurston compactification (cf. Theorem 18 and §4 in [7]).

4.3 Hyperboloid model in hyperbolic geometry This realization gives the hyperboloid model of Teichmüller space Tg as follows (cf. §4 in [7]): Let CBO  G C g be the pre-image of the image of Bonahon’s embedding under the projection G C g  f0g ! P G C g . Then the image of the mapping Tg 3 y 7! Ly 2 CBO  G C g looks like a hyperboloid f˛ 2 CBO j i.˛; ˛/ D 2 2 .g  1/g with respect to the quadratic form i.; / and the boundary of the cone CBO is the “light cone” MF D f˛ 2 G C g j i.˛; ˛/ D 0g:

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[41] H. Miyachi, Unification of the extremal length geometry on Teichmüller space via intersection number. Math. Z., to appear. [42] H. Miyachi, A differential formula for extremal length. In In the tradition of Ahlfors-Bers, VI, Contemp. Math 590, Amer. Math. Soc., Providence, RI, 2013, 137–152. [43] Y. Minsky, Teichmüller geodesics and ends of hyperbolic 3-manifolds. Topology 32 (1993), 625–647. [44] Y. Minsky, Extremal length extimates and product regions in the Teichmüller space. Duke Math. J. 83 (1996) 249–286. [45] K. Rafi, A combinatorial model for the Teichmüller metric. Geom. funct. anal. 17 (2007), 936–959. [46] M. Rees, An alternative approach to the ergodic theory of measured foliations on surfaces. Ergodic Theory Dynam. Systems 1 (1981), no. 4, 461–488 (1982). [47] M. Rieffel, Group C  -algebra as compact quantum metric spaces. Doc. Math. 7 (2002), 605–651. [48] H. Royden, Automorphisms and isometries of Teichmüller space. In Advances in the theory of Riemann surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Stud. 66, Princeton University Press, Princeten, NJ, 1971, 369–383. [49] K. Strebel, Quadratic differentials. Ergeb. Math. Grenzgeb. (3) 5, Springer, Berlin 1984. [50] O. Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale. Abh. Preuß. Akad. Wiss., math.-naturw. Kl. 1939 (1940), No. 22. [51] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. 19 (1988), 417-431. [52] W. P. Thurston, Three-dimensional geometry and topology. Volume 1, Princeton University Press, Princeten, NJ, 1997. [53] W. Veech, Gauss measures for transformations on the space of interval exchange maps. Ann of Math. 115 (1982), 201–242. [54] C. Walsh, The horoboundary and isometry group of Thurston’s Lipschitz metric. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume IV, EMS Publishing House, Zürich 2014, 327–353. [55] M. Wolf, On realizing measured foliations via quadratic differentials of harmonic maps to R-trees. J. d’Analyse, 68 (1996), 107–120.

Chapter 5

Compactifications of Teichmüller spaces Ken’ichi Ohshika

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thurston compactification . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definition of the Thurston compactification . . . . . . . . . . . . 2.2 Earthquakes and the Thurston compactification . . . . . . . . . . 2.3 Actions of surface groups on R-trees . . . . . . . . . . . . . . . 3 Teichmüller compactification . . . . . . . . . . . . . . . . . . . . . . 3.1 Definition of the Teichmüller compactification . . . . . . . . . . 3.2 Non-extendability of actions of mapping class groups: the case of the Teichmüller boundary . . . . . . . . . . . . . . . . . . . . . 4 The Bers compactification . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Definition of the Bers compactification . . . . . . . . . . . . . . 4.2 End invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Non-extendability of actions of mapping class groups: the case of the Bers boundary . . . . . . . . . . . . . . . . . . . . . . . . . 5 The rigidity of mapping class group actions . . . . . . . . . . . . . . . 5.1 Two kinds of reduced spaces . . . . . . . . . . . . . . . . . . . . 5.2 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction Teichmüller spaces are often compared to hyperbolic spaces. Hyperbolic n-space has a natural compactification whose boundary is identified with an .n  1/-dimensional sphere at infinity. The metric structure on hyperbolic space is reflected on the sphere at infinity as a conformal structure. A lot of properties of the hyperbolic space can be understood by studying this sphere at infinity with its conformal structure. Therefore, it is natural to expect that there should be a compactification of Teichmüller space which is useful for studying the structure of Teichmüller space. Teichmüller space first appeared in the world of mathematics as a space of quasiconformal deformations of a Riemann surface. By work of Teichmüller himself, it is

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known that between two homeomorphic closed Riemann surfaces, there is a unique quasi-conformal map, called the Teichmüller map, which attains the infimum of the maximal dilatations. Teichmüller maps determine a metric on Teichmüller space, which is called the Teichmüller metric. The Teichmüller compactification is defined to be the “ray” compactification with respect to the Teichmüller geodesic rays. We regard Teichmüller space as a cone consisting of geodesic rays starting from some fixed basepoint, and by adding a point at infinity to each geodesic ray, we get the Teichmüller compactification. This compactification turns out to be dependent on the basepoint by work of Kerckhoff, as we shall explain in §3.2. This dependence on basepoints makes it impossible for the mapping class group action to extend continuously to this compactification. Bers considered an analytic embedding of Teichmüller space into the space of holomorphic quadratic differentials using simultaneous uniformisation and Schwarzian derivatives. This embedding is also regarded as sending Teichmüller space into the space of quasi-Fuchsian groups as a slice over some fixed basepoint in Teichmüller space. Taking the closure of the image of this embedding, we get the Bers compactification. This compactification is closely related to the theory of Kleinian groups and is a very interesting object to study. Still, as in the case of the Teichmüller compactification, this compactification depends on the basepoint. In fact, Kerckhoff and Thurston proved that there is no continuous extension of the action of the mapping class group to the Bers compactification, as we shall explain in §4.3. In contrast to these two kinds of compactification, the compactification introduced by Thurston has an action of the mapping class group which is a continuous extension of its action on Teichmüller space. This compactification is obtained by embedding Teichmüller space into an infinite-dimensional real projective space by considering the lengths of simple closed curves with respect to hyperbolic metrics. We shall give two ways to interpret this compactification: one using deformations of hyperbolic structures called earthquakes, and the other using Gromov limits and actions of surface groups on R-tress. Although there is no natural extension of the mapping class group to the Teichmüller compactification and the Bers compactifications, by considering their reduced boundaries, which are quotient spaces of the original boundaries, we can extend the action of the mapping class group continuously. This fact pinpoints what is the cause of the non-extendability of the mapping class group action. These reduced boundaries are non-Hausdorff and are hard to understand intuitively. Still, in the final section, we shall show that the entire auto-homeomorphism groups of such reduced boundaries coincide with the extended mapping class group. Throughout this chapter, surfaces are assumed to be closed and orientable, and to have genus greater than 1. Although all the theories can be generalised to the case when surfaces have punctures or are non-orientable, our assumption here makes the description substantially simpler. The generalisation to the case when surfaces have infinite genus are more problematic and needs a separate treatment.

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Acknowledgement. The author would like to express his gratitude to Athanase Papadopoulos for his valuable comments on the first version of the manuscript. This work was partially supported by JSPS grant-in-aid for scientific research (A) 22244005.

2 Thurston compactification 2.1 Definition of the Thurston compactification Although chronologically the Thurston compactification was invented after those of Bers and Teichmüller, we shall explain it first here because the other compactifications can be understood better if compared with this one. Let S be an oriented closed surface of genus g  2, and T .S / its Teichmüller space. A point of T .S/ can be regarded as a marked hyperbolic structure on S , determined up to isotopy. For m 2 T .S/ and a non-contractible simple closed curve c on S , we denote by lengthm .c/ the length of the closed geodesic homotopic to c, with respect to m regarded as a hyperbolic structure. Let  be the set of isotopy classes of non-contractible simple closed curves. We define RÃC to be the set of nonnegative functions on , endowed with the weak topology. By P RÃC , we denote its projectivisation, that is, the quotient space of RÃC n f0g obtained by collapsing each ray to a point, and let  W RÃC n f0g ! P RÃC be the projection. Now, we can consider an embedding S of T .S/ into RÃC by sending m 2 T .S / to .lengthm .s//s2Ã . Using elementary hyperbolic geometry, it is not so hard to see that the map  B S is injective. Thurston proved that the image of  B S is relatively compact and that its boundary can be interpreted geometrically. To understand the boundary of the image of  B S , we need to introduce the notion of measured and projective lamination on the surface S. We fix some hyperbolic metric on S . A geodesic lamination  on S is a (possibly empty) closed set consisting of disjoint simple geodesics. Geodesics constituting  are called leaves of . A measure defined on arcs transverse to leaves of  is called a transverse measure if it is invariant by homotopic moves of arcs along leaves. A geodesic lamination equipped with a transverse measure is called a measured lamination. We always assume that the support of the measure is the entire lamination. A simple closed geodesic with a positive weight is an example of a measured lamination. In fact, as we shall see later, weighted simple closed geodesics are dense in ML.S /. The set of all measured laminations on S is denoted by ML.S/. Its projectivisation .ML.S / n fg/=RC is called the projective lamination space, and denoted by P ML.S /. Its elements are called projective laminations. Here, fg denotes the empty lamination. We define a map ml W ML.S/ ! RÃC by ml ./.s/ D i.s; / for any  2 ML.S / and s 2 . Here i.s; / means the infimum of the measure of simple closed curves isotopic to s with respect to the transverse measure of . It is fairly easy to see that this

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number is actually realised by the transverse measure of the closed geodesic isotopic to s. By projectivising the map ml , we get a map pl W P ML.S / ! P RÃC . Thurston proved in the late 1970s the following remarkable theorem. Theorem 2.1 (Thurston). The image of  B S is relatively compact in P RÃC . Its boundary coincides with pl .P ML.S//. The closure  B S .T .S // [ pl .P ML.S // is homeomorphic to a sphere of dimension 6g  7. By this theorem, we regard P ML.S/ as the boundary of T .S / calling it the Thurston boundary, and T .S/ [ P ML.S/ as a compactification of T .S / which we call the Thurston compactification. Since the mapping class group acts continuously on P RÃC and this action is compatible with the action on T .S/, as an immediate corollary to this theorem, we get the following. Corollary 2.2. The action of the mapping class group on T .S / extends continuously to the Thurston compactification T .S/ [ P ML.S /. Thurston’s original proof of these results can be found in detail in the book by Fathi–Laudenbach–Poénaru [7]. In the following subsections, we shall give alternative approaches in two ways, the first using earthquakes and the second using R-trees. The fact that ML.S/ is homeomorphic to R6g6 , hence P ML.S / is homeomorphic to S 6g7 , is proved independently of the compactness of T .S / [ P ML.S /. The former fact can be proved either by using the correspondence between measured foliations and holomorphic quadratic differentials as in Hubbard–Masur [8], which we touch upon later, or by a topological argument as in Fathi–Laudenbach–Poénaru [7]. We shall not explain this part of the proof, and concentrate on compactness. One of the important applications of Corollary 2.2 is a classification of mapping classes. There are three types of mapping classes in this classification: periodic classes, reducible classes and pseudo-Anosov classes. The types are determined by how the class acts on T .S/ [ P ML.S/. A periodic class has a fixed point in T .S /, which implies that it is represented by a periodic diffeomorphism. A reducible class has a fixed point in P ML.S/ which is represented by a multi-curve. This means that it is represented by a diffeomorphism which fixes set-wise some essential multicurve on S . A pseudo-Anosov class has exactly two fixed points in P ML.S /, both of which are laminations without compact leaves. Such a class has an invariant axis in the Teichmüller space which is a geodesic with respect to the Teichmüller metric, which we shall explain in §3.

2.2 Earthquakes and the Thurston compactification As mentioned above, we shall give two ways to prove the compactness part of Theorem 2.1. The first way uses the earthquake deformation which was introduced by Thurston. (Cf. Thurston [29] and Kerckhoff [12].)

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Recall that we can regard positively weighted simple closed geodesics as measured laminations. The following fact, which was first proved by Thurston when he introduced the notion of measured lamination, is very important to understand measured lamination space. There is a combinatorial object called a train track which serves as something like a coordinate neighbourhood for measure lamination space. A train track is a C 1 -graph embedded in S. The edges are called branches, and vertices are called switches. Branches are tangent to each other at switches, and no switch is a stop: it has always branches in both sides. We assume that each component of the complement of a train track is neither a disc nor a monogon nor a bigon nor an annulus without corners. For more details on train tracks, we refer the reader to Casson–Bleiler [4] and Penner–Harer [25]. We observe that for a geodesic lamination  on S , every component of S n  is the interior of a complete hyperbolic surface with geodesic boundary (possibly with ideal vertices on the boundary.) Then we can construct a train track  and its tied neighbourhood N (see Papadopoulos [22] or Penner–Harer [25]) containing  in such a way that every leaf of  is transverse to the ties, by taking a nice regular neighbourhood of . We say that  carries  in this situation. If  is a measured lamination, then its transverse measure gives a weight system on the branches of  which satisfies the “switch condition”, i.e. the condition that the sum of the incoming weights is equal to that of the outgoing weights at every switch. (From now on we always assume weight systems to satisfy the switch condition.) Conversely, any weight system on a train track gives rise to a measured lamination. Using train tracks, we can show the following density of weighted simple closed curves in ML.S /. Lemma 2.3. The set of positively weighted simple closed geodesics is dense in ML.S /. Sketch of proof. We shall sketch a proof of this lemma based on the technique of train track approximation. We choose a train track  carrying a measured lamination  to be “admissible” in the terminology of Papadopoulos [22], or equivalently “recurrent” in the terminology of Penner–Harer [25], by extending and modifying the original train track if necessary. We are not going to define the admissibility or the recurrence formally here, but it was proved in [22] that if a train track is admissible, then there is a one-to-one correspondence between the weight systems and the measured laminations carried by the train track. Also, for an admissible train track, if a sequence fwi g of weight systems converges to a weight system w1 , then the laminations corresponding to wi converge to that corresponding to w1 , which was also proved in [22]. It is easy to see that if each coordinate of a weight system is rational, then the corresponding measured lamination is a rationally weighted multi-curve. Therefore we see that  is a limit of a sequence of rationally weighted multi-curves in ML.S /. A rationally weighted multi-curve can be approximated in P ML.S / by weighted simple closed geodesics

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by making them spiral around each component of the multi-curve proportionally to the weights. This shows that every measured lamination is a limit of weighted simple closed geodesics in ML.S/. The earthquake deformation is defined to be a limit of Fenchel–Nielsen twists around simple closed geodesics. Let  be a measured lamination, and fri ci g a sequence of weighted simple closed geodesics converging to  in ML.S /. Let i be the Fenchel– Nielsen twist around ci with amount of twisting ri length.ci /, i.e. we cut S along ci and paste back the two boundaries after twisting one of them by the amount ri length.ci / to the right. This deformation is regarded as a homeomorphism from T .S / to T .S /. The following lemma due to Thurston guarantees the existence of earthquake deformations. See Kerckhoff [12] for the proof. Lemma 2.4. Let i W T .S/ ! T .S/ be a Fenchel–Nielsen deformation associated to a weighted simple closed geodesic ri ci such that fri ci g converges to . Then i converges to a homeomorphism E W T .S/ ! T .S / uniformly on every compact set. We call this homeomorphism E the earthquake deformation along . The fundamental result on earthquake deformations is the following theorem due to Thurston stating that the earthquake deformation is bijective with respect to the parameter  ranging in ML.S/. Theorem 2.5 (Thurston). For any two distinct points g1 , g2 in T .S /, there is a unique measured lamination  such that g2 D E .g1 /. It is easy to verify that the earthquake deformation E has the following properties. Lemma 2.6. For any g 2 T .S/ and measured laminations ;  with i.; / D 0, we have lengthE .g/ ./ D lengthg ./. In particular, lengthE .g/ ./ D lengthg ./. On the other hand, for a measured lamination  intersecting  essentially, Kerckhoff showed the following variation formula. Lemma 2.7 (Kerckhoff [12]). We have @lengthE t  .g/ ./ @t

Z Z D

cos d d:

Here  denotes the angle formed by  and  at each point of  \ . We integrate cos  with the product of the transverse measures of ;  in the right hand side. The orientation of S determines the sign of the cosine. Lemma 2.6 can be regarded as a special case of Lemma 2.7 when  and  are disjoint.

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Proposition 2.8. Let  2 ML.S/ be a measured laminations and  a simple closed geodesic with i.; / > 0. Then for any > 0, there is a number T0 and a neighbourhood U of  such that for any 2 U and t > T0 , we have ˇ ˇ ˇ @lengthE t  .g0 / ./ ˇ ˇ  i. ; /ˇˇ < : ˇ

@t

Sketch of proof. The right hand side of the equation in Lemma 2.7 converges to i.; / if cos  converges to 1 uniformly on  \ . Therefore, what we have to show is that inf \ cos  converges locally uniformly to 1 on ML.S / as t ! 1. We shall see how inf \ cos  converges to 1 for fixed  and  assuming that  is also a simple closed geodesic. We consider the universal covering p W H2 ! S and Q Q in H2 of ; . The Fenchel–Nielsen twist along  is interpreted the preimages ; using a conjugate of the covering translation group of 1 .S / by a map defined on H2 as follows. Q and translating it by an isometry of H2 , we can Taking a component U0 of H2 n , Q Let Q 0 be assume that the origin 0 of the Poincaré disc model of H2 lies on U0 \ . the component of Q containing 0, and g an element of PSL2 R representing the free homotopy class of  whose axis is Q 0 . Since i.; / > 0, the point g.0/ is contained in a component U1 of H2 n Q distinct from U0 . Similarly, for any m  2, the point g m .0/ is contained in another component Um of H2 n Q which lies on the other side of U1 from U0 . We lift the Fenchel–Nielsen twist along  of length t length./ so that U0 is fixed, and denote it by  t . Then the action of  t on a point x 2 Um is the composition of length t length./ translations along all the leaves of Q crossing the geodesic segment on 0 between 0 and g m .0/. We denote the image of Um under this composition of translations by Um0 Let ag and bg be the absorbing and repelling endpoints of Q 0 under the action of g. The observation above shows that after performing the Fenchel–Nielsen twist along  of length t length./, the action of g is transformed to have the absorbing fixed point T 2 1 x0 in 1 mD1 Um , where x denotes the closure in H [ S1 . Similarly, we Tcan define 0 x0 the regions Um and Um for m  1, and the repelling fixed point lies in 1 mD1 Um . Let C and  be the leaves of Q which Q 0 intersects for the first time starting from 0 in the positive direction and in the negative direction respectively. Let aC be the endpoint of C in the direction of the lift of the Fenchel–Nielsen twist, and a that of  also in the direction of the twist. Now, as t ! 1, the region Um0 with m  2 converges to aC , and Um0 with m  2 converges to a . This shows that the angle formed by 0 and ˙ converges Q we see that the angle to 0. Moving the basepoint 0 to another component of H2 n , at each point of  \  goes to 0, and the convergence is uniform by the compactness of  and . The same argument applies even when  is not a simple closed geodesic, using the fact that the angle with  varies continuously on . Also, the above convergence can be made locally uniform as  varies. This completes the sketch of the proof of our proposition.

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As a corollary of this proposition, we get the following. Corollary 2.9. Let fi g be a sequence of measured laminations converging to 0 , and fti g a sequence of positive real numbers going to 1. Then for any g 2 T .S / and s1 ; s2 2 , we have lengthE t  .g/ .s1 / i.s1 ; 0 / i i D lim i!1 lengthE .s / i.s 2 ; 0 / t  .g/ 2 i i

provided that i.s2 ; 0 / > 0. Proof. Since this is obvious when i.s1 ; 0 / D 0 by Lemma 2.6, we can assume that i.s1 ; 0 / > 0. Then, Lemma 2.7 and the argument in the proof of Proposition 2.8 show that lengthE t  .g/ .s2 / goes to 1 as t ! 1 . Now our corollary is obtained i from Proposition 2.8 by applying de l’Hôpital’s rule. Now, we can start to show how we get Thurston’s compactification using earthquakes. Recall that we embedded T .S/ into P RÃC by  B S . Consider a sequence of points fgi g in T .S/ which does not have a convergent subsequence within T .S /. By fixing some hyperbolic metric g on S, we define the unit sphere U ML.S / of ML.S / to be f 2 ML.S/ j lengthg ./ D 1g. Evidently U ML.S / is homeomorphic to P ML.S/ under jU ML.S/. Then, by Theorem 2.5, there is a sequence i 2 U ML.S/ such that gi D E ti i .g/ with ti ! 1. Since U ML.S / is compact, we can take a subsequence so that fi g converges to a measured lamination  2 U ML.S/. Then, by Corollary 2.9,  B S .gi / converges to pl .Œ/. This shows that S .T .S // [ pl .P ML.S// is compact.

2.3 Actions of surface groups on R-trees We next turn to another way to show that T .S / is compactified by attaching P ML.S / as its boundary, which involves actions of 1 .S / on R-trees. Recall that an R-tree is a geodesic space in which for any two points, there is a unique simple arc (up to changing parametrisation) connecting them. The following was first proved by Morgan–Shalen [18]. More geometric proofs were given by Paulin [24] and Bestvina [2]. Theorem 2.10. Let fgi g be a sequence of points in T .S /. Then after taking a subsequence, there is an action  of 1 .S/ on an R-tree T such that for any s1 ; s2 2 , we have lengthgi .s1 / trlength..s1 // D lim ; i !1 trlength..s2 // lengthgi .s2 / where trlength..s// denotes the translation length of the conjugacy class of 1 .S / under the action of , and we define the fraction to be 1 when the denominator is 0.

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The proof of Morgan–Shalen is based on their general theory of compactifying character varieties using the space of valuations. Here, we shall outline a more geometric approach following Paulin and Bestvina. For each i, the hyperbolic structure gi determines an isometric action i of 1 .S / on H2 , which is unique up to conjugation by elements of PSL2 R. We fix a generator : : ; p g of 1 .S/. We can conjugate i so that the system f 1 ; :P P origin 0 attains the minimum of pkD1 d.x; i . k /.x// for x 2 H2 . Let us denote pkD1 d.0; i . k /.0// by Li . Then, it can be proved that trlength i . k /=Li is bounded for every k D 1; : : : ; p, and that there is an R-tree T on which 1 .S / acts by isometries in such a way that the translation length of k is equal to limi!1 trlength i . k /=Li . This R-tree T and the action of 1 .S/ is nothing but the Gromov–Hausdorff limit of the action of 1 .S/ by i on H2 rescaled by the factor L1 i . Moreover, by applying Jørgensen’s inequality, it can be shown that the stabiliser of every non-trivial arc in T is abelian. For such actions on R-trees, Skora [26] proved the following remarkable theorem. Theorem 2.11 (Skora). Let T be an R-tree and W 1 .S / ! Isom.T / an isometric action such that every non-trivial arc in T has abelian stabiliser. Then there is a measured lamination  to which is dual. We now explain what we mean by an action on an R-tree dual to a measured lamination. Let  be a measured lamination on the surface S. Let F be a measured foliation corresponding to , i.e. we have i.F ; c/ D i.; c/ for every simple closed curve c on S . Consider the universal covering p W H2 ! S , and lift F to a measured foliation Fz on H2 . We now consider the leaf space T of Fz and endow T with a metric by defining the distance x; y 2 T represented by leaves `x ; `y to be the infimum of measures (with respect to Fz ) of arcs with endpoints lying on `x and `y respectively. It is easy to check that T is an R-tree. Since 1 .S / acts on H2 via taking leaves to leaves and preserving the transverse measure of Fz , it acts on T by isometries. This action of 1 .S/ on T is said to be dual to . In this situation, for any two simple closed curves s1 ; s2 , we have trlength . .s1 // i.s1 ; / D : i.s2 ; / trlength . .s2 // Together with Theorem 2.10, this implies that if we embed T .S / into P RÃC by  B S , then any sequence in T .S/ which does not stay in a compact set has a subsequence converging to a point in pl .P ML.S//. This completes an alternative proof of Thurston’s compactification using R-tree actions.

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3 Teichmüller compactification 3.1 Definition of the Teichmüller compactification The Teichmüller compactification is defined as a ray compactification using Teichmüller geodesic rays. First, we recall that for two points .X; f / and .Y; g/ of Teichmüller space T .S /, their Teichmüller distance is defined to be the infimum of the maximal dilatations of quasi-conformal maps from X to Y homotopic to g Bf 1 . We fix .X; f / as a basepoint. Then it was proved by Teichmüller (and reformulated by Ahlfors and Bers) that for any .Y; g/, there is a holomorphic quadratic differential  on X such N that if we set  D k =jj with k < 1, and solve the Beltrami equation wzN D wz , then we get a quasi-conformal homeomorphism h W X ! Y homotopic to g B f 1 . The above  is called the Beltrami coefficient of the equation. The maximal dilatation 1 of h is equal to K D 1Ckk , and this realises the infimum of the maximal dilata1kk1 tions of quasi-conformal homeomorphisms homotopic to g B f 1 . Such a map is called a Teichmüller map. The Teichmüller distance from .X; f / to .Y; g/ is defined to be 12 log K , and this determines a metric structure on T .S /, which is called the Teichmüller metric. N For t 2 Œ0; 1/ and  D = with a fixed holomorphic quadratic differential , we consider the Beltrami coefficient t on X and the corresponding Teichmüller map h t W X ! Y t which is homotopic to g t B f 1 where g t W S ! Y t is a marking for Y t . Then the family f.Y t ; g t /g constitutes a geodesic ray with respect the Teichmüller metric, which we call a Teichmüller geodesic ray. The Teichmüller compactification of T .S / based at .X; f / is a compactification with respect to Teichmüller geodesic rays: Let .Yi ; gi / be a sequence of points in T .S /, and suppose that dT ..X; f /; .Yi ; gi // goes to 1, where dT denotes the Teichmüller distance, and that .Yi ; gi / is contained in a Teichmüller geodesic ray starting at .X; f / and corresponding to a holomorphic quadratic differential i . We may assume that the norm of i is equal to 1. We say that f.Yi ; gi /g converges to a ray corresponding to a holomorphic quadratic differential 1 (with unit norm) if fi g converges to 1 . Since the space of unit-norm holomorphic quadratic differentials is homeomorphic to S 6g7 , we get a compactification of T .S / whose boundary is homeomorphic to S 6g7 . Moreover there is a natural homeomorphism from this space of unit-norm quadratic differentials to P ML.S / sending a holomorphic quadratic differential to its horizontal foliation as was shown in Hubbard–Masur [8] (see below) and then converting it into a projective lamination.

3.2 Non-extendability of actions of mapping class groups: the case of the Teichmüller boundary Thus we have a compactification of T .S/ whose boundary coincides with the Thurston boundary. Still, as was shown by Kerckhoff, these two compactifications are quite different.

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Theorem 3.1 (Kerckhoff). The action of the mapping class group on T .S / does not extend continuously to the Teichmüller compactification. In particular, the Teichmüller compactification is distinct from the Thurston compactification. Kerckhoff proved this theorem using the fact that the topology of the boundary of the Teichmüller compactification depends on the basepoint .X; f /. For a holomorphic quadratic differential  on a Riemann surface X and a point z 2 X, the tangent vector @ @ v (expressed by the complex coordinates @x ; i @y ) at z is said to be horizontal if .v; v/ > 0 and vertical if .v; v/ < 0. By considering integral curves of horizontal vectors, we have a foliations on X with singularities at zeroes of , which we call the horizontal foliation of  and denote by Fh . The values of  for vertical vectors determine a transverse measure on the horizontal foliation; hence Fh is a measured foliation. Let c be an essential simple closed curve on S . Then, as was shown by Jenkins and Strebel (see e.g. Strebel [27]), there is a holomorphic quadratic differential fX.c/ on X whose horizontal foliation consists of non-singular leaves all homotopic to f .c/ and one or two singular leaves. Here we regard leaves passing through the same singular point as lying on one singular leaf. Moreover if there are two such holomorphic quadratic differentials, then one is obtained by multiplying the other by a positive scalar. Therefore, there is a unique Teichmüller geodesic ray starting from .X; f / corresponding to fX.c/ as above, which is independent of the choice of the holomorphic quadratic differential. We denote such a ray by r.X;f / .c/. Kerckhoff showed that for any two points .X; f / and .X 0 ; f 0 / in T .S /, the second ray r.X 0 ;f 0 / .c/ is convergent to the first r.X;f / .c/, that is, the ray r.X 0 ;f 0 / .c/ converges to a point at infinity corresponding to r.X;f / .c/ when viewed in the Teichmüller compactification based at .X; f /. What is remarkable is that if we consider a multi-curve instead of a simple closed curve c, the rays no longer have this kind of property. Let  be a positively weighted disjoint union of non-parallel simple closed curves, which we call a weighted multicurve in short. In the same way as in the case of a simple closed curve, the work of Jenkins–Strebel shows that there is a holomorphic quadratic differential fX./ whose horizontal foliation consists of non-singular leaves homotopic to components of  and singular leaves, such that the transverse measures of the union of leaves homotopic to components of  are proportional to the weight given to  . As before, if there are two such holomorphic quadratic differentials, one is a positive multiple of the other. As in the case of simple closed curve, we consider a geodesic Teichmüller ray corresponding to fX./ starting at .X; f /, which we denote by r.X;f / . /. Kerckhoff showed that in this setting, it may happen that r.X 0 ;f 0 / ./ is not convergent to r.X;f / . /. In fact, he proved that r.X 0 ;f 0 / ./ is convergent to r.X;f / . / if and only if the two holomorphic 0 quadratic differentials fX. / and fX0 ./ are “modularly equivalent” in the following sense. Since the horizontal foliation of fX. / consists of leaves parallel to components of f . / and singular leaves, by cutting X along singular leaves, we get a union of

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annuli each of which has a core curve homotopic to a component of f ./. For each component c of  (with the weights forgotten), we define wc .fX./ / to be the modulus of the obtained annulus whose core curve is homotopic to c. It should be noted that this number depends not only on the transverse measures of compact leaves, but also their lengths with respect to the singular Euclidean structure determined by the holomorphic 0 quadratic differential. The two holomorphic quadratic differentials fX./ and fX0 . / 0

are said to be modularly equivalent if .wc .fX./ //c is a multiple of .wc .fX0 . / //c . Kerckhoff proved the following lemma. Lemma 3.2. Let  be a weighted multi-curve on S . The geodesic Teichmüller ray r.X 0 ;f 0 / . / is convergent to the ray r.X;f / . / if and only if fX./ is modularly equiv0

alent to fX0 ./ . Now, we shall explain how Kerckhoff proved Theorem 3.1 using Lemma 3.2. First, we note that a mapping class Œh of S acts on T .S / by taking .X; f / to .X; f Bh1 /. Let  be a weighted multi-curve consisting of more than one component. It is fairly easy to find a mapping class Œh such that fX. / and fXBh1 . / are not modularly equivalent. By Lemma 3.2, this implies that r.X;f / ./ and r.X;f Bh1 / . / are not convergent. Take a sequence of simple closed curve fci g converging to  in P ML.S /. Then the rays r.X;f / .ci / converge to r.X;f / ./ uniformly on every compact set. As was remarked before, r.X;f Bh1 / .ci / is convergent to r.X;f Bh1 / .ci /. Therefore, if Œh acted on the Teichmüller compactification, then r.X;f / . / would be convergent to r.X;f Bh1 / . /, which contradicts our choice of h. Therefore, we see that the action of Œh on T .S / cannot extend continuously to the Teichmüller compactification. Kerckhoff also observed that this phenomenon of discontinuity disappears if we take a quotient of the Teichmüller boundary by collapsing holomorphic quadratic differentials whose horizontal foliations are the same forgetting the transverse measures. The quotient space thus obtained is called the unmeasured foliation space. He claimed that the action of the mapping class group on T .S / extends continuously to this quotient space. We can consider the same kind of space for measured lamination space, which is called the unmeasured lamination space. We shall study more this space and the action of the mapping class group on it in the final section.

4 The Bers compactification 4.1 Definition of the Bers compactification The Bers compactification is another compactification of Teichmüller space, which is extremely interesting from the viewpoint of Kleinian groups. By work of Ahlfors and Bers, it is known that for any two points m, n in T .S /, there is a Kleinian group G, which is unique up to conjugacy and is called a quasi-Fuchsian group corresponding

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to .m; n/, with the following properties. The group G is isomorphic to 1 .S / by an isomorphism , and the limit set ƒG of G is a Jordan curve on the Riemann sphere y n ƒG . The conformal structure on C y y Let 1 , 2 be the two discs constituting C C. induces those on 1 and 2 . For 2 , we define its conformal structure to be the one y Then 1 =G and 2 =G obtained by reversing the orientation of that induced by C. are conformally equivalent to .S; m/ and .S; n/ in such a way that the conformal equivalences induce the isomorphism . Let AH.S/ be the space of faithful discrete representations of 1 .S / into PSL2 C modulo conjugacy. Let qf W T .S/  T .S / ! AH.S / be a map taking .m; n/ 2 T .S /  T .S/ to the corresponding quasi-Fuchsian group which is regarded as an element of AH.S/ by using the inverse of the correspondence defined above. It is clear that this map is injective. By the continuity of the solutions of Beltrami equations with respect to Beltrami coefficients, we see that qf is continuous. We call this map qf the Ahlfors–Bers map, and denote its image by QF .S /. Now, we fix one of the two coordinates to be m0 and vary the other coordinate n. Then we get a map from T .S/ to QF .S/  AH.S /, which we call the Bers embedding of T .S /. The image qf .fm0 g  T .S// is called the Bers slice over m0 . The following remarkable result by Bers is one of the pioneering works in the field of Kleinian groups. Theorem 4.1 (Bers). The Bers slice is relatively compact in AH.S / for any m0 2 T .S /. The Kleinian groups lying on the boundary of the Bers slice are b-groups: each such group has a unique invariant component in its domain of discontinuity. Bers proved the relative compactness by giving an upper bound for the translation length of an element in a quasi-Fuchsian group in terms of its (extremal) lengths with respect to the upper and lower conformal structures. This kind of bound is also given using what is called Sullivan’s theorem, whose proof can be found in Epstein–Marden [6]. These imply that any sequence in the Bers slice converges as representations (after taking conjugates and passing to a subsequence). By using Jørgensen’s theorem [10], we see that the limit representation is also discrete and faithful. It is rather easy to see that the limit Kleinian group has an invariant component in its domain of discontinuity: since every group in the Bers slice has such a component and since its quotient by the group has a conformal structure depending only on the slice, these components converge to such a component of the limit group. This component must be unique since otherwise the limit group would be also quasi-Fuchsian. (It needs a bit of argument to show that if fni g is divergent in T .S /, then qf .m0 ; ni / cannot converge to a quasi-Fuchsian group.) The Bers compactification of T .S/ is defined to be the closure of qf .fm0 gT .S // in AH.S /, with T .S/ and qf .fm0 g  T .S // identified. Its boundary is called the Bers boundary of T .S/ based at m0 .

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4.2 End invariants Recent developments in the theory of Kleinian groups, which were started by work of Thurston, make it possible to parametrise the boundary of a Bers slice using an invariant consisting of conformal structures and “ending laminations”. Let  be a b-group with isomorphism W 1 .S/ ! . Let  be the domain of discontinuity of y Since  is a b-group,  has a unique component that is invariant under the  in C. action of , which we denote by 0 . The quotient 0 =  determines a point in T .S / by a marking corresponding to . In the case where  lies on the boundary of a Bers slice qf .fm0 g  T .S//, this point coincides with m0 . We may have that 0 D  , in which case  is said to be totally degenerate. Otherwise, . n 0 /=  determines a conformal structure on some (possibly disconnected) subsurface F of S . Ahlfors’ finiteness theorem implies that each frontier component of F represents a parabolic element of . Now, consider the hyperbolic 3-manifold H3 = . It is known by work of Thurston and Bonahon [28], [3] that H3 = is homeomorphic to S  R. We can embed S into H3 =  as a horizontal surface with respect to the parametrisation S  R which induces the isomorphism between 1 .S/ and 1 .H3 = / D . Since each frontier component of F corresponds to a primitive parabolic element, by Margulis’ lemma, we can take disjoint (open) Z-cusp neighbourhoods each of which has a core curve homotopic to one of the frontier components of F . We denote by M0 the complement of these cusp neighbourhoods in M . The ends of M0 consist of one below S , and the others above S , each of which faces a component of F or a component of S n F . Let † be a component of S n F . Bonahon showed in [3] the following three facts: (1) There is a sequence of essential simple closed curves fci g on † such that the closed geodesics ci homotopic to ci tend to an end of M0 . (By choosing the cusp neighbourhoods referred to above to be sufficiently small, the geodesics ci can be assumed to lie in M0 .) (2) The Hausdorff limit of the ci , regarded as closed geodesics with respect to some fixed hyperbolic metric on †, is a geodesic lamination with unique minimal component , each of whose complementary regions is either simply connected or an annulus containing a component of Fr†. (3) For any other sequence fci0 g on † with the same property (1) as fci g, the Hausdorff limit of the ci0 has a unique minimal component, and it coincides with . The lamination  is called the ending lamination of an end facing †. The end invariant of a b-group  consists of the following three kinds of ingredients. (1) The components of FrF . The union of them is called the parabolic locus. (2) The conformal structures on 0 = (lower conformal structure) and . n 0 /=  (upper conformal structure). (3) The ending laminations lying on components of S n F .

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For more general finitely generated Kleinian groups, we can similarly define the above three elements: parabolic locus, conformal structures at infinity and ending laminations. In the case where the Kleinian groups are freely decomposable, we need some more care to define ending laminations, but we are not going into this, as it does not concern our groups in AH.S/. Thurston’s ending lamination conjecture, which was proved by Minsky [16] and Brock–Canary–Minsky [5], says that for any finitely generated Kleinian group, the homeomorphism type of the corresponding quotient hyperbolic 3-manifold and its end invariant determine the isometry type, that is, if two hyperbolic 3-manifolds are homeomorphic and the homeomorphism preserves the end invariants, then they are isometric. In the case of b-groups, this means that once the surface S is fixed, the end invariant defined above determines the representation up to conjugacy. Therefore, the Bers boundary based at m0 is parametrised by parabolic loci, upper conformal structures and ending laminations.

4.3 Non-extendability of actions of mapping class groups: the case of the Bers boundary As in the case of the Teichmüller compactification, we now turn to the extendability of the mapping class group action. Kerckhoff and Thurston [13] showed the following. Theorem 4.2 (Kerckhoff–Thurston). The action of the mapping class group on T .S / does not extend continuously to the Bers boundary based at any point m0 . In particular, the Bers compactification is distinct from the Thurston compactification. We shall explain how Kerckhoff and Thurston proved this theorem, restricting ourselves to the case when the genus of S is 2. The line of the argument for the general case is the same although the details are a bit more complicated. The action of the mapping class group extends to the Bers boundary continuously if and only if there is a continuous extension of the natural identification between two Bers slices qf .fm0 g  T .S // and qf .fh.m0 /g  T .S// to their Bers boundaries, for any orientation-preserving diffeomorphism h of S. We shall show that the Bers compactification does not have this property. Suppose that S has genus 2, and let c be an essential simple closed curve on S separating S into two once-punctured tori. The n-time Dehn twist around c, which we denote by cn , represents an element of the mapping class group. We consider a sequence of quasi-Fuchsian groups .Gn ; n / D qf .m0 ; .cn / n0 / in the Bers slice based at m0 2 T .S/ for some point n0 2 T .S /. The limit of f.Gn ; n /g in AH.S / is a b-group .; / on the Bers boundary based at m0 having connected parabolic locus, which is c, and upper conformal structures on the two components of S n c. To understand better the upper conformal structures, it is useful to consider geometric limits. For a sequence of Kleinian groups fGn g, its geometric limit G1 is a Kleinian group having the following two properties:

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(i) for any element g 2 G1 , there is a sequence fgn 2 Gn g converging to g; (ii) any convergent sequence fgni 2 Gni g has its limit in G1 . We consider a sequence of quasi-Fuchsian groups f.Gn ; i /g as in the previous paragraph, and take conjugates of the Gn so that fn g converges as representations into PSL2 C. It is known that such fGn g has a geometrically convergent subsequence, and the geometric limit contains the limit b-group as a subgroup. Now, the geometric limit of our sequence fGn g is easy to understand. The hyperbolic 3-manifold H3 =G1 is homeomorphic to S R with c f0g deleted. This means that H3 =G1 has two non-cuspidal ends. Both of them are geometrically finite, and they have conformal structures at infinity which correspond to m0 and n0 respectively. The end S  f1g has m0 and S  f1g has n0 as conformal structures. (Strictly speaking, the marking of the conformal structure on S  f1g can be defined only up to Dehn twists around c.) In particular G1 itself is also geometrically finite. The algebraic limit is contained in G1 as a subgroup coming from S  f1g. Therefore, the end S  f1g lifts to the algebraic limit, and corresponds to the end coming from the invariant component of  . Since every subgroup of a geometrically finite group is again geometrically finite, the algebraic limit  is geometrically finite. The lower end has a conformal structure at infinity equal to m0 as explained above. The upper end is split into two along a Z-cusp whose core curve is homotopic to c. Therefore the upper ends have two conformal structures at infinity both of which can be regarded as points in the Teichmüller space of a once-punctured torus, which we denote by T .T /. As we vary the points m0 , n0 inside T .S / and consider the algebraic limit of fqf .m0 ; .cn / n0 /g as above, we get a pair of points in T .T /  T .T / by considering the upper conformal structures. Thus we have an analytic map r W T .S/  T .S/ ! T .T /  T .T /. Now, suppose that the action of the mapping class group extends continuously to the Bers boundary. Then, as was remarked before, the base change between Bers slices by mapping classes should extend continuously to the boundaries. By a relatively easy argument, we can see that this implies that r defined above descends to an analytic map rN W .T .S/=M C G.S//  T .S/ ! T .T /  T .T /, where M C G.S / denotes the mapping class group of S. On the other hand, using a geometric argument involving Klein–Maskit combination, it can be shown that there is a diffeomorphism h such that r.m0 ; / W T .S/ ! T .T /  T .T / and r.h .m0 /; / W T .S / ! T .T /  T .T / are different. This shows that it is impossible to extend continuously the action of the mapping class group to the Bers boundary. As the proof of Theorem 4.2 shows, what causes the non-extendability of the mapping class group action is the existence of the quasi-conformal deformation space, T .T /  T .T / which appeared above, inside the Bers boundary. Therefore, it is natural to suspect that if we consider the quotient space collapsing each quasi-conformal deformation space into a point, we can extend the mapping class group action. According to McMullen [17], it was Thurston who conjectured that this was the case. Indeed, this turns out to be true as was shown in Ohshika [20].

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Theorem 4.3. Let @rb m0 T .S/ be the quotient space of the boundary of qf .fm0 gT .S // obtained by collapsing each quasi-conformal deformation space contained in it into a point. (We call this quotient space the reduced Bers boundary based at m0 .) Then the action of the mapping class group on T .S / extends continuously to @rb m0 T .S /.

5 The rigidity of mapping class group actions 5.1 Two kinds of reduced spaces We mentioned in §3.2 that if we consider the quotient space of the Teichmüller boundary by ignoring the transverse measure, then we get a continuous extension of the mapping class group action on T .S/ to the unmeasured foliation space. Similarly, as was shown in Theorem 4.3, the mapping class group action extends continuously to the reduced Bers boundary. On the other hand, as was explained in §4.2, the points lying on the Bers boundary are parametrised by end invariants. If we deform a point in the Bers boundary to another point quasi-conformally, then what changes is the upper conformal structure while the parabolic locus and the ending laminations remain the same. Therefore, each group in the reduced Bers boundary has a parameter which is the union of its parabolic locus and ending laminations, which is an unmeasured lamination on S . By identifying unmeasured laminations with unmeasured foliations, we can regard the reduced Bers boundary as a subset of unmeasured foliation space. Still, as was shown in [20], the topology on the reduced Bers boundary, which is the quotient topology coming from the Bers boundary, is distinct from the topology induced from the unmeasured foliation space. (The unmeasured foliation space or unmeasured lamination space has the quotient topology coming from the measured foliation space.) It should also be noted that neither of these two topologies is Hausdorff.

5.2 Rigidity The extended mapping class group of S is the group of isotopy classes of diffeomorphisms which may be orientation-reversing. It is a Z2 -extension of the mapping class group. It is easy to apply the above argument for the mapping class groups to the extended mapping class groups to see that the action of the extended mapping class group on T .S / extends continuously both to the unmeasured foliation space UMF .S / and the reduced Bers boundary @rb m0 T .S/. The unmeasured foliation space is identified with the unmeasured lamination space N ML.S /. Here we adopt the symbol N ML.S / instead of UML.S/, which is usually used, to avoid the confusion with the unit measured lamination space U ML.S /. In the following, we shall only talk about the unmeasured lamination space, but it can be replaced with the unmeasured foliation space everywhere. We can prove that conversely every self-homeomorphism of these spaces is induced from an extended mapping class.

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This fact in the case of unmeasured lamination space was shown by Papadopoulos [23] on a dense subset contained in the unmeasured lamination space, and the result was extended to the entire space by Ohshika [21]. Theorem 5.1. Let h W N ML.S/ ! N ML.S / be a homeomorphism. Then there is a diffeomorphism f W S ! S which induces h on N ML.S /. Such a diffeomorphism f is unique up to isotopy. For the case of reduced Bers boundary, a similar result was obtained in Ohshika [20]. rb Theorem 5.2. Let h W @rb m0 T .S/ ! @m0 T .S / be a homeomorphism. Then there is a diffeomorphism f W S ! S which induces h on @rb m0 T .S /. Such a diffeomorphism f is unique up to isotopy.

The proofs of both theorems use in an essential way the result of Ivanov, Korkmaz and Luo [9], [14], [15] saying that every simplicial automorphism of the curve complex is induced from a diffeomorphism. Another important ingredient in the proof is to measure non-separability of points. Recall that neither the unmeasured lamination space nor the reduced Bers boundary is Hausdorff. Moreover neither of them is T1 ; that is, there may be points which are not closed sets. For the unmeasured lamination space, Papadopoulos introduced the notion of adherence number. For x 2 N ML.S/, we say that x is adherent to y if every neighbourhood of x intersects every neighbourhood of y. A subset X in N ML.S / is said to be a complete set of adherence if every point in X is adherent to every point of X. The adherence number of x is the supremum of the cardinalities of complete adherence sets containing x. It is clear that adherence numbers of points are preserved under homeomorphisms, and Papadopoulos used this fact to show that every homeomorphism of N ML.S / induces a simplicial automorphism of the curve complex. We introduced in [20] and [21] another notion concerning the non-separability of the space, which is called the adherence height. Let X be a topological space, which is either the unmeasured lamination space or the reduced Bers boundary. A point x is said to be unilaterally adherent to another point y (distinct from x) if every neighbourhood of y contains x. We define the adherence height of x to be the maximal length of a sequence x D x0 ; x1 ; x2 ; : : : such that xi is unilaterally adherent to xiC1 . It can be shown that the adherence height is bounded by a constant depending only on the topological type of S, whether X is the unmeasured foliation space or the reduced Bers boundary. In [21], we used this notion to show that if two auto-homeomorphisms of S induce the same simplicial automorphism of the curve complex, then they must coincide on the entire N ML.S/. This implies Theorem 5.1. In [20], the notion of adherence height was used to show that every homeomorphism of @rb m0 T .S / preserves the set consisting of geometrically finite b-groups, and the number of cusps, which implies that it induces a simplicial automorphism of the curve complex. To show that two homeomorphisms inducing the same simplicial

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automorphism on the curve complex coincide, we used some deeper results in the theory of Kleinian groups, which we developed in [19].

References [1]

L. Bers, On boundaries of Teichmüller spaces and on Kleinian groups, I. Ann. of Math. (2) 91 (1970), 570–600.

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M. Bestvina, Degenerations of the hyperbolic space. Duke Math. J. 56 (1988), 143–161.

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F. Bonahon, Bouts des variétés hyperboliques de dimension 3. Ann. of Math. (2) 124 (1986), 71–158.

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A. Casson and S. Bleiler, Automorphisms of surfaces after Nielsen and Thurston. London Math. Soc. Stud. Texts 9, Cambridge University Press, Cambridge 1988.

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J. Brock, R. Canary, and Y. Minsky, The classification of Kleinian surface groups, II: The ending lamination conjecture. Ann. of Math. (2) 176 (2012), 1–149.

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D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces. In Analytical and geometric aspects of hyperbolic space, London Math. Soc. Lecture Note Ser. 111, Cambridge University Press, Cambridge 1987, 113–253.

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A. Fathi, F. Laudenbach, and V. Poénaru (eds.), Travaux de Thurston sur les surfaces. Astérisque 66—67 (1979).

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J. Hubbard and H. A. Masur, Quadratic differentials and foliations. Acta Math. 142 (1979), 221–274.

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N. V. Ivanov, Automorphisms of complexes of curves and of Teichmuüller spaces. In Progress in knot theory and related topics, Travaux en Cours 56, Hermann, Paris 1997, 113–120.

[10] T. Jørgensen, On discrete groups of Möbius transformations. Amer. J. Math. 98 (1976), 739–749. [11] S. Kerckhoff, The asymptotic geometry of Teichmüller space. Topology 19 (1980), 23–41. [12] S. Kerckhoff, The Nielsen realization problem. Ann. of Math. (2) 117 (1983), 235–265. [13] S. Kerckhoff and W. Thurston, Noncontinuity of the action of the modular group at Bers’ boundary of Teichmüller space. Invent. Math. 100 (1990), 25–47. [14] M. Korkmaz, Automorphisms of complexes of curves on punctured spheres and on punctured tori. Topology Appl. 95 (1999) 85–111. [15] F. Luo, Automorphisms of the complex of curves. Topology 39 (2000), 283–298. [16] Y. Minsky, The classification of Kleinian surface groups, I. Models and bounds. Ann. of Math. (2) 171 (2010), 1–107. [17] C. McMullen, Rational maps and Kleinian groups. In Proceedings of the International Congress of Mathematicians (Kyoto, 1990), Vol. II, Math. Soc. Japan, Springer, Tokyo 1991, 889–899. [18] J. W. Morgan and P. Shalen, Valuations, trees, and degenerations of hyperbolic structures, I. Ann. of Math. (2) 120 (1984), 401–476.

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[19] K. Ohshika, Divergence, exotic convergence, and self-bumping in the quasi-Fuchsian spaces. Preprint, arXiv:1010.0070. [20] K. Ohshika, Reduced Bers boundaries of Teichmüller spaces. Preprint, arXiv:1103.4680 [21] K. Ohshika, A note on the rigidity of unmeasured lamination space. Proc. Amer. Math. Soc. 141 (2013), 4385–4389. [22] A. Papadopoulos, Réseaux ferrovaires, difféomorphismes pseudo-Anosov et automorphismes sympléclique de l’homologie d’une surface. Prépublications mathématiques d’Orsay, 83-03. [23] A. Papadopoulos, A rigidity theorem for the mapping class group action on the space of unmeasured foliations on a surface. Proc. Amer. Math. Soc. 136 (2008), 4453–4460. [24] F. Paulin, Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Invent. Math. 94 (1988) 53–80. [25] R. Penner and J. Harer, Combinatorics of train tracks. Ann. of Math. Stud. 125, Princeton University Press, Princeton, NJ, 1992. [26] R. Skora, Splittings of surfaces. J. Amer. Math. Soc. 9 (1996), 605–616. [27] K. Strebel, Quadratic differentials. Ergeb. Math. Grenzgeb 5, Springer, Berlin 1984. [28] W. Thurston, The geometry and topology of 3-manifolds, Lecture Notes, Princeton University Press, Princeton, NJ, 1978; available at http://www.msri.org/publications/books/gt3m/. [29] W. Thurston, Earthquakes in two-dimensional hyperbolic geometry. In Low-dimensional topology and Kleinian groups, London Math. Soc. Lecture Note Ser. 112, Cambridge University Press, Cambridge 1986, 91–112.

Chapter 6

Arc geometry and algebra: foliations, moduli spaces, string topology and field theory Ralph M. Kaufmann

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The basic idea . . . . . . . . . . . . . . . . . . . . . . 1.2 Windowed surfaces with partial measured foliations . . 1.3 The spaces of weighted arcs . . . . . . . . . . . . . . . 1.4 Different pictures for arcs . . . . . . . . . . . . . . . . 1.5 Quasi-filling families, arc graphs and dual ribbon graphs 1.6 Foliation picture . . . . . . . . . . . . . . . . . . . . . 2 The gluing and the operad structures . . . . . . . . . . . . . . 2.1 Standard gluing for foliations . . . . . . . . . . . . . . 2.2 Cyclic operad structure: the scaling approach of [31] . . 2.3 Chains and homology . . . . . . . . . . . . . . . . . . 2.4 Singular homology and singular chains . . . . . . . . . 2.5 Open/cellular chains . . . . . . . . . . . . . . . . . . . 2.6 Modular structure: the approach of [32] . . . . . . . . . 2.7 S 1 action . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Twist gluing . . . . . . . . . . . . . . . . . . . . . . . 2.9 Variations on the gluings . . . . . . . . . . . . . . . . . 3 Framed little discs and the Gerstenhaber and BV structures . . 3.1 Short overview . . . . . . . . . . . . . . . . . . . . . . 3.2 (Framed) little discs and (spineless) cacti . . . . . . . . 3.3 Cellular structure . . . . . . . . . . . . . . . . . . . . . 3.4 The BV operator . . . . . . . . . . . . . . . . . . . . . 3.5 The associator . . . . . . . . . . . . . . . . . . . . . . 4 Moduli space, the Sullivan-PROP and (framed) little discs . . 4.1 Moduli spaces . . . . . . . . . . . . . . . . . . . . . . 4.2 Operad structure on moduli spaces . . . . . . . . . . . . 4.3 The Sullivan quasi-PROP . . . . . . . . . . . . . . . . 5 Stops, stabilization and the Arc spectrum . . . . . . . . . . . 5.1 Stops: adding a unit . . . . . . . . . . . . . . . . . . .

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5.2 Adding a unit in the arc formalism . . . . . . . . . . . . . 5.3 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Generalization to all of Arc . . . . . . . . . . . . . . . . 5.5 Stabilization and moduli space . . . . . . . . . . . . . . . 5.6 Stabilization and adding a unit. The E1 and Ek structures 5.7 CW decomposition and [i products . . . . . . . . . . . . 6 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Deligne’s conjecture . . . . . . . . . . . . . . . . . . . . 6.3 The cyclic Deligne conjecture . . . . . . . . . . . . . . . 6.4 Moduli space action and the Sullivan-PROP also known as string topology action . . . . . . . . . . . . . . . . . . . 7 Open/closed version . . . . . . . . . . . . . . . . . . . . . . . A Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Operads and PROPs . . . . . . . . . . . . . . . . . . . . A.2 Standard operads and their algebras . . . . . . . . . . . . A.3 Hochschild cohomology . . . . . . . . . . . . . . . . . . A.4 Frobenius algebras . . . . . . . . . . . . . . . . . . . . . A.5 Reference for symbols . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction There has long been an intricate relationship between foliations, arc complexes and the geometry of Teichmüller and moduli spaces [49], [12]. The study of string theory as well as that of topological and conformal field theories has added a new aspect to this theory, namely to study these spaces not only individually, but together all at once. The new ingredient is the idea to glue together surfaces with their additional data. Physically, this can for instance be viewed as stopping and starting time for the generation of the world-sheet. Mathematically, the general idea of gluing structures together in various compatible ways is captured by the theory of operads and PROPs [41], [3], [40]. This theory was originally introduced in algebraic topology to study loop spaces, but has had a renaissance in conjunction with the deepening interaction between string theory and mathematics. The Arc operad of [31] specifically provides the mathematical tool for this approach using foliations. Combinatorially, the underlying elements are surfaces with boundaries and windows on these boundaries together with projectively weighted arcs running in between the windows. Geometrically these elements are surfaces with partial measured foliations. This geometric interpretation is the basis of the gluing operation. We glue the surfaces along the boundaries, matching the windows, and then glue the weighted arcs by gluing the respective foliations. The physical interpretation

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is that the mentioned foliations are transversal to the foliation created by the strings. The details of this picture are given in [32]. The gluing operation, which is completely natural from the foliation point of view, yields a surface based geometric model, for a surprising abundance of algebraic and geometric structures germane to loop spaces, string theory, string topology as well as conformal and topological field theories. Surprisingly this also includes higherdimensional structures such as the little k-cubes, associahedra, cyclohedra and Dbranes. This is true to the slogan that one only needs strings. It gives for instance rise to models for the little discs and framed little discs operads, moduli space and the Sullivan-PROP. These models exist on the topological, the chain and the homology levels. On the chain and homology these operads and PROPs correspond to Gerstenhaber, BV algebras, string topology operations and CFT/string field theory operations. One characteristic feature is that they are very small compared to their classical counterparts. Topologically this means that they are of small dimension. On the chain level this means that they are given by a small cellular model. A classical result is that the little discs operad detects two-fold loop spaces, consequently so does the arc operad. The classical theory about loop spaces goes further to state that k-fold loop spaces are detected by the little k cubes or any Ek operad. This generalized to k D 1. By using a stabilization and a unital fattening of a natural suboperad of Arc one obtains a surface model for all these operads. A consequence is a new infinite loop space spectrum coming from the stabilized unital fattened Arc operad. Another consequence of the foliation description are natural actions of the chains on the Hochschild cohomology of associative or Frobenius algebras, lifting the Gerstenhaber algebra structure on the cohomology. This type of action was conjectured by Deligne and has been a central theme in the last decade. One important application is that this type of action together with the fact that the little discs are formal as an operad implies Kontsevich’s deformation quantization. There is a vast extension of this chain level action, which gives a version of string topology for simply connected spaces. For this the surface boundaries are classified as “in” or “out” boundaries. This is the type of setup algebraically described by a PROP. In particular, as a generalization of the above results there is a PROP, the SullivanPROP, which arises naturally in the foliation picture. Again there is a CW model for it and its chains give an action on the Hochschild co-chains of a Frobenius algebra extending the previous action. This action further generalizes to a model for the moduli space of surfaces with marked points and tangent vectors at these marked points as it is considered in conformal field theory and string field theory. Both actions are given by a discretization of the Arc operad and their algebra and combinatorics are geometrically explained by foliations with integer weights. Finally, there is an open/closed version of the whole theory. This generalizes the actions as well. On the topological level one consequence of this setup is a clear

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geometric proof of the minimality of the Cardy–Lewellen axioms for open/closed topological field theory using Whitehead moves. Thus the Arc operad and its variations provide a wonderful, effective, geometric tool to study and understand the origin of these algebraic structures and give new results and insights. The explicit homotopy BV equation provided by the arc operad given in Figure 11 or the geometric representation of the classical [i products in Figure 17 may serve as an illustration. We expect that this foliation geometry together with the operations of gluing will provide new results in other fields such as cluster algebras, 2 C 1-dimensional TFTs and any other theory based on individual moduli spaces.

Scope The scope of the text is a subset of the results of the papers [31], [22], [26], [33], [32], [28], [27], [25], [29] and [32]. It is the first time that all the various techniques developed in the above references are gathered in one text. We also make some explicit interconnections that were previously only implicit in the total body of results. The main ones being the stabilization of Arc and the arc spectrum and the analysis of the S 1 equivariant geometry. For a more self-contained text, we have added an appendix with a glossary containing basic notions of operads/PROPs and their algebras as well as Hochschild cohomology and Frobenius algebras.

Layout of the exposition In this theory there are usually two aspects. First and foremost there is the basic geometric idea about the structure, and then secondly there is a more technical mathematical construction to make this idea precise. This gives rise to the basic conundrum in presenting the theory. If one first defines the mathematically correct notions, one has to wait for quite a while before hearing the punch line. If one just presents the ideas, one is left with sometimes a formidable task to make mathematics out of the intuitive notions. We shall proceed by first stating the idea and then giving more details about the construction and, if deemed necessary, end with a comment about the finer details with a reference on where to find them. The text is organized as follows: In the first section, we introduce the spaces of foliations we wish to consider. Here we also give several equivalent interpretations for the elements of these spaces. There is the foliation aspect, a combinatorial graph aspect and a dual ribbon graph version. The basic gluing operation underlying the whole theory is introduced in Section 2 as are several slight variations needed later. This leads to the Arc operad, which is a cyclic operad. We also study its discretization and the chain and homology level operads. For the homology level, we also discuss an alternative approach to the gluing which

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yields a modular operad structure on the homology level. We furthermore elaborate on the natural S 1 -actions and the resulting geometry. The chain and homology levels for instance yield new geometric examples for the so-called K-modular operads. Section 3 contains the explicit description of the little discs and framed little discs in this framework. Technically there are suboperads of Arc that are equivalent, that is quasi-isomorphic, to them. This includes an explicit presentation of the Gerstenhaber and BV structures and their lift to the chain level. In this section we give an explicit geometric representation for the bracket and the homotopy BV equation, see Figures 9 and 11. Section 4 contains the generalization to the Sullivan-(quasi)-PROP which governs string topology and the definition of the rational operad given by the moduli spaces. There are some fine points as the words “rational” and “quasi-” suggest which are fully explained. Another fine point is that the operads as presented do not have what is sometimes called a unit. This is not to be confused with the operadic unit that they all possess. This is why we call operads with a unit “pointed”. For the applications to string topology and Deligne’s conjecture the operads need not be pointed. However, for the applications to loop spaces this is essential. The main point of Section 5 is to give the details of how to include a unit and make the operad pointed. This allows us to fatten the Arc to include a pointed E2 operad and hence detect double loop spaces. The other basic technique given in this section is that of stabilization. The result of combining both adding a unit and stabilization leads to the Ek operads, explicit geometric representatives for the [i products (see Figure 17) and a new spectrum, the Arc spectrum. The various chain level actions are contained in Section 6. The first part is concerned with Deligne’s conjecture and its A1 , the cyclic and cyclic A1 generalization. These are given by actions given by a dual tree picture. The section also contains the action of the chain level Sullivan-PROP and that of moduli space on the Hochschild cochains of a Frobenius algebra. For this we introduce correlation functions based on the discretization of the foliations. As a further application we discuss the stabilization and the semi-simple case. We close the main text in Section 7 with a very brief sketch of the open/closed theory.

Conventions We fix a field k. For most constructions any characteristic would actually do, but sometimes we use the isomorphism between Sn invariants and Sn co-invariants in which case we have to assume that k is of characteristic 0. There is a subtlety about what is meant by Gestenhaber in characteristic 2. We will ignore this and take the algebra over the operads in question as a definition.

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When dealing with operads, unless otherwise stated, we always take H .X / to mean H .X; k/, so that we can use the Künneth theorem to obtain an isomorphism H  .X  Y; k/ ' H .X/ ˝k H .Y /.

1 The spaces 1.1 The basic idea As any operad the Arc operad consists of a sequence of spaces with additional data, such as symmetric group actions and gluing maps. There are two ways in which to view the spaces: Geometric version. The spaces are projectively weighted families of arcs on surfaces with boundary that end in fixed windows at the boundary considered up to the action of the mapping class group. An alternative equivalent useful characterization is: Combinatorial version. The spaces are projectively weighted graphs on surfaces with boundaries, where each boundary has a marked point and these points are the vertices of the graph, again considered up to the action of the mapping class group. The geometric version can be realized by partial measured foliations which make the gluing natural, while the combinatorial version allows one to easily make contact with moduli space and other familiar spaces and operads, such as the little discs, the Sullivan-PROP, etc.

1.2 Windowed surfaces with partial measured foliations We will now make precise the geometric version following [31]. s be a fixed oriented topological surface 1.2.1 Data and notation. Let F D Fg;r of genus g  0 with s  0 punctures and r  1 boundary components, where 6g  7 C 4r C 2s  0. Also fix an enumeration @1 ; @2 ; : : : ; @r of the boundary components of F once and for all. Furthermore, in each boundary component @i of F , fix a closed arc Wi  @i , called a window. The pure mapping class group PMC D PMC.F / is the group of isotopy classes of all orientation-preserving homeomorphisms of F which fix each @i  Wi pointwise (and fix each Wi setwise). Define an essential arc in F to be an embedded path a in F whose endpoints lie in the windows, such that a is not isotopic rel endpoints to a path lying in @F . Two arcs are said to be parallel if there is an isotopy between them which fixes each @i  Wi pointwise (and fixes each Wi setwise).

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An arc family in F is the isotopy class of a non-empty unordered collection of disjointly embedded essential arcs in F , no two of which are parallel. Thus, there is a well-defined action of PMC on arc families. 1.2.2 Induced data. Fix F . There is a natural partial order on arc families given by inclusion. Furthermore, there is a natural order on all the arcs in a given arc family as follows. Since the surface is oriented, so are the windows. Furthermore we enumerated the boundary components. This induces an order, by counting the arcs by starting in the first window in the order they hit this window, omitting arcs that have already been enumerated and then continuing in the same manner with the next window. This procedure also gives an order 0 and its PMC quotient We also consider the de-projectivized version jKg;r s s Dg;r D Ag;r  R>0 .

1.4 Different pictures for arcs Depending on the circumstances there are different completely equivalent pictures which we can use to be closer to intuition. There are the following choices for the windows. I Disjointly embedded arcs with endpoints in windows. II Shrinking the complement of the (open) window to a point. The two endpoints of the window then are identified and give a distinguished point on the boundary. Arcs still do not intersect pairwise and avoid the marked points on the boundary. This version is particularly adapted to understand the S 1 action (see 2.7) and the operads yielding the Gerstenhaber and BV structures. III Shrinking the window to a point. The arcs may not be disjointly embedded at the endpoints anymore, but they form an embedded graph. This is a version that is

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very useful in combinatorial descriptions, e.g. a dual graph approach for moduli spaces. These are depicted in Figure 1. We also may choose the following different pictures for the arcs as we discuss in 2.1 in greater detail. A Arcs with weights. B Bands of leaves with (transversal) width. C Bands of leaves with width filling the windows. The arcs with weights are the quickest method to construct the relevant spaces, the bands-of-leaves picture is what makes the operadic gluing natural. It also greatly helps elucidate the S 1 action and the discretization that acts on the Hochschild complexes. The cases I, II, III A and I B are depicted in Figure 1, the cases I C and II C are in Figure 2. (In the figures u, v, w denote positive real weights.)

uvw

uvw

u v w

uvw

IA

II A

III A

IB

Figure 1. I A. Arcs running to a point on the boundary. II A. Arcs running to a point at infinity. III A. Arcs in a window. I B. Bands in a window.

w u v w

I C

v u

II C

Figure 2. I C. Bands ending on an interval. II C. Bands ending on a circle.

0 , see Figure 3. The 0-simplices are given by (the 1.4.1 Example. Consider K0;2 isotopy class of) a straight arc and all its images under a Dehn twist. Thus the 0 skeleton can be identified with Z. It is possible to embed two arcs which differ by one Dehn twist. Calling these one cells Ii iC1 , if the first arc is the i -fold Dehn twist

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1 1 0

0 1

0

1

0 Figure 3. The space K0;2 D R as a simplicial space. We indicated two 0-simplices and a connecting 1-simplex.

2 0 of 0, we see that jK0;2 j D R. PMC.F0;2 / is generated by the Dehn twist and hence 0 1 A0;2 D R=Z D S . This S 1 is what underlies the BV geometry, see §3.

1.4.2 Elements as weighted arc families. A weight function wt on an arc family ˛ is a map that associates to each arc of ˛ a positive real number. There is a natural scaling action by R>0 on the set of weight functions on ˛ and we denote by Œwt the equivalence class of a given weight function wt under this action. An element a 2 jAsg;r j in the realization of Asg;r lies in a unique open simplex. If ˛ is the arc family indexing this simplex, then using the enumeration of arcs, we can identify the barycentric coordinates with weights on the arcs of ˛. In this picture, a codimension-one boundary is given by sending one of the weights to zero. We are free to think of the barycentric coordinates as a projective class Œwt of a positive weight function wt on the arcs. In this fashion a D .˛; Œwt/. s j  R>0 are naturally pairs .˛; wt/ of an arc In this picture, the elements of jKg;r family together with a weight function. s as pairs Taking the quotient by PMC, we get a description of elements of Dg;r .Œ˛; wt/ where Œ˛ denotes the PMC orbit of ˛. Further taking the quotient with respect to the R>0 action elements of Asg;r are pairs .Œ˛; Œwt/. s , 1.4.3 Weights at the boundary. Taking up the picture above, given .˛; wt/ 2 Dg;r we define the weight wt.@i / at the boundary i of ˛ to be the sum of the weights of the ends of the arcs incident to @i . Notice that in this count, if an arc has both ends on @i its weight counts twice in the sum. s Definition 1.1. A weighted arc family .Œ˛; wt/ 2 Dg;r or .Œ˛; Œwt/ 2 Asg;r is called exhaustive if wt i .˛/ ¤ 0 for all i. s to be the subsets of We set Arcsg .r  1/  Asg;r and DArcsg .r  1/  Dg;r exhaustive elements.

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We furthermore set Arc.n/ D qg;s Arcsg .n/; DArc.n/ D qg;s DArcsg .n/ where q is the coproduct given by disjoint union, and finally Arc D qn Arc.n/ and DArc.n/ D qn DArc.n/. The natural Sr action descends both to Arc.r  1/ and to DArc.r  1/.

1.5 Quasi-filling families, arc graphs and dual ribbon graphs We call an arc family quasi-filling if the complementary regions are polygons which contain at most one marked point. y 1.5.1 Dual (ribbon) graph. Let .˛/ be the dual graph in the surface of ˛. This means there is one vertex for every component of F n ˛ and an edge for each arc of ˛ connecting the two vertices representing the two regions on either side of the arc. If the graph is quasi-filling, this graph is again an embedded (up to isotopy) naturally ribbon graph. The cyclic order at each vertex is induced by the orientation of the surface. The cycles of the ribbon graph are naturally identified with the boundary components of F . This identification also exists in the non-quasi-filling case. Here O the set of oriented edges or flags of .˛/ is partitioned into cycles, or, in other words, it is partitioned into a disjoint union of sets each with a cyclic order. There is an additional structure of a marking where a marking is a fixed vertex for every cycle. This vertex is the vertex corresponding to the region containing the complement of the window. Combinatorially, the vertices have valence  2 with only the marked vertices possibly having valence 2. Given an element a 2 DArc, we also obtain a metric on the dual graph of the underlying arc family, simply by keeping the length of each edge. The geometric realization is obtained by gluing intervals of these given lengths together at the vertices. 1.5.2 Arc graph. It is sometimes convenient to describe the arc families simply as a graph. The basic idea is as follows: given an arc family ˛ on F we define its graph .˛/ to be the graph on F obtained by shrinking each window Wi to a point vi . The vi are then the vertices and the arcs of ˛ are the edges. We can think of .˛/ as embedded in F . Again there is some fine print. First the graph only has an embedding up to homotopy. Secondly by changing the window, we changed our initial data, which is fine, but then the arcs are not disjointly embedded anymore. A rigorous geometric interpolation of the two pictures is given in [31]. As an abstract graph, we can also let the vertices be given by the Wi and the edges be given by the set of arcs of ˛. In §1.6.2, we give a geometric construction of this space. In the situation s D 0 and in the case of a quasi-filling ˛ the data of the marked y is equivalent to ˛, since one can obtain ˛ by reversing the dualization. ribbon graph 

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1.6 Foliation picture s If .Œ˛; wt/ D .Œfa0 ; a1 ; : : : ; ak g; wt/ 2 Dg;r is given by weights .w0 ; w1 ; : : : ; wk / 2 kC1 RC , then we may regard wi as a transverse measure on ai , for each i D 0; 1; : : : ; k to determine a “measured train track with stops” and a corresponding “partial measured foliation”, as considered in [49]. This works as follows. Fix some complete Riemannian metric  of finite area on F , suppose that each ai is smooth for , and consider for each ai the “band” Bi in F consisting of all points within -distance wi of ai . Since we can scale the metric  to , for  > 1, we will assume that these bands are pairwise disjointly embedded in F , and have their endpoints lie in the windows. The band Bi about ai comes equipped with a foliation by the arcs parallel to ai which are at a fixed -distance to ai , and this foliation comes equipped with a transverse measure inherited from ; thus, each Bi can be regarded as a rectangle of width wi and some irrelevant length. The foliated and transversely measured bands Bi , for i D 0; 1; : : : ; k, combine to give a “partial measured foliation” of F , that is, a foliation of a closed subset of F supporting an invariant transverse measure (cf. [49]). The isotopy class in F rel @F of this partial measured foliation is independent of the choice of metric .

1.6.1 Partial parametrization at the boundary. For i D 1; 2; : : : ; r, consider @i \ `k  j D0 Bj , which is empty if ˛ does not meet @i and its intersection with Wi is otherwise a collection of closed intervals in Wi with disjoint interiors. Collapse to a point each component complementary to the interiors of these intervals in Wi to obtain an interval, which we shall denote @i .˛ 0 /. Each such interval @i .˛ 0 / inherits an absolutely continuous measure i from the transverse measures on the bands. If @i .˛ 0 / is not empty, scaling the measure to have total weight one, this gives a unique measure preserving map of ci˛ W @i .˛ 0 / ! S 1 where S 1 has the Haar measure. Further collapsing the complement of the interior of Wi to a point, we get a space Si1 .˛/ and ci˛ induces a map from this quotient to S 1 which is a measure preserving homeomorphism that maps the image of the endpoints of Wi to 0 2 S 1 D R=Z. We call this map the parameterized circle at @i . A pictorial representation can be found in Figure 2. 1.6.2 Loop graph of an arc family: a geometric construction of the dual graph. The loop graph of a weighted arc family a 2 DArc is the space obtained from qi Si1 =  where  is the equivalence relation which is the transitive closure of the symmetric relation p  q if p and q are the endpoints of a leaf. The loop is invariant under the PMC action and we call the resulting space L.a/. There are natural maps li W Si1 ! L.a/, the images are called the i -th circle or lobe. The 0-th circle is also called the outside or output circle, while the circles for i ¤ 0 are called the input circles. The loop of the graph is homeomorphic to the geometric realization of the dual graph with its metric. The circles correspond to the cycles and the marked point on each cycle is the image of 0.

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2 The gluing and the operad structures 2.0.3 Basic idea. Think of the elements of Asg;r as partial measured foliations modulo common scaling. The most natural way to do this is in terms of foliations as derived from the theory of “train tracks” (cf. [49]) In this picture, two weighted exhaustive arc families can be naturally composed by fixing one boundary component on each of the surfaces and glue naturally if one glues the underlying surfaces along a pair of fixed boundary components. Concretely on the condition that the weights on the two boundaries agree one produces a weighted arc family on the glued surface from two given arc families by viewing them as foliations. If the families are exhaustive, this can always be achieved by scaling. If one starts with projective weights, one only chooses representatives which satisfy the condition. This gluing yields the sought after operadic structure.

2.1 Standard gluing for foliations s 2.1.1 Basic idea. Given two weighted arc families .˛; wt/ in F D Fg;mC1 and t 0 0 .ˇ; wt / in F D Fh;nC1 , construct the respective foliation. Now picking one boundary component on each surface, we can glue them and the respective foliations if they have the same weights. This is well defined up to the action of PMC. More precisely, if the two foliations have the non-zero same weights say wti .˛/ D wt 0 .ˇ/ we can glue them to give a foliation on the surface obtained by gluing the sCt boundary i of F to the boundary i of F 0 . Identifying the glued surface with FgCh;mCn 0 0 we obtain the weighted family .˛; wt/ Bi .ˇ; wt /. If we have two exhaustive weighted families whose weights do not agree, we can use the R>0 action to make them agree and then glue. That is in general: .Œ˛; wt/ Bi .Œˇ; wt0 / WD wti .˛/.Œ˛; wt/ B0i wt0 .Œˇ/.ˇ; wt0 / and this descends to Arc.

2.1.2 Gluing weighted arc families. Given two weighted arc families .˛; wt/ in s t Fg;mC1 and .ˇ; wt0 / in Fh;nC1 so that i .@i .˛ 0 // D 0 .@0 .ˇ 0 //, for some 1  i  m, we will define a weighted arc family .; wt00 / WD .˛; wt/Bi .ˇ; wt0 / for each 1  i  m sCt on the surface FgCh;mCn as follows: s , First, let’s fix some notation: let @i denote the i -th boundary component of Fg;mC1 0 t and let @0 denote the 0-th boundary component of Fh;nC1 . We glue the boundaries together using the maps to S 1 given above. This yields a surface X homeomorphic to sCt , where the two curves @i and @00 are thus identified to a single separating FgCh;mCn sCt , curve in X. There is no natural choice of homeomorphism between X and FgCh;mCn s t but there are canonical inclusions j W Fg;mC1 ! X and k W Fh;nC1 ! X . We enumerate the boundary components of X in the order @0 ; @1 ; : : : ; @i1 ; @01 ; @02 ; : : : @0n ; @iC1 ; @iC2 ; : : : @m : s The punctures are enumerated simply by enumerating the ones on Fg;mC1 first.

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sCt Choose an orientation-preserving homeomorphism H W X ! FgCh;mCn which preserves the labeling of the boundary components as well as those of the punctures, if any. In order to define the required weighted arc family, consider the partial measured s t foliations G in Fg;mC1 and H in Fh;nC1 corresponding respectively to .˛ 0 / and .ˇ 0 /. By our assumption that i .@i .˛ 0 // D 0 .@0 .ˇ 0 //, we may produce a corresponding partial measured foliation F in X by identifying the points x 2 @i .˛ 0 / and y 2 @0 .ˇ 0 / if ci.˛/ .x/ D c0.ˇ / .y/. The resulting partial measured foliation F may have simple closed curve leaves which we must simply discard to produce yet another partial measured foliation F 0 in X . The leaves of F 0 thus run between boundary components of X and therefore, as in the previous section, decompose into a collection of bands Bi of some widths wi , for i D 1; 2; : : : ; I , for some I . Choose a leaf of F 0 in each such band Bi and associate to it the weight wi given by the width of Bi to determine a weighted arc family .ı 0 / sCt in X which is evidently exhaustive. Let . 0 / D H.ı 0 / denote the image in FgCh;mCn under H of this weighted arc family. sCt Lemma 2.1. The PMC.FgCh;mCn /-orbit of . 0 / is well-defined as .˛ 0 / varies over s s /-orbit of weighted arc families in Fg;mC1 and .ˇ 0 / varies over a a PMC.Fg;mC1 t t PMC.Fh;nC1 /-orbit of weighted arc families in Fh;nC1 .

Proof. Suppose we are given weighted arc families .˛20 / D .˛10 / s for  2 PMC.Fg;mC1 / and

and

.ˇ20 / D

.ˇ10 /

t 2 PMC.Fh;nC1 /, respectively, as well as a pair sCt H` W X` ! FgCh;mCn

s of homeomorphisms as above together with the pairs j1 ; j2 W Fg;mC1 ! X` and t k1 ; k2 W Fh;nC1 ! X` of induced inclusions, for ` D 1; 2. Let F` , F`0 denote the partial measured foliations and let .ı`0 / and .`0 / denote the corresponding weighted sCt , respectively, constructed as above from .˛`0 / and arc families in X` and FgCh;mCn 0 .ˇ` /, for ` D 1; 2. Let c` D j` .@0 / D k` .@0i / X` , and remove a tubular neighborhood U` of c` in X` to obtain the subsurface X`0 D X`  U` , for ` D 1; 2. Isotope j` ; k` off of U` in s t ! X`0 and k`0 W Fh;nC1 ! X`0 with the natural way to produce inclusions j`0 W Fg;mC1 disjoint images, for ` D 1; 2. The mapping class  induces a homeomorphism ˆ W X10 ! X20 supported on 0 s j1 .Fg;mC1 / so that j20 B  D ˆ B j10 , and induces a homeomorphism ‰ W X10 ! X20 t / so that k20 B D ‰ Bk10 . Because they have disjoint supports, supported on k10 .Fh;nC1 ˆ and ‰ combine to give a homeomorphism G 0 W X10 ! X20 so that j20 B  D G 0 B j10 and k20 B D G 0 B k10 . We may extend G 0 by any suitable homeomorphism U1 ! U2 to produce a homeomorphism G W X1 ! X2 .

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By construction and after a suitable isotopy, G maps F1 \ X10 to F2 \ X20 , and there is a power of a Dehn twist along c2 supported on the interior of U2 so that K D B G also maps F1 \ U1 to F2 \ U2 . K thus maps F10 to F20 and hence .ı10 / to .ı20 /. It follows that the homeomorphism sCt sCt H2 B K B H11 W FgCh;mCn ! FgCh;mCn sCt maps .10 / to .20 /, so .10 / and .20 / are indeed in the same PMC.FgCh;mCn /-orbit.

Notice that although the gluing is local on the boundaries, there is a global effect of gluing the leaves together. For instance there can be bands which have both ends on the same boundary. If these are split, they may recursively cut other bands. An example of such a gluing is given in Figure 4. Alternatively, one can describe the

c

a b

d

b

e

d

f

a)

a

b)

c

b

d

e

c)

b

d

d)

Figure 4. a) The arc graphs which are to be glued assuming the relative weights a, b, c, d and e as indicated by the solid lines in c). b) The result of the gluing (the weights are according to c). c) The combinatorics of cutting the bands. The solid lines are the original boundaries, the dotted lines are the first cuts, and the dashed lines represent the recursive cuts. d) The combinatorics of splitting and joining flags.

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gluing procedure purely combinatorially, see [23] for the details. For this one uses a least common partition of the unit interval, duplicates each edge for every cut and then glues the flags or half edges if they are indexed by the same subinterval. Remark 2.2. An alternative to discarding the simple closed curve leaves is to enlarge the space Asg;r to include them. This would be in the spirit of V. Jones’ planar algebras [18]. We however do not take this route and the applications such as to string topology do not exhibit these type of curves. s there is a natural action of the symmetric 2.1.3 Symmetric group actions. On Kg;r group of r elements Sr which permutes the labels 0; : : : ; r 1 enumerating the boundary components. It contains a subgroup Sr1 which only permutes the labels 1; : : : ; r keeping 0 fixed. Like above, after renumbering, we have to choose a homeomorphism to the standard surface. In the figures, this is usually suppressed.

2.1.4 Comments on the details, see [31]. Notice that since we fixed the surfaces s Fg;r the gluing actually depends a priori on a choice of homeomorphism of the glued surface. These choices become irrelevant after passing to PMC quotients. Other s and compatible morphisms of surfaces glued possibilities are to choose models Fg;r from these back to the chosen models. 2.1.5 Partial/colored operad structure Proposition 2.3. The gluings together with the symmetric group action permuting the s labels give a (cyclic) partial operad structure to the spaces D.r  1/ WD qg;s Dg;r . Moreover this partial operad structure is an R0 colored operad. Here R0 is considered with the discrete topology. Proof. Notice that Sn naturally acts on D.n/ via permuting the labels 1; : : : ; n on the boundaries. Moreover SnC1 acts by permuting the labels on the boundaries 0; : : : ; n. The gluings if defined are associative and symmetric group equivariant; for the precise definition of the various actions, see the Appendix. The important point is that the gluing did not depend on the name of the boundaries. This is a straightforward check. The additional equation for a cyclic operad is also easy to check. The only obstruction to gluing is that the weights on the two boundaries which are glued are the same. The procedure also works if the two boundaries are both not hit by any arc. Thus assigning the color wt.@i /  0 to each i we obtain an R0 colored operad.

2.2 Cyclic operad structure: the scaling approach of [31] In the gluing operation above, we could compose two weighted arc families if they had the same weight at the designated boundaries. We can get rid of this restriction

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if we consider exhaustive families by using the R>0 scaling action. Given exhaustive arc families .Œ˛; Œwt/ 2 Arc.n/ and .Œˇ; Œwt0 / 2 Arc.m/ .Œ˛; wt/ Bi .Œˇ; wt0 / WD wti .˛/.Œ˛; wt/ B0i wt0 .Œˇ/.ˇ; wt0 /:

(2.1)

Theorem 2.4 ([31]). The spaces DArc.n/ form a cyclic operad and this cyclic operad structure descends to Arc.n/, which comprise the Arc operad. Proof. One has to recheck the associativity for this case, but it again works out [31]. Remark 2.5. Notice that if we start in Arc there are unique representatives .Œ˛; wt/ 2 D.n/ and .Œˇ; Œwt/ 2 D.m/ such that wt.@i / D wt0 .@0 / D 1 and we could have used these to define gluing directly on Arc, but that would have not allowed us to lift the operad structure to DArc. 2.2.1 Discretization: the suboperad of positive integer weights/multiarcs Proposition 2.6. The arc families in DArc with positive integer weights form a cyclic suboperad. Using the inclusion N  R, they also form an N colored cyclic suboperad of the R-colored version of DArc. A useful pictorial realization of an arc family with positive integer weights is to replace an arc of weight k by k parallel arcs. Alternatively, one adds k  1 parallel copies to the arc after say fixing a small rectangular neighborhood of the original arc. We will call these multi-arc families. These multi-arc families are what is used in the string topology and moduli space actions. There they will appear in their N colored version. Another relevant operad structure is the one sums of these elements given as follows: Given an exhaustive arc graph ˛, with arcs e1 ; : : : ek , let ˛ .n1 ;:::;nk / 2 DArc for ni 2 N be defined by wt.ei / D ni , ˛ N D qnE 2N k ˛ nE :

(2.2)

Furthermore, for two exhausting arc graphs we set ˛ N Bi ˇ N D

0 X

˛ nE Bi ˇ mE ;

(2.3)

.E n;m/ E

P where 0 runs over the pairs .E n; m/ E such that wt.@i .˛ nE // D wt.@0 .ˇ mE //, that is, those pairs for which the N-colors match. Proposition 2.7. The compositions Bi are operadic. Furthermore dropping the superscript N they give an operad structure to the collection of exhaustive arc graphs ˛ where the operad degree is the number of boundaries of ˛.

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2.3 Chains and homology One basic question is how operads behave with respect to functors of homology and various chain functors. It is the homology level that gives the algebra and the chain level basically the “algebra up to homotopy” level which is relevant for applications from Deligne’s conjecture to field theory. 2.3.1 Operads and functors: technical details. The general answer to the question of what kind of functor pushes forward an operad structure is that it should be a weak monoidal one. Let us denote this functor as F W C ! D where both C and D are monoidal categories with a product ˝. The condition of being weakly monoidal means among other things (see e.g. [20]) that there are natural morphisms F .X; Y / W F .X /˝ F .Y / ! F .X ˝ Y /. For our operads this means that we get compositions BF i by using the sequence of maps F .O.n/;O.m//

F .Bi /

BF ! F .O.n/˝O.m// ! F .O.mCn1//: i W F .O.n//˝F .O.m// 

2.4 Singular homology and singular chains If we take (singular) homology with coefficients in a field k then the Künneth formula guarantees us that the functor H . ; k/ is a strong monoidal functor, that is, there is an isomorphism H .O.n/; k/ ˝ H .O.m/; k/ ' H .O.n/  O.m/; k/, and this shows that the homology is again a cyclic operad. For singular chains the Eilenberg–Zilber theorem provides us with a weak monoidal functor on the chain level. Corollary 2.8. The singular chains and the homologies of DArc.n/ and Arc.n/ form cyclic operads. 2.4.1 Other chains. There may be other chain models besides singular chains we might want to use. For singular chains one has to use the Eilenberg–Zilber theorem. This is for instance easier to track in cubical chains. Mostly we will be interested in either singular or cellular chains. Throughout we will denote singular chains by S and cellular chains by CC . When dealing with CW complexes one has the extra bonus of proving that the topological compositions indeed give rise to cellular maps. Additionally in some particular cases one may obtain special chain models that work for a given operad in a special situation, although there is no general a priori guarantee that the construction is valid. This then of course can and has to be checked a posteriori. We will just use Chain to denote any operadic chain model. In all the models we consider, one has families representing the chains. We will from now on treat families and leave open the specification of a particular chain model. We will mostly use singular or cellular chains in the following.

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2.5 Open/cellular chains Arcsg .n/ (unlike DArc) is a subspace of the CW complex Asg;r . That complex has cells CŒ˛ indexed by classes of arc graphs ˛. For Asg;r and the partial gluing structure, we can use cellular chains. Some of the suboperads/PROPs of Arc actually have homotopy equivalent CW models whose cellular chains are models for them on the chain level. These will give us the solution to Deligne’s conjecture and it generalizations up to and including string topology. Unfortunately Arc itself is not outright a CW complex, since the condition of being exhaustive is not necessarily stable under removing arcs. We can however consider the complex CC .Asg;r ; Asg;r n Arcsg .r  1//. Alternatively, Arcsg .r  1/ is also the disjoint union of open cells and hence filtered by using the dimension of cells Arcsg .r  1/ D qŒ˛WŒ˛ is exhaustive CP Œ˛

(2.4)

where CP denotes the open cell. And although we cannot use cellular chains, we can work with the free Abelian group generated by the open cells which is denoted by Co .Arcsg .r  1//. Each generator is given by an oriented cell. Such a cell is given in turn by an arc graph. The dimension is the number of arcs minus 1. There is a differential, which deletes arcs, as long as the result is still exhaustive. In order to get an operad structure on Co .Arc/, we recall the following facts from [23]: As sets, we have (2.5) CP Œ˛ Bi CP Œˇ  D q2I.Œ˛;Œˇ / CP Œ  where I.Œ˛; Œˇ/ is a finite index set of arc graphs on the glued surfaces [23] running through all the graphs that appear as the underlying graphs of the composed families. If ˛ has k arcs and ˇ has l arcs then, if two conditions are met, for any weights, generically the number of arcs in .Œ˛; Œwt/; .Œˇ; Œwt 0 / is kCl 1. The two conditions are that (1) there are no closed loops and (2) that not both arc families are twisted at the boundary at which they are glued. In these cases, the dimension of the composed cell drops. Overall the composition respects the filtration by dimension. Moreover, the “bad” part, that is the locus, where the families glue together to form families with less than the expected graphs is of codimension at least 1, if the two families are not twisted at the boundary simultaneously. In that case the number of arcs in the composition generically already has one less arc than expected. In the top dimension the composition map is bijective. The best way to treat the operadic structure is to pass to the associated graded Gr Co .Arc/ of Co .Arc/. In [23] we showed that Theorem 2.9. The Abelian groups Co .Arc/.n/ D

a g;s

Co .Arcgs .n//

Chapter 6. Arc geometry and algebra

and Gr Co .Arc/ D

a g;s

GrCo .Arcgs .n// D

a

273

CC .Asg;r ; Asg;r n Arcsg .r  1//

g;s

are cyclic operads. Here, the compositions are given by X ˙CP .i / CP .˛/ Bk CP .ˇ/ D

(2.6)

i2I

where ˙ is the usual sign corresponding to the orientation and I is a subset of I.˛; ˇ/. More precisely: (1) I is empty if both families are twisted at the glued boundaries, (2) I runs over the arc graphs which are not obtained by erasing arcs, and (3) in the graded case Gr, the arc family also has the expected maximal number of arcs. 2.5.1 Discretizing the chain level. When dealing with the chain level and the actions, there are two modifications for the discretization (2.3). The first is to add appropriate signs. The signs are given in [24] and are discussed in the appendix. Here we will just use the fact that such appropriate signs exist. The second is to adjust the gluing to fit with that of relative chains, that is to use the modification (1) or both modifications (1) and (2) of Theorem 2.9. On the discrete side, there is again a filtration by the degree of the underlying arc family. For the action we will use the following. Theorem 2.10 ([24]). For ˛ an arc graph with k arcs, let X ˙˛ n P .˛/ WD

(2.7)

n E 2N k

with the appropriate sign. Then for two arc graphs ˛ and ˇ we have: P .˛ Bi ˇ/ D ˙P .˛/ Bi P .ˇ/, where ˙ denotes again the appropriate sign and the two Bi are taken with the same modification. In particular using both modifications for the gluing P gives an operad morphism from Co .Arc/ to the operad of N-weighted arc families (modified with appropriate signs, see [24]. Using the respective associated grading on the discrete P gives an operad morphism from CC .Asg;r ; Asg;r n Arcsg .r  1// to the associated graded. Remark 2.11. Here ˛ Bi ˇ is the gluing given by gluing the open cells and enumerating the indexing set of the resulting cells. A priori this can be any gluing between two boundaries that are hit, but also between two empty boundaries. With the obvious modification this applies as well to gluing an empty boundary to a non-empty one by erasing.

2.6 Modular structure: the approach of [32] One can ask whether there is a modular operad structure for Arc. The challenge is to add self-gluings. One can readily see that the partial operad structure on the

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D.n/ is indeed modular. One can glue any two boundary components on connected or disconnected surfaces as foliations in the above manner, if the weights on the two boundaries agree. It is not possible in general, however, to scale in order to obtain self-gluings without restrictions, at least at the topological level. The reason for this is that the R>0 action scales the weights on all boundaries simultaneously, so if they do not agree one cannot change them to agree merely by the action. There is however a flow, which one can use, to make them agree. This is a very intricate procedure which even works in families. It is contained as one result in [32]. We will content ourselves with just stating the main result as it pertains to the discussion here. Theorem 2.12. The homologies H .DArc.n// D H .Arc.n// form a modular operad using g as the genus grading. Moreover, it is induced from a modular operad structure up to canonical homotopies on the chain level. That structure is obtained from the R>0 colored topological operad structure on DArc.n/ via flows.

2.7 S 1 action We have already seen that A00;2 ' S 1 . Since all the elements are exhaustive in this case, we obtain that Arc20 .1/ D A00;2 . We now show that this is even true as groups, where the composition in Arc is given by B1 . 2.7.1 Group structure. Pick two elements t , s with s; t 2 Œ0; 1/ as depicted in Figure 5. Gluing them together there are two situations, namely (i) s C t  1 or (ii) 1 < s C t < 2. In the first case we immediately see that t B1 s D sCt ; in the second situation we see that after gluing the outer two strands become parallel and indeed t B1 s D sCt1 .

0

1 1

1 s

0

1 1

Figure 5. The CW complex A00;2 D S 1 . We indicated the base point and a generic element.

Now via gluing there is an .S 1 /nC1 -action on each Arcsg .n/. Let us enumerate the Cartesian product of .S 1 /nC1 as having factors 0; : : : ; n. Then the factors i D 1; : : : n act via i .t /.Œ˛; Œwt/ D Œ˛; Œwt/Bi t and the 0 component acts as 0 .t /.Œ˛; Œwt/ D

t B0 Œ˛; Œwt/.

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s 1

1 s

1

s

D

1

I

1 1 s

II Figure 6. I. The identity. II. The arc family ı yielding the BV operator.

2.7.2 S 1 action as twisting and moving the base point. If we look at the arc picture II B or C, in the enumeration in § 1.4, we can nicely describe the geometry of this action. It simply moves the basepoint around the boundary in the direction of its orientation. The distance is given by the transverse measure of the bands as in §2.1. If the basepoint moves into a band, it simply splits it. Definition 2.13. An arc family is called untwisted at a boundary i if no two arcs are parallel after removing the base point of the boundary I in picture II. Otherwise it is called twisted at i. The elements of Asg;r and Arcsg .n/ are twisted or untwisted if their underlying arc families are. We will also say that an arc family is twisted or untwisted if arcs become parallel or not after removing all basepoints at the boundaries. In the pictures II B and C we can see the twisting more explicitly. An element is twisted at i if the basepoint of @i is inside a band which is not split at the other end. An example of a twisted arc family is given by t . This is even the general case in the following sense. Lemma 2.14. Any element of Arcsg .n/ lies in the orbit of an untwisted element under the .S 1 /nC1 -action. Proof. Just use the action to slide the basepoints at the different boundaries out of any band they might be inside of. In this way we obtain an element which is untwisted at each boundary. Now it can happen that there are arcs which become parallel only after removing both the basepoints of the boundaries they run between. In this case, we can move the points in sync outside of the band. Proposition 2.15. There is an action of .S 1 /nC1 on Arc.n/. Proof. The action is given by .0 ; : : : ; n /˛ D . .. 0 B1 ˛/ B1 1 / / Bn n /. The fact that this is indeed an action follows from the associativity of the operadic compositions. We denote the coinvariants by Arc.n/S 1 .

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2.8 Twist gluing There is an additional gluing we can perform which yields an odd structure on the chain and the homology level. This is inspired by string field theory and it gives rise to a second type of Gerstenhaber and BV structure on the homology and chain levels. For this we let be the chain given by I ! Arc.1/, t 7! t . Notice that this chain represents a generator of the homology H 1 .Arc00 .1//. Definition 2.16. Given two chains ˛ 2 S .Arc.n// and ˇ 2 S .Arc.m// we define the twist gluing ˛ i ˇ to be the chain obtained from the map n  I  m ! Arc.m C n  1/ given by ˛ Bi B1 ˇ. Theorem 2.17. On the chain and homology level the operations Bi induce the structure of an odd cyclic operad. Furthermore, H .Arc.n// is an odd modular operad, also known as a K-modular operad. Likewise the chains on the S 1 co-invariants Arc.n/.S 1 / form an odd cyclic operad and H .Arc.n/S 1 / is an odd modular operad. Proof. It is easy to see that on the chain level the twist gluing is of degree 1, which is precisely what we need for the odd versions of the operadic structures. This immediately shows the first two claims. Notice that the S 1 coinvariants do not form an operad by themselves. To define the gluings, we simply choose representatives, glue and take coinvariants again. This is independent of choices, since any two lifts differ by an S 1 action on the boundaries and these get absorbed into the family. L Corollary L 2.18. The direct sums of cyclic group coinvariants n .H .Arc.n///CnC1 and n L Chain.Arc.n//CnC1 carry a Gerstenhaber bracket and that of SnC1 coinvariants n H .Arc.n//SnC1 carries a BV operator. An analogous result holds for the S 1 coinvariants.

2.9 Variations on the gluings We have already deviated a bit from the original gluing to obtain the modular structure on homology. There are several other variations on the basic gluing, which are necessary and helpful. Usually these do not alter the picture on the level of homology. 2.9.1 Local scaling. In the gluing of DArc we scaled both surfaces in order to obtain an associative structure. To make the two weights match, we could also just locally scale the width of only those bands incident to @i .F / and/or those incident to @0 .F /. In fact for gluing disjoint surfaces this is exactly done by the first type of flow in [32]. What happens in this case is that the gluings are not associative any longer, but there is a homotopy between the two different ways to compose. This guarantees a bona fide associative structure on homology. It is rather surprising that in several situations,

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277

notably that of string topology, there are already strictly associative chain models, – see §3.3 and §4.3.2 below. 2.9.2 Erasing. We have not discussed how to glue a boundary of weight 0 with one of non-zero weight. The natural idea is to simply erase all bands incident to the boundary which is glued. Indeed this is sometimes the answer. There are two caveats however. First, this operation is again not associative. Second one has to be careful that iterating such gluings one does not obtain an empty arc family. One such undesirable situation occurs when one tries to use this type of gluing to extend the operad structure to Asg;r . Then one could obtain an empty family, but adding it would entail making the spaces contractible and hence kill all homology information. However this type of gluing is used in string topology. The trick here is that the empty family does not appear due to the conditions that are placed on the graphs to make them part of the Sullivan-PROP. 2.9.3 Wilting. The last modification is that instead of erasing, one lets the leaves wilt. Technically this can be formalized by adding wilting weights at the boundary. A wilting weight is an assignment of a length in R0 to each interval of @i n Wi . The source of the map ci is then the full interval with the induced measure. For the source of the map L , we contract the intervals just as before. The effect on the maps li is that they are stationary on these intervals. A foliation interpretation is as follows: on top of the foliation on the surface, we also consider a compatible germ of a trivial transverse measured foliation of a neighborhood of each boundary. Here trivial means that all leaves are homotopic to a meridian of the cylinder. Compatible means that the restriction of the given partial measured foliation of the surface is a sub-foliation. The other leaves are called wilted leaves. Notice that the weight can be zero, which means that the band is empty. For the gluing of these foliations we use the modified maps ci and proceed as before. Upon gluing regular leaves to wilted leaves, the leaves wilt and are erased from the surface foliation, but kept for the foliations near the boundaries which are not glued. This gluing is used in section 5.1 to add units.

3 Framed little discs and the Gerstenhaber and BV structures 3.1 Short overview One extremely important feature of the Arc operad is that it contains several suboperads that are quasi-isomorphic to classically important operads. The main ones are spineless cacti which are equivalent to the little discs, cacti which are equivalent to the framed little discs and the corrolas which are equivalent to the tight little intervals suboperads.

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The first two structures are responsible for Gerstenhaber and BV algebras on the homology level. This is what gives rise to string topology brackets and operators as well as solutions to various forms of Deligne’s conjecture. In our approach we get a version of these algebras up to homotopy on the cell level which has all homotopies explicitly given. One nice upshot is the new symmetry of the BV equation which now manifests itself as a completely symmetric 12 term identity which is geometrically nicely described by a Pythagorean like triangle, see Figure 11. On the topological level these operads give rise to loop space structures. Going a bit further the stabilization of the Arc operad gives rise to a filtered sequence of so-called Ek , k 2 N [ f1g, operads. These detect k-fold loop spaces respectively infinite loop spaces, see the Background section 3.1.1 below. Without the stabilization Arc contains a E2 in the form of spineless cacti and an E1 operad in the form of corollas, both which are defined below. There is one technical detail, namely, whether or not to include a 0 component in the operad. We will call operads with such a component pointed.1 For the chain level and the algebraic structures it is enough to have the non-pointed version of E2 . For the topological level and e.g. loop space detection it is necessary to have the pointed versions. For this one has to enlarge the setup by “fattening” the operads. The details are given in §5.1. Theorem 3.1. The Arc operad contains suboperads Cor, Cact and Cacti. Cor is an E1 operad that is it is equivalent to the little intervals, Cact is an E2 operad equivalent to the little discs, Cacti are equivalent to the framed little discs. This is as non-pointed operads. Corollary 3.2. (1) On the topological level: Any algebra over the group completion of Fat Cor has the homotopy type of a loop space. Any algebra over the group completion of Fat Cact has the homotopy type of a double loop space. Any algebra over the group completion of Fat Arc has the homotopy type of a double loop space. (2) On the homology level: Any algebra over H .Cact/ is a Gerstenhaber algebra. Any algebra over H .Cacti/ is a BV algebra. Any algebra over H .Arc/ is a BV algebra. (3) On the chain level: Any algebra over Chain.Cor/ is a homotopy associative algebra. Any algebra over Chain.Cacti/ is a homotopy Gerstenhaber algebra. Any algebra over Chain.Cacti/ is a homotopy BV algebra. Any algebra over Chain.Arc/ is a homotopy BV algebra. (4) Cellular chains: For Cor, Cact and Cacti there exist CW-models where the up to homotopy structures of the relevant algebras are given by explicit chains.

1Another common name for these are unital operads. This is however confusing, since this could also mean that there is a unit in the 1 component of the operad. This is the case for all the operads we consider.

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279

3.1.1 Background. One role of linear operads, that is those based on (complexes of) vector spaces, is that they can encode certain algebraic structures. Among these are associative, commutative, Lie, but also more complicated algebras like pre-Lie, Gerstenhaber algebras and BV algebras. We will say that an operad represents a type of algebra if the algebras over this operads are precisely of the given type. For a vector space to be an algebra over an operad means that for each element of the operad there is an associated multi-linear operation and these operations are compatible with all the operad structures. Moreover some of these linear operads are actually the homology of a topological operad. This provides the geometric reason for the appearance of certain types of algebras. If the topological operad acts on a space at the topological level, the homology of this operad acts on the homology of this space. This provides algebraic structures on these homologies. In this type of setup it is clear that one can replace the operad by a different one if they have the same homology operad. The correct notion for this type of equivalence is the one induced by quasi-isomorphism. One of the questions that arises is to what extent linear actions can be lifted to the chain or topological level. On the chain level we are dealing with a dg structure and the algebras are of the type of the algebra over the homology, but only up to homotopy. The two classical examples we will consider are the little discs and the framed little discs. Theorem 3.3 ([5], [14]). An algebra is a Gerstenhaber algebra if and only if it is an algebra over the homology of the little discs operad D2 . An algebra is a Gerstenhaber algebra if and only if it is an algebra over the homology of the framed little discs operad f D2 . On the topological level the relevant theorems are: Theorem 3.4 ([53], [3], [41]). A connected space has the homotopy type of a loop space if and only if it is an algebra over the A1 operad with base point. If a connected space is an algebra over an Ek operad, it has the homotopy type of a k-fold loop space, for k 2 N [ f1g with base point. Here the Ek are the little k-cubes operads and being an Ek operad means that the operad is equivalent to an Ek operad. For instance the little discs are an E2 operad. There is a subtlety here wether or not to include the base point, see §5.1. The Arc operad itself contains E0 , E1 and E2 operads as well as the framed versions. To obtain the higher Ek operads one can stabilize as in [27] and §5.1. We will now make the structures present in Arc explicit. Note that this gives an explicit chain level version of these operads which in turn gives explicit 1 or better “up to homotopy”-versions of the respective algebras. That is for example an explicit notion of Gerstenhaber algebra up to homotopy or BV algebra up to homotopy. These explicit up to homotopy versions are extremely adapted to describe natural actions. This is

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one of the “miracles” of the theory: “The geometry of foliations chooses the correct algebraic model”. The Sullivan quasi-PROP is a rigorous incarnations of the idea of Chas–Sullivan on string topology. This is a PROP up to homotopy, which contains homotopy versions of the two suboperads above. It is designed to furnish even more operations on homology of loop spaces. This meshes well with the above results. In particular, we will exhibit such an action by using the Hochschild co-chain approach. The main Theorem being Theorem 3.5 ([17], [6]). If M is a simply connected manifold then H .LM / ' HH  .S  .M /; S .M //: The fact that the suboperads act are versions of Deligne’s Hochschild conjecture. We will give the details below.

3.2 (Framed) little discs and (spineless) cacti The operad of framed little discs appears as a suboperad as follows. Definition 3.6. The operad Cacti is the suboperad of DArc given by the surfaces with weighted arc families, which satisfy (1) g D s D 0, (2) there are only arcs which run from @0 to @i , where i ¤ 0. The operad Cact called spineless cacti operad is the suboperad of Cacti where additionally (3) for any two arcs e1 , e2 incident to @i with e2 0 , where the R>0 factors simply keep track of wt.@i /. Dropping the R>0 factors, the spaces loose their operadic structure, since the condition (*) is not preserved upon gluing. The way out is to use the local scaling version of the gluing. For two elements ˛ 2 Cacti1 .n/ and ˇ 2 Cacti1 .m/ we define ˛ B1i ˇ to be the weighted arc family obtained by scaling all weights of arcs incident to @i of ˛ homogeneously by the factor m. Proposition 3.11 ([21]). The operation B0i preserves the condition . / and it also preserves the conditions of spineless cacti. Therefore, B0i W Cact1 .n/  Cact1 .m/ ! Cact1 .m C n  1/ and B0i W Cacti1 .n/  Cacti1 .m/ ! Cacti1 .m C n  1/. Theorem 3.12 ([21]). Both Cact1 and Cacti are CW complexes.

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Chapter 6. Arc geometry and algebra

The operations Bj0 are symmetric group invariant, associative up to homotopy and cellular. Moreover, the induced operations CC .Bj0 / induce a bona fide operad structure on the collection of cellular chains. Finally the induced operad structure on homology agrees with the one induced from DArc. 3.3.1 Explicit representatives for the bracket and the BV equation. The points in Arc00 .1/ D Cacti.1/ are parameterized by the circle, which is identified with Œ0; 1, where 0 is identified to 1. As stated above, there is an operation associated to the family ı. For instance, if F1 is any arc family F1 W k1 ! Arc00 , ıF1 is the family parameterized by I  k1 ! Arc00 with the map given by the picture by inserting F1 into the position 1. By definition,

D ı 2 C1 .1/: In C .2/ we have the basic families depicted in Figure 8 which in turn yield operations on C .

s b a

b

a

s

1

a

1 1

The dot product

b

s

The star

s

1

ı.a; b/

Figure 8. The binary operations.

To fix the signs, we fix the parameterizations we will use for the glued families as follows: we say the families F1 , F2 are parameterized by F1 W k1 ! Arc00 and F2 W k2 ! Arc00 and I D Œ0; 1. Then F1 F2 is the family parameterized by k1 k2 ! Arc00 as defined by Figure 8 (i.e., the arc family F1 inserted in boundary a and the arc family F2 inserted in boundary b). Interchanging labels 1 and 2 and using as the explicit chain homotopy given in Figure 9 yields the commutativity of up to chain homotopy: d.F1 F2 / D .1/jF1 jjF2 j F2 F1  F1 F2 :

(3.1)

Notice that the product is also associative up to chain homotopy. Likewise F1 F2 is defined to be the operation given by the second family of Figure 8 with s 2 I D Œ0; 1 parameterized over k1  I  k2 ! Arc00 .

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By interchanging the labels, we can produce a cycle fF1 ; F2 g as shown in Figure 9 where now the whole family is parameterized by k1  I  k2 ! Arc00 , fF1 ; F2 g WD F1 F2  .1/.jF1 jC1/.jF2 jC1/ F2 F1 : ab

2 1

s

s

1 1

1

2

2

fa; bg

1

1 1

s

s

2 1 .jaj C 1/.jbj C 1 / . 1/b  a

Figure 9. The definition of the Gerstenhaber bracket.

Remark 3.13. We have defined the following elements in C : ı and D ı in C1 .1/;

in C0 .2/, which is commutative and associative up to a boundary. and f; g in C1 .2/ with d. / D  and f; g D  . Note that ı; and f; g are cycles, whereas is not.

3.4 The BV operator The operation corresponding to the arc family ı is easily seen to square to zero in homology. It is therefore a differential and a natural candidate for a derivation or a higher order differential operator. It is easily checked that it is not a derivation, but it is a BV operator.

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Proposition 3.14. The operator satisfies the relation of a BV operator up to chain homotopy:

2  0;

.abc/  .ab/c C .1/jaj a .bc/ C .1/jsajjbj b .ac/  .a/bc jaj

 .1/ a .b/c  .1/

jajCjbj

(3.2)

ab .c/:

Thus, any Arc algebra and any Arccp algebra is a BV algebra. Lemma 3.15. ı.a; b; c/  .1/.jajC1/jbj bı.a; c/ C ı.a; b/c  ı.a/bc:

(3.3)

Proof. The proof is contained in Figure 10. Let a W ka ! Arc00 , b W kb ! Arc00 and c W kc ! Arc00 be arc families, then the two parameter family filling the square is 1

1

s

c

s

a b

1

Œjaj C 1/jbj . 1/bı.a/c

1

ı.a/bc

t c

1

a 1

b

t

.1

1

s.1 .1

c

st

a

t/

s/.1

1

s/t

c b

t/

1

t

t

a b

1

ı.a; b/c Œjaj C 1/jbj . 1/bı.a/c

1 c a

s 1

s

b

1

ı.a; b; c/

Figure 10. The basic chain homotopy responsible for BV.

1

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parameterized over I  I  ka  kb  kc . This family gives us the desired chain homotopy. Given arc families a W ka ! Arc00 , b W kb ! Arc00 and c W kc ! Arc00 , we consider the two parameter families given in Figure 11, where the families in the rectangles are the depicted two parameter families parameterized over I  I  ka  kb  kc and the triangle is not filled. Its boundary is the operation ı.abc/. 1 c 1 1

1

c

a

a

t

1

b 1

s.1

s .1

c

1 a

1

c

s

s

s

1

.1

b s.1 .1

1

b

t/ s/.1

1

s/t

c 1

a b s

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1

c

b

1 c

t

b

t c 1

a b

1 1 1

1 b 1

1

c 1

t/

1 a

a 1

t/

c 1

s/t

s

c

t/ s/.1

b

s s

c

s

1

.1

a

1 1

c

1 1

1 b c t 1 t a 1

1 s.1

s

a b

1

.1

b 1

1

a

b 1

t/

1

1

c

b 1

a b a

c 1

a

t/ s/.1

s/t

a b

t t

t

1 1 1

b 1

.1

1 c

t b

a

c

b

s s

1

1

t

1 a

t c t a 1

s s a 1

Figure 11. The homotopy BV equation.

From the diagram we get the chain homotopy for BV. The threefold operation consists of three terms from the boundary of the inner triangle, and this is homotopic to nine terms given by the outside sides of the three rectangles. This makes the BV equation a highly symmetric twelve term equation.

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Remark 3.16. The fact that the chain operads of Arc and as we show below Cact.i / or Cact 1 .i / all possess the structure of a G(BV) algebra up to homotopy means that for any algebra V over them the algebra as well as HomV have the structure of G(BV). If one is in the situation that one can lift the algebra to the chain level, then the G(BV) will exist on the chain level up to homotopy. Remark 3.17. We would like to point out that the symbol  in the standard super notation of odd Lie-brackets fa  bg, which is assigned to have an intrinsic degree of 1, corresponds geometrically in our situation to the one-dimensional interval I .

3.5 The associator It is instructive to do the calculation in the arc family picture with the operadic notation. For the gluing B1 we obtain the elements in C2 .2/ presented in Figure 12 to which we apply the homotopy of changing the weight on the boundary 3 from 2 to 1 while keeping everything else fixed. We call this normalization.

1 s

1 1

2 1

s

B1

1

t 2

t

t

D s

1

1 2

1

1

t

t

 1

s

1 2

s

1 1

t

1

s

3

3 2

1

Figure 12. The first iterated gluing of .

Unraveling the definitions for the normalized version yields Figure 13, where in the different cases the gluing of the bands is shown in Figure 14. The gluing B2 in arc families is simpler and yields the gluing depicted in Figure 15 to which we apply a normalizing homotopy – by changing the weights on the bands emanating from boundary 1 from the pair .2s; 2.1  s// to .s; 1  s/ using 2s; 1Ct .1  s// for t 2 Œ0; 1. pointwise the homotopy . 1Ct 2 2 Combining Figures 13 and 15 while keeping in mind the parameterizations we can read off the pre-Lie relation: F1 .F2 F3 /  .F1 F2 / F3  .1/.jF1 jC1/.jF2 jC1/ .F2 .F1 F3 /  .F2 F1 / F3 /

(3.4)

which shows that the associator is symmetric in the first two variables and thus following Gerstenhaber [G] we obtain: Corollary 3.18. fF1 ; F2 g satisfies the odd Jacobi identity.

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1

2s 3

1

1

3

2

1

2s

1

1 2s

2

2s

1

2s

3

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2

1

1

1

1

t

1

2s t

1

1

2

1 2s C t

t

3

II 2s < t

1

1

1 t t 3 2 1 1

IV t < 2s < t C 1

t

1 t 2 3 1 1

3

1

1

3 1

1

t

2s

2.1 s/ 2s .tC1/

t

2

3

2

1

1

1

VI 2s > t C 1

1

2

VII s D 1

1

III 2s D t

1

1

t

V 2s D t C 1 1

2

1

2

IsD0

3

1

t

t

3

2

t

t 2s

3

2 2

1

1

2

3 1 2s 1 1

2.1 s/

2

2s 1

3

1

3

1

1

Figure 13. The glued family after normalization.

I sD0 t

1

2

1

II 2s < t t

t

1

1

2s

2.1

s/

IV t < 2s < t C 1

III 2s D t t

t

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1

2s

2.1

s/

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1

t

2s

1

2.1

t

s/

V 2s D t C 1

VI 2s > t C 1

t

t

2s

1

1

2.1

t

s/

2s

Figure 14. The different cases of gluing the bands.

1

1

VII sD1 t

2.1

t

s/

1

2

1

t

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1 1 s

2 1

1 1

s

ı2

1

t 2 1

t

D

2s

2 t

1 2.1

1

s/

t

3 1



s

t

1

2 1

s t

3 1

Figure 15. The other iteration of .

4 Moduli space, the Sullivan-PROP and (framed) little discs One of the applications of the arc operad is to CFT and string topology. In principle, moduli space is a “suboperad” of the arc operad and the Sullivan-PROP is a quasiPROP generalization that works for a partial compactification of a subset of the arc operad in its ambient spaces Asg;r . This quasi-PROP is also a generalization of two bona fide suboperads of the arc operad which are equivalent to the well known little discs and framed little discs operads. These operads are responsible for the preeminent algebraic structures found in CFT and string topology, the Gerstenhaber bracket and the BV operator. In order to set up everything completely rigorously for the compositions a little finesse is needed.

4.1 Moduli spaces 1;:::;1 Let Mg;r;s be the subset of elements of Asg;r whose arc families are quasi-fillings. Here the superscript 1 is repeated r times. 1;:::;1 . From the description in terms of the dual ribbon If s D 0, we simply write Mg;r graphs 1.5.1 the following theorem can be obtained using Strebel differentials (see e.g. [23]) 1;:::;1 Theorem 4.1. The space Mg;r is proper homotopy equivalent to the moduli space of Riemann surfaces of genus g with n marked points and a tangent vector at each point modulo the free and proper scaling action of R>0 which scales all tangent vectors simultaneously. 1;:::;1 The homotopy equivalence can be lifted to the product with R>0 thus lifting Mg;r 0 to Dg;r and reversing the quotient by R>0 on the moduli space side. In fact, using the hyperbolic approach, Penner was able to identify the moduli space 1;:::;1 for arbitrary s. Define the “moduli space” M D M.F / of corresponding to Mg;r;s the surface F with boundary to be the collection of all complete finite-area metrics of constant Gauss curvature 1 with geodesic boundary, together with a distinguished

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point pi in each boundary component, modulo push forward by diffeomorphisms. There is a natural action of RC on M by simultaneously scaling each of the hyperbolic lengths of the geodesic boundary components. 1;:::;1 is proper homotopy equivalent to the quotient Theorem 4.2 ([47]). The space Mg;r;s M=RC . 1;:::;1 Theorem 4.3. If 3g  2n C 3 > 0 the .S 1 /r coinvariants of the subspace Mg;r is homeomorphic to the moduli space Mg;r .

4.2 Operad structure on moduli spaces A natural question to ask is whether the operad structure on Arc can be restricted to the 1;:::;1 quasi-filling families given by the subspaces Mg;r;s . This is not true on the nose. In fact on a codimension-1 set, the gluing of two quasi-filling families might take us to a non-quasi-filling family. Generically this does not happen, though. A careful analysis was given in [23]. The upshot is that if ˛ has k arcs and ˇ has l arcs then generically ˛ Bi ˇ has k C l  1 arcs. And in this case, essentially by an Euler-characteristic argument, the resulting family is again quasi-filling. In order for the number of arcs on ˛ Bi ˇ to drop we need that two of the points which form the boundary of the bands in the construction §2.1.2 coincide. We introduced new terminology for this type of situation. A rational (cyclic) operad is an operad structure on a dense open subset. 1;:::;1  Arc.r  1/ forms a rational Theorem 4.4. The collection M.r  1/ WD qg Mg;r;s cyclic operad.

Things really work out on the chain level after passing to the associated graded. 4.2.1 Cell level for the moduli spaces. As in the case of the arc operad the moduli space is the disjoint union of open cells CP .Œ where now there is one cell for any given quasi-filling Œ. We let Co .Arc0# / be the subgroup of Co .Arc/ generated by the cells corresponding to quasi-filling arc families with no punctures and write Gr Co .Arc0# / for the image of this subgroup on Gr Co .Arc/. Notice that on this subset the cells indexed by an y The arc graph  can be equivalently thought of as indexed by the ribbon graph . y differential that removes arcs in  acts on  by contracting the corresponding edge. Definition 4.5. Following the usual arguments [36], [47], [48], [10], the graph complex of marked ribbon graphs is the Hopf algebra whose primitive elements are connected marked ribbon graphs and whose productPis the disjoint union. Its differential is given by the sum of contracting edges d  D e2E 0 ./ ˙=e, where E 0 ./ is the subset of edges e such that the topological type of  coincides with that of =e and the sign is the usual sign.

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0 /; d / is isomorphic to the graph Theorem 4.6 ([23]). The complex .Gr Co .Arc0# /.Fg;r complex of marked ribbon graphs. Both can be used to compute H  .PMC.Fg;s 0 //, 1;:::;1 the cohomology of the pure mapping class group and the spaces H  .Mg;r;0 /. The induced operad structure on the collection 0 / Gr Co .Arc0# /.r  1/ D qg;s Gr Co .Arc0# /.Fg;r 1;:::;1 /. is a cyclic dg-operad structure which descends as a cyclic operad to H  .Mg;r;0

These are the cellular operads which give rise to the Hochschild actions for moduli space, see §6.4.

4.3 The Sullivan quasi-PROP The arc operad or even the spaces Asg;r are inherently symmetric in all boundaries. This symmetry was a bit broken by designating the boundary 0 as special. The idea is that this is the output boundary, while the other n boundaries are the input boundaries. The full symmetry is restored in the cyclic setting. Keeping with the in- and output picture, we can add additional information by s . Technically this marks the move specifying input and output boundaries on Fg;r from operads to PROPs. In the PROP setting one composes, by gluing all inputs of one element to all outputs of another if their number matches. This setup is used to describe the string topology of Chas and Sullivan [8], [9], [56], [55], [57], [7]. Furthermore in the PROPic setting one usually does not demand that the surfaces are connected. This ensures the existence of so-called horizontal compositions (see §4.3.3 below and the appendix). To spell this out in our situation, using the standard models, we can consider disjoint unions Fgs11;r1 q qFgsk1 ;rk and consider their PMCs which are the products of the individual PMCs. The number of boundaries r of such a not necessarily connected surface is just the sum of the ri . An arc family on such a disjoint union is just the disjoint union of arc families on the individual surfaces. The slightly subtle points are (1) that the disjoint union is not strictly symmetric monoidal and (2) the enumeration of the boundaries. Given a possibly non-connected surface as above with r boundaries, we also fix n, m such that n C m D r and now separately enumerate n of the boundaries from 1 to n calling them input boundaries and enumerate the remaining m boundaries from 1 to m calling them output boundaries. We let D.n; m/ be the set of PMC orbits of weighted arc families on such surfaces. Technically, we again enumerate components by the total genus and the total number of punctures to get a break down of the space into finite-dimensional spaces and then take the colimit. Moreover, the components are indexed by k and further by tuples .g1 ; : : : ; gk /, .s1 ; : : : ; sk /, .r1 ; : : : ; rk /. To get a CW complex we could quotient out by a global scaling action. The space D.n; m/ has an Sn  Sm action which permutes the input and the output boundaries. We let D 0 .n; m/ be the families on surfaces without punctures.

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4.3.1 The spaces Definition 4.7. The spaces of Sullivan type arc families ull .n; m/  D 0 .n; m/ are the subspaces which satisfy that (1) arcs only run from inputs to the outputs or from the outputs to outputs, (2) all input windows are hit. We furthermore define the following subspaces: (1) the strict Sullivan arc families ull st .n; m/ which is the subset of families with arcs only from in to out boundaries. (2) ull 1 .n; m/  ull st .n; m/ where the condition is that the weight of each of the n input boundaries is 1. These spaces have the following properties: (1) ull .n; m/, has an Sn  Sm action which permutes the input and the output boundaries (separately). (2) ull .n; m/ retracts onto ull st .n; m/ simply by scaling the weights of all the arcs from output to output to zero. (3) The dual ribbon graphs of the arc graphs of ull .n; m/ are Sullivan chord diagrams in the sense that the cycles corresponding to the in boundaries can be disjointly embedded up to finitely many points of intersection. These circles are joined by edges corresponding to the arcs going from outputs to outputs. These are not present in ull st .n; m/. Remark 4.8. Notice that as the arc families neither have to be quasi-filling nor exhaustive, these dual graphs do not necessarily determine the topological type of even the number of boundaries of the surface they lie on. One can add this extra information if one chooses to do so. We will continue with the arc graphs since these are unambiguous as the surface they lie on carries this extra information. Notice that if DArc.n; m/ again denotes the exhaustive families in D.n; m/, then ull .n; m/ ª DArc.n; m/, and ull .n; m/ « DArc.n; m/ but also ull .n; m/ \ DArc.n; m/ ¤ ;. In the same fashion the quasi-filling families M.n; m/  D.n; m/ have non-zero intersection but have no containment relation with DArc.n; m/. 4.3.2 A CW model: ull 1 Proposition 4.9. The space ull.n; m/ deformation retracts to ull st .n; m/ and this in turn deformation retracts onto a smaller subspace ull 1 .n; m/ which is a CW complex.

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The cells are indexed by the classes of arc graphs Œ˛.2 And the dimension of a cell is given by the jŒ˛j  n. To see that these are cells and yield a CW complex, we can proceed as before. Given Œ˛ with n inputs, the arc families with that type and non-zero weights given the restriction are a product of n open simplices. The face maps are given as before by identifying the face with the open simplex corresponding to the family where an arc has been removed. Notice that since the total weight on each input boundary is one, the condition of all input boundaries being hit is stable under taking the boundary of the simplex – some arc always remains. The retraction is simply given as follows. First we can retract by scaling the weights of arcs going form outputs to outputs as before. Since there are no arcs from input to input boundary each remaining arc is incident to a unique boundary component. For each boundary component we now simultaneously scale the weights of all the arcs incident to it to make their sum equal to one. We can do this at each boundary separately or we can do it at all the boundaries at once. 4.3.3 (Quasi)-PROPic gluing. As mentioned above a PROP P is similar to an operad, but there are two main differences. The first is that there is a simultaneous gluing of all inputs to all outputs B W P .n; m/ ˝ P .m; p/ ! P .n; p/ which is associative and equivariant with respect to the various symmetric group actions. The intuitive example is again based on a vector space V over a field k: HomV .n; m/ WD Homk .V ˝n ; V ˝m / with composition. The composition will be given by a local scaling version of the scaling and its extension by erasing. As we have mentioned before, the local scaling version usually does not produce associative structures and this happens here as well. We do get a structure that is associative up to homotopy, however, which is what we defined to be a topological quasi-PROP. The second difference is that in the proper definition of a PROP there is also a horizontal composition  W P .n; m/ ˝ P .k; l/ ! P .n C k; m C l/ again associative and compatible with B and the symmetric group actions. For HomV the horizontal composition is given by the tensor product. In our case of ull as well as in most topological examples we add a horizontal composition by taking it to be the disjoint union. For this reason, we enlarged the spaces to include not necessarily connected components. The composition in ull is given as follows. Let a 2 ull .n; m/ and b 2 ull .m; p/ then we first prepare b so that the input boundary i of b has the same weight as the output boundary i of a. We do this by using the local scaling action as before. This action is naturally given by a flow which scales all the weights of the arcs incident to a given input boundary in the same fashion. Again we can prepare all input boundaries simultaneously, since each arc that hits an input boundary hits a unique such boundary. After this preparation step the weights on the boundaries 1; : : : ; n of 2 We

keep the same notation also in the non-connected case.

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a and b that are to be glued coincide and we just glue the surfaces and the foliations at all these boundaries. This type of gluing is not associative, but the fact that the preparation step is given by a flow ensures that there is a homotopy between two different ways of associating. This is basically done by using the flow in the reverse direction. The condition that there is no arc from outputs to outputs is preserved, since we only glue together arcs that run from inputs to outputs. Also neither the flow nor the gluing changes the total weights on the input boundaries of a so that the spaces ull 1 .n; m/ are stable. Theorem 4.10 ([23]). The composition B W ull.m; n/  ull.n; p/ ! ull.m; p/ is homotopy associative, symmetric, group invariant and compatible with the horizontal composition q W ull.m; n/  ull.k; p/ ! ull.m C k; n C p/; that is these spaces form a topological quasi-PROP. The subspaces ull 1 .m; n/ form a topological sub-quasi-PROP. The cellular chains CC .ull 1 / form a (strict) PROP. The last statement is not straightforward, it relies on an analysis of the gluing maps as in the case of cacti. Corollary 4.11. The Sn  Sm modules H .ull.n; m/; k/ form a PROP.

5 Stops, stabilization and the Arc spectrum 5.1 Stops: adding a unit 5.1.1 The little discs operad case. When considering the little discs operad D2 .n/, one has to be a bit careful whether or not one considers it pointed or not. In practice this means that one either includes D2 .0/ in the sequence of the D2 .n/ or not. D2 .0/ is just the big discs without any little discs inside. Notice that an element in the 0 component of an operad has no inputs. Gluing it into another element decreases the number of inputs by one. In the particular case of D2 , the zero component D2 .0/ is just a point and composition with it just erases the little disc it is glued into. This point is taken to be the base point of the operad. We will call D2 with the 0 component the little discs with base point. On the homology level, the inclusion of H .D0 ; k/ D k has the effect that the algebras over the little discs with base point are unital Gerstenhaber algebras. The unit of the algebra is just the image of 1 2 k, while without the base point the algebras are not required to have a unit. This is a general phenomenon. Including a contractible 0 component, mostly just a (base) point, to an operad whose algebras are some known type of algebra, restricts the algebras over it to be unital.

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On the topological level, especially for detecting loop spaces, the base point is needed as most of the construction works in pointed topological spaces. Indeed in Theorem 3.4 the versions of the Ek operads with base points are needed.

5.2 Adding a unit in the arc formalism We will show how to add a unit in this sense to the suboperads GTree of Arc as in [27] and also give the generalization, to the PROP ull .n; m/. The latter generalization has not formally appeared. GTreeg .n/  Arc0g .n/ is defined to be the suboperad which has arcs only running from the input boundaries i to the output boundary 0. This is the condition (2) of Definition 3.6. It was called the operad of Chinese trees in [31], the G stands for “higher genus”. We take GTree.n/ D qg GTree.n/. This is the same space as ull st .n; 1/=R>0 . There is an operadic inclusion of Cacti into GTree as the components of genus 0. The straightforward generalization to higher genus of Cact is the suboperad LG T ree which is comprised of the arc families satisfying the condition (3) of Definition 3.6. 5.2.1 Basic idea. As a first approximation to the unit, we could add a 0 component to Cacti by setting Cacti.0/ D pt. This point could represents a disc, without an arc family considered as having no inputs, but an output. If we simply erase the foliation upon gluing the result will not be associative. It can be seen that it is homotopy associative, by using a flow argument. That would be enough for the chain and homology level, but to get access to the topological theorems about loop spaces, one needs to have a strict operad structure. This is achieved through the process of wilting. So the unit will be a disc with wilted leaves on the boundary, see §2.9.3. The scaling action of R>0 is retained and thus we can assume the measure or the weight to be 1. When gluing in these discs, the leaves that are wilted are glued in the same fashion as in the standard gluing see §2.9.3. If we add this point then by gluing, we have to allow replacing any sub-band of leaves of an element of GTree by wilted leaves. In particular, to extend units to all of GTree, we also have to add one point for each genus GTreeg .0/ D pt represented by the surface of genus g with one output boundary and only wilted leaves at that boundary. 5.2.2 Details and results. For any of the spaces considered thus far, we define the fattened version by allowing wilting weights on the outputs as defined in §2.9.3 and denote the result by adding superscript Fat on the left. Proposition 5.1 ([27]). The spaces Fat

GTree.n/

together with Fat

GTree.0/ D qg Fat GTreeg .0/

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where GTreeg .0/ consists of a point which represents a surface of genus g with one boundary and a wilting weight (scaled to 1) on that boundary, form an operad under the gluing of §2.9.3. The reason why this works is that there are only arcs from the inputs to the output. Upon gluing some of these may wilt, but then the wilted leaves are again only part of the foliation near the output. In the same manner one can check that the conditions that (1) all the input boundaries are hit and that (2) unless the surface is just a disc, there is at least one arc, are stable under the gluing. Remark 5.2. We could also lift GTree to DArc, then the 0 component of genus g would be R>0 , representing the choice of allowing any weight for the wilted foliations. We would simply get a homotopy equivalent operad. In this spirit, we let Fat Cact and Fat Cacti be the respective operads whose arc families are in Cact respectively Cacti for n > 0 and whose 0 component is R>0 . Theorem 5.3. Fat Cact is equivalent to the pointed version of the little discs. is equivalent to the pointed version of the framed little discs.

Fat

Cacti

The first part of the theorem is contained in [27] and the second follows similarly. Corollary 5.4.

Fat

Cact as well as GTree detect double loop spaces.

5.2.3 Cacti with stops. If we regard the map loop L as defined in 2.9.3 the subspaces Cact and Fat Cacti get a very nice geometrical interpretation as cacti with stops. This just means that the parameterization of the outside circle may be constant for certain intervals. This point of view was explained in [30] and used by Salvatore [51] to provide the details of the announcement made by McClure and Smith [44]. This is the topological version the cyclic Deligne conjecture. Fat

5.2.4 PROP version. The same type of analysis leads to the PROP version of the above proposition. Proposition 5.5. The spaces PROP.

Fat

ull st .m; n/ as well as ull Fat .m; n/ form a quasi-

These statements have not appeared so far, but they follows in the same manner as the operadic counterparts.

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5.2.5 The E1 case. We have treated the full E2 case. For reference we also give the E1 restriction. The restriction is given by Fat Cor. It is easily seen that Cor are isomorphic to the tight little intervals, that is, partitions of the unit interval, while Fat Cor are isomorphic to the little intervals operad, which is the same as the little 1-cubes. 5.2.6 Further generalizations. One can also fatten the Arc as a cyclic operad to obtain the unital cyclic operad Fat Arc. When adding the necessary families obtained by gluing, one quickly realizes by gluing two cylinders that one can produce a cylinder whose boundaries both are only hit by wilted leaves. This then allows one given any arc family to let all its leaves wilt resulting in a surface with only wilted leaves. So in addition to the wilting weights the condition of Arc that every boundary is hit by an arc is changed to every boundary is hit by an arc or at least has a non-vanishing wilting weight. Proposition 5.6. The spaces t Fat obtained by adding wilting weights and using the above condition form a cyclic operad Fat Arc called the unital arc operad. The proof is straightforward.

5.3 Stabilization Stabilization is a process in which the Arc operad or its suboperads are glued together along the non-quasi filling families. We will call such families unstable. 5.3.1 Basic idea. The basic idea is that given an unstable family, we just delete any topology from the complementary region. This is easy to grasp if we only have one boundary component in a complementary region. In that case, which we call a genus defect, we wish to just replace the complementary region with a disc. The other case that can appear is that there are more boundary components, we call this a boundary defect. Again, we wish to just forget about such defects. This idea can be made rigorous by taking a colimit over a system of maps, which introduce boundary and genus defects. The result is then that the operad structure descends and we obtain the stabilized operad. If we restrict to the tree-like setting and add an identity, the resulting operad contains an E1 sub-operad. This fact leads to loop space detection and the Arc spectrum. 5.3.2 Technical details. As shown in [27], for GTree all the unstable elements can be obtained by gluing an unstable element from GTree.1/ to a quasi-filling one. That is, every a 2 GTree can be decomposed as a1 B1 a0 with a1 2 GTree.1/ and a0 quasi-filling, and we can furthermore decompose a1 into a sequence of standard generators and a

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quasi-filling element. The standard generators are Ta B1 Hb B1 Ta and Ta B1 G B1 Ta , where Ta is an element in GTree0 .1/ and G; Ha are given in Figure 16. Gluing on g these generators to the boundary 0 gives maps stH .a; b/ W GTreeg .n/ ! GTreegC1 .n/ g and stG .a/ W GTreeg .n/ ! GTreegC1 .n/.

1

a

1

a

Figure 16. The two basic unstable arc graphs G and Ha .

Definition 5.7. We define t GTree.n/ WD colim GTree.n/ where the colimit is taken g over the system of maps  generated by stGg .a/ and stH .b; c/ with a; b 2 Œ0; 1/ and c 2 .0; 1/. We will denote the image of a subspace by the prefix t , e.g. t LG T ree. Theorem 5.8 ([27]). The operad structure of GTree descends to t GTree. Moreover t LG T ree is a suboperad. Furthermore, the elements in t GTree have a unique quasi-filling representative. The proof goes through a standard form argument using the decomposition mentioned above.

5.4 Generalization to all of Arc The arguments of [27] generalize to the full arc operad by using the colimit over g gluing at all boundaries. So for i D 1; : : : ; n let stGg .i I a/ and stH .i I b; c/ be the maps of Arcg .n/ ! ArcgC1 .n/ given by ˛ 7! ˛ Bi .Ta B1 G B1 Ta / and ˛ 7! ˛ Bi .Tb B1 Hc B1 Tb /. For stGg .0I a/ we use the same definition as for stGg .a/, now g .0I a; b/. extended to all of Arc and likewise for stH Theorem 5.9. The spaces t Arc.n/ WD colim Arc.n/ where the colimit is taken over g the system of maps  generated by stGg .i I a/ and stH .i I b; c/ with a; b 2 Œ0; 1/ and c 2 .0; 1/ form an operad called the stabilized arc operad.

5.5 Stabilization and moduli space The situation about representatives is more complicated, the stabilization with respect to the two types of elements above still allows for non-quasi filling representatives.

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The new type of degeneracy comes from being able to find closed curves that do not intersect any of the arcs. We call a maximal choice of a system of such curves which are mutually non-intersecting a curve degeneracy system. If we cut along the curve degeneracy system, we get elements in the spaces Asg;r , since there will be some boundaries which are not hit. This is a partial operadic decomposition if we allow to glue along empty boundaries. It also becomes operadic, if we allow boundaries with only wilting leaves. By shrinking the curves in the degeneracy system they become double points and the representatives live in a Deligne–Mumford 1;:::;1 . The precise details will be given elsewhere, and they type setup of the space Mg;r;s should be compared to [39].

5.6 Stabilization and adding a unit. The E1 and Ek structures Combining the two procedures, stabilization and adding a unit, we end up with the operad versions Fat t . Here it is inessential in which order we do the two procedures. If we fatten first, then we wish to point out that in the stabilization step all elements in operad degree 0 become identified to the disc with wilted leaves on the boundary. That is, Fat t GTree.0/ D pt is a point and so is Fat t Arc.0/. For the latter operad, the resulting fattened spaces Fat t Arcs .n/ are contractible to the representative given s with no arcs, but some constant wilting weight on the boundary. Notice by F0;nC1 that these surfaces are fixed points under the SnC1 action. If we restrict to s D 0, the spaces Fat t Arc0 .n/ are contractible, but as the Sn action is not free we do not get E1 operads. Staying within Fat t GTree however we obtain an E1 operad and along with it a filtration by Ek operads. We define Fat LG T ree analogously to its non-thickened counterpart. Theorem 5.10 ([27]). The operad structure of Fat GTree descends to Fat t GTree and Fat t LG T ree is a suboperad. Using the same arguments as in loc. cit. one obtains: Theorem 5.11. The operad structure of Fat Arc descends to Fat t Arc and Fat t GTree is a suboperad. The most important structure theorem is then the following. Theorem 5.12 ([27]). There exists a filtration of Fat t LG T ree by suboperads Fat t LG T reek . The operads Fat t LG T reek are Ek operads and the operad Fat t LG T ree is an E1 operad. The proof uses Fiedorowicz’s theorem, see Berger, Theorem 1.16. For this one constructs so-called cellular Ek operads k 2 N [ f1g and according to [1] these are Ek operads.

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Corollary 5.13. The sub-operads ft LG T reek .n/; n > 0g with zero wilted weights are equivalent to fCk .n/; n > 0g where Ck are the non-pointed k cubes and the operad ft LG T ree; n > 0g is an E1 operad without a 0-term. Moreover Fat t LG T ree acts on Fat t Arc and thus: Theorem 5.14. The group completion of Fat t Arc is homotopy equivalent to an infinite loop space and hence gives rise to an infinite loop space spectrum, the Arc spectrum. An analogous statement holds for Fat t GTree as stated in [27]. 5.6.1 The Ek -operads in detail: the hemispherical decomposition of S 1 . In order to identify the Ek suboperads, one uses so-called degeneracy maps. These are simply given by gluing in the element of t GTree.0/ into the i -th position. In particular, if one glues into all but the i th and j th position, one obtains a map ij W t LG T ree.n/ ! t LG T ree.2/. This space retracts to S 1 . We let Ek˙ be the upper and lower hemisphere of S k  S 1 . As is well known these are the cells of a CW decomposition of S 1 . Proposition 5.15 ([27]). t LG T ree.2/ retracts to S 1  R2>0 . The preimage of Ek˙ R2>0 under the retraction are arc families whose underlying graphs for EkC R2>0 are those given in Figure 17, while those for Ek  R2>0 are simply the image under interchanging the labels 1 and 2. The factor of R2>0 is simply the weight wt.@i / on the boundaries 1 and 2. This implies that Fat t LG T ree.2/ retracts onto S 1 by scaling the wilting weights to zero. T Definition 5.16. tLG T reek .n/ D i;j 2f1;:::;ng ij1 .S k  R2>0 /. This means these are those stabilized arc families which land in S k  R2>0  S 1  R2>0 under all the maps i;j . We set Fat t LG T reek .n/ to be those arc families which lie in t LG T reek .n/ after forgetting the wilting weights, or, equivalently, those families which under the above retraction land in S k . The above theorem also asserts that these are indeed suboperads.

5.7 CW decomposition and [i products For t LG T ree, there is a CW model, given simply again by normalizing the weight on the input boundaries to be 1. Since Fat t LG T ree.n/ retracts onto its suboperad t LG T ree.n/ for n > 0, we also get a chain model for the fattened version. If we want to include a 0 component on the chain level, we simply take it to be k, the ground field, represented by the disc with the empty graph and the action given by erasing arcs.

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This CW model for t LG T ree.2/ is exactly S 1 in its hemispherical decomposition. The cells of the upper hemispheres give the chain representatives for the [i products. These are given in Figure 17. Recall that [iC1 is a homotopy between [i and 12 .[i /. This is again an incarnation of S 1 with S 0 being the two orders of the multiplication . This is made explicit in Figures 18 before the stabilization and 19 after applying stabilization. This may also serve as a good general example of how the stabilization works.

1

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2

2

Figure 17. The [i operations for i even and i odd. These are also the cells for upper hemispheres of S 1 in its hemispherical decomposition.

t

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Figure 18. The [2 operation and its boundary before stabilization and stable representatives of the boundary components.

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s 1

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1

Figure 19. The [2 operation after stabilization.

6 Actions 6.1 Algebras One of the main applications of the Arc operad and its derivates are chain and homology level actions on the Hochschild complexes of an algebra A. There are several types of algebras one considers. The algebras are either taken to be strictly associative or A1 . The latter is an algebra with a multiplication 2 W A ˝ A ! A, a differential d and it is associative up to homotopy with all higher homotopies explicitly given by higher multiplications n W A˝n ! A. We also take the algebras to be unital for simplicity. The next choice is if these algebras have a suitable duality. In the associative case this means that the algebras are Frobenius algebras, that is they have a nondegenerate symmetric (even) bilinear form h ; i which is invariant: ha; bci D ha; bci. In the A1 case one postulates the symmetry of all the n : ha0 ; n .a1 ; : : : ; an /i D han ; n .a0 ; : : : an1 /i. We remark that we can also use the category of graded algebras

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and their graded duals where we can get a duality if the graded pieces are finitedimensional. Since the algebras are unital, we will use the following notation: Z a WD ha; 1i (6.1) where 1 is the unit. In this notation, the symmetry and invariance become cyclicity Z Z (6.2) a0 a1 : : : an D a1 : : : an a0 : The techniques for cases with or without a duality are a bit different. Without a duality, we have an asymmetry between inputs and outputs. To a given action or better an algebra V over an operad or a PROP, we need a map in Hom.V ˝n ; V / or Hom.V ˝n ; V ˝ m/ for each element of the operad or PROP. This asymmetry is reflected in the geometry by restricting the arc families we can use to give cellular actions. The actions in this case are given by flow charts. If there is a duality, we can also ask that V has such a duality which is compatible. This means that we can construct maps in Hom.V ˝nC1 ; k/ or Hom.V ˝nCm ; k/ R which we call correlation functions. These are defined with the help of the co-unit . We can actually define these correlation functions even in the absence of a duality, but without it there is no way to compose unless we choose extra data, such as special elements or propagators in physics parlance in V ˝ V which are otherwise provided by the Casimir element of the non-degenerate bilinear form, see Appendix A.1.9 for the formulas.

6.2 Deligne’s conjecture Deligne’s conjecture is the following statement. Theorem 6.1. There is an operadic cell model of the little discs operad that acts on the Hochschild cochains CH  .A; A/ of an associative algebra in such a way that its induced action of the homology of the little discs operad gives rise the known Gerstenhaber algebra structure on HH  .A; A/. The theorem has been proved in many variants [35], [58], [44], [62], [37] [43], [2]. Notice that is has two main statements. (1) There is a chain level action which induces the Gerstenhaber structure on (co)homology, and (2) the chain level operad is an operadic chain model for the little discs. The second statement means that (a) the chains compute the right homology and (b) that the induced structure on homology coincides with the one from the little discs. In some of the variants the second statement is not proven, we will call such a solution to the problem a weak solution.

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Theorem 6.2. The operad of spineless cacti are equivalent to the little discs and its chain model CC .Cact1 / solves Deligne’s conjecture. We give an outline of the proof. First, notice that indeed CC .Cact 1 / is an operadic chain model for spineless cacti; for details see [21]. The action will be given by a flow chart. 6.2.1 Intersection graph trees. Given a cell indexed by an arc family Œ˛ its flow chart .Œ˛/ is given by the intersection graph of the dual graph. The intersection graph is a bipartite rooted planar tree. Here “rooted planar tree” which is sometimes also called planted planar tree means that there is a marked vertex called root and at each vertex there is a cyclic order of the adjacent edges and a linear order at the root.3 A dual graph of an element of Cact is a ribbon graph which has one cycle that contains all edges in exactly one orientation; call this cycle the outside cycle or loop – the one corresponding to the boundary 0. The intersection graph for such a graph has black vertices corresponding to the vertices of the graph. It has white vertices corresponding to the cycles of the graph except for the outside cycle. There is an edge connecting a white and a black vertex if the respective vertex lies on the respective cycle. For an example, see Figure 20. 4

3

1 R1−s−v

s v

2 t

R2−t

3

4 R3

5

R4 4

R3

5

2

3

R5 R5

5 v R2−t 2 1 s R1−s−v

t

1

R4

I

II

III

Figure 20. I. An element of Cact. II. Its dual graph. III. Its intersection graph.

This graph can be shown to be a tree. The cyclic order at each vertex is induced by the ribbon graph structure. For the black vertices this is the identical order and for the white vertices this is the order induced on the directed edges of a cycle. The root is taken to be the vertex which we called the global zero. That is the one corresponding to the marked point on the boundary 0. Finally, the linear order at this root vertex is given by saying that its first edge is the one corresponding to the first arc on the boundary 0. Since each other vertex has a unique edge pointing towards the root, declaring this to be the last edge gives a linear order of the edges at each vertex. 3 It is enough to give a linear order on all vertices. At any non-root vertex there is a unique edge going towards the root. This edge is set to be the last one.

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Since this data only depends on the incidence conditions of the arcs on the boundary, it is clear that this tree only depends on the class of ˛. The white vertices are also numbered 1; : : : ; n corresponding to the boundary components they represent. 6.2.2 Flow chart of the tree. The action is simply given by decorating .Œ˛/ with elements fi 2 CH  .A; A/ and performing brace operations at each white vertex and multiplication at each black vertex. More precisely, the action is defined for homogeneous elements and then extended by linearity. For the basis element of CC  .Cact.m// given by a cell indexed by Œ˛ the action on .f1 ; : : : ; fn / with fi 2 CH ni .A; A/, i D 1; : : : ; n is zero unless m D n. In the case where these numbers match: Decorate the white vertex i with fi . Now .Œ˛/ is planar and has a flow toward the root. The outermost vertices or leaves are white. We start with these functions. If the flow hits a black vertex, we multiply the incoming functions in the linear order given by this vertex. The product is the outgoing function. If the flow hits a white vertex i, we take the brace operation of fi with the product of all incoming functions (again in the order dictated by the linear order on the vertex) and make this the outgoing function. In the example given in Figure 20 the operation would be f1 ff2 ff3 f4 g; f5 g. 6.2.3 The A1 -version. The A1 version of Deligne’s conjecture was first proven in [37]. The action was given by means of a homotopy argument and was not explicit. The basic idea is that there is a naturally acting operad called the minimal operad and that this operad is quasi-isomorphic to the little discs. To build the quasi-isomorphism one goes through a very large model and proves existence of the quasi-isomorphisms by homotopy theory without having to construct them. In [33] the A1 version was proven in a minimal constructive way. The method used is to again employ flow charts given by planted, planar, two-colored, stable trees. The colors on the vertices are again black and white and stability means that each black vertex is at least 3-valent. The white vertices are also numbered. These form the minimal operad of [37] and their action is given by using the multiplication n at a black vertex that is .n  1/-valent and the brace operations at the white vertices. Theorem 6.3. (1) The trees above with fixed numbering from 1; : : : ; n index the cells of a CW complex K 1 .n/. (2) The collection of CW complexes K 1 .n/ are a cell model for the little discs operad. (3) The cellular chains of K 1 .n/ form an operad isomorphic to the minimal operad of [37] and hence give operations on CH  .A; A/ for any A1 algebra A in such a way that the induced action is the usual Gerstenhaber structure on HH  .A; A/. The proof of the second statement uses spineless cacti as a reference model. The tertium comparationis is third cell model called K ht , where ht stands for “height”. K ht is shown to be a subdivision of K 1 and retractable to spineless cacti.

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The connection to the arc picture and thus to moduli spaces, foliations and moduli spaces comes from K ht . It is given in the Appendix of [33]. The basic upshot is that the trees are again the intersection graphs of dual graphs of arc families. Again, the genus is zero and arcs are allowed to run from boundaries i D 1; : : : ; n to the boundary 0 or from 0 to 0. All boundaries i are hit and there is no arc from 0 to 0 that is homotopic to the union of another arc and a boundary. Tracing through the definitions, one sees that the arcs from 0 to 0 give rise to edges between two black vertices and the last condition ensures that there are no black vertices of valence less that 3. One upshot of this treatment is an arc indexed subdivision of Stasheff polytopes and a new subdivision of cyclohedra by cells indexed by arc families, which leads to a new explicit blowup procedure starting at a simplex, see [33].

6.3 The cyclic Deligne conjecture The cyclic Deligne conjecture, first proven in [26] using spineless cacti and their CW model Cact 1 , states that Theorem 6.4 ([26]). There is an operadic cell model of the framed little discs operad that acts on the Hochschild cochains CH  .A; A/ of a Frobenius algebra in such a way that its restriction to the operadic chain model for the sub-operad of the little discs operad gives rise the known Gerstenhaber algebra structure on HH  .A; A/. It was actually conjectured in [59]. A consequence of this statement is Corollary 6.5. For a Frobenius algebra HH  .A; A/ has the structure of a BV algebra, for which the induced bracket is the Gerstenhaber bracket. This statement was first proved by [45] without the chain level version. During the publication process of [26], several other versions of actions yielding a BV structure on homology providing weak solutions in the sense of §6.2 were produced in [61], [38], [11]. In some of these references actions of even bigger operads or PROPs were constructed. A very recent version is given in in [64]. The proof in [26] again uses a tree picture for the actions. We will give these actions in a different but equivalent guise when discussing the Sullivan-PROP in 6.4.4. This makes the fundamental role of foliations in the solution more apparent. In the Frobenius case we also have an isomorphism of complexes CH  .A; A/ and CC  .A/, the cyclic cohomology chain complex, see e.g. [24]. 6.3.1 The cyclic A1 -version. A solution to the A1 generalization of the cyclic conjecture (and not just a weak one) was announced in [28] and just fully proven in [65]. The method of proof is to extend the action of [33] to the case of trees with spines of marks as in [26].

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Theorem 6.6 ([65]). The cyclic A1 conjecture holds. The weak statement of this can be found in [38].

6.4 Moduli space action and the Sullivan-PROP also known string topology action 6.4.1 Correlation functions. Fix a commutative unital Frobenius algebra A with R multiplication  and pairing h ; i. Set a WD ha; 1i and let e be the Euler element of A, that is, e D  .1/ where is the adjoint of , see the Appendix. The actions will be given on CH WD CH  .A; A/. Now since A is a Frobenius algeof the Hochschild bra, then CH n .A; A/ ' A˝nC1 so that we may use an isomorphism L ˝n is the cochains with the tensor algebra CH  .A; A/ ' TA, where TA D 1 nD1 A reduced tensor algebra. Furthermore in the graded sense TA is Frobenius by using the tensor product of the Frobenius algebra structures. Thus suitably dualizing, we can represent any ˆ 2 Hom.CH ˝n ; CH ˝m / as an element Y 2 Hom.TA˝nCm ; k/ and vice versa. The multi-linear maps Y are called correlators and are fixed on homogenous element of TA. We will define basic correlators Y depending on the cells of Asg;r . For homogenous elements i 2 TA; i D 0; : : : ; r these will be multilinear maps to k denoted by h0 ; : : : ; r iŒ˛ for any cell given by a PMC class Œ˛. 6.4.2 Basic idea. The action is roughly given as follows: fix ˛ an arc graph with k s and fix homogenous i 2 TA; i D 0; : : : ; r  1. arcs on some Fg;r (1) Duplicate edges so that the number of incoming edges at the vertex i D deg.i /. We sum over all possibilities to do this, if this is not possible then the operation is zero. (2) Assume the i are pure tensors. Pull apart the edges and decorate the pieces of the boundary with the elements of . Cut along all the edges of the graph and call the set of disjoint pieces of surface P . Let I.F / be the index set of the components aj of the i decorating edges belonging to a piece S 2 P and let .S / be the Euler characteristic of the surface S . (3) For each S integrate over the product over all the boundary decorations and a factor of e .S/C1 . (4) Multiply together all the local contributions of (3). An example is given in Figure 21.

308

Ralph M. Kaufmann hi

hi Cj C1 hi Cj C2

g0 f0

g0

1

h0 f1

fi

gj C1

2 g1

fi

hi Cj

h0

gj hi Cj

h1

hi C1

hi

f0

gj hi Cj C1

I

II

Figure 21. A partitioned arc graph with decorations by elements of A and one of its decorated polygons. The bold line corresponds to the bold edge.

0 6.4.3 Technical details. Given any arc graph ˛ on F D Fg;r , let e1 ; : : : ; ek be its arcs enumerated in their order. For any tuple of positive numbers n D .n1 ; : : : ; nk /, ni 2 N>0 , let ˛ n be the arc graph obtained from ˛ by replacing each edge ei by ni parallel arcs. This can be for instance done in some rectangle with spine ei . Consider @F n .˛ n \ @F / D qr1 iD0 qj 2Ji Iij , which is a disjoint union of intervals or simply the whole boundary components, and Ji indexes the components sitting inside @i . The set Ji has a natural cyclic order, which we upgrade to a linear order by declaring the unique interval containing the marked point of @i to be the first interval. 0 n ˛ n D qF 2P F . Each surface F 2 P has a boundary in which Likewise, let Fg;r each boundary component is a 2k-gon whose sides alternate between the arcs and pieces of the boundary, or simply the whole boundary, if the boundary has no incident arc. N Consider homogenous elements i D j 2Ji 2 A˝Ji . For S 2 P let B.S/ be the subset of those boundary indices .i; j / such that Iij is a part of the boundary of S, Z Y aij e .S /C1 h0 ; : : : r1 iS WD .i;j /2B.S /

and finally set h0 ; : : : ; r1 iFg;r 0 ;˛ WD

Y

h1 ; : : : ; n iS

S2P

Z

where h1 ; : : : ; n ip D

Y

ai e .S /C1 :

i2I.S /

Combining these correlators with the map P yields:

(6.3)

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Theorem 6.7. Let A be a Frobenius algebra and let CH  .A; A/ be the Hochschild complex of the Frobenius algebra. Then the cyclic chain operad of the open cells of Arc acts on CH  .A; A/ via correlation functions. In the same fashion all the suboperads, di-operads and sub-PROPs of [23] act. 1;:::;1 of In particular the total graph complex Gr Co .Arc0# / of the moduli spaces Mg;r;s pointed curves with fixed tangent vectors at each point acts. 6.4.4 String topology. For string topology there is a modification of the above basic correlators , which amounts to inserting the unit element in certain places, depending on the choice of in and out boundaries. Consider @F n .˛ n \ @F / D qr1 iD0 qj 2Ji Iij as above. These elements are called angles in [24]. On each boundary there is one angle that contains the marked point. This angle is called the outer angle. If Iij has as its endpoints the endpoints of two parallel arcs of ˛ n (and is not outer) it is called a partitioning angle. All the other angles, that is neither outer and nor partitioning, are called inner angles. These are the angles corresponding to the angles of ˛ that are not outer angles. Let F i=o be a surface together with designated in and out boundaries, as in the Sullivan-PROP situation. Given i 2 CH ni .A; A/ we let Q i D  for all in boundaries i , and for an out boundary k, given k 2 CH nk .A; A/, we let Q k 2 CH jJk j .A; A/ be the element obtained from k by inserting the unit 1 in all the positions j for which Ikj is an inner angle. This is accomplished by the use of degeneracy maps. Given k D i1 ˝ ˝ ini set sj .i / D i1 ˝ ˝ ij 1 ˝ 1 ˝ ij ˝ ˝ ini . That is Q k D sj1 : : : sjl k , where j1 < < jk are the positions of the inner angles: (6.4) h0 ; : : : ; r1 iF i=o ;˛ WD hQ 0 ; : : : ; Q r1 iF ;˛ : Notice that this expression is 0 unless ni D jJi j for all in boundaries and nk D jJk j the number of inner angles. An example of this type of decoration is given in Figure 22. Theorem 6.8 ([24]). The Y.˛/ defined in equation (6.4) give operadic correlation functions for the chain model CC .ull 1 / of the Sullivan-PROP ull and induce a dg-action of the dg-PROP CC .ull 1 / on the dg-algebra CH  .A; A/ of reduced Hochschild co-chains for a commutative Frobenius algebra A. By restricting to genus 0 and ull .n; 1/ one gets an action of Cacti; this together with the quasi-isomorphism of Cacti and the framed little discs yields: Theorem 6.9. The cyclic Deligne conjecture holds. The connection to string topology is as follows. Let M be a simply connected compact manifold M , denote the free loop space by LM and let C .M / and C  .M / be the singular chains and (co)-chains of M . We know from [18], [6] that C .LM / D

310

Ralph M. Kaufmann 0

0

1 1

2 2

Figure 22. Examples of the partitioned families yielding [, Bi in the string topology/Deligne setting and t and i in the moduli setting. The outer angles are the ones with the dot, the bold angles are the non-partitioning inner angles are marked in bold.

CH  .C  .M; C .M /// and H .LM / ' HH  .C  .M /; C .M //. Moreover C  .M / is an associative dg algebra R with unit, differential dRand an integral (M was taken to be a compact manifold) W C  .M / ! k such that d! D 0. By using the spectral sequence and taking field coefficients we obtain operadic correlation functions Y for Cacti on E 1 D CH  .H; H / which converges to HH  .C  .M // and which induces an operadic action on the level of (co)-homology. Except for the last remark, this was established in [26]. Theorem 6.10. When taking field coefficients, the above action gives a dg action of a dg-PROP of Sullivan chord diagrams CC .ull 1 / on the E 1 -term of a spectral sequence converging to H .LM /, that is, the homology of the free loop space of a simply connected compact manifold and hence induces operations on this loop space. Remark 6.11. There is also a lift to co-cycles of a dg-algebra whose cohomology is Frobenius, see [24]. 6.4.5 Moduli space and Arc action. The basic correlators compose completely algebraically, see A.1.9. This will give an algebraic action, if one leaves out the modification in the operad composition that the closed leaves are erased as discussed in Remark 2.2. Also in this version, one uses no extra signs in the discretization. Notice that in the string topology gluing, no closed leaves can appear. For the case of the relative chains of Arc or in the moduli space case, namely Gr Co .Arc0# /, we need the signs. In order to get a map of operads, we therefore first have to account for the signs in the operad Hom.CH  /. This is implicitly done in string topology and the various versions of Deligne’s conjecture. Here if one wishes to phrase the fact that the action exists in operadic terms, the target of the morphism of operads is not Hom.CH  / but the so-called brace operad Brace which is formed by certain subspaces of the endomorphism operad, but has different sign rules. This is what we called a twisted Hom-operad structure. Moreover in our solution to Deligne’s

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conjecture Brace is in a sense the tautological recipient of the operad map, since it is the isomorphic image of CC .Cact/. This is however only a posteriori and not true in all solutions of Deligne’s conjecture – albeit this type of statement is a special feature in all forms of the conjecture coming from chains of the Arc operad. A priori Brace is defined to be spanned by the multiplication and brace operations. The signs then come from a new natural grading. Since any Hom.V / operad is already graded this is actually a bi-grading. The theorem below gives identification of such a Hom-operad for the moduli space case. That is, there is an isomorphic image of Gr Co .Arc0# / of subspaces MC S in Hom.CH  .A// which is naturally (bi)graded, thus providing the signs. Furthermore, in the case at hand, on the geometric side there is also a modification in the combinatorial gluing coming from the use of cellular chains, namely that if the dimension of the chain is not additive under the gluing, the result is 0. On the geometric side this was handled by passing to the associated graded (see §2.5) and likewise on the algebraic side we analogously pass to the associated graded. Theorem 6.12. Let A be a Frobenius algebra and let CH  D CH  .A; A/ be the Hochschild complex of the Frobenius algebra. There are subspaces MC S  Hom.CH  /, defined in terms of natural operations below, such that for all ˛ n for ˛ a quasi-filling, the correlation functions (6.3) suitably dualized are elements of MC S. Furthermore MC S has a natural grading, such that the correlation functions Y .P .˛// for ˛ 2 Gr Co .Arc0# / yield an operad morphism of cyclic operads to GrMC S, viz. they are operadic correlation functions with values in GrMC S. Pushing forward the differential, this action becomes dg. Remark 6.13. Again there is a lift to the chain/cycle level. Remark 6.14. Unlike in the previous cases, the differential is not the natural one on the Hochschild side. This is actually the case for the arc graphs with arcs running only from input to output. We give the details below. Although the details have to be spelled out, it is fairly straightforward to obtain: Claim 6.15. The above theorem also holds in the modular operad setting. And, by dualization, the analogous theorem holds for the PROP setting, that is, for the complex of quasi-filling families with input and output markings.

6.4.6 MC S and the operations of Cact. For completeness we will briefly define MC S. It is given as the subspace generated by three types of operations and certain

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permutations. These are the maps: Z Z Z W TA ! k; .a1 ˝ ˝ an / D a1 : : : an ; ˙ W TA ! TA ˝ A ˝ TA; X .a1 ˝ ai1 / ˝ ai ˝ .aiC1 ˝ ˝ an /; a1 ˝ ˝ an 7! i

n W A˝n ˝ A˝n ! k; .a1 ˝ ˝ an ˝ b1 ˝ ˝ bn / D ˙

YZ

.ai bi /;

i

where TA is the tensor algebra and the sign is the usual Koszul sign. To explain the shuffles, one introduces a new monoidal structure  for bimodules [24]. For TA this amounts to the definition TA  TA D TA ˝ A ˝ TA. This means that ˙ is a co-product. ˙ W TA ! TA  TA. This type of co-product also naturally appeared in [43]. The correlators Y .˛ n / then can be factorized as n

Y .˛ / W TA

˝n

!

r O

Ami C1 ! k

(6.5)

i D0

where mi is the number of inner angles at the boundary iR. The shuffles are then the shuffles of factors TA and A in the middle part. Note that is cyclically invariant, so that only the cyclic order of the tensors matters. More precisely for any ˛ n with ˛ a quasi-filling, let mi be the number of inner angles at boundary i , ni be the entries of n; then Y .a / D p



O 2Comp.˛/

Z ˝ 

r O iD0

ni 1



B B

n O

˙mi

(6.6)

iD0

where R we identify the complementary regions Comp.˛/ of ˛ with a subset of those of ˛ n , is applied to the cyclically ordered decorations of the fixed polygonal region  as in 6.3, the iterated coproduct defined by ˙l W TA ! TAlC1 is the iteration of ˙ given by .˙ ˝ .idA ˝ idTA /˝l / B ˙ ˝ .idA ˝ idTA /˝l1 / B B .˙ ˝ idA ˝ idTA / B ˙,  a shuffle of the factors A and TA in the image of the ˙ operations, that is, Nis r mi C1 . i D0 A MC S is then the space dual to the one Pgenerated by operations of the type (6.6). The degree of such an operation is l D 12 .mi C 1/  1. If the operation comes from some ˛ then this is exactly the dimension of the cell given by ˛. For the basic cells from Cact given by the diagrams in Figure 22 we obtain the operations B



t W .A ˝ A{˝n /  .A ˝ A{˝m / ! A ˝ A ˝ .A{˝n  A{˝m / ! A ˝ A{˝nC1Cm (6.7)

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and i .f; g/.a1 ; : : : ; anCmC2 / (6.8) D f .a1 ; : : : ; ai1 ; ai g.aiC1 ; : : : aim /aiCmC1 ; aiCmC2 ; : : : ; anCmC2 /: These type of operations have also appeared in other work on the Deligne conjecture. Proposition 6.16. The operations of the suboperad Cact correspond to the operations t and i induced by „2 as defined in [43]. 6.4.7 Stabilization and the action. Given the results of the previous paragraph, we can ask when does the action pass to the stabilization. The answer [25] is that this is the case if and only if A is a normalized semi-simple Frobenius algebra. Semi-simple means that multiplication ei ej D ıij ei . It follows R ei with theP Pthere are generators that 1 D i ei and if i D ei then e D iR i ei . The algebra is normalized if in addition e D 1 2 A which is equivalent to all i D 1. Any semi-simple Frobenius algebra can be normalized by rescaling the metric. Moreover, if the algebra is just semi-simple then all unstable correlators are completely determined by the stable ones. In both cases the action of the Sullivan-PROP on the homology level is of course trivial, since the Hochschild cohomology of a semi-simple algebra is trivial. The chain level gives such a preferred trivialization. But there is an interesting action of the moduli spaces. This could be related to a similar story of stabilization discussed by Teleman [60]. The connection is given by Gromov–Witten invariants with semisimple quantum cohomology such as that of projective spaces and conjecturally a class of Fano varieties.

7 Open/closed version There is an open/closed version of the whole story. This is given in details in [32] and [29]. For this one introduces marked points on the boundaries and brane labels for these as well as marked points in the interior. Here the brane label can be ; to indicate the closed sector or some element of an indexing set to indicate an open brane label. One major difference is that in that setting, basically due to the Cardy equation, one cannot restrict to the case of no marked point in the interior. The theory again is completely natural from the foliation aspect. Geometrically there is one simple rule in the background. Points with the closed label are considered as marked points in the surface, but points with an open label are considered as deleted from the surface. One upshot is a short topological proof of the minimality of the Cardy/Levellen axioms.

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Acknowledgements. It is my foremost pleasure to thank my co-author Bob Penner who worked with me on many of the results of this chapter. My thanks also goes to Muriel Livernet whose input was very valuable in the beginning of the endeavor. Throughout the years there have been numerous useful discussions with fellow mathematicians; thank you to all. Special thanks goes to Ralph Cohen, Ezra Getzler, Yuri I. Manin, Jim McClure, Anton Kapustin, Maxim Kontsevich, Jim Stasheff, Dennis Sullivan and Sasha Voronov. A very special thanks goes to Athanase Papadopoulos for seeing me through the writing of this article. The work presented here has benefitted from visits to the Max-Planck-Institute in Bonn, the Institut des Hautes Etudes Scientifiques in Bure-sur-Yvette, the Institute for Advanced Study in Princeton, the MSRI and the University of Hamburg, and the support of the Humboldt Foundation. The author also thankfully acknowledges support from NSF DMS-0805881.

A Glossary A.1 Operads and PROPs An operad basically formalizes combinations of flow diagrams. The individual pieces have n inputs and 1 output. Let C be one of the following: (chain complexes of dg) vector spaces with tensor product, topological spaces with Cartesian product or chain complexes of Abelian groups with tensor product over Z. In general C will be a symmetric monoidal category. Definition A.1 (Short definition). An operad in C is a sequence of objects O.n/ 2 C together with an Sn -action on O.n/ and morphisms Bi W O.n/ ˝ O.m/ ! O.m C n  1/;

for i D 1; : : : n;

which are equivariant for the symmetric group actions and associative. Remark A.2. In the full definition, C should be a symmetric monoidal category. The categories above are such categories. A.1.1 Functors and operads Remark A.3. It is clear that (weak) symmetric monoidal functors transform operads to operads. The ones that we care about are H and S as well as CC (cellular chains for CW complexes with cellular maps). A.1.2 Standard example. The standard example is the one of multivariable functions. From this example one can also make precise how the associativity and symmetric group equivariance can be written in formulas. We will make this concrete in

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two examples. Let X be a locally compact Hausdorff space. We let Hom.X /.n/ WD C.X n ; X /; an element is just a continuous function f of n variables. f Bi g just substitues g in the i -th variable of f . If f has n variables and g has m variables then f Bi g has n C m  1 variables. The Sn -action is given by permuting the variables of the function. Given two permutations n and m there is a unique permutation m Bi n of the new n C m  1 variables such that m .f / Bm .i / n .g/ D m Bi n .f Bi g/. The interested reader is referred to e.g. [40] or encouraged to work out the combinatorial formula. The associativity states that if we have three functions f , g, h then it does not matter in which way we make the substitutions. Writing down the explicit formula is again a bit subtle, since the indexing of the variables changes. The reader is encouraged to work out the combinatorial formula which is in general given by A.1.3 Associativity. For three elements opk 2 O.k/; op0l 2 O.l/ and op00m 2 O.m/, 8 0 00 ˆ 3) points, 0; 1; 1; a1 ; : : : ; an3 . Let ' W K  y be a holomorphic motion of E over K with basepoint 0 . For each 2 K , E!C we have a Riemann surface of type .0; n/: R WD C n f0; 1; 1; '. ; a1 /; : : : ; '. ; an3 /g: Since '.; ai / is holomorphic on K , the map K 3 7! R makes a holomorphic family of Riemann surfaces of type .0; n/ over K and we have a monodromy E W 1 .K ; 0 / ! Mod.0; n/ for the holomorphic family. Then, we have the following: y a holomorphic motion Theorem 5.5 ([2]). Let E be a closed set and ' W K E ! C y over K if and of E over K . Then ' can be extended to a holomorphic motion of C 0 only if the monodromy E 0 for any finite subset E of E is trivial. Proof. From Proposition 5.4 we may assume that E is a finite set. y be a holomorphic motion. Suppose Let E D fz1 ; : : : ; zn g and let ' W K E ! C y that the holomorphic motion ' W K  E ! C can be extended to a holomorphic y over K . Then, from Proposition 5.3 we see that the holomorphic motion 'O of C motion is extended to a holomorphic motion of E over  and it is also extended to y over  by Theorem 5.2. Thus the monodromy must be a holomorphic motion of C trivial.

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Conversely, suppose that the monodromy E W 1 .K / ! Mod.0; n/ is trivial. Let K be a Fuchsian group acting on  which represents K and  W  ! K a holomorphic universal covering, with .0/ D x0 . We define a holomorphic motion y by ˆ D   .'/ WD '../; /. So, we have ˆW   E ! C ˆ.x; z/ D '..x/; z/ for all .x; z/ 2   E: zW  By Slodkowski’s theorem (Theorem 5.2), there exists a holomorphic motion ˆ y y z C ! C such that ˆ extends ˆ. Therefore, there exists a basepoint preserving holomorphic map f W  ! M.C/ such that z ˆ.x; z/ D ˆ.x; z/ D w f .x/ .z/

for .x; z/ 2   E:

 defines a holomorphic map Fz from  to the The correspondence  3 x 7! Œw y with n punctures). Since the Teichmüller space T .0; n/ (the Teichmüller space of C z monodromy E is trivial, the holomorphic map F satisfies f .x/

Fz B g D Fz for any g 2 K . Hence, it can be projected to a holomorphic map F from K D = K to T .0; n/. Then, the holomorphic map F can be extended a holomorphic map  K such that of . Indeed, for any point  2 K there exists a sequence fxk g1 k limk!1 xk D  (since K is AB-removable, it has no interior points). Now we consider the Carathéodory (pseudo-)distance. For a complex manifold M , the Carathéodory distance cM on M is defined by cM .a; b/ D sup d .f .a/; f .b// f

for a; b 2 M , where the supremum is taken over all holomorphic functions f from M to the unit disk  and d is the hyperbolic distance on . Since K isAB-removable, the Carathéodory distance cK on K is equal to c jK , which is the restriction of the Carathéodory distance on  to K . It is well known that c D d . Therefore, the sequence fxk g1 is a Cauchy sequence with respect kD1 to cK . By the above definition of the Carathéodory distance, immediately we have cT .0;n/ .F .xk /; F .x` //  cK .xk ; x` /

.k; ` 2 N/

for the holomorphic map F W K ! T .0; n/, where cT .0;n/ is the Carathéodory is also a Cauchy sequence in T .0; n/ with distance on T .0; n/. Thus, fF .xk /g1 kD1 respect to cT .0;n/ and it converges to a point in T .0; n/ because of the completeness of the Carathéodory distance in Teichmüller space (cf. [31]). Therefore, F can be extended to a holomorphic map of . Since the extended holomorphic map gives a holomorphic motion of E over  which extends ', it follows from Proposition 5.3 y over K . that ' can be extended to a holomorphic motion of C

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6 Flat structures on surfaces and Teichmüller curves. We begin with the definition of a flat structure (cf. [9], [13]). Definition 6.1. Let X0 be a hyperbolic Riemann surface of type .g; n/. A flat structure u on X0 is an atlas of X0 satisfying the following conditions: (1) local coordinates of u determines the same orientation as the complex structure of X0 ; (2) for coordinate neighborhoods .U; z/ and .V; w/ with U \ V 6D ;, the transition function is of the form w D ˙z C c in z.U \ V / for some c 2 C. A pair .X0 ; u/ of a Riemann surface X0 and a flat structure u on X0 is called a flat surface. By the definition of the flat structure, we may consider some notions in Euclidean geometry, such as area, horizontal and vertical segments etc., on a flat surface. In our setting, we consider only flat structures with area one. On a flat surface .X0 ; u/, we have a natural SL.2; R/ action, because the flat structure is locally the Euclidean 2-space. We consider the subgroup SL.X0 ; u/ of matrices A 2 SL.2; R/ stabilizing the flat surface .X0 ; u/. The image PSL.X0 ; u/ of SL.X0 ; u/ via the projection to PSL.2; R/ is called the Veech group of the flat surface. Hence, the Veech group acts on the hyperbolic plane. A flat structure u on X0 of unit area determines a holomorphic quadratic differR ential  with X0 jj D 1 and conversely the trajectories of a holomorphic quadratic R differential  gives a flat structure of X0 with area X0 jj. N is a Beltrami For a given holomorphic quadratic differential  on X0 , t =jj differential on X0 for t 2 . The Beltrami differential defines a quasiconformal mapping on X0 , and the map gives a point in T .X0 /. Thus, we have a map F from  to T .X0 /. The image F ./ is called a Teichmüller disk and we denote it by D . It is known that the set of orbits of .X0 ; u/ by SL.2; R/ is identified with the Teichmüller disk D for  which is associated to .X0 ; u/. Thus, each element A 2 SL.X0 ; u/ yields a Teichmüller map associated to . By the definition of the Veech group, the Teichmüller map preserves the flat structure. Hence it induces an element of the mapping class group. This implies that the Veech group PSL.X0 ; u/ can be regarded as the group of mapping classes fixing the Teichmüller disk D associated to the flat structure. Furthermore, there is a natural identification between the Veech group and a Fuchsian group acting on the unit disk (see [13], §2). From now on, we identify the Veech group and the Fuchsian group. Now, suppose that the quotient D =PSL.X0 ; u/ is of finite hyperbolic area when the Teichmüller disk is identified with the hyperbolic plane. Then we denote by V the

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Veech group acting on the Teichmüller disk D and by R the quotient space D =V . We call R the Veech surface in M.g; n/ (See [9], [37] and [38] for details). Let C be a Riemann surface of finite type and f W C ! M.g; n/ a holomorphic map from C to the moduli space of closed Riemann surfaces of genus g > 1. If f is a local isometry with respect to the hyperbolic metric on C and the Teichmüller metric on M.g; n/, then .C; f / is called a Teichmüller curve in M.g; n/. For example, if ˛ W C ! R is a finite (smooth) covering of a Veech surface R , then for the natural embedding … W R ! M.g; n/, .C; … B ˛/ is a Teichmüller curve. The following proposition says that the converse is also true. Proposition 6.2. Let .C; f / be a Teichmüller curve. Then there exists a Veech surface R such that C is a finite covering of R . Proof. Let .C; f / be a Teichmüller curve. From the definition, f W C ! M.g; n/ is holomorphic and locally isometric. Then, there exists a holomorphic map F W  ! T .g; n/ which is a lift of f . Thus, there exists a Fuchsian group  on  representing C such that …0 B F B  D …0 B F for every  2  and for the natural projection …0 W T .g; n/ ! M.g; n/. Hence we have a monodromy F W  ! Mod.g; n/ satisfying (6.1) F B  D F . / B F: We shall show that F W  ! T .g; n/ is globally isometric. Namely, we will show that d .z1 ; z2 / D dT .F .z1 /; F .z2 // for any z1 ; z2 2 . We may assume that z1 D 0 and z2 > 0. Let r2 be the hyperbolic geodesic connecting 0.D z1 / and z2 in . Since F is a local isometry, there exists a neighborhood U1 WD fjzj < ıg of 0 such that F jU1 is an isometry. Therefore, L1 WD F .r2 \ U1 / is a Teichmüller geodesic arc. Indeed, if it is not geodesic, then there exists a geodesic arc ˛ connecting F .0/ and F .ı/. Both L and ˛ have the same length. Thus, we conclude that ˛ D L, by the uniqueness of the Teichmüller geodesic. Let 1 be a holomorphic quadratic differential on X0 which corresponds to L1 . Now, we define a holomorphic map F1 from  to T .g; n/ as follows: for each z 2 , we set z WD zj1 j=1 . Then, z is a Beltrami differential on X0 which depends holomorphically on z 2 . Therefore, it gives a quasiconformal map fz from X0 to Xz WD fz .X0 /. We define F1 .z/ WD ŒXz ; fz  2 T .g; n/: By a theorem of Teichmüller, the map F1 is holomorphic and isometric. In particular, F .z/ D F1 .z/ for z 2 r2 \ U1 . Thus, we conclude that F D F1 and it is a global isometry. Next, we consider D WD F ./. Since F is holomorphic and isometric, D is a Teichmüller disk. From (6.1), F ./  Mod.g; n/ preserves the Teichmüller disk D. Moreover, the monodromy F is an isomorphism. Indeed, if it is not, then there exists  2  n fidg such that F ./ D id. On the other hand, from (6.1), we have F ..0// D F ./.F .0// D F .0/:

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This contradicts the injectivity of F . Let V be the subgroup of mapping classes fixing D. As before, V is regarded as a Fuchsian group on . The group V contains F ./ and =  D C is a Riemann surface of finite type. Thus, R D =V is also a Riemann surface of finite type. In other words, V is the Veech group for D and C D =  D D= F ./ is a finite covering of the Veech surface R. Next, we show a rigidity theorem for Teichmüller curves. Let C be a Riemann surface of finite type and f W C ! M.g; n/ a non-constant holomorphic map. Now, we consider a lift F W  ! T .g; n/ of f . We denote by

F W GC ! Mod.g; n/ the monodromy of F , where GC is a Fuchsian group on  representing C . Two holomorphic maps fi W Ci ! M.g; n/ .i D 1; 2/ are said to be geometrically equivalent if there exists a quasiconformal map w W C1 ! C2 such that

F1 D F2 B w

(6.2)

up to conjugation, where w is a natural isomorphism from GC1 onto GC2 induced by w. Then we have the following rigidity theorem. Theorem 6.3. Let C0 be a Riemann surface of finite type and f0 W C0 ! M.g; n/ a Teichmüller curve. Suppose that there is a holomorphic map f W C1 ! M.g; n/ such that .C1 ; f1 / is geometrically equivalent to the Teichmüller curve .C0 ; f0 /. Then, .C1 ; f1 / D .C0 ; f0 / up to conformal automorphisms of C0 . Proof. Let f1 W C1 ! M.g; n/ be a holomorphic map which is geometrically equivalent to the Teichmüller curve f0 W C0 ! M.g; n/. Let w W C0 ! C1 be a quasiconformal map and w W GC0 ! GC1 an isomorphism induced by w. For any g 2 GCi , we have Fi B g D Fi .g/ B Fi .i D 0; 1/ and

F0 D F1 B w : Let g 2 GC0 be a hyperbolic element and Ag its axis in . Since F0 W  ! T .g; n/ is an isometry, F0 .Ag / is a Teichmüller geodesic. Moreover, we have F0 .Ag / D F0 .g.Ag // D F0 .g/.F0 .Ag //: Thus, F0 .g/ fixes the Teichmüller geodesic F0 .Ag /. On the other hand, we have dT .F0 .z/; F0 .g/n .F0 .z/// D dT .F0 .z/; F0 .g n .z/// D nd .z; g.z// ! 1 .n ! 1/ for z 2 Ag . This implies that F0 .g/ 2 Mod.g; n/ is of infinite order. Thus the mapping class F0 .g/ is hyperbolic in the sense of the Thurston–Bers classification

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of mapping classes (cf. [5]). Thus, for z 2 Ag we have d .z; g.z// D dT .F0 .z/; F0 .g/.F0 .z/// D

inf

p2T .g;n/

dT .p; F0 .g/.p//:

On the other hand, for gw WD w .g/, we see that d .z; gw .z//  dT .F1 .z/; F1 .gw .z/// from the decreasing property of the Kobayashi distance. It follows from (6.2) that dT .F1 .z/; F1 .gw .z/// D dT .F1 .z/; F0 .g/.F1 .z///: Therefore, we conclude that for any hyperbolic element g 2 GC0 , inf d .z; g.z//  inf d .z; gw .z//:

z2

z2

This means that the length of any closed geodesic ˛ in C0 is not greater than that of the closed geodesic homotopic to w.˛/. Hence, from a theorem of Thurston [36], we conclude that C0 D C1 and w W C0 ! C1 is homotopic to a conformal map h. Furthermore, the geometric equivalence implies that two holomorphic maps f0 and f1 B h induce the same monodromy 0 . Therefore, the conclusion follows from Theorem 3.1. The following rigidity is an immediate consequence of the above theorem. Corollary 6.4 (McMullen [23]). Let .C; f / be a Teichmüller curve in M.g; n/. Then, there is no non-trivial deformation of .C; f /. Namely, if f t W C t ! M.g; n/ .t 2 / is a holomorphic deformation with f0 D f , then f t D f for all t 2 . We shall end this section after showing a finiteness theorem for Teichmüller curves. Let g 0 , n0 be non-negative integers. A Teichmüller curve .C; f / is said to be of type .g 0 ; n0 / if C is a Riemann surface of type .g 0 ; n0 /. Theorem 6.5. Let g 0 , n0 be non-negative integers with 2g 0  2 C n0 > 0. Then there are only finitely many Teichmüller curves of type .g 0 ; n0 / in M.g; n/. Proof. Suppose that there are infinitely many Teichmüller curves f.Ci ; fi /g1 iD1 of type .g 0 ; n0 / in M.g; n/. Let ˛i be the shortest simple closed curves on Ci .i D 1; 2; : : :/ and denote by `i the hyperbolic length of ˛i . We consider a Fuchsian group Gi acting on the unit disk  so that Ci D =Gi , and a lift Fi W  ! T .g; n/ of fi with monodromy i . Since fi W Ci ! M.g; n/ is locally isometric, we have dT .Fi .z/; i .gi /.Fi .z/// D d .z; gi .z// D `i ; where gi 2 Gi is a hyperbolic element for ˛i and z is a point on the axis of gi . Hence, we have a. i .gi // WD inf dT .p; i .gi /.p///  `i : p2T .g;n/

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As we have seen in the proof of the previous theorem, i .gi / is a hyperbolic mapping class in Mod.g; n/. Then, Penner [28] shows that a. i .gi // 

log 2 : 12g  12 C 4n

Thus we have `i 

log 2 : 12g  12 C 4n

Since `i is the length of the shortest geodesic on Ci , it follows from a compactness theorem (cf. [4]) that the Riemann surfaces Ci .i D 1; 2; : : :/ are contained in a compact subset of the moduli space M.g 0 ; n0 /. Taking a subsequence of fCi g1 iD1 if necessary, we may assume that Ci converges to a Riemann surface C as i ! 1 in M.g 0 ; n0 /. It is not hard to see that .Ci ; fi / and .Cj ; fj / are geometrically equivalent for sufficiently large i; j . Thus, we have a contradiction from Theorem 6.3. Remark 6.6. If we do not restrict the type of Teichmüller curves, the number of Teichmüller curves could be infinite. In fact, McMullen [22] shows that there are infinitely many Veech surfaces in M.g; 0/ for g D 2; 3 or 4. For further discussion on Teichmüller curves, see [22], [25] and their references. Acknowledgement. The author thanks Professor A. Papadopoulos for his careful reading of the manuscript and valuable suggestions. He also thanks Y. Shinomiya for his valuable comments. This work was supported by JSPS Grant-in-Aid for Scientific Research (B) 22340028 and Grant-in-Aid for Challenging Exploratory Research 23654024.

References [1]

N. A’Campo, A. A’Campo-Neuen, L. Ji, and A. Papadopoulos, A commentary on Teichmüller’s paper Veränderliche Riemannsche Flächen (Variable Riemann surfaces) Deutsche Math. 7 (1944), 344–359, in Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume IV, EMS Publishing House, Zürich 2014, 805–814.

[2]

M. Beck, Y. Jiang, S. Mitra, and H. Shiga, Extending holomorphic motions and monodromy. Ann. Acad. Sci. Fenn. 37 (2012), 53–67.

[3]

L. Bers, On boundaries of Teichmüller spaces and on Kleinian groups I. Ann. of Math. 91 (1970), 570–600.

[4]

L. Bers,A remark on Mumford’s compactness theorem. Israel J. Math. 12 (1972), 400-407.

[5]

L. Bers, An extremal problem for quasiconformal mappings and a theorem by Thurston. Acta Math. 141 (1978), 73–98.

[6]

L. Bers, Holomorphic families of isomorphisms of Möbius groups. J. Math. Kyoto Univ. 26 (1986), 73–76.

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[7]

L. Bers and H. L. Royden, Holomorphic families of injections. Acta Math. 157 (1986), 259–286.

[8]

C. J. Earle and R. S. Fowler, Holomorphic families of open Riemann surfaces. Math. Ann. 270 (1985), 249–273.

[9]

C. J. Earle and F. P. Gardiner, Teichmüller disks and Veech’s F -structures. in Extremal Riemann surfaces (San Francisco, CA, 1995), Amer. Math. Soc., Providence, RI, 1997, 165–189.

[10] C. J. Earle, I. Kra, and S. L. Krushkal’, Holomorphic motions and Teichmüller spaces. Trans. Amer. Math. Soc. 343 (1994), 927–948. [11] C. J. Earle and P. L. Sipe, Families of Riemann surfaces over the punctured disk. Pacific J. Math. 150 (1991), 79–96. [12] F. P. Gardiner, Approximation of infinite dimensional Teichmüller spaces. Trans. Amer. Math. Soc. 282 (1984), 367–383. [13] F. Herrlich and G. Schmithüsen, On the boundary of Teichmüller disks in Teichmüller and in Schottky space. In Handbook of Teichmüller Theory (A. Papadopoulos, ed.), Vol I, EMS Publishing House, Zürich 2007, 293–349 . [14] J. H. Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, 2006. [15] Y. Imayoshi, Holomorphic families of Riemann surfaces and Teichmüller spaces. In Riemann surfaces and related topics (1978 Stony Brook Conference), Princeton University Press, Princeton, NJ, 1981, 277–300. [16] Y. Imayoshi, A construction of holomorphic families of Riemann surfaces over the punctured disk with given monodromy. In Handbook of Teichmüller theory(A. Papadopoulos, ed.), Vol. II, EMS Publishing House, Zürich 2009, 93–130. [17] Y. Imayoshi and H. Shiga, A finiteness theorem for holomorphic families of Riemann surfaces. in Holomorphic functions and moduli II, Math. Sci. Res. Inst. Publ. 11, Springer, New York 1988, 207–219. [18] Y. Imayoshi and M. Taniguchi, Introduction to Teichmüller spaces, Springer, Tokyo 1992. [19] R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps. Ann. Sci. École. Norm. Sup. 16 (1983), 193–217. [20] V. Markovic, Biholomorphic maps between Teichmüller spaces. Duke Math. J. 120 (2003), 405–431. [21] C. McMullen, Renormalization and 3-manifolds which fiber over the circle. Princeton University Press, Princeton, NJ, 1996. [22] C. T. McMullen, Prym varieties and Teichmüller curves. Duke Math J. 133 (2006), 569–590. [23] C. T. McMullen, Rigidity of Teichmüller curves. Math. Res. Lett 16 (2009), 647–649. [24] S. Mitra and H. Shiga, Extensions of holomorphic motions and holomorphic families of Möbius groups. Osaka J. of Math. 47 (2010), 1167–1187. [25] M. Möller, Finiteness results for Teichmüller curves. Ann. Inst. Fourier (Grenoble) 58 (2008), 63–83.

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[26] S. Nag, The complex analytic theory of Teichmüller spaces. Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York 1988. [27] J. Noguchi, A higher-dimensional analogue of Mordell’s conjecture over function fields. Math. Ann. 258 (1981/82), 207–212. [28] R. C. Penner, Bounds of least dilatations. Proc. Amer. Math. Soc. 113 (1991), 443–450. [29] H. L. Royden, Automorphisms and isometries of Teichmüller space. In Advances in the theory of Riemann surfaces, Ann. of Math. Studies 66, Princeton University Press, Princeton, NJ, 1970, 369–383. [30] G. Shabat, The complex structure of domains covering algebraic surfaces. Funct. Anal. Appl. 11 (1977), 135–142. [31] H. Shiga, On analytic and geometric properties of Teichmüller spaces. J. Math. Kyoto Univ. 24 (1984), 441–452. [32] H. Shiga, On the monodromies of holomorphic families of Riemann surfaces and modular transformations. Math. Proc. Cambridge Philos. Soc. 122 (1997), 541–549. [33] H. Shiga and H. Tanigawa, On the Maskit coordinates of Teichmüller spaces and modular transformations. Kodai Math. J. 12 (1989), 437–443. [34] Z. Slodkowski, Holomorphic motions and polynomial hulls. Proc. Amer. Math. Soc. 111 (1991), 347–355. [35] O. Teichmüller, Veränderliche Riemannsche Flächen. Deutsche Math. 7 (1944), 344–359. [36] W. P. Thurston, Minimal stretch maps between hyperbolic surfaces. Preprint, 1998. [37] W. Veech, Dynamics over Teichmüller space. Bull. Amer. Math. Soc. (N.S.) 14 (1986), 103–106. [38] W.Veech, Moduli spaces of quadratic differentials. J. d’Analyse Math. 55 (1990), 117–171. [39] S. A. Wolpert, The Fenchel–Nielsen deformation. Ann. of Math. 115 (1982), 501–528.

Part B

Representation spaces and generalized structures, 2

Chapter 12

The deformation of flat affine structures on the two-torus Oliver Baues

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The theorem of Benzécri . . . . . . . . . . . . . . . . . . . . . 3 Locally homogeneous structures and their deformation spaces . . 4 Construction of flat affine surfaces . . . . . . . . . . . . . . . . . 5 The classification of flat affine structures on the two-torus . . . . 6 The topology of the deformation space . . . . . . . . . . . . . . Appendix A. Conjugacy classes in the universal covering group of GL.2; R/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Example of a two-dimensional geometry where hol is not a local homeomorphism . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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461 467 469 491 505 512

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1 Introduction A flat affine structure on a smooth manifold is specified by an atlas with coordinate changes in the group of affine transformations of Rn . A manifold together with such an atlas is called a flat affine manifold. Equivalently, a flat affine manifold is a smooth manifold which has a flat and torsion-free connection on the tangent bundle. A particular class of examples is furnished by Riemannian flat manifolds, but the class of flat affine manifolds is much larger. The study of flat affine manifolds has a long history which can be traced back to the local theory of hypersurfaces and Cartan’s projective connections. Global questions were first studied in the context of Bieberbach’s theory of crystallographic groups, and they have gained renewed interest in a more general setting by Ehresmann’s theory of locally homogeneous spaces, and more recently in Thurston’s geometrisation program which shows the importance of locally homogeneous structures in the classification of manifolds. Flat affine manifolds are affinely diffeomorphic if they are diffeomorphic by a diffeomorphism which is locally an affine map in the coordinate charts. The universal

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covering space of a flat affine manifold admits a local affine diffeomorphism into affine space An D Rn which is called the development map; its image is an open subset of Rn , called the development image. The development map and image provide rough invariants for the classification of flat affine manifolds. Benzécri [12] showed that a closed oriented surface which supports a flat affine structure must be diffeomorphic to a two-torus, thereby confirming in dimension two a conjecture of Chern that the Euler characteristic of a compact flat affine manifold must be zero. The flat affine structures on the two-torus and their development images were partially classified by Kuiper [51] in 1953. The classification was completed by independent work of Furness–Arrowsmith [28] and Nagano–Yagi [60] around 1972. Their works show that the flat affine structures on the two-torus fall into four main classes which have development image the plane A2 , the halfspace, the sector, or the once-punctured plane. The moduli space of flat affine structures is by definition the set of flat affine structures up to affine diffeomorphism. More precisely, the group Diff.T 2 / of all diffeomorphisms of the two-torus T 2 acts naturally on the set of flat affine structures on T 2 . The set of orbits classifies flat affine two-tori up to affine diffeomorphism; it is called the moduli space. The deformation space is the set of all flat affine structures divided by the action of the group Diff 0 .T 2 / of diffeomorphisms which are isotopic to the identity. This action classifies flat affine structures on T 2 up to isotopy, or, equivalently, affine two-tori with a marking. The deformation space has a natural topology, which it inherits from the C 1 topology on the space of development maps. In this chapter, our aim is to describe the topology of the deformation space D.T 2 ; A2 / of all flat affine structures on the two-torus. The development process gives, for each flat affine two-torus, a natural homomorphism h W Z2 D 1 .T 2 / ! Aff.2/ of the fundamental group of the torus to the plane affine group Aff.2/ D Aff.R2 /. This homomorphism is called the holonomy homomorphism and it is defined up to conjugacy with an affine map. The holonomy thus gives rise to a continuous open map hol W D.T 2 ; A2 / ! Hom.Z2 ; Aff.2//=Aff.2/ from the deformation space to the space of conjugacy classes of homomorphisms, which is called the holonomy map. By a general theorem of Thurston and Weil concerning deformations of locally homogeneous structures on manifolds, this map has an open image. As such, the deformation space of flat affine structures is the natural analogue of the Teichmüller space of conformal structures, or, equivalently, constant curvature Riemannian metrics, on surfaces. Its construction is completely analogous to the definition of the Teichmüller space for flat Riemannian metrics on the two-torus, or hyperbolic constant curvature 1 metrics on surfaces Mg , g  2. In these classic situations, both the Teichmüller space Tg and its quotient the moduli space are Hausdorff spaces. The Teichmüller space of flat metrics T1 is diffeomorphic to R2 , and the Teichmüller space of hyperbolic metrics Tg , g  2, is diffeomorphic to R6g6 . More-

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over, the corresponding holonomy map topologically identifies Tg with an open subset of the quotient space Hom.Z2 ; Isom.R2 //=Isom.R2 /, for g D 1, or, respectively, a component of the space Hom.g ; PSL.2; R//= PSL.2; R/, g  2. Here the analogy with the classical theory breaks down, and neither of these facts is true for the deformation space of flat affine structures. In fact, the group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus, namely, the action of the affine group Aff.2/ on the homogeneous space X D R2 D Aff.2/= GL.2; R/; has non-compact stabiliser GL.2; R/, and therefore the underlying geometry on X is highly non-Riemannian. This is illustrated by the fact that various kinds of flat affine structures, with sometimes strikingly distinct geometric properties, are supported on the two-torus. A fact which can be seen already from the various possible development images for flat affine structures, and which is also reflected in the structure and topology of the deformation space. Here phenomena arise which are completely different from the case of constant curvature metrics or conformal structures on surfaces. Another salient difference stems from the fact that the local model of the deformation space of flat affine structures, namely, the character variety Hom.Z2 ; Aff.2//=Aff.2/; arises as a quotient space of an algebraic variety by a non-reductive group action. The properties of such actions and their invariant theory are generally poorly understood. The case of deformation of complete flat affine structures bears the closest resemblance to the classical situation. A flat affine structure is called complete if the development map is a diffeomorphism, a property which in the Riemannian situation is always guaranteed. A flat affine two-torus is complete if and only if its development image is the affine plane A2 . The deformation space of complete affine structures on the two-torus was studied recently in [2], [7]. It is shown there, for example, that the holonomy map identifies the space of complete structures Dc .T 2 ; A2 / with a locally closed subspace of the space of homomorphisms Hom.Z2 ; Aff.R2 //, and, moreover, the space Dc .T 2 ; A2 / is homeomorphic to R2 . However, the topology of the moduli space of complete flat affine structures, which, with respect to appropriately chosen coordinates for Dc .T 2 ; A2 /, is homeomorphic to the quotient space of R2 by the natural action of GL.2; Z/, is highly singular. This chapter is devoted to the study of the global and local structure of the space of deformations of all flat affine structures on the two-torus. The deformation space of all flat affine structures is much larger than the deformation space of complete flat affine structures. Indeed, the deformation space Dc .T 2 ; A2 / of complete flat affine structures on the two-torus forms a closed two-dimensional subspace in the deformation space of all structures D.T 2 ; A2 /, which itself is a space of dimension four. In the general situation, the holonomy map hol W D.T 2 ; A2 / ! Hom.Z2 ; Aff.2//=Aff.2/ for the deformation space of flat affine structures is no longer a homeomorphism

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onto its image. That is, there exist flat affine structures on the two-torus which have the same holonomy group and which have dramatically different geometry. (Compare, in particular, Example 5.5 in this chapter.) Moreover, the holonomy image in Hom.Z2 ; Aff.2// contains singular orbits for the affine group Aff.2/, which in turn give rise to non-closed points in the deformation space D.T 2 ; A2 /. This also shows that the deformation space is not a Hausdorff space. It is a four-dimensional and connected space which has an intricate topology and it supports various substructures arising from the different types of affine flat geometries on the two-torus. As our main result on the local structure of the space of deformations of flat affine structures we prove in this chapter that the holonomy map hol is a local homeomorphism onto its image. That is, at least locally the topology of the deformation space D.T 2 ; A2 / is fully controlled by the character variety. We remark that this is not a general phenomenon for deformation spaces of locally homogeneous structures, not even on surfaces. Indeed, in Appendix B of this chapter, we specify a two-dimensional homogeneous geometry whose deformation space of structures on the two-torus has a holonomy map hol which locally near certain structures is a branched covering. Examples of deformation spaces of flat conformal structures on three-dimensional manifolds where the holonomy map hol is not locally injective at exceptional points were found previously by Kapovich and are discussed in [45]. The chapter is organized as follows. In Section 2 we give a self-contained proof of Benzécri’s theorem which states that a closed orientable flat affine surface is diffeomorphic to the two-torus. In Section 3 we describe the deformation theory of compact locally homogeneous manifolds, including its foundational results and give basic examples. Section 4 discusses several methods to construct flat affine surfaces and introduces the main classes of flat affine structures on the two-torus. In Section 5 we prove the main classification theorem for flat affine structures on the two-torus in detail, including the crucial and nontrivial fact that the development map of such a structure is always a covering map. Finally, in Section 6 we put the pieces together in order to prove that the holonomy map for the deformation space of flat affine structures on the two-torus is a local homeomorphism to the character variety. In addition, Appendix A gives an account on conjugacy classes in GL.2; R/ and in its universal covering group. In Appendix B we describe a two-dimensional homogeneous geometry such that the holonomy map for its deformation space of structures on the two-torus is not everywhere a local homeomorphism. Acknowledgement. The author wishes to thank Wolfgang Globke, Bill Goldman and Athanase Papadopoulos for their interest, advice and support during the long gestation of this article. I thank Athanase Papadopoulos especially for inviting this project as a contribution to Volume IV of the “Handbook of Teichmüller theory”, and Bill Goldman for sharing his insight on the subject. Most pictures in the article were created by Wolfgang Globke with the software Omnigraffle for Macintosh.

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The following and other similar pictures illustrate convergence of development maps in the deformation space of flat affine structures on the two-torus. Each development map gives rise to a tiling of an open domain in affine space which is deformed with the change of development maps.

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Figure 1. Tiled sectors approaching the standard plane.

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Figure 3. Tiled sectors approaching a halfplane of type C2 .

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Figure 4. Tiled sectors approaching a halfplane of type C1 .

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Figure 5. Tiled halfplanes approaching the standard plane.

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2 The theorem of Benzécri Let M be a closed oriented surface of genus g. Then the Gauß–Bonnet theorem [41] expresses the Euler characteristic .M / D 2  2g as an integral over the Gauß curvature of any Riemannian metric on M . In particular, a flat Riemannian closed surface M has Euler characteristic zero, and therefore it is diffeomorphic to a two-torus. If M is a closed flat affine surface, the Gauß– Bonnet theorem does not apply, since the corresponding flat connection is possibly non-Riemannian. However, the strong topological restriction applies to flat affine surfaces, as well: Theorem 2.1 (Benzécri, [12]). Let M be a closed flat affine surface. Then M has Euler characteristic zero. Proof. First we remark that the sphere S 2 does not admit a flat affine structure. In fact, since S 2 is simply connected, the development image of a flat affine structure on S 2 would be compact and open in R2 , which is absurd. z ! M be the uniNow we assume that M has genus g, g  1. Let p W M z is a flat affine manifold which is diffeomorphic to R2 . versal covering. Then M z along its consecutive sides Moreover, M is obtained by gluing a 4g-gon P  M a1 ; b1 ; a1 ; b1 ; : : : ; ag ; bg ; ag ; bg , with side pairing transformations gai , gbi such that gai ai D ai and gbi bi D bi . These transformations are subject to the single cycle relation Y gai gbi ga1 gb1 D idMz i i iD1;:::;g

z ! and generate the discontinuous group of deck transformations of the covering p W M z . Note however M . In particular, the side pairing transformations are affine maps of M z is a closed oriented topological disc with piecewise smooth that the polygon P  M boundary. (The construction may be carried out, in Euclidean geometry if g D 0, respectively hyperbolic geometry, for g  2, such that the edges of P are geodesic segments. See, for example, [64].) Let xQ 0 denote the vertex of P belonging to the sides a1 and bg . Let v ¤ 0 be a tangent vector at xQ 0 . Now choose a non-vanishing vector field V along the boundary of P such that V .xQ 0 / D v, and, furthermore, such that V restricted to ai (resp. bi ) is related to V restricted to ai (resp. bi ) by the corresponding side pairing transformation. (We can obtain such a V by constructing vector fields along the closed curves p ai , p bi which coincide at x0 D p.xQ 0 /.) Next we extend V to a vector field X on P , which has an isolated singularity in the interior of P . The index of the vector field X at the singularity may be calculated as the turning number of the restriction of X to the positively traversed boundary of P , see [41]. For this, recall that the turning number .V / 2 Z of a closed non-vanishing vector field

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V W I ! A2  0 is defined by the equation .V / 2 D .1/  .0/; where  W I ! R is any lift of the map I ! S 1 , t 7! V .t /=jV .t /j. Now, since the flat z is simply connected, M z has a global parallelism which identifies affine manifold M z z each tangent space Tx M with TxQ0 M . Therefore, we can choose any scalar product in z to compute the index of X by the above formula. TxQ0 M z , they Since the side pairing maps gai and gbi are affine transformations of M preserve antipodality of any two vectors V .s/ and V .t /. This implies that the turn of V restricted to the positively traversed curve a1 b1 a1 b1 is less than 2. And consequently, j.V /j < g. Our construction implies that the vector field X on P projects to a vector field on M . Therefore, by the Poincaré–Hopf theorem [41], [58], the index of X equals the Euler characteristic .M / of M . We thus obtain the estimate j.M /j D j2  2gj < g: This implies g D 1. Benzécri’s theorem was generalised by Milnor [57] to the more general Theorem 2.2. Let E be a flat rank two vector-bundle over a closed orientable surface Mg , g  1, then j.E/j < g. Here, .E/ denotes the evaluation of the Euler-class e.E/ 2 H 2 .M; Z/ on the fundamental homology class of M . In case of the tangent bundle E D TMg , the equality .TMg / D .Mg / (see Section 11 of [59]) implies Benzécri’s theorem. Wood [76] interpreted Milnor’s result in the context of circle-bundles. See [37] for a recent survey on Benzécri’s theorem, Milnor’s inequality and related topics. Generalizations to higher dimensions. Weak analogues of the Milnor–Wood estimate for the Euler-class of higher-dimensional manifolds were subsequently given by Sullivan [68] and Smillie in his doctoral thesis [66]. See also [18] for a recent contribution in this realm. The Chern conjecture asserts that any compact flat affine manifold should have Euler characteristic zero. Kostant and Sullivan [50] observed that every compact and complete flat affine manifold has Euler characteristic zero. There are some additional affirmative results under various assumptions on the holonomy group, see for example [31]. The original conjecture, however, remains a difficult open problem. Another fruitful generalization of the Milnor–Wood inequality concerns the representation theory of surface groups into higher-dimensional simple Lie groups, see [19] for a survey.

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3 Locally homogeneous structures and their deformation spaces z ! M . Let X be Let M be a smooth manifold and fix a universal covering space p W M a homogeneous space for the Lie group G, on which G acts effectively. The manifold M is said to be locally modeled on .X; G/ if M admits an atlas of charts with range in X such that the coordinate changes are locally restrictions of elements of G. A maximal atlas with this property is called an .X; G/-structure on M . The manifold M together with an .X; G/-structure is called an .X; G/-manifold, or locally homogeneous space modeled on .X; G/. A map between two .X; G/-manifolds is called an .X; G/-map if it coincides with the action of an element of G in the local charts. If the .X; G/map is a diffeomorphism it is called an .X; G/-equivalence and accordingly the two manifolds are called .X; G/-equivalent.

3.1 .X; G /-manifolds, development map and holonomy Every .X; G/-manifold comes equipped with some extra structure, called the develz ! M the universal opment and the holonomy. Via the covering projection p W M covering space of the .X; G/-manifold M inherits a unique .X; G/-structure from M . We fix x0 2 M , and a local .X; G/-chart at x0 . The corresponding development map of the .X; G/-structure is the .X; G/-map z !X DW M which is obtained by analytic continuation of the local chart. z , there exists a unique element h.ˆ/ 2 G For every .X; G/-equivalence ˆ of M such that D B ˆ D h.ˆ/ B D:

(3.1)

z via deck transformations. The fundamental group 1 .M / D 1 .M; x0 / acts on M This induces the holonomy homomorphism h W 1 .M; x0 / ! G which satisfies D B  D h./ B D;

for all  2 1 .M; x0 /.

(3.2)

Note that, after the choice of the development map (which corresponds to a choice z of x0 ), of a germ of an .X; G/-chart in x0 and also the choice of a lift xQ 0 2 M the holonomy homomorphism h is well defined. Therefore, the .X; G/-structure on M determines the development pair .D; h/ up to the action of G, where G acts by left-composition on D, and by conjugation on h. Specifying a development pair is equivalent to constructing an .X; G/-structure on M :

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z ! X which satisfies (3.2), for Proposition 3.1. Every local diffeomorphism D W M some h W 1 .M; x0 / ! G, defines a unique .X; G/-structure on M , and every .X; G/structure on M arises in this way. 3.1.1 Compactness and completeness of .X; G /-manifolds. An important special case arises if the development map is a diffeomorphism. Recall the following definition: Definition 3.2 (Proper actions). A discrete group  is said to act properly discontinuously on X if, for all compact subsets   X , the set  D f 2  j   \  ¤ ;g is finite. More generally, if  is a locally compact group, and  is required to be compact, then the action is called proper. Example 3.3 (.X; G/-space forms). Let  be a group of .X; G/-equivalences acting properly discontinuously and freely on X . Then X=  is a manifold which inherits an .X; G/-structure from X. If X is simply connected the identity map of X is a development map for X= . If the development map is a covering map onto X, the .X; G/-manifold M will be called complete. Simple examples (cf. the Hopf tori in Example 4.8) show that compactness of M does not imply completeness. Example 3.4 (Compactness and completeness). If G acts properly on X then every compact .X; G/-manifold is complete. In general, the relation between the properties of the G-action on X, and the completeness properties of compact .X; G/-manifolds is only vaguely understood. See [21] for a striking contribution in this direction in the context of flat affine manifolds. Further discussion of .X; G/-geometries and the properties of the development process may be found in [20], [70]. It may well happen that an .X; G/-geometry does not admit (non-finite) proper actions (see [47], [52], [43]) or no compact .X; G/-manifolds at all [11]. Example 3.5 (Calabi–Marcus phenomenon). Let A2  0 be the once- punctured affine plane. It is easily observed that every discrete subgroup of SL.2; R/ which acts properly on A2  0 must be finite, see Figure 6. This is called the Calabi–Markus phenomenon. It follows that the homogeneous space .A2  0; SL.2; R// has only quotients by finite groups. Therefore, a complete space modeled on .A2  0; SL.2; R// cannot be compact. In fact, we will remark in Example 3.11 below that there do not exist compact .A2  0; SL.2; R//-manifolds at all.

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Figure 6. Dynamics of a hyperbolic rotation and a shearing acting on A2  0.

Prominent .X; G /-structures on surfaces. Let Mg denote an orientable surface of genus g. In the context of this paper, the following .X; G/-structures play a prominent role. Example 3.6. (1) .S2 ; O.2//, spherical geometry, g D 0. (2) .R2 ; E.2//, plane Euclidean geometry, g D 1. (3) .H2 ; PSL.2; R//, plane hyperbolic geometry, g  2. (4) .R2 ; Aff.2//, plane affine geometry, g D 1. (5) .P 2 .R/; PSL.3; R//, plane projective geometry, g  0. Every compact orientable surface of genus g  2 supports hyperbolic structures. Also every compact surface supports a projective structure, see [22]. By Benzécri’s theorem the only compact surfaces which support a flat affine structure are the twotorus and the Klein bottle. The classification of flat affine structures on the two-torus was completed in the 1970s, see Section 5 of this article. Subsequently, Bill Goldman in his undergraduate thesis [30] classified projective structures on the two-torus in 1977. Note that every Euclidean or hyperbolic compact surface is complete (compare Example 3.4). The majority of flat affine structures on the two-torus are not complete but the development map of a flat affine structure on the two-torus is always a covering onto its image (see Theorem 5.1). The development map of a projective structure on a surface may not even be a covering [22]. 3.1.2 .X; G /-subgeometries. We may relate different locally homogeneous geometries by inclusion as follows. Definition 3.7. Let .X; G/ and .X 0 ; G 0 / be homogeneous spaces and W G 0 ! G a homomorphism together with a -equivariant local diffeomorphism o W X 0 ! X. Then we say that .X 0 ; G 0 / is subjacent to or a subgeometry of .X; G/. The subgeometry is called full if the map o is surjective onto X . The subgeometry is called a covering of geometries if o W X 0 ! X is a regular covering map with group of deck transformations precisely the kernel of . If .X 0 ; G 0 / is a subgeometry of .X; G/ then X 0 is an .X; G/-manifold with development map o W X 0 ! X. Note also that o is a covering map onto its image, since o is an

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equivariant map of homogeneous spaces. The group G 0 then acts as a group of .X; G/equivalences of X 0 , so that X 0 is, in fact, a homogeneous .X; G/-manifold.

A

Example 3.8. Let p W A2  0 ! A2  0 be the universal covering of the once-puncf R/ ! GL.2; R/ the universal covering group of tured affine plane A2  0, and GL.2; GL.2; R/. Then the subgeometry

A

f R// ! .A2  0; GL.2; R// p W .A2  0; GL.2; is a full subgeometry and, indeed, it is a covering of geometries. If .X 0 ; G 0 / is a subgeometry of .X; G/ then, in particular, every .X 0 ; G 0 /-manifold with development map D 0 inherits naturally an .X; G/-manifold structure with development map o B D 0 . This observation provides a useful tool to construct .X; G/-manifolds. Assume, for instance, that G 0 acts properly on X 0 and  0  G 0 is a discrete subgroup. Then  0 nX 0 is an .X 0 ; G 0 /-manifold which inherits an .X; G/-structure via o. The following special case is of particular importance: Definition 3.9 (Étale .X; G/-representations). If G 0 acts on X 0 with finite stabilizer then an inclusion of geometries W G 0 ! G as above is called an étale representation of G 0 into .X; G/. If is étale with open orbit .G 0 /x0 the group manifold G 0 inherits via the orbit map 7 .g 0 /x0 ; o W G 0 ! X; g 0 ! a natural .X; G/-structure which is invariant by left-multiplication of G 0 . In particular, if  0  G 0 is a discrete subgroup then the coset space  0 nG 0 inherits an .X; G/manifold structure. Example 3.10 (Geometries subjacent to the punctured plane). The affine automorphism group of the once-punctured affine plane A2   0 is the linear group 2  0; GL.2; R/ has full subgeometries GL.2; R/. The homogeneous geometry A  2  2 A  0; GL.1; C/ and .A  0; SL.2; R//. Note that the first one arises from an étale affine representation of the abelian Lie group R2 . Further subgeometries, which are not full, are defined by the abelian étale Lie subgroups C1 and B, which are listed in (2) and (3) of Example 4.2. Of course, all these homogeneous spaces define particular subgeometries of plane affine geometry, as well. 3.1.3 Existence of compact forms. A compact manifold M , which is locally modeled on .X; G/, will be called a compact form for .X; G/. Given a homogeneous space .X; G/, it is possibly a difficult problem to decide if it has compact form.

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Example 3.11 (.A2  0; SL.2; R// has no compact form). By the Calabi–Markus phenomenon (see Example 3.5), .A2  0; SL.2; R// has only quotients by finite groups. Since .A2  0; SL.2; R// is a subgeometry of plane affine geometry, Benzécri’s theorem (Theorem 2.1) and the classification of flat affine structures with development image A2  0 (see Theorem 5.1 ) imply the stronger result that there is no compact locally homogeneous surface modeled on the homogeneous space .A2  0; SL.2; R//.   On the contrary, the spaces .A2  0; GL.2; R// and A2  0; GL.1; C/ evidently have complete compact forms. For example, every lattice   GL.1; C/ acts properly discontinuously and freely on A2  0, which thus gives rise to a compact flat affine ı manifold A2  0 . Benzécri’s theorem implies that every orientable compact form of the space .A2  0; GL.2; R// is diffeomorphic to the two-torus, and in particular it has abelian fundamental group Z2 . Example 3.12 (Compact forms of .A2  0; GL.2; R//). The classification theorem asserts that the development map of a flat affine structure on the two-torus is a covering map onto the development image (cf. Proposition 5.8). In particular, every compact locally homogeneous surface modeled on the homogeneous spaces .A2  0; GL.1; C// or .A2  0; GL.2; R// is diffeomorphic to the two-torus and either it is complete (which is always true for .A2  0; GL.1; C//-structures) or its development image is a sector of halfspace in A2 (see Section 4.1). In Section 4.3, we describe the construction of all compact .A2  0; GL.2; R//manifolds which are complete. The classification theorem for all structures, including the non-complete case, is stated in Section 5.2.

3.2 Convergence of development maps The space of .X; G/-development maps for the manifold M is the set Dev.M / D Dev.M; X; G/ 1

of all local C -diffeomorphisms z !X DW M which, for some h 2 Hom.1 .M /; G/, and, for all  2 1 .M /, satisfy D B  D h. / B D: We endow the space of development maps with the compact C 1 -topology. In this topology, a sequence of smooth maps converges if and only if it and all its derivatives (computed in local coordinate charts) converge uniformly on the compact subsets of z . In particular, Dev.M / thus becomes a Hausdorff second countable topological M space.

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3.2.1 Convergence of holonomy. Let M be compact. Then 1 .M / is finitely generated, and we equip Hom.1 .M /; G/ with the topology of pointwise convergence. Then the map hol W Dev.M / ! Hom.1 .M /; G/; D 7! h; is continuous, since G has the C 1 -topology of maps on X . The main theorem on deformations of .X; G/-structures (see Theorem 3.15 below) asserts that a small deformation of holonomy induces a deformation of development maps. That is, the map hol admits local sections. By compactness of M , the convergence of development maps is controlled on a fundamental domain and by the holonomy. z be an open subset with Fact 3.13 (Holonomy determines convergence). Let U  M z compact closure such that p.U / D M , where p W M ! M is the universal covering. Then a sequence of development maps Di 2 Dev.M / with holonomy hi converges to a development map D if and only if the restrictions of Di to U converge to D and the homomorphisms hi converge to the holonomy h of D. A particular property of the C 1 -topology is that it does not control the behavior of maps outside compact sets. This allows for possibly unexpected phenomena: Example 3.14 (Openness of embeddings fails). Let Deve .M / be the subset of develz be a compact fundamental domain opment maps which are injective. Let K  M for the action of 1 .M /. By Lemma 1.3 in Chapter 2 of [39], the set of development maps which are injective on K is open with respect to the C 1 -topology. In particular, it is open with respect to the C 1 -topology. However, the global behavior of development maps is controlled by the holonomy. Therefore, even if M is compact Deve .M / may not be an open subset in Dev.M /. On the two-torus there are injective development maps in Dev.T 2 ; A2 ; Aff.2// which contain a non-trivial covering map in every small neighborhood. This is even true for the development of the standard translation structure, cf. Section 6.4.1 and Figure 7.

3.2.2 Deformation of development maps. If M is compact then, as observed by Thurston [71], building on earlier work of Weil [74], a small deformation of holonomy in the space of homomorphisms Hom.1 .M /; G/ induces a deformation of .X; G/development maps. Before stating the theorem precisely, we discuss the z, Action of diffeomorphisms of M on development pairs. Let x0 2 M and xQ 0 2 M p.xQ 0 / D x0 , be basepoints and ˆ 2 Diff.M; x0 / a basepoint-preserving diffeomorz 2 Diff.M z ; xQ 0 /. The group Diff.M; x0 / of basepoint-preserving phism with lift ˆ z diffeomorphisms then acts on development pairs, by mapping D 2 Dev.M / to D B ˆ. We let Diff 1 .M; x0 / denote the subgroup of Diff.M; x0 / consisting of diffeomorphisms which are homotopic to the identity by a basepoint-preserving homotopy, and Diff 0 .M; x0 / the identity component of Diff.M; x0 / (that is, the subgroup of elements

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which are isotopic to the identity). Then the action of Diff 1 .M; x0 / and its subgroup Diff 0 .M; x0 / on the set of development maps Dev.M / leaves the holonomy invariant, since Diff 1 .M; x0 / acts trivially on 1 .M; x0 /. See [20], [31], [53], [14] for a more detailed discussion of the following: Theorem 3.15 (Deformation theorem, Thurston et al.). Let M be a compact manifold. Then the induced map hol W Diff 0 .M; x0 /n Dev.M / ! Hom.1 .M /; G/

(3.3)

which associates to a development map its holonomy homomorphism is a local homeomorphism. The theorem states that the map hol W Dev.M / ! Hom.1 .M /; G/ is continuous and open. In addition, hol locally admits continuous sections. Such a section is called a development section. More specifically, it is proved (see below) that every convergent sequence of holonomy maps lifts to a convergent sequence of development maps, and two nearby development maps with identical holonomy are isotopic by a basepoint-preserving diffeomorphism. Therefore, a sequence of points in the quotient space Diff 0 .M; x0 /n Dev.M / is convergent if and only if there exists a corresponding lifted sequence of development maps which converges. The main idea in the proof of Theorem 3.15 due to Weil [74] is easy to grasp. Here we sketch the construction of the development section in the particular case of flat affine two-tori. In addition, we consider only tori which are obtained by gluing polygons in the plane (cf. Section 4.2). A similar approach is also valid for non-homogeneous tori which are obtained as quotients of the universal covering affine manifold of A2  0 (cf. Section 4.3), and, in fact, in the general case of arbitrary .X; G/-manifolds, compare [53], [74]. A somewhat different approach to this result is explained in [31] and the recent survey [36] on locally homogeneous manifolds.

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Proof of Theorem 3.15. Let M be a flat affine two-torus and Do W R2 ! A2 , ho W Z2 ! Aff.2/ a development pair for M . Let h 2 Hom.Z2 ; Aff.2//,  0, be a small deformation of ho . To obtain the development section, we construct a curve of development maps D with holonomy h , which converges to Do in the compact C 1 -topology. For this, we assume that the development pair of M is represented as the identification space of a polygon P in affine space. In fact, as explained in [2] (Section 2), P can be chosen to be a quadrilateral contained in A2 , which is glued along its sides by the generators 1 , 2 of 1 .T 2 / D Z2 using the holonomy images ho .i / 2 Aff.2/. The generators satisfy cycle relations and certain gluing conditions. Next we fix a diffeomorphism of the standard unit square in R2 with P . Using h, this extends 1 .T 2 /-equivariantly to a smooth covering ı R2 ! Xx D .P  / ; where the identification space Xx is a flat affine manifold which is obtained as the disjoint union of the polygons P ,  2 , glued along their edges as determined by the side pairings ho .i /. Here  D ho .Z2 / is the holonomy group of M . Moreover, the inclusion P ! A2 extends to a development map x W Xx ! A2 : D The composition of both maps yields the desired development map D W R2 ! A2 with holonomy ho . The space Xx is the holonomy covering space of M , see Proposition 2.1 of [2] for a detailed account. The development section D may now be obtained in a similar manner. In fact, for small > 0, P can be deformed continuously to a quadrilateral P , which satisfies the gluing conditions with respect to h . (See Figure 8 for an illustration.) This gives rise to a series of identification spaces Xx D .P   /=  , and corresponding development maps D W R2 ! A2 with holonomy h . By the above Fact 3.13, the developments maps D converge to Do in the C 1 -topology. The above construction of the development section is illustrated in Figures 8 and 9. 3.2.3 Topological rigidity of development maps. Although local rigidity holds by the deformation theorem, it may fail globally. If the map hol in (3.3) is not injective (as happens in the case of flat affine two-tori, see the basic Examples 4.9, 4.15 and also Section 6 for further discussion), there do exist non-isomorphic .X; G/-manifolds with the same holonomy homomorphism h. On the contrary, if the domain of discontinuity  for the holonomy group h./,  D 1 .M /, on X is large then the development is uniquely determined. This is the case, for example, if  D X and h./ is the holonomy of a compact complete .X; G/-manifold. z ! X be the development map Example 3.16 (Discontinuous holonomy). Let D W M for an .X; G/-structure on the compact manifold M with holonomy homomorphism h.

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P

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Figure 8. The fundamental polygon P and its development deform with the holonomy.

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Figure 9. A family of development maps for the once-punctured plane A2  0 collapses to the development process of an affine half-plane.

If h./ acts properly discontinuously and freely with compact quotient on X then D is a covering map onto X . (In fact, D is a covering, since the local diffeomorphism on compact manifolds M ! X= h./ induced by D is a covering map.) It follows z ! X with holonomy homomorphism h is that every other development map D 0 W M z / is a diffeomorphism which centralizes of the form D 0 D D B ˆ, where ˆ 2 Diff.M the deck transformation group .

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A more involved argument allows to show that D is determined by h./ if the Hausdorff dimension of X   is small, see [38].

A

f C.2; R//-forms are Example 3.17. The development maps of compact .A2  0; GL rigid, see Section 5.2, Theorem 5.6. We remark that the domain of discontinuity for the holonomy group of such a manifold can be a proper open subset of A2  0.

A

3.3 Deformation spaces of .X; G /-structures Let S.M / D S.M; X; G/ denote the set of all .X; G/-structures on M . The group Diff.M / of all diffeomorphisms of M acts naturally on this set such that two .X; G/structures are in the same orbit if and only if they are .X; G/-equivalent. The set of all .X; G/-structures on M up to .X; G/-equivalence is called the moduli space M.M / D M.M; X; G/ of .X; G/-structures. Definition 3.18. The deformation space for .X; G/-structures on M is the quotient space D.M / D D.M; X; G/ D S.M; X; G/=Diff 1 .M / of equivalence classes of .X; G/-structures up to homotopy. Thus, two .X; G/-structures define the same point in D.M / if they are equivalent by an .X; G/-equivalence which is homotopic to the identity of M . The moduli space M.M / is the quotient space of the deformation space D.M / by the group of homotopy classes of diffeomorphisms of M . Remark 3.19. There is some inconsistency in the literature about the definition of the deformation space. Many authors define D.M / to be the space of structures up to isotopy. If M is a surface (two-dimensional manifold) two homotopic diffeomorphisms are isotopic, by classical results of Dehn, Nielsen, and Baer (see for example [67]). Therefore, in this case, these two definitions coincide. The corresponding fact fails in higher dimensions, even for tori, see [42]. We observe that the Lie group G acts by left-composition on the space of development maps. This action is continuous and free, and the set of .X; G/-structures naturally identifies with the quotient by the action of G, that is, S.M; X; G/ D G nDev.M; X; G/: Indeed, if g 2 G and D 2 Dev.M; X; G/ is a development map then g B D is another development map for the same .X; G/-structure on M . This exhibits the deformation space as a double quotient space D.M; X; G/ D G nDev.M; X; G/= Diff 1 .M /:

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The C 1 -topology on the set of .X; G/-structures is the quotient topology inherited from Dev.M; X; G/. (Thurston [70] (Chapter 5) also gives a direct description of the topology on S.M / in terms of convergence of sets of local charts which define the elements of S.M /, see [20], 1.5.1.) The deformation space and the moduli space carry the quotient topology inherited from the set of .X; G/-structures. 3.3.1 Orientation components of the deformation space. Let X be a G-space which is orientable. We let G C denote the normal subgroup of orientation-preserving elements of G. Now assume that M is an .X; G/-manifold which is orientable. We fix an orientation for M . Then there is a disjoint decomposition Dev.M; X; G/ D DevC .M; X; G/ [ Dev .M; X; G/;

(3.4)

where DevC .M; X; G/ and Dev .M; X; G/ denote the closed (and open) subspaces. Since M is orientable, the components of the decomposition (3.4) are preserved by the action of Diff 1 .M; x0 / on development maps. Furthermore, the action of G C on development maps preserves the components. Therefore, the deformation space D.M; X; G C / is decomposed into two disjoint open and closed subsets, the orientation components, D.M; X; G C / D DC .M; X; G/ [ D .M; X; G/:

(3.5)

Note that every orientation-reversing element of G exchanges the orientation components of Dev.M; X; G/ and therefore also of D.M; X; G C /. Hence, if G contains orientation-reversing elements then the subgeometry .X; G C / ! .X; G/ induces a homeomorphism DC .M; X; G/ D.M; X; G/: 3.3.2 The topology of the deformation space. The following classical and fundamental example gives a role model for the investigation of the properties of deformation spaces for locally homogeneous structures. Example 3.20 (Teichmüller space Tg ). Let G C D PSL.2; R/ be the group of orientation-preserving isometries of the hyperbolic plane H2 and M D Mg a surface of genus g, g  2. By the uniformization theorem, the Teichmüller space Tg of conformal structures on a surface Mg , g  2, may be considered as the deformation space of constant curvature 1 metrics, that is, Tg D DC .Mg ; H2 ; PSL.2; R//: The space Tg is homeomorphic to R6g6 . Recall that the mapping class group Mapg D Diff C .Mg /=Diff 0 .Mg / Š OutC .g / is the group of isotopy classes of orientation-preserving diffeomorphisms of a surface. This group acts properly discontinuously on Tg , and the moduli space of conformal

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structures M.Mg / D Tg = Mapg is a Hausdorff space. (See, for example, [1], [23], [64] and other chapters of this handbook [19], [35]). In general, however, the topology on the moduli space and the deformation space can be highly singular, as we can see, in particular, from Examples 3.26 and 3.29 below. The local properties of the deformation space are reflected in the character variety Hom.1 .M /; G/=G, which is the space of conjugacy classes of representations of 1 .M / into G. The holonomy map on the deformation space. Since S.M; X; G/ has the quotient topology from development maps, the holonomy (3.3) induces a continuous map hol W S.M; X; G/ ! Hom.1 .M /; G/=G;

which gives rise to the map hol W D.M / ! Hom.1 .M /; G/=G:

(3.6)

The continuous map hol associates to a homotopy class of .X; G/-structures on M the corresponding conjugacy class of its holonomy homomorphism h. By the deformation theorem (Theorem 3.15), hol is furthermore an open map. The map hol thus encodes a good picture of the topology on D.M /: Example 3.21 (Teichmüller space Tg is a cell). The holonomy image of hyperbolic structures on Mg , g  2, is the subspace Homc .g ; PSL.2; R// of the space Hom.g ; PSL.2; R// which consists of injective homomorphisms with discrete image. The space Homc .g ; PSL.2; R// has two connected components [33]. The  components HomC c .g ; PSL.2; R// and Hom c .g ; PSL.2; R// arise from the orientation of development maps. The group PSL.2; R/ acts freely and properly (by conjugation) on HomC c .g ; PSL.2; R//; the quotient space being homeomorphic to R6g6 . (See, for example, [64], Theorem 9.7.4). By completeness of hyperbolic structures on Mg , every development map is a diffeomorphism. The topological rigidity of development maps (cf. Example 3.16) implies that the induced map hol

Tg D DC .Mg ; H2 / ! HomC c .g ; PSL.2; R//= PSL.2; R// is a homeomorphism. In general, it seems difficult to decide if the map hol is a local homeomorphism, as well. Indeed, Kapovich [45] constructed examples of deformation spaces such

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that the map hol is not everywhere a local homeomorphism. We construct such a counterexample for the deformation space of a two-dimensional geometric structure on the two-torus in Appendix B. In the case of flat affine two-tori though, we shall show that hol is a local homeomorphism (see Section 6.5). The induced map of a subgeometry. Let o W .X 0 ; G 0 / ! .X; G/ be a subgeometry with W G 0 ! G the associated homomorphism (see Section 3.1.2). There is an associated map Dev.M; X 0 / ! Dev.M; X /;

D 0 7! D D o B D 0 ;

(3.7)

and a map on homomorphisms Hom.1 .M /; G 0 / ! Hom.1 .M /; G/;

h0 7! h D B h0 ;

where h D hol.D/. These maps allow to relate the deformation spaces in a commutative diagram of the form D.M; X 0 / 

D.M; X/

hol

hol

/ Hom.1 .M /; G 0 /=G 0 

(3.8)

/ Hom.1 .M /; G/=G.

Note that the properties of the induced map D.M; X 0 / ! D.M; X / can vary wildly with various types of subgeometries. In general, the induced map need not be injective nor surjective. Recall the notion of covering of geometries from Definition 3.7. We shall require the following lemma: Lemma 3.22. If o W .X 0 ; G 0 / ! .X; G/ is a covering of geometries then the induced map on deformation spaces D.M; X 0 / ! D.M; X / is a homeomorphism. Proof. Indeed, since o is a covering the above map (3.7), D 0 7! D, on development maps descends to a Diff.M /-equivariant map on the sets of structures S.M; X 0 / ! S.M; X / which is a homeomorphism. 3.3.3 The topology on the space of .X; G /-structures. The topology on the space S.M; X; G/ is rather well behaved. In fact, S.M; X; G/ is a Hausdorff and metrizable topological space. This can be seen by representing an .X; G/-structure on M as an

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integrable higher order structure in the sense of Ehresmann (cf. [46], Section I.8). We discuss two important examples now: Example 3.23 (S.M; H2 ; PSL.2; R//). The space of hyperbolic structures on a surface M is homeomorphic to the space of hyperbolic (constant curvature 1) Riemannian metrics with the C 1 -topology on the space of Riemannian metrics. It can also be equipped with the structure of a contractible Fréchet manifold, see [23]. Similarly, the space S.M; R2 ; E.2// of flat Euclidean structures is homeomorphic to the space of flat Riemannian metrics on M . In the case of flat affine structures, the action of the affine group on development maps admits a global slice: Example 3.24 (S.M; An / is Hausdorff). Let Dev.M; An / be the set of development maps for flat affine structures on M . We choose a base frame Ex0 on An and a frame z , respectively, and let FmQ 0 on M Devf .M; An / D Devf .M; FmQ 0 ; Ex0 / denote the set of frame-preserving development maps. Since Aff.n/ acts simply transitively on the frame bundle of An , there is a well defined continuous retraction Dev.M; An / ! Devf .M; An /, and, in fact, there is a homeomorphism Dev.M; An / Aff.n/  Devf .M; An /: This shows that the quotient Dev.M; An /=Aff.n/ is homeomorphic to the subspace Devf .M; An / and the affine group Aff.n/ acts properly on the set of development maps. In particular, the space of flat affine structures S.M; An / is a Hausdorff space. Another way to understand the topology on S.M; An / is to identify flat affine structures with flat torsion-free connections on the tangent bundle of M . These form a space of sections of a quotient of the bundle of 2-frames over M , see [46], Proposition IV.7.1. In Section 6.1 of this chapter we employ this approach to study flat affine structures on the two-torus. 3.3.4 The subspace of complete .X; G /-structures. Let Dc .M / denote the subset of the deformation space D.M / which consists of complete .X; G/-space forms (that is, the subspace corresponding to development maps which are diffeomorphisms). We denote with Homc .1 .M /; G/ the set of all injective homomorphisms 1 .M / ! G, such that the image acts properly discontinuously on X. We call Homc .1 .M /; G/ the set of discontinuous homomorphisms. The holonomy homomorphisms belonging to the elements of Dc .M / form an open subset of Homc .1 .M /; G/. In fact, by the rigidity of development maps belonging to discontinuous holonomy homomorphisms (cf. Example 3.16), a small deformation of holonomy, which remains in the domain of discontinuous homomorphisms, lifts to a deformation of complete .X; G/-manifold

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structures on M . Therefore, Theorem 3.15 implies that the restricted map hol W Diff 0 .M; x0 /n Devc .M / ! Homc .1 .M /; G/

is a local homeomorphism. Then the following result is easily observed (see also [3]): Theorem 3.25. Let M be a smooth compact manifold such that the natural homomorphism Diff.M /=Diff 1 .M / ! Out.1 .M // is injective. Then the induced map hol W Dc .M / ! Homc .1 .M /; G/=G is a homeomorphism onto its image. Note that the assumptions of the theorem are satisfied, for example, if X is contractible. Example 3.26 (Complete flat affine structures on T 2 ). The holonomy image of development maps for complete flat affine structures on the two-torus is Homc .Z2 ; Aff.2//, that is, it consists of all injective homomorphisms with properly discontinuous image. As is shown in [2], Section 4.4, this is a locally closed subset of Hom.Z2 ; Aff.2//, defined by algebraic equalities and inequalities, and it has two connected components. Moreover, the conjugation action of the group Aff.2/ on Homc .Z2 ; Aff.2// is orbit equivalent to its restriction to the subgroup GL.2; R/. The latter group acts freely and properly on Homc .Z2 ; Aff.2// and the quotient space is homeomorphic to R2 . Since hol

Dc .T 2 ; A2 / ! Homc .Z2 ; Aff.2//=Aff.2/ is a homeomorphism, the deformation space of complete flat affine structures Dc .T 2 ; A2 / is homeomorphic to R2 . As is shown in [2], [7], natural coordinates can be chosen such that the action of MapC .T 2 / D SL.2; Z/ on Dc .T 2 ; A2 / corresponds to the standard representation of SL.2; Z/ on R2 . 3.3.5 Deformation of lattices (A. Weil, 1962). Let G be a simply connected Lie group and o  G a cocompact lattice. We put Mo D G= o ; z o D G of Mo . Let where o acts by left-multiplication on the universal cover M .X; GL / D .G; GL / be the homogeneous geometry which is defined by the action of G on itself by left-multiplication. Since the action of G on itself is proper, every .G; GL /-manifold is complete. Hence D.Mo ; G/ D Dc .Mo ; G/ and the holonomy image of D.Mo ; G/ is contained in the space of lattice homomorphisms HomL .o ; G/ D f W o ,! G j .o / is a lattice in Gg:

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We call the space of conjugacy classes of lattice homomorphisms DL .o ; G/ D HomL .o ; G/=G the deformation space of the lattice o . The holonomy map hol W D.Mo ; G/ ! DL .o ; G/

(3.9)

therefore locally embeds D.Mo ; G/ as an open (and closed) subspace of the deformation space of o . This is the original setup which is studied in the seminal paper [74] by André Weil. Fundamental results on the nature of the involved spaces HomL .o ; G/ and DL .o ; G/ are obtained in the foundational papers [74], [75], [73], see also [14]. For a recent contribution in the context of solvable Lie groups G, see [8]; the examples which are constructed in Section 2.3 of [8] show that there exist deformation spaces of the form D.Mo ; G/, which have infinitely many connected components. Rigidity of lattices and action of the automorphism group of G. Note that the group Aut.G/ of automorphisms of G has natural actions on the space of development maps Dev.Mo ; G/ and on HomL .o ; G/. Indeed, let 2 Aut.G/, and D W G ! G a development map for an .G; GL /-structure on Mo with holonomy 2 HomL .o ; G/. Then the composition B DW G ! G is a development map with holonomy B . These actions descend to actions on D.Mo ; G/, DL .o ; G/ respectively, such that (3.9) becomes an equivariant map. Example 3.27 (Rigid lattices). A lattice o is called rigid in G if Aut.G/ acts transitively on DL .o ; G/. For example, lattices in nilpotent Lie groups G, or lattices in simple Lie groups G not locally isomorphic to SL.2; R/ are rigid, see [72]. In these two cases we then have identities  D.Mo ; G/  ! DL .o ; G/ Aut.G/=Inn.G/;

for any lattice o  G. Here, Inn.G/ denotes the group of inner automorphisms of G. More generally, we call a lattice o smoothly rigid, if the holonomy map (3.9) is a homeomorphism, that is, if D.Mo ; G/ D DL .o ; G/. For example, lattices in solvable Lie groups are smoothly rigid, by a theorem of Mostow; but there do exist solvable Lie groups which admit non-rigid lattices. See [72], or [8] and the references therein for specific examples. The deformation spaces of the form D.Mo ; G/ play an important role in the analysis of general deformation spaces, since many geometric structures arise from étale representations. An illustrative example is given by the stratification of the space of deformations of flat affine structure on the two-torus which is studied in detail in Section 6.4.

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The induced map of an étale representation. Let G 0 be a simply connected Lie group and o  G 0 a cocompact lattice. We put Mo D G 0 = o . Let us assume for simplicity that o is smoothly rigid as well. Now let .X; G/ be a homogeneous space and W G 0 ! G be an étale representation (see Definition 3.9). Then the orbit map which is associated to an open orbit of G 0 defines a subgeometry o W .G 0 ; GL0 / ! .X; G/ which in turn gives rise to a map (3.8) of deformation spaces D.Mo ; G 0 / ! D.Mo ; X; G/; that is, we obtain a map DL .o ; G 0 / D HomL .o ; G 0 /=G 0 ! D.Mo ; X; G/: This map factors over the action of the normalizer NG . / of .G 0 / in G, that is, we have an induced map HomL .o ; G 0 /= NG . / ! D.Mo ; X; G/: We remark that, if o is rigid in G 0 then HomL .o ; G 0 /= NG . / D Aut.G 0 /=N; where N  Aut.G 0 / denotes the image of NG . / in Aut.G 0 /. 3.3.6 Dynamics of the G -action on Hom.; G /. In Examples 3.21 and 3.26 above, the map hol is a homeomorphism, and the corresponding deformation spaces are Hausdorff. These properties hold in particular if the holonomy image in Hom.; G/=G is obtained as a quotient by a proper group action. In fact, if G acts properly (and freely) on the image of hol, then, by the slice theorem (cf. [61]), the projection map Hom.; G/ ! Hom.; G/=G admits a section near every holonomy homomorphism. It then follows from Theorem 3.15 that hol W D.M / ! Hom.; G/=G is a local homeomorphism. Example 3.28 (Subvariety of stable points). If G is a reductive linear algebraic group, then, by a general fact on representations of such groups, there exists a Zariski-open subset of stable points in Hom.; G/, where G acts properly. Recall that, for any representation of G on a vector space, or any action of G on an affine variety V , a point x 2 V is called stable if the orbit Gx is closed and dim Gx D dim G. The set of stable points may be empty though. For the action of G on Hom.; G/ it is non-empty if there are points 2 Hom.; G/ such that ./ is sufficiently dense in G. In the specific context where  is abelian (or solvable), Hom.; G/ has no stable points (as follows from [44], Theorem 1.1). See [44], [31] for further discussion of these facts and for some applications.

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One cannot expect D.M / to be a Hausdorff space, in general. In fact, the image of hol in Hom.1 .M /; G/=G may contain non-closed points. In this situation also D.M / has non-closed points. The following example is due to Bill Goldman: Example 3.29 (Non-closed points in D.T 2 ; A2 /). Let  



A D ; where > 1. 0 Then M D hA in A2  0 is a flat affine two-torus, which has an infinite cyclic holonomy group generated by A (see also Example 4.12). Let  denote a corresponding holonomy homomorphism for M . Since the A , ¤ 0, are all conjugate elements of GL.2; R/, the closure of the GL.2; R/-orbit of 1 2 Hom.Z2 ; GL.2; R// contains the holonomy homomorphism 0 . Therefore, the orbit Œ 1  is not closed in Hom.Z2 ; Aff.2//=Aff.2/. By Corollary 6.10, M1 defines a non-closed point in the deformation space. Observe that 0 is the holonomy of the Hopf torus H . By Theorem 3.15 there exists a corresponding family of development maps D with holonomy  which converges to the development map of the Hopf torus M0 D H . We observe that these development maps belong to affine structures which are isotopically equivalent to the tori M . Hence, the closure of M1 in the deformation space contains the Hopf torus H . (To see explicitly how the development maps for the tori M converge to the Hopf torus in the deformation space, we may use the constructions in Section 4.3 in this chapter. In fact, we construct M as a quotient space M D TA ;id;2 of A2  0, as in Example 4.15. Then we deform the development D D D0 of M0 as in the proof of Theorem 3.15 to obtain a sequence of development maps D W A2  0 ! A2 for TA ;id;2 which converges to D0 .)

A

A

3.3.7 Dynamics of the Diff 0 .M /-action on .X; G /-structures. In favorable cases, the topology on D.M / may be determined by constructing slices for the action of Diff 0 .M / on S.M; X; G/. The study of the action of Diff 0 .M / on S.M; X; G/ may then be used to deduce information on the topology (diffeomorphism groups carry the C 1 -topology) of Diff 0 .M /, or, vice versa, on the topology of S.M; X; G/. The theory of slices for actions of diffeomorphism groups on spaces of Riemannian metrics was developed by Palais and Ebin [24]. Recall that a continuous action of Diff.M / on a space  is called proper if the map Diff.M / !  , .g; s/ 7! .g s; s/ is proper. If the action is proper, the quotient space is Hausdorff (cf. [17], III, Section 4.2). Example 3.30 (Diff.Mg / acts properly). The group of diffeomorphisms of a closed surface Diff.Mg / acts properly on the space of conformal structures S.Mg ; H2 ; PSL.2; R//;

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if g  1. In particular, the identity component Diff 0 .Mg / acts properly and freely. Moreover, the projection map S.Mg ; H2 ; PSL.2; R// ! Tg is a trivial Diff 0 .Mg /-principal bundle. Since S.Mg ; H2 ; PSL.2; R// and Tg are contractible, this implies at once that the group Diff 0 .M / is contractible. These results were shown in [23], Section 5 D. Similar results hold for the space T1 of conformal structures (flat Riemannian metrics) on the two-torus. In fact, Diff.T 2 / acts properly on the space S.T 2; R2; E.2//, and the moduli space of such structures is a Hausdorff space. However, the action of Diff 0 .T 2 / is not free, since every flat Riemannian structure on T 2 has S 1  S 1 acting as a group of isometries. In this situation, we may replace Diff.T 2 / with the subgroup Diff.T 2 ; x0 /. Indeed, S 1  S 1 is a deformation retract of Diff 0 .T 2 / and the group Diff 0 .T 2 ; x0 / is contractible (cf. [23]). In general, the action of Diff 1 .M / on a space of structures S.M; X; G/ need neither be free nor proper, as we show in the following examples. Action of Diff.T 2 / on the space of flat affine structures. In the case of flat affine structures on the two-torus, the action of Diff 0 .T 2 / on the set of all affine structures S.T 2 ; A2 / is not proper, for otherwise D.T 2 ; A2 / would be a Hausdorff space. But, in fact, as we show in Example 3.29, D.T 2 ; A2 / has singularities. An interesting in-between case arises when restricting to the subspace Sc .T 2 ; A2 / of complete flat affine structures. This case bears some resemblance to the case of conformal structures, although here the action of Diff.T 2 / on the set of structures Sc .T 2 ; A2 / is not proper. However, the action of the subgroup Diff 0 .T 2 / on Sc .T 2 ; A2 / is proper. Example 3.31 (Action of Diff.T 2 / on Sc .T 2 ; A2 /). Observe first that every complete flat affine structure on T 2 is homogeneous and the identity component of its automorphism group acts simply transitively. This follows from the classification given in Theorem 5.1. Therefore, like in the case of Euclidean structures, Diff 0 .T 2 ; x0 / acts freely on Sc .T 2 ; A2 / and Dc .T 2 ; A2 / D Sc .T 2 ; A2 /=Diff 0 .T 2 ; x0 /: Since hol W Diff 0 .M; x0 /n Devc .T 2 ; A2 / ! Homc .Z2 ; Aff.2// locally admits continuous equivariant sections (see the discussion before Theorem 3.25), it follows that the map Sc .T 2 ; A2 / ! Dc .T 2 ; A2 / is a locally trivial principal bundle for Diff 0 .T 2 ; x0 /. (It is also a universal bundle, since Sc .T 2 ; A2 / is contractible, as we see in Proposition 3.32 below.) This already implies that Diff 0 .T 2 ; x0 / acts properly on Sc .T 2 ; A2 /. On the other hand,

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Diff.T 2 ; x0 / does not act properly, since the action of the (extended) mapping class group of the two-torus Diff.T 2 ; x0 /=Diff 0 .T 2 ; x0 / Š GL.2; Z/ on the deformation space Dc .T 2 ; A2 / D R2 is not properly discontinuous. (See also Example 3.26). In the previous example a slightly stronger result holds. Indeed, by the proof of Corollary 4.9 in [2] the projection map Homc .Z2 ; Aff.2// ! Dc .T 2 ; A2 / admits a global (continuous) section. Since the space Dc .T 2 ; A2 / D R2 is contractible, we may use the covering homotopy theorem to conclude that there exists a continuous section s W Dc .T 2 ; A2 / ! S.T 2 ; A2 /. This shows that the above principal bundle is trivial. (For an explicit construction of such a section, refer to Section 6.2.2 of this chapter.) A typical application is: Proposition 3.32. The space Sc .T 2 ; A2 / of complete flat affine structures on the two-torus is contractible. Proof. The group Diff 0 .T 2 ; x0 / acts freely on the set of complete flat affine structures. By the above, invariant sections exists for this action of Diff 0 .T 2 ; x0 /. It follows that there is a homeomorphism Sc .T 2 ; A2 / Diff 0 .T 2 ; x0 /  Dc .T 2 ; A2 /: In particular, since both Diff 0 .T 2 ; x0 / and Dc .T 2 ; A2 / are contractible, the space Sc .T 2 ; A2 / is contractible. See Section 6.1 for a description of S.T 2 ; A2 / and Sc .T 2 ; A2 / as subsets of the affine space of torsion-free flat affine connections of T 2 .

3.4 Spaces of marked structures Let M0 be a fixed smooth manifold. A diffeomorphism f W M0 ! M , where M is an .X; G/-manifold is called a marking of M . Two marked .X; G/-manifolds .f; M / and .f 0 ; M 0 / are called equivalent if there exists an .X; G/-equivalence g W M 0 ! M such that g B f 0 is homotopic to f . Let DM.M0 ; X; G/ denote the set of classes of marked .X; G/-manifolds. By composing with f , every local .X; G/-chart for the marked manifold .f; M / extends to the development map of an .X; G/-structure on Mo . This correspondence descends to a bijection of DM.M0 ; X; G/ with the deformation space D.M0 ; X; G/. We can thus topologize the space of classes of markings with the topology induced from the C 1 -topology on development maps.

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Example 3.33 (Teichmüller metric on Tg ). Classically, Teichmüller space Tg is represented as a space of marked conformal structures on Riemann surfaces. Let S0 be a closed Riemann surface of genus g. A marking of a Riemann surface R is an orientation-preserving quasi-conformal homeomorphism f W S0 ! R. Two marked surfaces .f; R/ and .f 0 ; R0 / are equivalent if there exists a biholomorphic map h W R0 ! R such that h B f 0 is homotopic to f . Teichmüller space Tg is the set of classes of marked surfaces. The infimum of dilatations K.`/, where ` W R ! R0 is a quasiconformal map homotopic to f 0 B f , defines the Teichmüller distance of .f; R/ and .f 0 ; R0 / in Tg : dT .Œf; R; Œf 0 ; R0 / D inf log K.`/: With the metric topology induced by dT , Tg is homeomorphic to the Fricke space F.S0 / D Homc .g ; PSL.2; R//= PSL.2; R/; as defined in Example 3.21. See [1] and [62] in Vol. I of this handbook for reference on this material. Therefore, the topology defined by Teichmüller’s metric coincides with the topology on Tg , which is defined by the convergence of development maps for hyperbolic structures. We may also consider various refined versions of classes of marked .X; G/manifolds and corresponding deformation spaces. .X; G /-manifolds with basepoint. Fix a basepoint x0 2 X and write X D G=H , where H D Gx0 . The space DMp .M0 ; X; G/ of basepointed marked structures z0; m is defined as follows. Let p W .M Q 0 / ! .M0 ; m0 / be a fixed universal cover. A marking of .M; m/ is a based diffeomorphism f W .M0 ; m0 / ! .M; m/. Two marked basepointed .X; G/-manifolds are equivalent if there exists an .X; G/-equivalence g W .M 0 ; m0 / ! .M; m/ such that g B f 0 is homotopic to f by a basepoint-preserving homotopy. Let Devp .M0 / be the set of basepoint-preserving development maps. For every based local .X; G/ chart ', defined near m0 , there exists a unique development map D for M0 , which extends ' B f B p from a neighborhood of m Q 0 . This correspondence induces a homeomorphism  DMp .M0 ; X; G/  ! Diff 1 .M0 ; m0 /nDevp .M0 /=H:

Example 3.34 (Homogeneous .X; G/-structures). Note that the natural (forgetful) map DMp .M; X; G/ ! DM.M; X; G/ is surjective, but it is usually not injective. In fact, let D be a development map. Then for the classes G B D B Diff 1 .M; m/ and G B D B Diff 1 .M / to coincide it is necessary that the group of .X; G/-equivalences acts transitively on M . That is, D is the development map of a homogeneous .X; G/structure. We let Devh .M; X; G/ denote the set of development maps of homogeneous .X; G/-structures. Note further that, in general, Diff 1 .M0 ; m0 / is a proper subgroup of all basepointpreserving diffeomorphisms which are freely homotopic to the identity. The difference

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is obtained by the natural action of 1 .M0 ; m0 / on based homotopy classes of maps. However, if 1 .M0 ; m0 / is abelian, the inclusion is an isomorphism. Example 3.35. Let Dh .T 2 ; A2 / be the deformation space of homogeneous flat affine structures on the two-torus. Then Dh .T 2 ; A2 / D Diff 1 .T 2 ; x0 /nDevh .T 2 /=Aff.2/: In particular, since every complete affine two-torus is homogeneous, Dh .T 2 ; A2 / contains the subspace of complete flat affine structures Dc .T 2 ; A2 / D Diff 1 .T 2 ; x0 /nDevc .T 2 /=Aff.2/: 3.4.1 Framed .X; G /-manifolds. The holonomy of a marked .X; G/-manifold is a G-conjugacy class of homomorphisms. To get rid of the dependence on the conjugacy class, one introduces framed structures. The holonomy theorem implies that the deformation space of framed structures is a locally compact Hausdorff space. We shall discuss only the particular simple case of .An ; Aff.n//-manifolds. Example 3.36. Let m 2 M . A frame Fm for a flat affine structure on M is a choice of basis of the tangent space Tm M . The pair .M; Fm / is called a framed flat affine manifold. Fix a frame Fm0 for M0 , as well, and call a frame-preserving diffeomorphism .M0 ; Fm0 / ! .M; Fm / a marking of .M; Fm /. Two marked framed flat affine manifolds .f; M; Fm / and .f 0 ; M 0 ; Fm0 0 / are called equivalent if there exists a frame-preserving affine diffeomorphism g W .M 0 ; Fm0 0 / ! .M; Fm / such that g B f 0 is based homotopic to f . The set of classes is denoted DMf .M0 ; A2 /. Let us fix a base frame Ex0 on affine space An . Given a marked framed flat affine manifold .f; M; Fm /, there exists a unique frame-preserving affine chart for M , which z 0 ; FzmQ / ! is defined near m. This chart lifts to a unique development map D W .M 0 n .A ; Ex0 /. The correspondence descends to a bijection DMf .M0 ; An / D Diff 1;f .M0 ; Fm0 /nDevf .M0 /; where Diff 1;f .M0 ; Fm0 / denotes the group of frame-preserving diffeomorphisms which are based homotopic to the identity. By the deformation theorem, there is a map hol W DMf .M0 ; An / ! Hom.1 .M; m0 /; Aff.n//

which is continuous and which is a local homeomorphism onto its image. This shows that DMf .M0 ; An / is a locally compact Hausdorff space. As is apparent from Example 3.24, the natural map DMf .M0 ; An / ! DM.M0 ; An /

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is surjective, and it factors over DMp .M0 ; An /. The following tower of maps thus sheds some light on the topology of the deformation space of flat affine structures: DMf .M0 ; An /  DMp .M0 ; An /

(3.10)

 D.M0 ; An / D DM.M0 ; An /.

Note that the group GLC .n; R/ D Diff 1 .M0 ; m0 /=Diff 1;f .M0 ; Fm0 / acts on DMf .M0 ; An /, such that the first projection map in the tower is the quotient map for this action. In particular, DMp .M0 ; An / arises as the quotient space of a locally compact Hausdorff space by a reductive group action. (Note also that the holonomy map is equivariant with respect to the conjugation action of GLC .n; R/ on holonomy homomorphisms.) Example 3.37 (Homogeneous framed flat affine structures on the torus). As we have seen already in Example 3.35 above, the lower map in the tower (3.10) is a bijection on homogeneous flat affine structures, that is, Dh .T 2 ; A2 / D DMp;h .T 2 ; A2 /. The deformation space of homogeneous structures is thus obtained as a quotient by an action of GLC .2; R/: Dh .T 2 ; A2 / D DMf ;h .T 2 ; A2 /= GLC .2; R/: We shall further study this quotient space in Section 6.2. Observe that the action of GLC .2; R/ is not free, since, in fact, the Hopf tori have non-trivial stabilizers. On the other hand, as follows from the discussion in Example 3.26, GLC .2; R/ acts freely on the subspace of complete affine structures, and the map DMf ;c .T 2 ; A2 / ! Dc .T 2 ; A2 / is a trivial GLC .2; R/-principal bundle.

4 Construction of flat affine surfaces A flat affine manifold is called homogeneous if its group of affine automorphisms acts transitively. Homogeneous flat affine manifolds may be constructed from étale affine representations of two-dimensional Lie groups in a straightforward way. Compact examples can be derived from étale affine representations of the two-dimensional group manifold R2 by taking quotients with a discrete uniform subgroup. Every flat affine surface constructed in this way is then a homogeneous flat affine torus. In fact, an easy argument (see Section 5) shows that all homogeneous flat affine surfaces are obtained in this way. Therefore, all homogeneous flat affine tori are affinely diffeomorphic

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to quotients of abelian Lie groups with left-invariant flat affine structures. This also relates homogeneous affine structures on tori to two-dimensional associative algebras, a point of view which will be discussed in Section 6. In Section 4.1, we describe the classification of abelian étale affine representations on A2 . By the above remarks, this amounts to a rough classification of homogeneous flat affine tori. A genuinely more geometric approach is to construct flat affine surfaces by gluing patches of affine space along their boundaries. The affine version of Poincaré’s fundamental polygon theorem allows to construct flat affine tori by gluing affine quadrilaterals along their sides. The flat affine two-tori thus obtained depend on the shape of the quadrilateral and also on the particular affine transformations which are used in the gluing process. This, in turn, gives natural coordinates for an open subset in the deformation space of flat affine structures on the two-torus. As it turns out, the flat affine tori which are obtained by gluing an affine quadrilateral along its sides are all homogeneous, and they form a dense subset in the deformation space of homogeneous flat affine structures on the torus. This material is explained in Section 4.2. To construct all flat affine two-tori, it is required to glue more general objects. In the following sections, Section 4.3 and Section 4.4, we discuss in detail a construction method for flat affine tori with development image A2  0, which builds on the idea of cutting flat affine surfaces into simple building blocks. Here flat affine tori are constructed by gluing several copies of half annuli in A2  0, or cutting the surface into affine cylinders. Equivalently, these tori are obtained by gluing certain strips which are situated in the universal covering space of A2  0, and which project to annuli in A2  0. In this way also non-homogeneous examples of flat affine tori arise. As follows from the main classification theorem, which will be proved in Section 5, the above construction methods exhaust all flat affine two-tori.

4.1 Quotients of flat affine Lie groups If a Lie group G has an étale action (cf. Definition 3.9) on affine space we call it an étale affine Lie group. An étale affine Lie group carries a natural left invariant flat affine structure, and, thus, for every discrete subgroup   G, the quotient space nG inherits the structure of a flat affine manifold. If G is abelian the resulting flat affine structure is homogeneous. The following result will be established in the course of the proof of the classification theorem (see Section5.3): Proposition 4.1. Every homogeneous flat affine two-torus is affinely diffeomorphic to a quotient of an abelian étale affine Lie group. Up to affine conjugacy there are six types T, D, C1 , C2 , B, A of étale abelian subgroups in the affine group Aff.2/. Both the plane and the halfplane admit two distinct simply transitive abelian affine actions T, D, and C1 , C2 respectively.

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Example 4.2 (Affine automorphisms of development images). (1) (The plane A2 ) The groups 80 80 19 19 < 1 0 u = < 1 v u C 12 v 2 = A T D @0 1 v A and D D @0 1 v : ; : ; 0 0 1 0 0 1 are abelian groups of affine transformations which are simply transitive on the plane. (2) (The half space H ) Let H be the half space y > 0. Then 80 9 1 < ˛ z v ˇ = ˇ Aff.H / D @ 0 ˇ 0A ˇ ˛ ¤ 0; ˇ > 0 : ; 0 0 1 is its affine automorphism group. The subgroups ² ³ exp.t/ z C1 D  GL.2; R/ 0 exp.t / and

80 19 0 v = < 1 C2 D @0 exp.t / 0A  Aff.2/ : ; 0 0 1

are simply transitive abelian groups of affine transformations on H . The half spaces .x; y/, y > 0 and y < 0 are open orbits for the groups Ci . (3) (The sector Q) Let Q denote the upper right open quadrant. Then ² ˇ ³ a 0 ˇ 0 B D Aff.Q/ D ˇ a > 0; b > 0  GL.2; R/ 0 b is an abelian, simply transitive linear group of transformations of Q. (4) (The punctured plane A2  0) ²  cos  A D exp.t/  sin 

sin  cos 

³  GL.2; R/ D Aff.A2  0/

is an abelian linear group, which is simply transitive on A2  0. Let G  Aff.2/ be one of the above groups and   G a lattice. Then  acts properly discontinuously and with compact quotient on every open orbit U  A2 of G and the quotient space M D n U is a flat affine two-torus. The group  is a discrete abelian subgroup of Aff.2/ and, by choosing an appropriate fundamental domain, its action defines a tessellation of the open domain U . In fact, a convex affine quadrilateral may be chosen as fundamental domain. See Figure 10 and Figure 11 for some examples. Since G is abelian and

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centralizes , the affine action of G on U descends to M . Thus, G acts on M by affine transformations, and M is a homogeneous flat affine manifold.

0

0

0

0

0

0

Figure 10. Tesselations of homogeneous affine domains of type T, D, C1 .

0

0

0

0

0

0

Figure 11. Tesselations of homogeneous affine domains of type B, C1 , A.

For further reference we note the following: Lemma 4.3 (Normalisers of étale affine groups). (1) The étale affine groups A, B have index two and eight in their normalizers in Aff.2/. The quotients are generated by the reflections         0 1 1 0 1 0 0 1 2 GL.2; R/, respectively, ; ; : 1 0 0 1 0 1 1 0 (2) The normalizers in Aff.2/ of the étale affine groups C1 , C2 are 80 19 ² ³ < ˛ 0 v = ˛ z  GL.2; R/; @ 0 ˇ 0A  Aff.2/ 0 ˇ : ; 0 0 1 respectively. (3) The normalizer in Aff.2/ of the étale affine group D is the semi-direct product generated by D and the linear group 80 2 9 1 b 0 ˇ < d = ˇ N D D @ 0 d 0A ˇ d 2 R ; b 2 R : : ; 0 0 1 In the case of the group A, which is not simply connected, we can consider, more z!A z of A. The covering homomorphism A generally, the universal covering group A

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z into an étale affine Lie group. Let  be a lattice in A z . Then turns A z M D n A z ! A2  0 at any point inherits a flat affine structure, for which the orbit map A 2 x 2 A  0 is a development map. The holonomy group h./  A is the image of  z ! A. Since  is central in A z , the group A z acts on M by affine under the covering A transformations. In particular, as before, M is a homogeneous flat affine two-torus. Note that, in this case, it may also happen that the holonomy h./ is not discrete in Aff.2/, see Figure 12.

0

0

0

0

0

0

0

0

0

Figure 12. Development process with non-discrete holonomy group.

If the holonomy is cyclic, as is the case for Hopf tori (Example 4.8), M cannot be constructed by gluing an affine quadrilateral which is contained in the development z, image. However, M may always be obtained by gluing a strip which is situated in A see Example 4.21.

4.2 Affine gluing of polygons Let P  A2 be polygon with S its set of sides. Let fgS 2 Aff.2/ j S 2 S g be a set of affine transformations pairing the sides of P and let M be the corresponding identification space of P . We say that the affine gluing criterion1 holds if, for each vertex x 2 P with cycle of edges S1 ; : : : ; Sm , the cycle relation gS1 gSm D 1 holds, and furthermore the corners at x of the polygons gS1 gSi P , i D 1; : : : ; m, add up subsequently to a disc, while intersecting only in their consecutive boundaries. This disc then provides an affine coordinate neighborhood in the identification space of P defined by the pairing of sides. If the gluing criterion is satisfied the identification space M inherits the structure of a flat affine manifold from P . The following result is the analogue of Poincaré’s fundamental polygon theorem (cf. [55] for the classical version) for gluing flat affine surfaces: 1 See

[2] for more details on this definition.

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Proposition 4.4 (see [2], Proposition 2.1). If the affine gluing criterion holds then the group   Aff.2/ generated by the side-pairing transformations fgS j S 2 Sg acts properly discontinuously and with fundamental domain P on a flat affine surface Xx which develops -equivariantly onto an open set U in A2 . The inclusion of P into Xx identifies M and the orbit space  nXx . It follows that M inherits a natural flat affine structure from P . In fact, the surface Xx is the holonomy covering space of M and the group  is the holonomy group of M . Note also that the construction of Xx is sketched in the proof of Theorem 3.15. The situation is pictured in the following commutative diagram of maps: Xx

x D

 M D nXx

/ U  A2 /

 nU  :

Example 4.5. Figure 13 shows how to glue a trapezium T with angle ˛ < . The sides S1 and S3 are glued with a homothety. S2 and S4 are glued with a rotation of angle ˛. The developing image is U D A2  0. It is tessellated by the translates of T if and only if m˛ D 2 for an integer m. If the angle ˛ is rational, p˛ D q2, the development Xx ! A2  0 is a finite cyclic covering of degree q. Otherwise the development is an infinite cyclic covering and Xx is simply connected. s
A


˛ A
--A A
 S3 AKA S2 S4

-- A

A S1

Figure 13. Gluing a trapezium in A2 .

4.2.1 Gluing affine quadrilaterals. Theorem 2.1 implies that the gluing criterion imposes strong restrictions on the possible combinatorial types of polygons and pairings. However, flat affine two-tori are easily obtained by gluing an affine quadrilateral P in the way indicated in Figure 14. The equivalence class of the flat affine manifold thus obtained depends on the affine equivalence class of P and the particular side pairing transformations chosen, see [2], Section 3. The gluing conditions for such a pairing are easily verified:

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-6

--

6

Figure 14. Gluing a torus.

Lemma 4.6 ([2], Lemma 3.2). Let A; B 2 Aff.2/, A.0; 0/ D .1; 0/, A.0; 1/ D p, B.0; 0/ D .0; 1/, B.1; 0/ D p. The side pairing transformations fA; A1; B; B 1 g for the polygon with vertices P D ..0; 0/; .1; 0/; p; .0; 1// satisfy the gluing conditions if and only if det l.A/ > 0 and det l.B/ > 0 (where l denotes the linear part of an affine transformation) and ŒA; B D Id. We remark further: Proposition 4.7. Every flat affine two-torus obtained by gluing a quadrilateral P on its sides is homogeneous. Conversely, if M is a homogeneous flat affine two-torus with non-cyclic affine holonomy group, then M may be obtained by gluing a quadrilateral. Proof. For the proof that the gluing torus of P is homogeneous, we have to appeal to some of the facts which are explained in Section 5. In particular, we use Proposition 5.2 and Proposition 5.3. The gluing conditions imply that the minimal connected abelian subgroup N which contains the holonomy of M is at least two-dimensional. Therefore, N is one of the two-dimensional abelian subgroups listed in Example 4.2. By Proposition 5.3, N acts on the development image of M . Let U be an open orbit for N , which is one of the domains of Example 4.2. Then, by convexity of the homogeneous domain U , the polygon P is contained in U . The construction of M and its development process show that the development image of M is covered by the holonomy translates of P . Since the holonomy is contained in N , it follows that the development image of M is contained in the open orbit U . On the other hand, by Proposition 5.3, the development image contains U . This implies that the development image of M equals the open orbit U . Therefore, N acts transitively on the development image, and thus also on M . In particular, it follows that M is homogeneous. We omit the proof of the converse statement. A special case is treated in Section 4.6 of [2]. 4.2.2 The gluing variety. By Lemma 4.6, the set of side pairings V D f.p; A; B/g  R2  R6  R6 ; which satisfy the gluing conditions for the (convex) quadrilateral P D . .0; 0/; .1; 0/; p; .0; 1/ / form a semi-algebraic subset of R2  R6  R6 . It is easily computed that V is fourdimensional and that the set of solutions with respect to a fixed P is of dimension two.

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We let p W V ! R2 denote the projection to the first factor. Note that the projection of V to the matrix factors R6  R6 defines an embedding V ,! Hom.Z2 ; Aff.2// as a subset of the holonomy image. We call the set V the gluing variety of quadrilaterals. Embedding into the deformation space. Observe that the gluing of a quadrilateral P naturally constructs a framed affine two-torus. If we choose a diffeomorphism of the unit-square with P , the development process of the gluing extends this diffeomorphism to the development map of a marked framed affine two-torus (see Example 3.36). This defines a continuous map V  p1 .P / ! DMf .T 2 ; A2 /; where p W V ! R2 is the projection to the first factor. We may furthermore choose a natural identification of the unit square with P (for example, by decomposing any quadrilateral into two triangles and using affine identifications of the triangles). Then, using (3.10), we obtain a continuous (open) embedding  W V ! DMf .T 2 ; A2 / to the space of classes of framed flat affine tori. Note that  is a section of the holonomy map hol W DMf .T 2 ; A2 / ! Hom.Z2 ; Aff.2//. Since V also defines a slice for the GL.2; R/-orbits on Hom.Z2 ; Aff.2//, the map  descends to an embedding V ! DMp .T 2 ; A2 /, whose image consists of homogeneous structures. Therefore, by the discussion in Example 3.37, the gluing variety V embeds as a locally closed subset of the deformation space D.T 2 ; A2 /.

4.3 Tori with development image A2  0 Here we discuss how to glue flat affine tori from annuli which are contained in the oncepunctured plane or the universal covering flat affine manifold of the once-punctured plane. 4.3.1 Hopf tori and quotients of A2  0. The simplest examples of flat affine tori with development image A2  0 are obtained by gluing closed annuli along their boundary curves. Example 4.8 (Hopf tori). Let A , > 0, be a dilation with scaling factor , and  D h A i the subgroup of GL.2; R/ generated by A . Then  acts properly discontinuously on A2  0 and the quotient space   H D  n A2  0 is a compact flat affine two-torus H , which is called a Hopf torus. The Hopf torus H is obtained by gluing a closed annulus A  A2  0 of width

along its boundary circles, see Figure 15.

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Figure 15. Gluing of annuli: Hopf torus, non-homogeneous flat affine tori.

The geometric construction of the Hopf tori H may be refined as follows: Example 4.9 (Finite coverings of Hopf tori). Let Xk ! A2  0 be a k-fold covering flat affine manifold. Then we may lift the action of A on A2  0 to a properly discontinuous action of an affine transformation A;k of Xk . The quotient spaces H;k D hA;k i nXk are flat affine manifolds, which are k-fold covering spaces of H . Geometrically, Xk is a topological annulus with a flat affine structure which is obtained by cutting A2  0 at a radial line and then gluing k copies of A2  0 along this geodesic ray. A geodesic in a flat affine manifold is a curve which corresponds to a straight line in all affine coordinate charts. Thus, correspondingly, the manifolds H;k are obtained by gluing k copies of H at a closed geodesic. Note that the family of Hopf tori H;k gives a simple example of a family of distinct flat affine manifolds which have identical holonomy homomorphism. Example 4.10 (Finite quotients of Hopf tori). Let R˛ be a rotation with angle ˛ D pq  a rational multiple of . Then the finite group of rotations of order 2q generated by R˛ acts without fixed points on A2  0 and on the Hopf tori H;k . Therefore, the quotient spaces H;˛;k D hR˛ i nH;k are flat affine two-tori. Since A is in the center of GL.2; R/, H is a homogeneous flat affine manifold with affine automorphism group Aff.H / D GL.2; R/= : Hence, its finite coverings H;k are homogeneous, as well. Similarly, H;˛;k are homogeneous flat affine two-tori, with Aff.H;˛ /0 isomorphic to GL.1; C/= , except for ˛ D . In the latter case Aff.H; / D PGL.2; R/= .

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Expanding holonomy. Non-homogeneous quotients of A2  0 may be constructed by using expanding elements of GL.2; R/. A matrix A 2 GL.2; R/ is called an expansion if it has real eigenvalues 1 ; 2 > 1. (A1 is then called a contraction.) Every expansion acts properly discontinuously on A2  0, see Figure 16. This motivates the following: Definition 4.11 (Expanding elements). A matrix A 2 GL.2; R/ is called expanding if it acts properly on A2  0 and every compact subset of A2  0 is moved to infinity by its iterates Ak , k ! 1. Note that, if A is expanding, it is either an expansion, or a product of an expansion with R , or it is conjugate to a product of a dilation and a rotation.

Figure 16. Dynamics of expanding elements in GL.2; R/.

Figure 17. Dynamics of non-expanding elements in GL2 .R/.

Example 4.12 (Tori with expanding holonomy). If A is an expanding element then the quotient space HA D hAin A2  0 is a flat affine two-torus with development image A2  0. If A is an expansion, the torus HA is obtained by gluing an annulus AA  A2  0, as indicated in Figure 15. Note that, if A is an expansion which is not a dilation then HA is a flat affine torus, which is not homogenous.

B

4.3.2 Quotients of A2  0. We consider the universal covering flat affine manifold

A

q W A2  0 ! A2  0

A

of the open domain A2  0 in A2 . Let Aff.A2  0/ be its group of affine diffeomorphisms.

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Universal covering of GL.2 ; R/. The development q induces a surjective homomorphism

A

p W Aff.A2  0/ ! Aff.A2  0/ D GL.2; R/;

A

which exhibits Aff.A2  0/ as the universal covering group of GL.2; R/. Let R 2 GL.2; R/ denote rotation by . The center of the group

A

f R/ Aff.A2  0/ D GL.2; is therefore generated by an element  which satisfies p. / D R , and the kernel of p is generated by  2 . (cf. Section 6.5.)

A

Polar coordinates. We let .r; / denote polar coordinates for A2  0. Then  W .r; / 7! .r;  C /: f R/ of the rotation group is a subgroup More generally, the universal covering SO.2; f R/ which acts by translations in the -direction. of GL.2; Elements with non-zero rotation angle. Let B 2 GLC.2; R/ have positive eigenvalues. After conjugation, we may assume that B preserves the horizontal coordinate axis in A2 . We let Bz D Bz0 denote the lift of B to A2  0 which preserves the line  D 0. It follows that Bz preserves all horizontal strips

A

x ` D f .r; / j `     .` C 1/  g  and their boundary components. We observe that (the group generated by) any other lift z k ¤ 0; Bzk D  k B;

A

acts properly on A2  0. f C.2; R/. We say that Bz has Definition 4.13 (Non-zero angle of rotation). Let Bz 2 GL 2 a non-zero rotation angle if Bz acts properly on A  0, and for every compact subset the  coordinates are unbounded under the iterates Bz k , k ! 1.

A

The property to have non-zero rotation is an affine invariant, that is, it is invariant f C.2; R/. In particular, Bz ¤ 1 has non-zero angle of rotation if by conjugation in GL z f and only if B is conjugate to an element of SO.2/ or B has positive eigenvalues and k Bz D Bzk D  Bz0 , k ¤ 0, as above.

B

S x ` be the successive union of k Proper actions on A2  0. Let Hx0k D `D0:::k  x ` is a closed halfspace with the x ` . Note that the development image of  strips  origin 0 removed, and the development image of Hx0k is A2  0, k  1 (compare also Figure 23).

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Example 4.14 (Affine cylinders without boundary). Consider the quotient flat affine manifolds XB;k D hBzk i n A2  0; k  1:

A

These are open affine cylinders which are obtained by gluing Hx0k along its two incomplete boundary geodesics. The development image of XB;k is A2  0, and its holonomy group is generated by Rk B. Let A be an expansion which commutes with B and Az D Az0 the lift of A which preserves the line  D 0. Then Az acts properly on XBz .

A

Example 4.15 (Quotients of A2  0). We obtain the quotient affine torus z TA; zB z D TA;B;k D hAi nXB;k ; k ¤ 0: The holonomy homomorphism of TA;B;k is determined by A, B and the parity of k. Example 4.16 (Affine cylinders without boundary, general case). Let Bz be an element f C.2; R/ which has non-zero rotation. Then the quotient flat affine manifolds of GL

A

z n A2  0 XBz D hBi are open cylinders which are obtained by gluing a strip

A

Hx˛ D f.r; / j 0    ˛g

in A2  0 along its two boundary geodesics. Now let B 2 GL.1; C/ and Bz a lift with non-zero rotation, and A 2 GL.1; C/ an expanding element. Then Az acts properly on XBz if and only if Az and Bz generate a f C/. lattice in GL.1; Example 4.17. The quotient flat affine torus z TA; zB z D hAi nXB;k z ; is a homogeneous flat affine two-torus with holonomy in GL.1; C/.

4.4 Affine cylinders with geodesic boundary We show that by gluing flat affine cylinders whose boundary curves are incomplete geodesics we may construct flat affine two-tori with development image A2  0. This yields another construction of the manifolds TA;B;k which have been introduced in Example 4.15. As a matter of fact, as a key step in the course of the proof of Theorem 5.1, we shall show that all non-homogeneous flat affine tori may be obtained in this way.

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Example 4.18 (Affine cylinders with geodesic boundary). Let Hx0 be the closed upper halfplane with the origin 0 removed. If A is an expansion then CA D hAi nHx0 is topologically an annulus with two boundary components, and it is also a flat affine manifold with (incomplete) geodesic boundary, see Figure 18 and Figure 19. We call CA an affine cylinder.2

Figure 18. Gluing of flat affine cylinders CA .

Figure 19. Affine cylinder CA3 with geodesic boundary.

S x ` , and Az the lift of A, which preserves the More generally, let Hx0k D `D0:::k  z nHx k is called an affine line  D 0. Then the manifold with boundary CAk D hAi 0 cylinder. 4.4.1 Gluing of flat affine cylinders. Let A be an expansion and B 2 GLC.2; R/, commuting with A, such that B has positive eigenvalues. Then, as shown by Examz nA2  0 and yields the quotient flat ple 4.15, every lift Bzk , k  1 acts properly on hAi affine torus z nXB;k : TA;B;k D hAi

A

Thus, geometrically, the flat affine torus TA;B;k is constructed by gluing the flat affine cylinder CAk along its two boundary geodesics using the transformation Bzk , see Figure 20. Remark 4.19. Note that M D TA;B;k is a homogeneous flat affine two-torus if and only if its holonomy group h./ D hA; Bi is contained in the group of dilations, in which case Aff.M / is a finite covering group of PSL.2; R/, and M is a Hopf torus. 2 Benoist

[10] calls CA an annulus.

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Figure 20. Gluing a flat affine torus TA;B;3 .

Remark 4.20. Similarly, if A is an expansion and h./ D hA; Bi is a discrete group of rank two then h./ acts properly discontinuously and with compact quotient either on H or on Q. The corresponding torus is obtained by gluing TA or TA;˛ , ˛ < . Finally, we remark that a homogeneous flat affine torus with development image A2  0 may be constructed by gluing cylinders if and only if it admits a closed (noncomplete) geodesic: Example 4.21 (Homogeneous tori with a closed geodesic). Let A be a dilation, and B 2 GL.1; C/. Then every lift of B to A2  0 is of the form

A

Bzk W .r; / 7! . r;  C ˛/;

˛ D ˛0 C 2k;

A

with ˛0 2 Œ0; 2/. If ˛ ¤ 0, we define XB;k D hBzk i n A2  0, and h D hAz i nXB;k : T;B;k

A

Let Hx˛ D f.r; / j 0    ˛g be a strip in A2  0, and C;˛ D hAz i nHx˛ : h Then, T;B;k is a homogeneous flat affine two-torus which is obtained by gluing the cylinder C;˛ with Bzk .

Figure 21. Affine cylinders C;˛ , ˛ < 2.

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5 The classification of flat affine structures on the two-torus The classification of flat affine structures on the two-torus was carried out by Kuiper [51] and completed by Nagano–Yagi in [60], and independently also by Furness and Arrowsmith in [28]. Later on much of the work in [60] was clarified and beautifully generalised in Benoist’s paper [9]. In this section we describe the classification result in detail and explain its proof, following loosely along the lines of [60], and employing also the main ideas from [9], [10] to establish in Proposition 5.8 the crucial fact that the development map of a flat affine two-torus is always a covering map onto its image. Theorem 5.1. Let M be a flat affine two-torus. Then M is affinely diffeomorphic to either (1) a quotient of a simply connected two-dimensional affine homogeneous domain by a properly discontinuous group of affine transformations, or

A

(2) a quotient space of the universal covering A2  0 of the once-punctured plane. In particular, the universal covering flat affine manifold of M is affinely diffeomorphic to the affine plane A2 , the half-plane H , or the sector (quarter plane) Q, in the first case, and to A2  0, in the second case.

A

The first step in the proof of Theorem 5.1 consists of the determination of the open domains in A2 which appear as the development images of flat affine structures on the two-torus. This is done in Section 5.1. If M is homogeneous then the development map is a covering map. The main step is then the determination of the structure of flat affine two-tori with development image A2  0, which are not homogeneous. This is carried out in Section 5.3. We prove that such tori may be obtained by gluing affine cylinders with geodesic boundary. We deduce that also in this case the development map of M is a covering map onto its image. The following further consequences are implied by the theorem or its proof. Classification of divisible affine domains. An affine domain is called divisible if it admits a discontinuous affine action with compact quotient. Since they admit a simply transitive abelian group, all development images of flat affine structures on the twotorus are divisible by abelian discrete groups (isomorphic to Z or Z2 ). Conversely, by Benzécri’s theorem, every divisible plane affine domain is the development image of a flat affine structure on the two-torus. By Theorem 5.1, the universal covering of a flat affine two-torus is a homogeneous flat affine manifold, which covers a convex divisible homogeneous domain in A2 . The affine automorphism group of M (1) If M has development image A2 , the half-plane H , or the sector Q, then M is a homogeneous flat affine manifold. The connected component Aff.M /0 of the

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group Aff.M / of affine diffeomorphisms of M is a two-dimensional compact abelian Lie group, acting transitively and freely on M . (2) If the development image is the once-punctured plane A2  0 and M is homogef C.2; R/, as is the case neous, then the group Aff.M /0 is either a quotient of GL 0 for Hopf tori (Examples 4.8, 4.9), or Aff.M / is a quotient of GL.1; C/, as in Example 4.10. In either case, the action of GL.1; C/ on A2  0 descends to a transitive and free action of a two-dimensional compact abelian Lie group on M . (3) Otherwise, Aff.M /0 is a two-dimensional abelian connected Lie group which has a one-dimensional compact factor. In this case, M is not homogeneous, as in Example 4.12. (4) The affine automorphism group of M acts prehomogeneously on M , that is, it has only finitely many orbits on M . (5) The one-dimensional orbits of Aff.M /0 are non-complete geodesics in M along which M may be cut into flat affine cylinders. Homogeneous and complete flat affine tori (1) Every homogeneous flat affine two-torus M is affinely diffeomorphic to a quotient of an abelian étale affine Lie group of type T, D, C1 , C2 , B or A as listed in Example 4.2. (2) Every complete flat affine two-torus M is affinely diffeomorphic to a quotient of an abelian simply transitive affine Lie group of type T or D. In particular, M is also a homogeneous flat affine two-torus.

5.1 Development images The classification of development images is as follows: Proposition 5.2. Let M be a flat affine two-torus. Then the development image of M is either the affine plane A2 , the half-plane H , the sector (quarter plane) Q or the once-punctured plane A2  0, respectively. Proof of Proposition 5.2. Let h./  Aff.2/ be the holonomy group of M . Let N be the identity component of a maximal abelian subgroup of Aff.2/ which contains h./. Note that N contains the identity component of the Zariski-closure of h./. Therefore, h./ \ N is of finite index in h./. We let Nz denote the universal covering group of N . The first observation is that N acts on the development image, and it has only finitely many orbits on A2 . Proposition 5.3. The action of N on A2 lifts via the development map to an action z of M . Moreover, it follows that of Nz on the universal covering flat affine manifold M

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(1) N acts on the development image  of M . (2) N has only finitely many orbits on . Proof. Let Y be an affine vector field on R2 which is tangent to the action of N , and z via D. Since h./ commutes with N , the vector field Yx is let Yx denote its lift to M -invariant and projects to a vector field on M . Since M is compact, the flow of Yx is complete. Therefore, the action of N integrates to an action of the universal covering z , such that D.nx/ group Nz on M Q D nD.x/, where nQ 2 Nz and n D h.n/ Q is its image in N . This implies (1). To prove (2), we note that, up to affine conjugacy, every maximal abelian and connected subgroup N of Aff.2/ is either one of the abelian groups T, D, C1 ; C2 , B, A (as listed in Example 4.2), or the group 80 9 1 < 1 u v ˇ = ˇ N D @0 1 0A ˇ u; v 2 R : : ; 0 0 1 All of the groups appearing in Example 4.2 are simply transitive on an affine domain, and they have finitely many orbits on A2 . This shows (2). To complete the proof, we contend that h./ is not contained in the group N. We remark that the orbits of N are the horizontal lines on R2 . If h./ is contained in N, then by (1),  is a union of orbits. Thus horizontal lines define a one-dimensional foliation of , which is preserved by h./. This, in turn, defines a foliation on the manifold M , which has an open subset of the real line as its space of leaves. This is not possible, since M is compact: The space of leaves is a quotient of M , and therefore it is a compact and closed subset of the line as well. It follows that the development image  is a finite union of orbits of one of the connected abelian groups listed in Example 4.2. Since  is a connected open subset of A2 , as well, it follows that  must be one of the domains listed in Proposition 5.2. Conversely, as follows from Section 4.1, each of these domains appears as the development image of a homogeneous flat affine structure on the torus. This completes the proof of Proposition 5.2.

5.2 The classification of manifolds modeled on .A2  0; GL.2 ; R// As follows from the proof of Theorem 5.1, every compact manifold M modeled on .A2  0; GL.2; R// is either complete and the development image of M is A2  0, or M is (isomorphic to) a quotient of the open quadrant Q, or a quotient of an open half space H . In the first case

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M D  n A2  0; f C.2; R/ is a discontinuous subgroup, in the second case the affine where   GL holonomy group  is a discrete subgroup of B  GL.2; R/, C1  GL.2; R/ respectively.

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We arrive at the following classification theorem for .A2  0; GL.2; R//-manifolds which are complete (see also Corollary 5.11): Theorem 5.4. Let M be a compact complete .A2  0; GL.2; R//-manifold. If M is not homogeneous then it is isomorphic to a torus TA; zB z , as constructed in Section 4.4.1 and Example 4.15. Moreover, M is homogeneous if and only if it can be modeled on .A2  0; GL.1; C//. In particular, if M is not homogeneous then it is obtained by gluing flat affine f C.2; R/ has non-zero angle of cylinders CAk , where A is an expansion and Bz 2 GL rotation and commutes with A. Furthermore if A is a dilation, B cannot be conjugate to an element of GL.1; C/. Example 5.5 (Holonomy in GLC .2; R/ is not injective). This phenomenon already occurs for homogeneous flat affine manifolds which are quotients of the universal covz of the étale flat affine group A D GL.1; C/  GL.2; R/. ering flat affine Lie group A Here different lattices 1 and 2 of A determine non-isomorphic flat affine manifolds. However, different lattices may project to the same holonomy group in A. More striking examples arise as a consequence of the construction of the tori TA; zB z , as constructed in Section 4.4.1. In fact, for every non-complete homogeneous flat affine manifold modeled on C1 or B one can construct a non-homogeneous .A2  0; GLC .2; R//-manifold which has the same holonomy group in GLC .2; R/. These examples show in particular that the affine holonomy group does not determine the development image.

A

f C.2; R//-manifolds. Here we We can consider also the corresponding .A2  0; GL have:

A

f C.2; R//-manifolds are determined up to isoTheorem 5.6. All compact .A2  0; GL f C.2; R/. morphism by their holonomy group in GL Proof. For the complete manifolds the rigidity is shown in Example 3.16. In particular, f C.2; R/ complete and non-complete manifolds do not share the same holonomy in GL C (although they often do in GL .2; R/). In fact, the holonomy group of every complete manifold has an element with non-zero angle of rotation (cf. Definition 4.13), which is not possible if the development image is one of the domains H , Q. Similarly the non-complete examples are lattice quotients of a simply connected abelian Lie group contained in GLC .2; R/ which acts simply transitively on some open domain in A2 . Therefore, these manifolds are determined by their affine holonomy group. Also the following is an immediate consequence of the above classification result: f C.2; R/ be a non-finite discrete subgroup which is acting Corollary 5.7. Let   GL 2 properly on A  0. Then one of the following hold:

A

509

Chapter 12. The deformation of flat affine structures on the two-torus

(1)  is isomorphic to Z. Moreover, it is generated by an expansion, or it is generated by an element of non-zero rotation. (2)  is isomorphic to Z2 and it is conjugate to one of the subgroups constructed in Examples 4.15 or 4.17.

5.3 The global model spaces The classification theorem, Theorem 5.1, implies that there do exists four simply connected flat affine manifolds which appear as the universal covering space of a flat affine two-torus. These simply connected model spaces for two-dimensional compact flat affine manifolds are the plane A2 , the half-plane H , the sector Q, and A2  0, the universal covering space of the once-punctured plane.

A

This follows from the classification of development images once the following fact is established. Proposition 5.8. The development map of a flat affine two-torus M is a covering map. The main step in the proof of Proposition 5.8 relies on the decomposition of M into fundamental pieces, which are called bricks. This concept is due to Benoist [9]. The bricks in this case are flat affine cylinders with geodesic boundary. This brick decomposition for flat affine two-tori resembles the pants decomposition for closed hyperbolic surfaces (see [64]). It will also serve us in the parametrisation of the deformation space in Section 6.3.2. z be the universal cov5.3.1 The brick decomposition for flat affine two-tori. Let M z ering flat affine manifold of M ,   Aff.M / the group of covering transformations, z ! A2 the development map. Let N  Aff.2/ be the identity component and D W M of a maximal abelian subgroup containing h./, as in Section 5.1. By Proposition 5.3, z / lifts to an action of its universal the action of N on the development image D.M z , such that D is an equivariant map M z ! D.M z /. covering Nz on M z / is A2 , or the Homogeneous flat affine tori. In case the development image D.M half-plane H , or the sector Q, N is simply connected and acts simply transitively z /. It follows that Nz acts simply transitively on M z , and D is an equivariant on D.M z z local diffeomorphism. Hence, D W M ! D.M / is an affine diffeomorphism. If the development image is A2  0 and N is conjugate to GL.1; C/ then Nz acts simply z . It follows that D is a covering map. We thus have an affine covering transitively on M

A

D

A2  0 ! A2  0: This proves that the development map is a covering map for all homogeneous flat affine tori M .

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Inhomogeneous flat affine tori. We assume now that M is not homogeneous. Therefore, the development image of M is A2  0 and N is different from GL.1; C/. Then N equals either the group of diagonal matrices with positive entries B or the group C1 (compare Example 4.2). The open orbits of N on A2  0 are the open quadrant in the case N D B, or the open half space in the case N D C1 . In particular, in this case, N does not act transitively on A2  0, and therefore M is not a homogeneous flat affine torus. However, the orbits of N on M decompose M into finitely many pieces, the bricks, from which M is constructed. Proposition 5.9 (Brick Lemma for the flat affine two-torus). Let  D Nz xQ 0 be an x D g. x Then z , and  x the closure of . Let 0 D f 2  j   open orbit of Nz on M x ! A2  0 is a diffeomorphism onto its image. (1) D W  x 0 is a flat affine cylinder with geodesic boundary. (2) = Proof. Observe that N D Nz is simply connected. Put x0 D D xQ 0 . It follows that D W Nz xQ 0 ! N x0 is a diffeomorphism. The complement of N xQ 0 in its closure N xQ 0 consists of one-dimensional orbits for N , which are diffeomorphic to a ray in A2  0. z , N xQ 0 has precisely two such orbits in its cloSince D is a local diffeomorphism on M sure, which map to their corresponding orbits in A2  0, see Figure 22. It follows that D is injective on the closure N xQ 0 and, in fact, D W N xQ 0 ! A2  0 is a diffeomorphism onto its image. This proves (1). To prove (2), remark first that Nz has at most finitely many orbits on the compact z . Since manifold M . This implies that there are only finitely many orbits of  Nz on M z , it follows that every compact subset  of M z intersects only  acts properly on M finitely many orbits of Nz . In particular,  intersects only finitely many components x Therefore,   x is closed in M z . Hence,  x projects to a compact subset in M . of  . We may assume (by replacing M with a finite covering manifold if necessary) x \ x ¤ ;, then that h./ is contained in N . Note then that, if  2  such that   z Q D 1. Since  2 0 . In fact, since h./ 2 N , there exists nQ 2 N such that h. n/ x and  nQ  x \ x ¤ ;, we conclude that  nQ preserves D is a diffeomorphism on  x Thus  n both boundary components of . Q D , and therefore  D n1 2 Nz . x 0 is the image of  x in In particular, 0 D  \ Nz . Moreover, it follows that = z = . M DM x has two boundary components x 0 is compact. Since  We thus proved that = x which are geodesic rays, =0 must be a flat affine cylinder with geodesic circles as boundary components. This proves (2). x is a closed half space or a sector. This implies: The development image of  Proposition 5.10. If the development image is A2  0, then the holonomy h.0 / is generated by an expansion. Proof. The proof of the previous proposition implies that h.0 / is contained in N . Since 0 acts properly on N xQ 0 , and since D is a diffeomorphism onto its image N x0 ,

Chapter 12. The deformation of flat affine structures on the two-torus

˜x N ˜0

˜x N ˜2

N x0

N x2

D

511

˜x N ˜1 N x1

z. Figure 22. The orbits of Nz in M

it follows that h.0 / acts properly and with compact quotient on the orbit closure N x0 . This implies that h.0 / has positive eigenvalues on one-dimensional orbits, and a fortiori, by properness on N x0 , it must be a group of expansions of GL.2; R/ (see Figures 17 and 16). Final step in the proof. By the brick lemma (Proposition 5.9), M decomposes as x 0 , which are glued along their a finite union of copies of a flat affine cylinder = boundary geodesics. Therefore, there exists a finite union Hx of neighbouring copies of x in M z , such that the torus M is obtained by identifying the two boundary geodesics of  x z /. Since h.B/ z 2 GLC.2; R/ commutes H = 0 by an affine transformation Bz in Aff.M z with N , it follows that B D h.B/ is contained in N or B 2 R N . It follows that the development image of Hx must be A2  0 or Hx0 , the closed half space with the origin removed, respectively, see Figure 23. Hence, Hx is affinely diffeomorphic to one of the strips Hx0k , which are defined in Section 4.4. Let A 2 GLC.2; R/ be the expansion which generates 0 . Then Hx =0 is a flat affine cylinder CAk , as defined in Example 4.18. Therefore, M is affinely diffeomorphic to a flat affine torus TA;B;k constructed in Example 4.15. In particular, M is affinely diffeomorphic to a quotient of A2  0, by a properly discontinuous subgroup

A

z Bzk i  D hA;

A

of affine transformations in Aff.A2  0/. ˜ B

Ω1

θ = πk

Ω2

···

D

Ω2k

θ=0

x1 [

[  x 2k , B D R . Figure 23. Hx D 

B

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Corollary 5.11. Every non-homogenous flat affine two-torus M is affinely diffeomorphic to a two-torus TA;B;k (see Example 4.15), and M is obtained by gluing a flat affine cylinder CAk , where A 2 GLC.2; R/ is an expansion, and B 2 GLC.2; R/ has positive eigenvalues and commutes with A.

6 The topology of the deformation space In this section we describe the global and local structure of the deformation space D.T 2 ; A2 / of all flat affine structures on the two-torus. The deformation space decomposes into two overlapping subsets: the open subspace D.T 2 ; A2  0/ of structures modeled on the once-punctured plane A2  0 and the closed subspace Dh .T 2 ; A2 / of homogeneous flat affine structures. We describe the structure and topology of these two subspaces separately in Sections 6.3 to 6.4. In Section 6.5 we deduce our main result that the holonomy map for the deformation space D.T 2 ; A2 / is a local homeomorphism.

A

A

6.1 Flat affine connections In this subsection we introduce flat affine connections. These provide another point of view on flat affine structures, which turns out to be particularly useful in the study of homogeneous flat affine manifolds. An affine connection on the tangent bundle of M is determined by a covariant differentiation operation on vector fields which is an R-bilinear map r W Vect.M /  Vect.M / ! Vect.M /;

.X; Y / 7! rX .Y /;

and for f 2 C 1 .M /, satisfies rf X Y D f rX Y;

and

rX .f Y / D f rX Y C .Xf /Y

(where Xf 2 C 1 .M / denotes the directional derivative of f with respect to X ). The connection is torsion-free if and only if, for all X; Y 2 Vect.M /, rX Y  rY X D ŒX; Y ;

(6.1)

r

and it is flat if and only if the curvature tensor R vanishes. That is, if Rr .X; Y / D rX rY  rY rX  rŒX;Y  D 0:

(6.2)

6.1.1 Correspondence with flat affine structures. Specifying an affine structure on M is equivalent to giving a torsion-free flat affine connection on the tangent bundle of M . Indeed, let M be a flat affine manifold. Then the affine structure defines a unique torsion-free flat affine connection on M by pulling back the canonical affine connection on An (that is, the usual derivative on Rn ) via a development map. Conversely, given

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513

any torsion-free flat affine connection r on M , for each p 2 M , the exponential map for r at p is a connection-preserving diffeomorphism from an open subset of the tangent vector space Tp M (with the canonical flat affine connection) to a neighborhood of p, compare [48], VI. Theorem 7.2. This gives rise to an atlas of locally affine coordinates and therefore determines a unique flat affine structure on M . Thus, there is a natural one to one correspondence S.M; An /

! ftorsion-free flat affine connections on M g

(6.3)

of the set of flat affine structures S.M; A / with a set of affine connections. An affine connection is called complete if all of its geodesics can be extended to infinity. Under the correspondence (6.3) complete affine structures are in bijection with complete affine connections. n

Observe that the difference of two affine connections is a tensor field on M and therefore the set of all affine connections forms an affine space. Example 6.1 (Flat connections form a closed subset of an affine space). Let E denote the tangent bundle of the flat affine manifold M . Let r0 be the natural flat connection induced on M by its flat affine structure. We choose r0 as a basepoint in the space of all affine connections on M . Every torsion-free affine connection on M is of the form r D r0 CS , where S 2 .S 2 E  ˝ E/ is a vector valued symmetric form on M . The set of all torsion-free affine connections r on M is thus an affine space modeled on the vector space .S 2 E  ˝ E/. Every torsion-free flat affine connection on M is of the form r D r0 C S, where S is contained in the closed subset C of .S 2 E  ˝ E/ defined by the equation (6.2), which encodes the vanishing of curvature. The space of sections .S 2 E  ˝ E/ carries the C 1 -topology of maps. This defines a topology on the space of torsion-free affine connections. Proposition 6.2. The natural correspondence (6.3) of flat affine structures with flat torsion-free affine connections is a homeomorphism. In particular, the space of flat affine structures S.M; An / is homeomorphic to the closed subset C in the tensor space .S 2 E  ˝ E/ as described above. Proof. Let us fix a flat affine structure on M and let r0 be its compatible torsion-free flat affine connection. Let r D r0 C S be another torsion-free flat affine connection on M . In a local flat affine coordinate chart for M , S 2 C is represented by a set of functions ijk which are called Christoffel symbols for r, see [48], III. Proposition 7.10. We observe that the functions ijk also coincide with the coordinate representation of the tensor S . Therefore, a sequence rn of affine connections is convergent if and only if the corresponding Christoffel symbols converge in all local flat affine coordinate systems. Let Dn be a sequence of development maps and consider the corresponding sequence of flat affine connections rn . Since the Christoffel symbols for rn are polyno-

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mials in the first and second derivatives of Dn (see [48]), convergence of Dn implies convergence of rn . Therefore, the correspondence (6.3) is continuous. Conversely, for any torsion-free flat affine connection r on M , normal coordinate systems on M define compatible coordinate charts for the flat affine structure defined by r, see [48], VI. Theorem 7.2. Normal coordinate systems for r are determined by an ordinary differential equation whose solutions depend smoothly on the Christoffel symbols for r. Hence, the flat affine coordinate charts for r depend smoothly on r. This shows that the correspondence (6.3) is a homeomorphism. 6.1.2 Translation invariant flat affine connections. An affine connection r on the two-torus T 2 D S1  S1 is called translation invariant if the group S 1  S 1 acts by affine transformations. Let r0 be the natural Riemannian flat affine connection on T 2 . It is characterized by the property that the translation vector fields of the S 1  S 1 -actions are parallel. Then any other connection r D r0 C S is translation invariant if and only if the covariant derivatives of the translation vector fields of the S 1  S 1 -action are parallel with respect to r. This condition is satisfied, if and only if the Christoffel symbols S are constant functions in the flat coordinates for r0 . Therefore, the set of all translation invariant torsion-free flat affine connection is in bijection with the subset C .T 2 ; R/ of C which consists of all constant (that is, of all r0 -parallel) tensors contained in C . Remark 6.3. Equation (6.2) shows that C .T 2 ; R/ is a quadratic cone in the vector space of symmetric bilinear maps S 2 R2 ˝ R2 . Every element S 2 C .R/ represents a symmetric bilinear product

r W R2  R2 ! R2 ;

u r v WD S.u; v/;

which, for all u; v; w, satisfies the associativity relation .u r v/ r w D u r .v r w/: This product defines a left-invariant flat affine connection on the abelian Lie group R2 by extending the covariant derivative from left-invariant vector fields to all vector fields. Indeed, there is a general correspondence of associative, and more generally left-symmetric algebra products with left-invariant torsion-free flat affine connections on Lie groups, see for example [5], Section 5.1. Under this correspondence complete connections are represented by products which have the property that all maps v 7! u r v have trace zero (compare [5], Corollary 5.7). We summarize this discussion by the following: Corollary 6.4. (1) The set of all translation invariant flat affine connections on T 2 is homeomorphic to a four-dimensional homogeneous quadratic cone C .T 2 ; R/ in the six-dimensional vector space S 2 R2 ˝ R2 .

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(2) The subset of complete translation invariant flat affine structures on T 2 is homeomorphic to a two-dimensional homogeneous quadratic cone in the vector space R4 . In particular, one can deduce from (2) that the set of complete translation invariant flat affine connections on the two-torus is homeomorphic to R2 . In view of Lemma 6.6, this gives yet another proof of the fact (cf. Example 3.26) that the deformation space of complete affine structures on the two-torus is homeomorphic to R2 .

6.2 Translation invariant flat affine structures The usual representation of the two-torus T 2 D R2 =Z2 as a quotient of the vector group R2 by its integral lattice tacitly induces various extra structures. The translation action of the vector space R2 gives a simply transitive action of the abelian Lie group S 1  S 1 and the vector space structure on R2 descends to a compact abelian Lie group structure on T 2 . Similarly, the ordinary flat affine structure on R2 induces the natural Riemannian flat affine structure on T 2 which is invariant by the translation group S 1  S 1 . Definition 6.5. A flat affine structure on T 2 D S 1  S 1 is called translation invariant if the group S 1  S 1 acts by affine transformations. A translation invariant flat affine structure is thus compatible with the Lie group structure on T 2 . In particular, every flat affine torus with translation invariant flat affine structure is also a homogeneous flat affine torus. Note that the set of all translation invariant flat affine structures T .T 2 ; A2 / corresponds to the set of translation invariant flat affine connections C.T 2 ; R/ under the map (6.3). 6.2.1 Relation with homogeneous flat affine tori. Let M be a homogeneous flat affine two-torus, and Aff.M /0 the identity component of its affine automorphism group. By the classification Theorem 5.1, the following two cases occur: (1) Either the Lie group Aff.M /0 is isomorphic to S 1  S 1 and it develops to an action of an affine Lie group as listed in Example 4.2. (2) Or M is affinely diffeomorphic to a Hopf torus. In this case, Aff.M /0 contains a simply transitive group isomorphic to S 1  S 1 , and this subgroup develops to the action of an affine Lie group of type A. In particular, for every homogeneous flat affine two-torus M , the identity component Aff.M /0 of the affine automorphism group of M contains a two-dimensional compact abelian Lie group, which acts transitively and freely on M . This shows that every homogeneous flat affine two-torus is affinely diffeomorphic to a translation invariant flat affine torus.

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Recall that the diffeomorphism group Diff.T 2 / acts on the set of all flat affine structures, and two flat affine structures on T 2 are called homotopic if they are equivalent by a diffeomorphism of T 2 which is homotopic to the identity. Lemma 6.6. Every homogeneous flat affine two-torus is homotopic (isotopic) to a unique translation invariant flat affine two-torus. Proof. Let .f; M / be a marked homogeneous flat affine two-torus, where f W T 2 ! M is a diffeomorphism. By the above remarks, we may choose a Lie subgroup A of Aff.M /0 which acts simply transitively on M . The subgroup A is unique up to conjugacy in Aff.M /0 . We also choose a basepoint m0 2 M . This fixes the structure of a compact abelian Lie group on M which is isomorphic to A. Then there exists a unique isomorphism of Lie groups W T 2 ! M such that 1 B f is homotopic to the identity of T 2 . In other words, . ; M / and .f; M / are equivalent markings (cf. Section 3.4). By construction, the affine structure on T 2 induced by is translation invariant, and it is homotopic to the original homogeneous structure on T 2 , which is induced by .f; M /. It is also independent of the choice of basepoint since A acts transitively on M (compare also Example 3.34). Neither does it depend on the choice of the subgroup A in Aff.M /0 , since the conjugacy class of A in Aff.M /0 is uniquely determined. In particular, this argument implies that every two translation invariant structures which are homotopic do coincide. This shows uniqueness. The lemma asserts that every orbit of the identity component Diff 0 .T 2 / of the group of all diffeomorphisms Diff.T 2 / acting on homogeneous flat affine structures intersects the subset of translation invariant structures in precisely a single point. The proof also shows that on the subset of marked homogeneous tori which are in the complement of Hopf tori, we have a continuous projection onto translation invariant tori. This proves that outside the Hopf tori the subset of translation invariant flat affine structures on T 2 defines a slice for the action of Diff 0 .T 2 / on the set of all homogeneous flat affine structures. 6.2.2 Translation invariant development maps. We construct an explicit continuous section from the set of translation invariant flat affine structures to development maps. More specifically, we construct a continuous map E

T .T 2 ; A2 / D C .T 2 ; R/ ! Dev.T 2 ; A2 /;

S 7! DS ;

(6.4)

such that the development map DS defines an affine structure on T 2 which has associated flat affine connection r D r0 C S. The construction is based on the relation of translation invariant flat affine structures with the set of commutative associative algebra products on R2 as follows:

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Example 6.7 (Associated étale affine representation). For S 2 C .T 2 ; R/, and v 2 R2 we define an element   S.v; / v .v/ N D 2 aff.2/ 0 0 of the Lie algebra aff.2/ of the affine group Aff.2/. In fact, the map v 7! .v/ N is a Lie algebra homomorphism and the associated homomorphism of Lie groups D S W R2 ! Aff.2/;

v 7! .v/ D exp .v/; N

defines an affine representation of the Lie group R2 on A2 which is étale in 0 2 A2 (cf. Definition 3.9 and also the discussion in Section 2.1 of [6]). Let r be the translation invariant flat affine connection on R2 which is represented by S 2 C .T 2 ; R/. The orbit map of the étale representation S is DS D oS W R2 ! A2 ;

v 7! S 0;

and it is a development map for a translation invariant flat affine structure on T 2 D R2 =Z2 with associated affine connection r. (In fact, DS is also a frame-preserving development map. Compare Example 3.24 and Section 3.4.1.) Since, DS depends smoothly on S , E.S/ D DS defines the required continuous section. Note that the holonomy homomorphism h D hr W Z2 ! Aff.2/ for DS satisfies hr ./ D ./;

for all  2 Z2 :

We state without proof: Lemma 6.8. The continuous map C.T 2 ; R/ ! Hom.Z2 ; Aff.2/, r 7! hr , is locally injective.

6.3 The space of structures modeled on .A2  0; GL.2 ; R// Here we discuss in detail the subspace of the deformation space of flat affine structures on the two-torus which consists of structures which have the once-punctured plane as development image. Our main observation is that the holonomy map is a local homeomorphism on such structures. 6.3.1 The holonomy map. We first consider the subgeometry of structures which are modeled on the universal covering of the once-punctured plane. The topology of the deformation space of such structures is completely controlled by the holonomy map into the space of conjugacy classes of homomorphisms (the “character variety”):

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Theorem 6.9. The holonomy map

A

hol f C.2; R//=GL.2; f R/ D.T 2 ; A2  0/ ! Hom.Z2 ; GL

embeds the deformation space homeomorphically as an open connected subset of the character variety. f C.2; R/. Proof. Note that, since T 2 is orientable, the holonomy takes values in GL The map hol is injective, by Theorem 5.6. Since hol is also continuous and open, it is a homeomorphism onto an open subset. Connectedness of the deformation space will follow from the considerations in Section 6.3.2 below. Now we look at the deformation space of structures which is modeled on the oncepunctured plane. For such structures the holonomy map is not injective, as we already remarked in Example 5.5. However, as we show now at least locally the topology of the deformation space of .A2  0; GL.2; R// structures is fully controlled by the character variety: Corollary 6.10. The holonomy map hol

D.T 2 ; A2  0/ ! Hom.Z2 ; GLC .2; R//= GL.2; R/ is a local homeomorphism onto its image, which is a connected open subset in the character variety. Proof. Since the subgeometry

A

f R// ! .A2  0; GL.2; R// .A2  0; GL.2; is a covering, the induced map on the deformation spaces (cf. Section 3.1.2)

A

D.T 2 ; A2  0/ ! D.T 2 ; A2  0/ is a homeomorphism by Lemma 3.22. The commutative diagram (3.8) for the subgeometry takes the form   hol / f C.2; R//=GL.2; f R/ Hom.Z2 ; GL D.T 2 ; A2  0/

A





D.T 2 ; A2  0/

hol



(6.5)

/ Hom.Z2 ; GLC .2; R//= GL.2; R/:

Note that, by Theorem 6.9, the top horizontal map is a topological embedding. Furthermore, by Corollary A.8, the right vertical map is a local homeomorphism. We deduce that the bottom map hol for D.T 2 ; A2  0/ is locally injective, and therefore it is a local homeomorphism onto an open subset. In the situation of Corollary 6.10, all local topological properties of the deformation space are reflected in the character variety and also vice versa. For instance,

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singularities in the character variety give rise to singularities in the deformation space, as is the case in Example 3.29. This shows that D.T 2 ; A2  0/ is not Hausdorff, and it is not even a T1 -topological space. 6.3.2 Cartography of the deformation space. Important strata in the deformation space

A

D.T 2 ; A2  0/ f C.2; R/ on the image of the holonomy arise from the orbit types of the action of GL map f C.2; R//: hol W Dev.T 2 ; A2  0/ ! Hom.Z2 ; GL

A

We introduce several such strata and describe their topological relations with each other. We use this information to establish the connectedness of the deformation space.

A

Overview. According to the classification theorem, tori which are modeled on A2  0 fall into three main classes distinguished by their development images. Namely the classes are formed by structures which have development image equivalent to either

A

(1) the once-punctured plane A2  0 (“complete structures”), (2) or a sector Q, (3) or the open half space H .

A

The structures with development image A2  0 comprise non-homogeneous flat affine tori and the homogeneous structures which arise from (lifts of) étale representations of type A. The latter two strata arise from (the lifts of) étale representations of type B and C1 respectively (see Example 4.2 for notation). Therefore all corresponding tori in these two strata are homogeneous. Another decomposition of the deformation space is obtained when we consider the subset T of non-homogeneous structures and its complementary subspace Dh .T 2 ; A2  0/ consisting of homogeneous structures. The space of non-homogeneous structures can be decomposed into connected components parametrized by the level of a non-homogeneous structure. The subset of homogeneous structures is connected. The subspace Dh .T 2 ; A2  0/ contains a two-dimensional stratum H of Hopf tori as a distinguished subset. Non-homogenous structures are connected to homogeneous ones only along the space H of Hopf tori.

A

A

A

Hopf tori. Recall that a torus which is modeled on A2  0 is called a Hopf torus if its f C.2; R/. The holonomy homomorphisms of holonomy is contained in the center of GL f R/-conjugation marked Hopf tori form the closed subset of fixed points for the GL.2;

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A

action on the holonomy image of A2  0 - structures. Therefore, the Hopf tori form a closed subset H  D.T 2 ; A2  0/:

A

All Hopf tori are derived from the étale affine representation of type A as follows. Let o W R2 ! A2  0; .t; / 7! exp.t /.cos ; sin  /; be the orbit map associated to the representation A. For k1 ; k2 2 Z and 1 ; 2 > 0, let W R2 ! R2 be the linear map, which satisfies .e1 / D .log 1 ; k1 /;

.e2 / D .log 2 ; k2 /:

Then development maps of the form D D o B W R2 ! A2  0 define a two-parameter family of marked Hopf tori H1 ;2 ;k1 ;k2 ;

.log 1 /k2  .log 2 /k1 ¤ 0:

(See Section 3.3.5 for the general construction.) The corresponding holonomy homof R// satisfy morphisms h W Z2 ! Hom.Z2 ; GL.2; f C.2; R/: h.ei / D diag. i /  ki 2 GL (Here e1 , e2 denote generators of Z2 , and diag. / 2 A the diagonal matrix which has both diagonal entries equal to .) Observe that every marked Hopf torus is equivalent in the deformation space to precisely one of these tori. Forgetting about the marking, we note that every Hopf torus is affinely equivalent to a torus of the form H1 ;2 ;k;0 , where we call k D gcd.k1 ; k2 / ¤ 0 the level of H . For fixed k1 ; k2 2 Z, the set of all H1 ;2 ;k1 ;k2 parametrizes a closed (and also connected subset) of Hopf tori Hk1 ;k2 , and the subset of all Hopf tori decomposes as [ HD Hk1 ;k2  D.T 2 ; A2  0/:

A

.k1 ;k2 / ¤ 0

Non-homogeneous tori. We let

A

T  D.T 2 ; A2  0/ denote the subset of non-homogenous structures. Every marked manifold T which f C.2; R//-manifold to a represents an element of T is equivalent as an .A2  0; GL torus TA;B;k

A

as constructed in Corollary 5.11. Here A 2 GLC .2; R/ is an expansion and B 2 GLC .2; R/ is upper triangular and commuting with A. We call the number k 2 Zf0g

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the level of T , respectively the level of its class in T . Since the level cannot be zero, claim (2) of Lemma A.4 implies that the set T is indeed an open subset of the deformation space. Let Tk denote the set of all elements in T of level k. We have the disjoint decomposition [ Tk : TD k2Zf0g

Proposition A.5 implies that all subsets Tk and their complements are closed subsets of T . The closure of non-homogeneous tori. We show now that the boundary of the set of non-homogeneous structures T in the deformation space is formed by Hopf tori.

A

Proposition 6.11. The closure of Tk in D.T 2 ; A2  0/ is Tk [ Hk . Proof. Note first that the Hopf tori H1 ;2 ;k;0 2 Hk , are in the closure of the elements TA;B;k of Tk . (Just deform A and B to dilations.) It remains to show that every homogeneous torus Mo in the closure of Tk is a Hopf torus: Let M equivalent to TA;B;k be a marked non-homogeneous torus of level k ¤ 0 which is in the vicinity of Mo in the deformation space. Let h denote its holonomy homomorphism. We assume that h converges to ho in the space of conjugacy classes of homomorphisms. The f C.2; R/, i D 1; 2. The conjugacy holonomy group of M is generated by h .ei / 2 GL class C h .ei / is thus in the vicinity of the class C ho .ei /. Proposition A.5 implies that the projections p.ho .ei // 2 GLC .2; R/ are conjugate to an element of AN . Since Mo is homogeneous, Mo is either a Hopf torus or lev ho .e1 / D lev ho .e2 / D 0 (in which case Mo has development image H or Q). In the latter case, if M is close enough, we must have lev h .e1 / D lev h .e2 / D 0, again by Proposition A.5. This contradicts the fact that the level of the non-homogeneous torus M is different from zero. Therefore, Mo is a Hopf torus. Homogeneous tori modeled on A2  0. The subset of homogeneous structures

A

A

Dh .T 2 ; A2  0/  D.T 2 ; A2  0/ decomposes into three strata A, B and C1 , which are distinguished according to the development images A2  0, Q and H respectively. These structures arise from the étale affine representations of the abelian Lie group R2 of type A, B and C1 respectively. Moreover, it follows (using the construction in Section 3.3.5) that the three strata are continuous images of homogeneous spaces via maps

A

A

GL.2; R/=N ! D.T 2 ; A2  0/;

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where N describes the group of those automorphisms of R2 which are induced by the conjugation action of the normalizers in GL.2; R/ for the groups A, B and C1 . (The normalizers are listed in Lemma 4.3.) In particular, it follows that the strata A and B are images of connected four-dimensional manifolds, while C1 is a connected manifold of dimension three. Note that structures in A may be continuously deformed to structures in C1 , as follows from (4) of Lemma A.4. Compare also Figure 9. Similarly, structures in B can be deformed to structures in C1 , see Figure 4. This shows in particular that Dh .T 2 ; A2  0/ is connected.

A

Connectedness. The deformation space

A

A

D.T 2 ; A2  0/ D T [ Dh .T 2 ; A2  0/ is connected. Indeed, by Proposition 6.11 every marked non-homogeneous flat affine torus in T can be deformed to a Hopf torus contained in some Hk1 ;k2 . Since Hk1 ;k2

A

A

is a subset of the connected space D.T 2 ; A2  0/h it follows that D.T 2 ; A2  0/ is connected.

6.4 The subspace of homogeneous structures We describe now the properties of the subset Dh .T 2 ; A2 / of homogeneous flat affine structures as a subspace of the deformation space of all flat affine structures on T 2 . Since the non-homogeneous structures form an open subset the space Dh .T 2 ; A2 / is closed. The complement of Hopf tori Dh .T 2 ; A2 /  H forms a dense subset which is also open in the deformation space of all structures D.T 2 ; A2 /. We established in the previous subsections: Proposition 6.12. The set of all homogeneous structures Dh .T 2 ; A2 / is the continuous and injective image of the quadratic cone C .T 2 ; R/ under the map E in (6.4). The map is a homeomorphism in the complement of Hopf tori. In particular: Corollary 6.13. The deformation space Dh .T 2 ; A2 / of all homogeneous flat affine structures on the two-torus contains the complement of Hopf tori Dh .T 2 ; A2 /  H as a dense open subset, which is a Hausdorff space and homeomorphic to a Zariskiopen subset in a four-dimensional quadratic cone in R6 .

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Note that the space of complete affine structures Dc .T 2 ; A2 / forms a two-dimensional closed subcone which is homeomorphic to R2 , see [6]. 6.4.1 The action of the linear group on translation invariant structures and conjugacy of étale affine groups. The linear group GL.2; R/ naturally acts on the variety C .T 2 ; R/ of commutative and associative algebra products on R2 . The orbits of this action correspond to the isomorphism classes of algebra products. Since the section map E W C.T 2 ; R/ ! Dh .T 2 ; A2 / is a continuous bijection this constructs a natural induced action of GL.2; R/ on the deformation space of homogeneous structures Dh .T 2 ; A2 / which is continuous on the complement of Hopf tori. This group action may be used to reveal some of the topology of Dh .T 2 ; A2 / and the possible deformations of structures. Recall the classification of étale affine representations which is described in Section 4.1. Each orbit of GL.2; R/ in Dh .T 2 ; A2 / corresponds to exactly one of the affine conjugacy classes of abelian almost simply transitive groups of affine transformations on A2 . We label the orbits accordingly with the symbols A, B, C1 , C2 , D and T. The decomposition of Dh .T 2 / into the six orbit types of GL.2; R/ defines a natural stratification on Dh .T 2 / into manifolds which are homogeneous spaces of GL.2; R/, and each orbit is a subcone of Dh .T 2 /. Each such stratum may be also computed as the induced image of the subgeometry which is defined by the corresponding étale affine representation, see the examples in Section 6.3.2, as well as Section 3.3.5 and Lemma 4.3. The closure of each stratum consists of strata of lower dimensions and contains the unique closed stratum T, which is a point. There are two open strata of dimension four labeled A and B, which correspond to homogeneous flat affine structures whose development images are the punctured plane, and the sector respectively. In their closure are the three-dimensional orbits C1 and C2 , whose corresponding flat affine structures develop into the halfplane. The complete structures correspond to a two-dimensional orbit D and the translation structure T. We say that the orbit O1 degenerates to the orbit O2 if O2 is in the closure of O1 . Degeneration induces a partial ordering on the strata of Dh .T 2 ; A2 /, with the translation action the unique minimal point. By the theorem of Hilbert–Mumford if O2 degenerates to O1 then there exists a one-parameter group W R ! GL.2; R/ such that lim t !0 .t /o2 2 O2 . Therefore, every degeneration may be constructed explicitly as a limit of a curve of flat affine structures in the stratum. Moreover, every point of Dh .T 2 ; A2 / directly degenerates to the translation structure, compare, for example, Figure 7. The graph shown in Figure 24 describes all possible degenerations in the orbit stratification of Dh .T 2 ; A2 / with respect to the natural action of GL.2; R/. These degenerations are illustrated in Figures 1–5, Figure 7 and Figure 9.

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A

/

u

/

C1

D

/ o T

o C2

B

Figure 24. Degenerations of GL.2; R/-orbit types in the deformation space.

6.5 The deformation space of all flat affine structures on the two-torus Our main result is the following. Theorem 6.14. The holonomy map for flat affine structures on the two-torus hol W D.T 2 ; A2 / ! Hom.Z2 ; Aff.2//=Aff.2/ is a local homeomorphism onto an open connected subset of the character variety. The holonomy map for flat affine structures. For the proof of Theorem 6.14 we consider first the subgeometry of .A2  0; GL.2; R//-structures and its induced map on deformation spaces (cf. Section 3.1.2): Proposition 6.15. The induced map on deformation spaces D.T 2 ; A2  0/ ! D.T 2 ; A2 / is an embedding onto an open subset Uo of the space D.T 2 ; A2 /. Moreover, the holonomy map for D.T 2 ; A2 / restricts to a local homeomorphism on this subset. Proof. The commutative diagram (3.8) for the subgeometry takes the form D.T 2 ; A2  0/  D.T 2 ; A2 /

hol

hol

/ Hom.Z2 ; GLC.2; R//= GL.2; R/  / Hom.Z2 ; Aff.2//=Aff.2/:

The image of D.T 2 ; A2  0/ in D.T 2 ; A2 / consists of precisely those structures in D.T 2 ; A2 / whose linear part of the holonomy contains an expansion. Therefore, the image Uo of the induced map is open in D.T 2 ; A2 /. The left vertical map is clearly injective. Note further that the operation of taking the linear part of a homomorphism defines a continuous section of the right vertical map which is defined on the holonomy image hol.Uo /. The latter set is open in Hom.Z2 ; Aff.2//=Aff.2/. By Corollary 6.10, the upper map hol is a local homeomorphism. Therefore, the right vertical map is a topological embedding, and the lower map hol is a local homeomorphism on Uo . Below we construct an open neighborhood U1 of the translation structure T 2 D.T 2 ; A2 /, which has the following properties:

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(1) U1  Dh .T 2 ; A2 / is contained in the subset of homogeneous flat affine structures, (2) the restriction of the holonomy map hol W U1 ! Hom.Z2 ; Aff.2//=Aff.2/ is injective, (3) D.T 2 ; A2 / D Uo [ U1 . Together with Proposition 6.15 this shows that hol W D.T 2 ; A2 / ! Hom.Z2 ; Aff.2//=Aff.2/ is locally injective and therefore finishes the proof of Theorem 6.14. We observe the following refinement of Proposition 6.11: Proposition 6.16. The closure of the subset T of non-homogeneous structures in the deformation space D.T 2 ; A2 / consists of Hopf tori. Proof. Suppose there is a sequence of non-homogeneous marked tori Mi which converge in the deformation space D.T 2 ; A2 / to a flat affine torus M . Let hi W Z2 ! Aff.2/ be their corresponding holonomy homomorphisms. By Corollary 5.11, we may assume that the linear parts of the hi are contained in the group of upper triangular matrices AN [ AN , where AN is the index two subgroup with positive diagonal entries. Now if M is homogeneous and has development image different from the once-punctured plane, the linear parts of all hi are contained in AN for sufficiently large i . Let Di be a corresponding sequence of development maps for the Mi which converges to a development map D which represents M . Since M is not modeled on the once-punctured plane the development map D is injective. By Example 3.14, there is a neighborhood of D in the space of development maps such that D is injective on the fundamental domain for the standard action of Z2 on R2 . However, the development maps Di are not injective on this fundamental domain by construction of the non-homogeneous tori Mi , see Example 4.15. This contradicts the fact that the Di converge to D. The claim now follows from Proposition 6.11. Now the construction of U1 goes as follows: Following the notation in Appendix A, f C.2; R/ j j.g/j < g. By the proof of Proposition A.7, we may define U D fg 2 GL choose an open set f C.2; R//=GL.2; f R/; U  Hom.Z2 ; GL where U is of the form C .U  U / such that the projection U ! Hom.Z2 ; GLC.2; R//= GL.2; R/ is injective. The holonomy preimage hol1 .U / is a non-empty open subset of D.T 2 ; A2  0/ which contains certain homogeneous structures of type A, and the strata B and C1 . It corresponds to a non-empty open subset V of D.T 2 ; A2  0/

A

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such that the restriction hol W V ! Hom.Z2 ; GLC.2; R//= GL.2; R/ is injective. Let M D D.T 2 ; A2  0/  V be the complement (containing the nonhomogeneous flat affine tori and also homogeneous structures of type A). Then we observe that M is closed not only in D.T 2 ; A2  0/, but also in D.T 2 ; A2 /. (Indeed, this follows since the closure of the space T of non-homogeneous tori is contained in the space H of Hopf tori.) Now we put U1 D D.T 2 ; A2 /  M.

Appendix A. Conjugacy classes in the universal covering group of GL.2 ; R/ Let GLC .2; R/ be the group of 2  2 matrices with positive determinant, and let f C.2; R/ ! GLC .2; R/ p W GL be its universal covering group. Iwasawa decomposition.

Recall the Iwasawa decomposition GLC .2; R/ D KAN;

where K D SO.2; R/ is the subgroup of rotations, A is the group of diagonal matrices with positive entries, and N the group of unipotent upper triangular matrices. Furthermore, we let D be the central subgroup of GLC .2; R/ contained in A which consists of all elements of A with identical diagonal entries. Let Kz ! K be the universal covering of the rotation group. There is an induced Iwasawa decomposition z f C.2; R/ D KAN; GL f C.2; R/. where A and N are considered as subgroups of GL z Let Z be the subgroup of K, which is mapped by the covering projection onto f C.2; R/ consists of the fC1; 1g D fE2 ; R g  SO.2; R/. Note that the center of GL subgroup D extended by the group Z. We choose a generator  2 Kz for the infinite cyclic group Z. Then p./ D R . D E2 / and the element  2 2 Z generates the kernel of the covering projection p. The rotation angle function. We consider the diffeomorphism  W Kz ! R which satisfies .1/ D 0 and the relation   cos  sin  pD :  sin  cos 

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527

Using the Iwasawa decomposition we construct an angular map f C.2; R/ ! R  W GL f C.2; R/ has by extending  W Kz ! R. This means that .g/ D .k/, where g 2 GL decomposition g D kan. Geometrically, .g/ thus is the angle of rotation or polar angle for the image p.g/.e1 / of the first standard basis vector e1 . More specifically, f C.2; R/ on A2  0 (see Section 4.3.2), the function when considering the action of GL  can also be read off as the -coordinate of

A

A

g .r; 0/ D .s; .g// 2 A2  0: The following properties of the function  are easy to verify: f C.2; R/. Then Lemma A.1. Let g; h 2 GL (1) .g/ D 0 if and only if g 2 AN ; z . m / D m; (2) .kg/ D .k/ C .g/, for all k 2 K; (3) j.gh/  .g/  .h/j < ; (4) j.g/  .g 1 /j < ; z (5) j.gkg 1 /  .k/j < , for all k 2 K; (6) j.ghg 1 /j < , for all h 2 AN ;

A

Proof. Recall (see Section 4.3.2) that the action of AN on A2  0 preserves all lines .r; `/ 2 A2  0, where ` 2 Z, and the interior of all strips

A

A

x ` D f.r; / j `    .` C 1/g  A2  0:  Therefore, for any g 2 GLC.2; R/ with .g/ 2 .`; .` C 1// and h 2 AN , we have ` < .hgh1 / < .` C 1/: In particular, one deduces (5) and (6).

A.1 The induced covering on conjugacy classes Let G be a Lie group, and CG D fC.g/ D Ad.G/g j g 2 Gg its set of conjugacy classes. The set CG carries the quotient topology induced from G. Observe that the center of G acts on C G. Indeed, for any z 2 Z.G/, we have zC.g/ D C.zg/. Given a covering projection of Lie groups p W G 0 ! G, there is a natural induced surjective map on conjugacy classes C G 0 ! CG;

C.g/ 7! C.p.g//:

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The kernel  of the covering is a central subgroup of G 0 which acts on C G 0 . As a matter of fact, C G D C G 0 = is the quotient space of this action. Proposition A.2. The natural projection map on conjugacy classes f C.2; R/ ! C GLC.2; R/ C GL is a covering map. Proof. We consider the action of the kernel  D h 2 i of the covering map p on f C.2; R/. For this, let g 2 GL f C.2; R/ and consider its neighborhood C GL f C.2; R/ j j.g/  .h/j < g: U D U .g/ D fh 2 GL We also put CU D fC.h/ j h 2 U g for the corresponding neighborhood of C.g/ in the space of conjugacy classes. z g … ZD. Let V  U .g/ be a neighborhood Let us first assume that g 2 KD, of g, such that all its elements are conjugate to an element of Kz D. Let h 2 V . By using (5) of Lemma A.1, we deduce that, for all ` 2 C.h/, j.g/  .`/j <  C :

(A.1)

f C.2; R/ intersect We observe that the open subsets C V and  k C V D C  k V of C GL if and only if there exist elements h; ` 2 V such that  k ` 2 C.h/: If this is the case then, by the above estimate (A.1), we have j.g/  . k `/j D j.g/  k  .`/j <  C :

(A.2)

Furthermore, j.g/  .`/j < , since ` 2 U . If is small (A.2) is possible if and only if k 2 f0; 1; 1g. For small enough, this implies that all neighborhoods of the form  2k CV D C  2k V are mutually disjoint. Therefore, C V is a fundamental neighborhood of C .g/ for the action of  on C GLC.2; R/. Assume next that g 2 AN is upper triangular. Since g has real and positive eigenvalues, we may choose a small neighborhood V as above such that all its elements z are conjugate to an element of AN or of KD. In particular, for all h 2 V which are conjugate to an element of AN , we deduce from (6) of Lemma A.1 that the range of  on the conjugacy class C .h/ is contained in the open interval .; /. Consequently, .C . k h// is contained in ..k  1/; .k C 1//. It follows that all neighborhoods of the form  2k CV are mutually disjoint, and thus C V is a fundamental neighborhood of C .g/ for the action of . An analogous argument works for g with negative eigenvalues, that is, g 2 AN . f C.2; R/. This implies the propoTherefore  acts discontinuously and freely on C GL sition.

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Corollary A.3. The natural map on conjugacy classes f R/ ! C GL.2; R/ C GL.2; is a local homeomorphism. Proof. Indeed, local injectivity is implied by the commutative diagram f C.2; R/ C GL

/ C GLC.2; R/

 f R/ C GL.2;

 / C GL.2; R/:

Closures of sets of conjugacy classes. For any subset M  G we define C M to be the set of conjugacy classes of elements in M , and C M its closure in C G. We shall require the following lemma: Lemma A.4. With the above convention the following hold in the space of conjugacy f C.2; R/: classes C GL f C.2; R/ be a sequence of elements such that each gi is conjugate to (1) Let gi 2 GL an element of AN and such that the sequence j.gi /j converges to . Then the f C .2; R/. sequence gi leaves every compact subset of GL f C.2; R/. (2) C .AN / D C.AN / is a closed subset of C GL ®  1 1   1 1  ¯ (3) C N D CN D C 0 1 ; C 0 1 ; E2 consists of three conjugacy classes. S z C  k N , CD Kz D D C K. (4) C Kz D C Kz [ k

f C.2; R/ such that j.g/j D , and gN 2 GLC.2; R/ the projection of Proof. Let g 2 GL g. Clearly, by definition of , gN has at least one negative eigenvalue. The sequence gi can not have a subsequence convergent to g, since the corresponding gN i have positive eigenvalues. Thus (1) follows. f C.2; R/ which is the preimage To prove (2), we consider the subset C.AN /  GL C f of C .AN /. For g 2 GL .2; R/, let dis.g/ denote the discriminant of the characteristic polynomial of p.g/ 2 GLC .2; R/. Then g 2 C.AN / if and only if the following hold: i) j.g/j < , ii) dis.g/  0, iii) both eigenvalues of p.g/ are positive. In view of (1), the condition i) is closed. Therefore, C.AN / is a closed subset of f C.2; R/, proving (2). GL Regarding (4), remark first that the closure of C K in C GLC .2; R/ is contained in the union of C K and CN , and C  E2 N . Now here is an example of a sequence   p  sin 'p 1 cos ' C sin ' k' D sin ' cos '  sin '

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z and which, for ' ! 0 is of matrices, where k' is conjugate to the rotation R' 2 K, converging to   1 1 2 N: 0 1 z Since C Kz is invariant by left-multiplication Therefore, C N is in the closure of C K. z with  , the same is true for its closure. This shows that  k C N  C K. f C.2; R/ with p.g/ 2 AN  GLC.2; R/ is of the form  k go , Every element g 2 GL f where go 2 AN  GLC.2; R/. The integer lev g D k 2 Z is called the level of g. The notion of level is defined for the conjugacy class C .g/ of g. The following states that the level separates these conjugacy classes. In particular, the subset of conjugacy classes f C.2; R/  k CAN  C GL is closed. f C.2; R/ be a sequence such that each p.gi / is conjugate Proposition A.5. Let gi 2 GL to an element of AN . If the sequence of conjugacy classes C .gi / converges to C .h/ 2 f C.2; R/ then p.h/ is conjugate to an element of AN , and there exists i0 , such C GL that for all i  i0 , lev gi D lev h. Proof. The discriminant of the characteristic polynomial dis p.gi / and the eigenvalues of p.gi / are continuous functions on the conjugacy classes. Therefore, p.h/ is conjugate in GLC.2; R/ to an element of AN . By assumption, all p.gi / are contained in CAN . Therefore, we have gi 2  2ki C.hi / with hi 2 C AN . By Proposition A.2, f C.2; R/. In parthe group generated by  2 acts properly discontinuously on C GL ticular, there exists a neighbourhood CU of C .h/ such that C .gi / 2 C U implies lev gi D lev h. Incidentally, the assertion of Proposition A.2 fails to be true when considering the situation for the covering f R/ ! PGL.2; R/ D GL.2; R/=f˙E2 g: Pp W GL.2; Example A.6. We consider the induced map f R/ ! C PGL.2; R/ C GL.2;

(A.3)

f C.2; R/ with respect to the action on conjugacy classes. It is the quotient map of C GL f R/of the central subgroup Z D hi, generated by the element . Then the GL.2; C f .2; R/ is a lift of conjugacy class C .ga /, where ga 2 GL   0 a 2 GLC.2; R/; a ¤ 0; gN a D a 0 is fixed by translation with  . Indeed, C.ga / D C .ga / D C .ga /. Therefore, the map (A.3) cannot be a covering. It is not even a locally injective map: Indeed, in every

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531

neighborhood of ga there exist elements ga; projecting to matrices of the form  

a 2 GLC.2; R/; ¤ 0: gN a; D a

Then for the conjugacy classes in GL.2; R/, we have C p.ga; / D C p.ga; / and therefore C Pp.ga; / D C Pp.ga; / D C Pp.ga; /. But clearly, ga; and ga; f R/ unless D 0. Therefore, the map (A.3) is a twofold are not conjugate in GL.2; branched covering near ga .

A.2 Conjugacy classes of homomorphisms Let G be a Lie group. Recall that the evaluation map on the generators Hom.Z2 ; G/ ! G  G;

7! . .e1 /; .e2 // ;

identifies the space Hom.Z2 ; G/ of all homomorphisms Z2 ! G homeomorphically with an analytic subvariety of G  G. With respect to this map, the orbits of the conjugation action of G on Hom.Z2 ; G/ correspond to sets of the form C.g1 ; g2 / D f.gg1 g 1 ; gg2 g 1 / j g 2 Gg  G  G: We put X.Z2 ; G/ D Hom.Z2 ; G/=G for the space of conjugacy classes of homomorphisms Z2 ! G (also called the character variety). Given a covering homomorphism p W G 0 ! G there is a natural induced surjective map X.Z2 ; G 0 / ! X.Z2 ; G/: (A.4) Returning to our specific context we introduce the following extension of Proposition A.2: Proposition A.7. The induced map on conjugacy classes of homomorphisms f C.2; R//=GL f C.2; R/ ! Hom.Z2 ; GLC.2; R//= GLC.2; R/ Hom.Z2 ; GL is a covering map. Proof. Let Z.G/ denote the center of G. This representation of Hom.Z2 ; G/ as a subset of G  G gives rise to an action of Z.G/  Z.G/ on Hom.Z2 ; G/ which is determined by ..z1 ; z2 / / .ei / D zi .ei /; where zi 2 Z.G/. Moreover, it factors to an action of Z.G/  Z.G/ on the space of conjugacy classes X.Z2 ; G/. Let   Z.G 0 / denote the kernel of p W G 0 ! G. By the above,    acts on Hom.Z2 ; G 0 /, and the action factors to an action on the space of conjugacy classes

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X.Z2 ; G 0 /, and, as is easily verified, the natural map X.Z2 ; G 0 /=   ! X.Z2 ; G/ which is induced on the quotient is a homeomorphism. Therefore, (A.4) is a covering if and only if    acts discontinuously and freely on X.Z2 ; G 0 /. f C.2; R/ and G D GLC .2; R/. For any Here we consider only the case G 0 D GL 0 0 .g1 ; g2 / 2 G  G , choose open neighborhoods U .gi / as in the proof of Proposition A.2. As follows from this previous proof, the open neighborhood U .g1 ; g2 / D U .g1 /  U .g2 / projects to a fundamental neighborhood CU for the action of    on the set of all G 0 -orbits. Hence,    acts discontinuously on G 0 -orbits. Corollary A.8. The natural map on conjugacy classes of homomorphisms f C.2; R//=GL.2; f R/ ! Hom.Z2 ; GLC.2; R//= GL.2; R/ Hom.Z2 ; GL is a local homeomorphism.

Appendix B. Example of a two-dimensional geometry where hol is not a local homeomorphism Let .X; G/ be the homogeneous geometry which is defined by the natural action of PGL.2; R/ D GL.2; R/=f˙1g on the space X D P.A2  0/ D A2  0=f˙1g; that is, X is the quotient space of R2  f0g by the action of the center fE2 ; E2 g of SL.2; R/. The natural map .A2  0; GL.2; R// ! .P.A2  0/; PGL.2; R// is a covering of geometries in the sense of Definition 3.7. By Lemma 3.22, the induced map on deformation spaces D.T 2 ; A2  0/ ! D.T 2 ; P.A2  0//

(B.1)

is a homeomorphism. We claim that the holonomy for the deformation space D.T 2 ; P.A2  0// is not a local homeomorphism. For this we first recall the .A2  0; GL.2; R//-manifolds H; 2 ;k constructed in Example 4.10 (finite quotients of Hopf tori). Then we observe: Proposition B.1. The holonomy map hol

D.T 2 ; P.A2  0// ! Hom.Z2 ; PGL.2; R//=PGL.2; R/

Chapter 12. The deformation of flat affine structures on the two-torus

533

is a twofold branched covering near the image of a homogeneous flat affine torus H; 2 ;k under the map (B.1). Proof. The commutative diagram (3.8) for the subgeometry takes the form D.T 2 ; A2  0/

hol



 D.T 2 ; P.A2  0//

hol

/ Hom.Z2 ; GLC.2; R//= GL.2; R/  / Hom.Z2 ; PGL.2; R//=PGL.2; R/.

Note that, by Corollary 6.10, the top horizontal map is a local homeomorphism. Furthermore, by Example A.6, the right vertical map is a twofold branched covering near the holonomy homomorphism of every flat affine torus H; 2 ;k . We deduce that the bottom map hol for D.T 2 ; P.A2  0// is locally a twofold branched covering at the images of H; 2 ;k .

References [1]

W. Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Math. 820, Springer, Berlin 1980.

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[10] Y. Benoist, Tores affines. In Crystallographic groups and their generalizations (Kortrijk, 1999), Contemp. Math. 262, Amer. Math. Soc., Providence, RI, 2000, 1–37.

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[11] Y. Benoist and F. Labourie, Sur les espaces homogénes modéles de variétés compactes. Inst. Hautes Études Sci. Publ. Math. 76 (1992), 99–109. [12] J.-P. Benzécri, Variétés localement affines. Séminaire Ehresmann, 1959. [13] J.-P. Benzécri, Sur les variétés localement affines et localement projectives. Bull. Soc. Math. France 88 (1960), 229–332. [14] N. Bergeron and T. Gelander, A note on local rigidity. Geom. Dedicata 107 (2004), 111–131. [15] L. Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume (Erste Abh.). Math. Ann. 70 (1911), 297–336. [16] L. Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abh.). Math. Ann. 72 (1912), 400–412. [17] N. Bourbaki, Topologie générale. Chapitre III, Hermann, Paris 1960. [18] M. Bucher and T. Gelander, Milnor–Wood inequalities for manifolds locally isometric to a product of hyperbolic planes. C. R. Math. Acad. Sci. Paris 346 (2008), no. 11–12, 661–666. [19] M. Burger, A. Iozzi, and A. Wienhard, Higher Teichmüller spaces: from SL.2; R/ to other Lie groups. In In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume IV, EMS Publishing House, Zürich 2014, 539–618. [20] R. D. Canary, D. B. A. Epstein, and P. Green, Notes on notes of Thurston. In Analytical and geometric aspects of hyperbolic space, London Math. Soc. Lecture Note Ser. 111, Cambridge University Press, Cambridge 1984, 3–92. [21] Y. Carrière, Autour de la conjecture de L. Markus sur les variétés affines. Invent. Math. 95 (1989), 743–753. [22] S. Choi and W. M. Goldman, The classification of real projective structures on compact surfaces. Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 2, 161–171. [23] C. J. Earle and J. Eells, A fibre bundle description of Teichmüller theory. J. Differential Geom. 3 (1969), 19–43. [24] D. Ebin, The manifold of Riemannian metrics. In Global analysis, Proc. Sympos. Pure Math. XV, Amer. Math. Soc., Providence, RI, 1970, 11–40. [25] R. Fricke and F. Klein, Vorlesungen über die Theorie der automorphen Funktionen. B.G. Teubner, Leipzig 1897–1912. [26] D. Fried and W. M. Goldman, Three-dimensional affine crystallographic groups. Adv. in Math. 47 (1983), no. 1, 1–49. [27] D. Fried, W. M. Goldman, and M. W. Hirsch, Affine manifolds with nilpotent holonomy. Comment. Math. Helv. 56 (1981), no. 4, 487–523. [28] P. Furness and D. Arrowsmith, Locally symmetric spaces. J. London Math. Soc. (2) 10 (1972), 487–499. [29] P. Furness and D. Arrowsmith, Flat affine Klein bottles. Geom. Dedicata 5 (1976), no. 1, 109–115. [30] W. M. Goldman, Affine structures and projective geometry on manifolds. Undergraduate thesis, Princeton 1977.

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Chapter 13

Higher Teichmüller spaces: from SL.2 ; R/ to other Lie groups Marc Burger, Alessandra Iozzi, and Anna Wienhard

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540

I Teichmüller space and hyperbolic structures 2 Hyperbolic structures and representations . . . . . . . . . . . . . . . 2.1 Hyperbolic structures . . . . . . . . . . . . . . . . . . . . . . . 2.2 Representation varieties . . . . . . . . . . . . . . . . . . . . . 3 Invariants, Milnor’s inequality and Goldman’s theorem . . . . . . . . 3.1 Flat G-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Central extensions . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Description of H2 .1 .S/; Z/, a digression . . . . . . . . . . . 3.4 The Euler class . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Kähler form and Toledo number . . . . . . . . . . . . . . . . . 3.6 Toledo number and first Chern classes . . . . . . . . . . . . . . 3.7 Relations between the various invariants . . . . . . . . . . . . . 3.8 Milnor’s inequality and Goldman’s theorem . . . . . . . . . . . 3.9 An application to Kneser’s theorem . . . . . . . . . . . . . . . 4 Surfaces of finite type and the Euler number . . . . . . . . . . . . . . 4.1 Hyperbolic structures on surfaces of finite type and semiconjugations . . . . . . . . . . . . . . . . . . . . . . 4.2 The bounded Euler class . . . . . . . . . . . . . . . . . . . . . 4.3 Bounded Euler number and bounded Toledo number . . . . . . 4.4 Computations in bounded cohomology . . . . . . . . . . . . . 4.5 Hyperbolic structures and representations: the noncompact case 4.6 Relation with quasimorphisms . . . . . . . . . . . . . . . . . . II 5

542 . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

542 542 546 549 549 550 550 551 552 553 553 554 557 558

. . . . . .

. . . . . .

. . . . . .

558 560 563 567 569 572

575 Higher Teichmüller spaces Maximal representations into Lie groups of Hermitian type . . . . . . . . . 576 5.1 The cohomological framework . . . . . . . . . . . . . . . . . . . . . 576

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5.2 Maximal representations and basic geometric properties . . . . . 5.3 The structure theorem and tube type domains . . . . . . . . . . . 5.4 Tight homomorphisms, triangles and the Hermitian triple product 5.5 Boundary maps, rotation numbers and representation varieties . . 6 Hitchin representations and positive representations . . . . . . . . . . . 6.1 Hitchin representations . . . . . . . . . . . . . . . . . . . . . . . 6.2 Positive representations . . . . . . . . . . . . . . . . . . . . . . 7 Higher Teichmüller spaces – a comparison . . . . . . . . . . . . . . . 7.1 Boundary maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The symplectic group . . . . . . . . . . . . . . . . . . . . . . . 8 Anosov structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Definition, properties and examples . . . . . . . . . . . . . . . . 8.2 Quotients of higher Teichmüller spaces . . . . . . . . . . . . . . 8.3 Geometric structures . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Topological invariants . . . . . . . . . . . . . . . . . . . . . . . 9 Open questions and further directions . . . . . . . . . . . . . . . . . . 9.1 Positivity and other groups . . . . . . . . . . . . . . . . . . . . . 9.2 Coordinates and quantizations for maximal representations . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

581 584 586 590 592 593 594 596 596 598 598 598 601 606 609 611 611 612 612

1 Introduction Let S be a connected surface of finite topological type. The Teichmüller space T .S / is the moduli space of marked complex structures on S. It is isomorphic to the moduli space of marked complete hyperbolic structures on S, sometimes called the Fricke space F .S/. Associating to a hyperbolic structure its holonomy representation naturally embeds the Fricke space F .S / into the variety of representations Hom.1 .S /; PSU.1; 1//= PSU.1; 1/. This realization of the classical Teichmüller space as a subset of the representations variety is the starting point of this chapter. We begin in § 2.1 by defining the space Hyp.S / of hyperbolic structures on S and constructing in some detail the map   ı W Hyp.S/ ! Hom 1 .S /; PSU.1; 1/ ; as well as the embedding

  ı 0 W F .S/ D Diff C 0 .S/nHyp.S/ ! Hom 1 .S /; PSU.1; 1/ = PSU.1; 1/;

where Diff C 0 .S/ is the group of orientation-preserving diffeomorphisms that are homotopic to the identity. Then §§ 2.2 and 3 are devoted to various descriptions of the subset ı.Hyp.S//  Hom.1 .S/; PSU.1; 1//. When S is a compact surface, ı.Hyp.S // is: (1) the set of injective orientation-preserving homomorphisms with discrete image (see Theorem 2.5 and Corollary 2.13);

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(2) identified with one connected component of Hom.1 .S /; PSU.1; 1// (see § 2.2); (3) the (maximal value) level set of numerical invariants described in § 3 (see Theorem 2.2); (4) the solution set of a commutator equation (see (4.1)); (5) characterized in terms of bounded cohomology classes (see Corollary 4.5). When S is a noncompact surface of finite type, the description of ı.Hyp.S // is more involved, and the characterizations (1) and (2) do not hold in this case. In § 4 we define suitable (bounded cohomological) analogues of the numerical invariants described in § 3, which allow us to give in § 4.5 characterizations of ı.Hyp.S // for noncompact surfaces S, generalizing (3), (4) and (5) above. In the second part we ask how much of this “PSU.1; 1/ picture” generalizes to an arbitrary Lie group G. We discuss two classes of (semi)simple Lie groups for which one can make this question precise by defining, in very different ways, components (or specific subsets when S is not compact) of Hom.1 .S /; G/ that play the role of Teichmüller space. The terminology higher Teichmüller spaces, coined by Fock and Goncharov, has now come to mean subsets of Hom.1 .S/; G/, where G is a simple Lie group, which share essential geometric and algebraic properties with classical Teichmüller space considered as a subset of Hom.1 .S/; PSU.1; 1//. Up to now higher Teichmüller spaces are defined for two classes of Lie groups, namely for split real simple Lie groups, e.g. SL.n; R/, Sp.2n; R/, SO.n; n C 1/ or SO.n; n/ and for Lie groups of Hermitian type, e.g. Sp.2n; R/, SO.2; n/, SU.p; q/ or SO .2n/. The invariants defined in § 3 and § 4 can be defined for homomorphisms from 1 .S / with values in any Lie group G, but when G is a Lie group of Hermitian type these invariants are particularly meaningful. We describe how the basic objects available in the case of PSU.1; 1/ can be generalized to higher rank in § 5. Considering the maximal value level set of the numerical invariant thus constructed leads us to consider the space of maximal representations     Hommax 1 .S/; G  Hom 1 .S /; G ; some of whose properties are discussed in § 5. In particular we state a result (“structure theorem”) that describes the Zariski closure in G of the image of a maximal representation; a major part of §§ 5.4 and 5.5 then offers a guided tour showing the various aspects of the proof of the structure theorem. The space of maximal representations is an example of a higher Teichmüller space. Hitchin components and spaces of positive representations are other examples of a higher Teichmüller space, defined when G is a split real Lie group. We review the definition of these spaces shortly (§ 6), and then discuss important structures underlying both families of higher Teichmüller spaces (§ 7). In the case of compact surfaces the quest for common structures leads us to consider the concept of Anosov structures (§ 8). This more general notion provides an important framework within which one can start to understand the geometric significance of higher Teichmüller spaces, their

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quotients by the mapping class group (§ 8.2), the geometric structures parametrized by higher Teichmüller spaces (§ 8.3), as well as further topological invariants (§ 8.4). In § 9 we conclude by mentioning further directions of study. There are many aspects of higher Teichmüller spaces that we do not touch upon for lack of space and expertise. One of them is the huge body of work studying maximal representations from the point of view of Higgs bundles; such techniques give in particular precise information about the number of connected components of maximal representations, as well as information on the homotopy type of those components (see for example [6], [11], [12], [10], [41], [42], [52]). As a guide to the reader, we mention that the first part of this chapter and the discussion of maximal representations is very descriptive and should be read linearly, while starting from the definition of Hitchin representations the chapter is a pure survey. It was completed and submitted end of 2009; as a result the survey does not take into account later developments. Acknowledgments. This work was partially supported by the Swiss National Science Foundation project 2000021-127016/2 (A.I.) and by NSF Grant No. DMS-0803216 and NSF CAREER Grant No. DMS-0846408 (A.W.). We thank A. Papadopoulos for undertaking this project. We thank also O. Guichard and T. Hartnick for carefully reading a preliminary version of this chapter providing many helpful comments, W. Goldman for many bibliographical comments and F. Labourie for positive feedback. Our thanks go also to D. Toledo and N. A’Campo for useful discussions on this general topic over the years and to T. Delzant for helpful comments concerning central extensions of surface groups. Finally, we thank the referee for helpful comments.

Part I Teichmüller space and hyperbolic structures 2 Hyperbolic structures and representations 2.1 Hyperbolic structures In this section we review briefly how one associates to a hyperbolic structure on a surface a homomorphism of its fundamental group into the group of orientationpreserving isometries of the Poincaré disk, and how an appropriate quotient of the set of hyperbolic structures injects into the representation variety. Let D D fz 2 C W jzj < 1g be the unit disk endowed with the Poincaré metric and let G WD PSU.1; 1/ D SU.1; 1/=f˙Idg denote the quotient of SU.1; 1/ y D C [ f1g by linear fractional transformations by its center. The group G acts on C preserving D and hence can be identified with the group of orientation-preserving isometries of D. 4jdzj2 , .1jzj2 /2

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Given a surface S, that is, a two-dimensional smooth manifold,1 a hyperbolic metric on S is a Riemannian metric with sectional curvature 1. A .G; D/-structure on S is an atlas on S consisting of charts taking values in D, whose change of charts are locally restrictions of elements of G, [95]. Assuming from now on that S is orientable, we observe that by the local version of Cartan’s theorem, an orientation together with a hyperbolic metric on S determines a .G; D/-structure on S (the converse is also true and straightforward). Also, the hyperbolic metric is complete if and only if the same is true for the corresponding .G; D/-structure, i.e. if the developing map Sz ! D is a diffeomorphism. The group Diff.S/, and hence its subgroup Diff C .S / consisting of orientationpreserving diffeomorphisms of S , act on the set Hyp.S / of complete hyperbolic metrics on S in a contravariant way. In the sequel let Sz D D be a smooth oriented disk with a basepoint  2 D and let us fix once and for all a base tangent vector v 2 T D, v ¤ 0. By the correspondence between hyperbolic metrics and .G; D/-structures, let us also consider, for every h 2 Hyp.D/, the unique orientation-preserving isometry fh W .D; / ! .D; 0/ with dfh .v/ 2 RC e, where e D 1 2 C. If ' 2 Diff C .D/, then for any h 2 Hyp.D/, ' is by definition an orientation-preserving isometry between the hyperbolic metrics '  .h/ and h. Therefore c.'; h/ WD fh B ' B f'1  .h/ is an element of G. In this way we obtain a map c W Diff C .D/  Hyp.D/ ! G that verifies the cocycle relation

  c.'1 '2 ; h/ D c.'1 ; h/c '2 ; '1 .h/ :

Let now .S; / be a connected oriented surface with base point  and assume that z / D .D; /, let p W D ! S be the canonical projection and Hyp.S / ¤ ;. Let .S;  D fT W  2 1 .S; /g < Diff C .D/ the group of covering transformations. Then the pullback via p gives a bijection between Hyp.S/ and the set Hyp.D/ of -invariant elements in Hyp.D/. Furthermore, it follows from the cocycle identity that, for every h 2 Hyp.D/ , the map h W 1 .S; / ! G;

 7! c.T ; h/;

is a homomorphism with respect to which the isometry fh is equivariant. Thus we obtain the map ı, assigning to a hyperbolic structure its holonomy homomorphism   ı W Hyp.S/ ! Hom 1 .S; /; G ; h 7! p .h/ ; which has certain important equivariance properties that we now explain. 1 Note

that all manifolds here are without boundary. In particular a compact surface is necessarily closed.

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To this end, let N C be the normalizer of  in Diff C .D/. Then we have the diagram with exact line feg

/

/ NC



/ Diff C .S /

/ feg

a





 Aut 1 .S; / ; where  associates to every ' 2 N C the corresponding diffeomorphism of S obtained by observing that ' permutes the fibers of p; the fact that  is surjective follows from covering theory. The homomorphism a is the one associating to ' the automorphism a' of , or rather of 1 .S; /, obtained by conjugation. With these definitions, a computation gives   '  .h/ ./ D c.'; h/1 h a' . / c.'; h/; (2.1) for every ' 2 N C , h 2 Hyp.D/ and  2 . In view of (2.1), it is important to determine those ' 2 N C such that a' is an inner automorphism of . Let NiC be the subgroup consisting of all such diffeomorphisms. Then we have: Lemma 2.1. The map  induces an isomorphism nNiC Š Diff C 0 .S /; C where Diff C 0 .S/ is the subgroup of Diff .S / consisting of those diffeomorphisms that are homotopic to the identity.

Proof. If f W S ! S is homotopic to the identity, then by covering theory the conjugation of  by any lift fQ of f gives an inner automorphism of . Conversely, if 'T ' 1 D T1 for some  and all  , then the diffeomorphism 1 T ' W D ! D commutes with the -action on D; if we fix h 2 Hyp.D/ then the geodesic homotopy from T1 ' to IdD is -equivariant and hence descends to a homotopy between .T1 '/ D .'/ and IdS . Thus combining the inverse of  with a we obtain an injective homomorphism   ˛ / C Out 1 .S; / Map.S/ WD Diff C 0 .S/n Diff .S / of the mapping class group Map.S/ of S into the group of outer automorphisms of 1 .S /. It follows then from (2.1) that the map associating to h 2 Hyp.S / the class of the homomorphism Œp .h/  and taking values in the quotient Hom.1 .S; /; G/=G by the G-conjugation action on the target is invariant under the Diff C 0 .S /-action so C that finally we obtain a map from the Fricke space F .S / D Diff 0 .S /nHyp.S / to the representation variety   ı 0 W Diff C Œh 7! Œp .h/ ; 0 .S/nHyp.S/ ! Hom 1 .S; /; G =G;

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which is ˛-equivariant. Proposition 2.2. If S is a connected oriented surface admitting a complete hyperbolic structure, then ı 0 is injective. Proof. If h1 ; h2 2 Hyp.D/ are such that h1 and h2 are conjugate by g 2 G, then gfh1 is an orientation-preserving diffeomorit follows from the definitions that fh1 2 phism sending h1 to h2 , which furthermore is -equivariant; by the argument used in gfh1 / 2 Diff C Lemma 2.1, we get that .fh1 0 .S /. 2 We now describe the image of the homomorphism ˛ and of the map ı 0 in the case in which S is a compact oriented surface of genus g  2. This latter condition guarantees that the classifying map S ! B1 .S; /

(2.2)

is a homotopy equivalence; we use this fact to equip H2 .1 .S; /; Z/ with the canonical generator, image of the fundamental class ŒS  via the isomorphism H2 .S; Z/ ! H2 .1 .S; /; Z/ induced by (2.2). An isomorphism between the fundamental groups of two compact oriented surfaces S1 and S2 is said to be orientation-preserving if the generator of H2 .1 .S1 ; /; Z/ is mapped to the generator of H2 .1 .S2 ; /; Z/. Theorem 2.3. Let S1 and S2 be compact oriented surfaces of genus g  1. Then any orientation-preserving isomorphism 1 .S1 ; / ! 1 .S2 ; / is induced by an orientation-preserving diffeomorphism S1 ! S2 . Let us denote by AutC .1 .S; // the group of the orientation-preserving automorphisms of a compact orientable surface S of genus g  1, and by OutC .1 .S; // its quotient by the group of inner automorphisms. From Theorem 2.3 we conclude: Corollary 2.4 (Dehn–Nielsen–Baer Theorem, see [35] for a proof). If S is a compact orientable surface of genus g  2, the map   C ˛ W Map.S/ D Diff C .S/= Diff C 1 .S; / 0 .S / ! Out is an isomorphism. Let now Homd;i denote the subset of Hom consisting of all injective homomorphisms with discrete image. The following classical identification of the image of ı uses the Nielsen realization: Theorem 2.5. If S is a compact orientable surface of genus g  1, then the image of ı consists precisely of ˚     2 Homd;i 1 .S; /; G/ W im nD is compact and  is orientation-preserving :

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Remark 2.6. Since 1 .S/ is the fundamental group of a compact surface and  is a discrete embedding, the quotient im nD is automatically compact. Here, we include this property explicitly in order to stress the similarity with the definition of Hom0 .; G/ in § 2.2. As above, the representation  is orientation-preserving if the induced map  maps the generator of H2 .1 .S; /; Z/ to the generator of H2 .1 .im nD; /; Z/. Proof. Given h 2 Hyp.S/, it is immediate, by using fh , that h belongs to the above set. Conversely, apply Nielsen realization to the orientation-preserving isomorphism  to get an orientation-preserving diffeomorphism f W S ! im nD; if h D f  .hP /, where hP is the Poincaré metric on im nD, then one verifies that Œ D Œh . Remark 2.7. Contrary to what happens in the compact case, if S is a noncompact orientable surface of negative Euler characteristic, the inclusion       x  Homd;i 1 .S; /; PSU.1; 1/ ; ı Hyp.S/ [ ı Hyp.S/ where Sx denotes the surface S with the opposite orientation, is always proper. In fact, if  W 1 .S; / ! G is just discrete and injective, the surfaces im nD and S need not be diffeomorphic, although they have the “same” fundamental group. For example, the once-punctured torus and the thrice-punctured sphere have isomorphic fundamental groups F2 and admit complete hyperbolic structures. We will see in § 4 one way to remedy this problem.

2.2 Representation varieties In this section we review some basic properties of the set of discrete and faithful representations in Hom.; G/ in the general context of a finitely generated group  and a connected reductive Lie group G. One way to approach the problem of determining the image of Hyp.S / under the map ı is to equip Hom.; G/ with a topology. Quite generally if  is a discrete group and G is a topological group, Hom.; G/ inherits the topology of the product space G  . In case  is finitely generated with finite generating set F  , let p W FjF j !  be the corresponding presentation and R a set of generators of the relators ker p. Since every r 2 R is a word in FjF j , it determines a product map mr W G F ! G by evaluation on G. The map   Hom.; G/ ! G F ;  7! .s/ s2F ; identifies the topological space Hom.; G/ with the closed subset \r2R m1 r .e/  G F . In particular, Hom.; G/ is locally compact if G is so, and a real algebraic set if G is a real algebraic group. We record the following

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Proposition 2.8 ([4], [99]). If  is finitely generated and G is a real algebraic group, then Hom.; G/ has finitely many connected components and each of them is a real semialgebraic set, that is, it is defined by a finite number of polynomial equations and inequalities. Remark 2.9. Proposition 2.8 fails if G is not algebraic. An example for this, given in [50], is the quotient of the three-dimensional Heisenberg group by a cyclic central subgroup, where a simple obstruction class detects infinitely many connected components in the representation variety. In order to proceed further we assume that  is finitely generated, G is a Lie group and introduce (see [51], [98]) the following subset of Hom.; G/: Hom0 .; G/ D f 2 Homd;i .; G/ such that ./nG is compactg; where, as in § 2.1, Homd;i refers to the set of injective homomorphisms with discrete image so that Hom0 .; G/  Homd;i .; G/  Hom.; G/. The first result on the topology of Homd;i .; G/ requires a hypothesis on : Definition 2.10. We say that  has property .H/ if every normal nilpotent subgroup of  is finite. Observe that this condition is fulfilled by every nonabelian free group and every fundamental group of a compact surface of genus g  2. With this we can now state the following Theorem 2.11 ([51]). Let  be a finitely generated group with property .H/ and G a connected Lie group. Then Homd;i .; G/ is closed in Hom.; G/. Proof. The essential ingredient is the theorem of Kazhdan–Margulis–Zassenhaus ([91], Theorem 8.16) saying that there exists an open neighborhood U  G of e such that whenever ƒ < G is a discrete subgroup, then U \ ƒ is contained in a connected nilpotent group. We fix now such an open neighborhood and assume in addition that it does not contain any nontrivial subgroup of G; let also ` be an upper bound on the degree of nilpotency of connected nilpotent Lie subgroups of G. Let now fn gn1 be a sequence in Homd;i .; G/ with limit . We show that  is injective. For every finite set E  ker , we have that n .E/  U for n large, which, by Theorem 8.16 of [91], implies that for all k  `, the k-th iterated commutator of n .E/ is trivial, and the same holds therefore for E since n is injective. As a result, ker  is nilpotent and hence, by property .H/, finite; thus n .ker /  U for large n, which, by the choice of U, implies that n .ker / D e and hence that  is injective. We prove now that  is discrete. To this end, let L D ./ be the closure of ./; then L is a Lie subgroup of G and L0 is open in L. Let V be an open neighborhood of the identity on L0 with V  U; then ./ \ V is dense in V and V generates

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L0 , from which we conclude that ./ \ V generates a dense subgroup of L0 . For every finite set F 0   with .S/  V  U we have that n .S /  U for n large, which implies as before that for all k  l the k-th iterated commutator of F 0 , and hence of .F 0 /, is trivial; thus L0 is nilpotent and so is 1 .L0 / since  is injective. But then 1 .L0 / is finite and hence L0 D feg, which shows that ./ is discrete and concludes the proof. Next we turn to the set Hom0 .; G/ of faithful, discrete and cocompact realizations of  in G; this set was considered by A. Weil as a tool in his celebrated local rigidity theorem in which the following general result played an important role. Theorem 2.12 ([91]). Assume that  is finitely generated and that G is a connected Lie group. Then Hom0 .; G/ is an open subset of G. There are by now several approaches available: we refer to the paper by Bergeron and Gelander [5], where the geometric approach due essentially to Ehresmann and Thurston [95] is explained (see also [47], [25], [79]); this approach, based on a reformulation of the problem in terms of variations of .G; X /-structures leads to a more general stability result also valid for manifolds with boundary. We content ourselves with noticing the following consequence: Corollary 2.13. Assume that  is finitely generated, torsion-free and has property .H/. Assume that G is a connected reductive Lie group and that there exists a discrete, injective and cocompact realization of  in G. Then Hom0 .; G/ D Homd;i .; G/ and both sets are therefore open and closed, in particular a union of connected components of Hom.; G/. Proof. Let 0 2 Hom0 .; G/ and let K < G be a maximal compact subgroup; since by the Iwasawa decomposition X WD G=K is contractible and since 0 ./ acts on X as a group of covering transformations with compact quotient, we have that, for n D dim X, Hn .; R/ D Hn .0 ./; R/ ¤ 0. Therefore, if  W  ! G is any discrete injective embedding, we have that Hn ../nX; R/ does not vanish and hence ./nX is compact, thus implying that  2 Hom0 .; G/. Applying the preceding discussion to our compact surface S of genus g  2, we conclude using Theorem 2.5 and Corollary 2.13 that     ı Hyp.S/  Hom 1 .S; /; PSU.1; 1/ is a union of components of the representation variety Hom.1 .S; /; PSU.1; 1//.

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3 Invariants, Milnor’s inequality and Goldman’s theorem In this section we will discuss various aspects of a fundamental invariant attached to a representation  W 1 .S; / ! G, where G D PSU.1; 1/, namely the Euler number of . This leads to a quite different way of characterizing the image of   ı W Hyp.S/ ! Hom 1 .S; /; G in the case in which S is compact. This invariant can also be defined for targets belonging to a large class of Lie groups G, and this leads to natural generalizations of Teichmüller space (see the discussion in § 5 and § 7). In the sequel, S denotes a compact surface of genus g  2 and fixed orientation. z We drop moreover the basepoint in the notation 1 .S; / and we set D D S.

3.1 Flat G -bundles Given a connected Lie group G and a homomorphism  W 1 .S / ! G, we obtain, in the notation of § 2.1, a proper action without fixed points on D  G by    .x; g/ D T x; . /g whose quotient 1 .S/n.D  G/ is the total space G./ of a flat principal (right) G-bundle over S, where the projection map comes from the projection D  G ! D on the first factor. Given a G-bundle E over S, the first obstruction to find a continuous section of E ! S lies in H2 .S; 1 .G//. Namely, let K be a triangulation of S ; choose preimages in E for the vertices of K and extend this section over the 0-skeleton of K to the 1skeleton by using that G is connected; for each 2-simplex  we have thus a section over its boundary @. Using the flat connection, this section of E over @ can be deformed into a loop lying in a single fixed fiber; identifying this fiber with G we get for every  a free homotopy class of loops in G and hence a well-defined element c./ 2 1 .G/, since the latter is abelian. The map c is a simplicial 2-cocycle on K with values in 1 .G/ and hence defines an element in H2 .S; 1 .G// depending only on . In this way we obtain a map     o2 W Hom 1 .S/; G ! H2 S; 1 .G/ that assigns to  the obstruction o2 ./ 2 H2 .S; 1 .G// of the flat G-bundle G./. An important observation is that if 1 and 2 lie in the same component of Hom.1 .S /; G/, then the associated G-bundles G.1 / and G.2 / are isomorphic. As a result, the invariant o2 is constant on connected components of Hom.1 .S /; G/. (See also [46] for a discussion of characteristic classes and representations.)

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3.2 Central extensions An invariant closely related to the one defined above is obtained by considering the central extension of G given by the universal covering p

/G z

/ 1 .G/

feg

/G

/ feg;

where the neutral element is taken as basepoint. A homomorphism  W 1 .S/ ! G then gives a central extension  of 1 .S / by 1 .G/ in the familiar way ˚  z W . / D p.g/ :  D .; g/ 2 1 .S/  G Observing now that the isomorphism classes of central extensions of 1 .S / by 1 .G/ are classified by H2 .1 .S/; 1 .G//, we get a map     c2 W Hom 1 .S/; G ! H2 1 .S /; 1 .G/ : So far the discussion in §§ 3.1 and 3.2 applies to any connected Lie group G. In case G D PSU.1; 1/, we get a canonical generator of 1 .G/ from the orientation of D  C; by considering the loop  is  e 0 Œ0; 1 ! PSU.1; 1/; s 7! ; 0 e is we identify 1 .PSU.1; 1// with Z; we will denote by t 2 1 .PSU.1; 1// the image of 1 2 Z.

3.3 Description of H2 .1 .S /; Z/, a digression Let g  1 be the genus of S. Then 1 .S/ admits as presentation D

a1 ; b1 ; : : : ; ag ; bg W

g Q

E Œai ; bi  D e :

iD1

The orientation of S is built in, in that, when drawing the lifts to Sz of the loop a1 b1 a11 b11 a2 b2 : : : ag1 bg1 , one gets a 4g-gon whose boundary is traveled through in the positive sense. Now define D E g Q xg WD A1 ; B1 ; : : : ; Ag ; Bg ; z W ŒAi ; Bi  D z and Œz; Ai  D Œz; Bi  D e :  iD1

xg surjects onto 1 .S/ with kernel the cyclic subgroup generated by z, This group  which incidentally is central. In order to see that z has infinite order, observe first that

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x1 is isomorphic to the integer Heisenberg group  80 9 1 < 1 x z = @0 1 y A W x; y; z 2 Z : ; 0 0 1 by

0 1 1 1 0 A1 7! @0 1 0A ; 0 0 1

0 1 1 0 0 B1 7! @0 1 1A ; 0 0 1

0

1 1 0 1 z! 7 @0 1 0A : 0 0 1

Then conclude by considering the surjection xg !  x1  xg gives a central extension of obtained by sending Ai and Bi to e for i  2. Thus  xg  its image in H2 .1 .Sg /; Z/, we have the following 1 .S / by Z; denoting by Πxg . Proposition 3.1. H2 .1 .Sg /; Z/ D ZΠIn fact if 0

/Z

/ƒ

i

/ 1 .Sg /

/ feg

Q is any central extension by Z, take lifts ˛j ; ˇj 2 ƒ of aj ; bj : then jgD1 Œ˛j ; ˇj  is independent of all choices and the image under i of a well-defined n 2 Z. Using the Baer product of extensions [1] one shows, by induction on n, that xg : Œƒ D n Œ Now we come back to the invariant associated in § 3.2 to  2 Hom.1 .S /; G/ and use the identification 1 .G/ ! Z; t 7! 1; to get c2 ./ 2 H2 .1 .S/; Z/. In terms of central extension we have then that xg ; c2 ./ D z2 ./Œ where z2 ./ 2 Z is defined by the formula g Y

Œ˛i ; ˇi  D t z2 ./ ;

iD1

z are lifts of .a1 /; .b1 /; : : : ; .ag /; .bg /. where ˛1 ; ˇ1 ; : : : ; ˛g ; ˇg 2 G

3.4 The Euler class The following discussion is specific to the case where G D PSU.1; 1/; it takes as point of departure the observation that the action of G by homographies on D gives an action

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on the circle @D bounding D, by orientation-preserving homeomorphisms. It will become apparent that considering homomorphisms with values in G as homomorphisms with target the group HomeoC .S 1 / of orientation-preserving homeomorphisms of the circle gives additional flexibility. Let us take the quotient ZnR of R by the group generated by the integer translations T .x/ D x C 1 of the real line, and consider, as model of S 1 , HZC .R/ D ff W R ! R W increasing homeomorphisms commuting with T g: We then obtain the central extension 0

/Z

/ H C .R/ Z

p

/ HomeoC .S 1 /

/0

that realizes HZC .R/ as universal covering of the group HomeoC .S 1 /, the latter being endowed with the compact open topology. One obtains a section of p by associating to every f 2 HomeoC .S 1 / the unique lift fN W R ! R with 0  fN.0/ < 1. The extent to which f 7! fN is not a homomorphism is measured by an integral 2-cocycle given by fN B gN D f B g B T .f;g/ ; where T is the image in HZC .R/ of the generator 1 2 Z. The Euler class is then the cohomology class e 2 H2 .HomeoC .S 1 /; Z/ defined by . Definition 3.2. The Euler number e./ of a representation  W 1 .S/ ! HomeoC .S 1 / is the integer h .e/; ŒSi obtained by evaluation of the pullback  .e/ 2 H2 .1 .S /; Z/ of e on the fundamental class ŒS, or rather on its image under the isomorphism   H2 .S; Z/ ! H2 1 .S /; Z considered in (2.2).

3.5 Kähler form and Toledo number Contrary to § 3.4, the viewpoint we present here emphasizes the fact that the Poincaré disk D is an instance of a Hermitian symmetric space with G-invariant Kähler form !D WD

dz ^ d zN ; .1  jzj2 /2

where G D PSU.1; 1/. Given a homomorphism  W 1 .S / ! G, consider then the bundle with total space the quotient D./ WD 1 .S /n.D D/ of D D by the properly discontinuous and fixed point free action .x; z/ WD .T x; . /z/, and with basis S D 1 .S /nD. Since the typical fiber D is contractible, one can construct, adapting the procedure described in § 3.1, a continuous and even a smooth section. Equivalently there is a smooth equivariant map F W D ! D. As a result, the pullback F  .!D / is

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a 1 .S /-invariant 2-form on D that gives a 2-form on S denoted again, with a slight abuse of notation, by F  .!D /. The Toledo number T./ of the representation  is then Z 1 T./ WD F  .!D /: 2 S Recall that we have fixed an orientation on S once and for all. Remark 3.3. One verifies, using again geodesic homotopy, that any two -equivariant smooth maps D ! D are homotopic and hence, by Stokes’ theorem, one concludes 2 .S; R/ is independent of F . This that the de Rham cohomology class ŒF  .!D / 2 HdR shows that T./ is independent of the choice of F .

3.6 Toledo number and first Chern classes Let L ! D be a Hermitian complex line bundle over the Poincaré disk D and G 0 a finite covering of PSU.1; 1/ acting by bundle isomorphisms on L; then the curvature form L is a G 0 -invariant 2-form on D. Given a representation  W 1 .S / ! G 0 and a smooth equivariant map F W D ! D, 1 .S / acts by bundle automorphisms on F  L ! D and, by passing to the quotient, we get a complex line bundle L./ over 1 S . Then 2{ F  L descends to a 2-form !L./ on S, which, by Chern–Weil theory, represents the first Chern class of L./, i.e. Z   !L./ 2 Z: (3.1) c1 L./ D S

Applying this to specific line bundles we obtain integrality properties for the Toledo number. Namely, let  ! D  CP 1 be the restriction of the tautological bundle over 1 !D and  2 D 1i !D . The group PSU.1; 1/ CP 1 and  2 be its square. Then  D 2i 2 acts by isomorphisms on  , and (3.1) implies Z Z 1  F !D D  !2 D c1 .2 / 2 Z: T ./ D 2 S S The group SU.1; 1/ acts naturally on , so for representations  W 1 .S / ! SU.1; 1/ the relation in (3.1) gives Z Z 1  F !D D 2 ! D 2c1 . / 2 2  Z: T ./ D 2 S S In particular, a representation  W 1 .S/ ! PSU.1; 1/ lifts to SU.1; 1/ if and only if its Toledo number is divisible by 2.

3.7 Relations between the various invariants For G D PSU.1; 1/, we identify in the sequel 1 .G/ with Z as described in § 3.2 and obtain for a representation  2 Hom.1 .S /; G/ the obstruction class o2 ./ 2

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H2 .S; Z/ and the class c2 ./ 2 H2 .1 .S/; Z/; using the specific description of the xg , we get the invariant z2 ./ 2 Z by setting latter in terms of central extensions as ZŒ x c2 ./ D z2 ./Œg . Then ho2 ./; ŒSi D z2 ./; (3.2) (see [84], Lemma 2, and [102]). Turning to the Euler class, we observe that the injection PSU.1; 1/ ,! HomeoC .S 1 / is a homotopy equivalence as both groups retract on the (common) group of rotations. Therefore the restriction ejPSU.1;1/ 2 H2 .PSU.1; 1/; Z/ classifies the universal covering of PSU.1; 1/ and hence for  W 1 .S/ ! PSU.1; 1/ we have

xg ;  .e/ D z2 ./Œ

which implies that

e./ D h .e/; ŒS i D z2 ./:

(3.3)

To relate the previous invariant to the Toledo number we will recall the very general principle that invariant forms on a symmetric space form a complex, with 0 as derivative, which equals the continuous cohomology of the connected group of isometries. In our special case of the Poincaré disk, this takes the form

2 .D/G Š Hc2 .G; R/; where, given !D , we get a continuous cocycle Z 1 !D ; c.g1 ; g2 / WD 2 .0;g1 .0/;g1 g2 .0// where .0; g1 .0/; g1 g2 .0// denotes the oriented geodesic triangle having vertices at the points 0; g1 .0/; g1 g2 .0/ 2 D. We call the resulting class G the Kähler class. In fact, it is not difficult to show that under the change of coefficients Hc2 .G; Z/ ! Hc2 .G; R/  the Euler class e goes to the Kähler class G . If  2 1 .S /; G/, this implies that Z 1  F  .!D / D T./: (3.4) e./ D h .e/; ŒSi D 2 S

3.8 Milnor’s inequality and Goldman’s theorem In his seminal paper [84], J. Milnor treated the problem of characterizing those classes in H2 .S; Z/ that are Euler classes of flat principal GLC 2 -bundles. The fact that, in general, there are restrictions on the characteristic classes of flat principal G-bundles and in particular on o2 ./ 2 H2 .S; 1 .G// comes from the following observation: o2 is constant on connected components of Hom.1 .S /; G/ and the latter is a real algebraic set when G is a real algebraic group, thus possesses only finitely many

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connected components (see Proposition 2.8). To get explicit restrictions, however, is not a trivial matter. In the case of G D PSU.1; 1/-bundles this restriction, known as the Milnor–Wood inequality, is the following: Theorem 3.4 ([84], [102]). Let  2 Hom.1 .S /; G/ and let g be the genus of S. Then ˇ˝ ˛ˇ ˇ o2 ./; ŒS ˇ  2g  2: In light of subsequent generalizations of this inequality it is instructive to give an outline of the original arguments. Consider the retraction  ³ ²  {s 0 e W s 2 R=Z r W PSU.1; 1/ ! K D ˙ 0 e {s given by decomposing g D r.g/h.g/ as a product of a rotation r.g/ with a Hermitian matrix h.g/. Now lift r to the universal covering

C

rQ W PSU.1; 1/ ! R in such a way that r.e/ Q D 0. Then: n Q where t is the generator of 1 .PSU.1; 1//; (1) r.t Q g/ D n C r.g/, (2) r.g Q 1 / D r.g/; Q ˇ ˇ ˇ (3) r.ab/ Q  r.a/ Q  r.b/ Q ˇ
0 such that • for any e in   .E C /m and for any t > 0 one has k' t ek' t m  A exp.at /kekm ;   • for any e in  .E /m and for any t > 0 one has k't ek't m  A exp.at /kekm ; where k  k is any continuous norm on O./. Remark 8.2. The definition of Anosov structure does not depend on the choice of the hyperbolic metric on S. Definition 8.3. A representation  W 1 .S/ ! G is said to be a .PC ; P /-Anosov representation if O./ carries an Anosov structure. The conditions on  are equivalent to requiring that  .T 1 S / is a hyberbolic set for the flow ' t . Stability of hyperbolic sets implies stability for .PC ; P /-Anosov representations: Proposition 8.4 ([70]). The set of .PC ; P /-Anosov representations is open in Hom.1 .S /; G/: Since we fixed a hyperbolic structure on S we can equivariantly identify the boundary @1 .S / with the boundary S 1 D @D of the Poincaré disk as we did in § 4.1. Proposition 8.5 ([70]). To every Anosov representation  W 1 .S / ! G there are associated continuous -equivariant boundary maps ˙ W S 1 ! G=P˙ ; with the property that for all distinct t; t 0 2 S 1 , we have .C .t /;  .t 0 // 2 O. Moreover, for every element  2 1 .S/  f1g with fixed points  ˙ 2 S 1 , the point ˙ . C / is the unique attracting fixed point of ./ in G=P˙ and ˙ .  / is the unique repelling fixed point of ./ in G=P˙ .

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Sketch of proof. Since the existence of these boundary maps plays an important role in some results discussed above, let us sketch how these maps are obtained. Recall that T 1 Sz is naturally identified with the space of positively oriented triples in S 1 , via the map T 1 Sz ! .S 1 /.3C / ; v 7! .vC ; v0 ; v /; where v˙ are the endpoints at ˙1 of the unique geodesic gv determined by v and v0 is the unique point mapped to the basepoint of v under the orthogonal projection to the geodesic gv and such that .vC ; v0 ; v / is positively oriented. The existence of a continuous section  of O./ is equivalent to the existence of a -equivariant continuous map F W T 1 Sz ! O. The Anosov condition (1) on  is equivalent to F being ' t -invariant. In particular, the map F only depends of .vC ; v / 2 .S 1  S 1 /  diag DW .S 1 /.2/ . Thus we have a map F D .C ;  / W .S 1 /.2/ ! G=PC  G=P : It is not difficult to see that due to the contraction properties of the geodesic flow (see Anosov condition (2)) the map C .vC ; v / only depends on vC and  .vC ; v / only depends on v , and that ˙ satisfy the above properties. The property of a representation  W 1 .S / ! G being .PC ; P /-Anosov is indeed (almost) equivalent to the existence of such continuous boundary maps. Proposition 8.6 ([57]). Let  W 1 .S/ ! G be a Zariski dense representation and assume that there exists -equivariant continuous boundary maps ˙ W S 1 ! G=P˙ such that (1) for all t; t 0 2 S 1 distinct, we have .C .t /;  .t 0 // 2 O, and (2) for all t 2 S 1 , the two parabolic subgroups stabilizing C .t / and  .t / contain a common Borel subgroup. Then  is a .PC ; P /-Anosov representation. The Anosov section  can be easily reconstructed from the boundary map using the identification T 1 Sz Š .S 1 /.3C / . Let us list several consequences of the existence of such boundary maps; for proofs see [70] and [57]. (1) The representation  is faithful with discrete image. (2) For every  2 1 .S/  f1g the holonomy . / is conjugate to a (contracting) element in H D PC \ P . (3) The orbit map 1 .S/ ! X;  7! . /x0 for some x0 2 X is a quasiisometric embedding with respect to the word metric on 1 .S / and any G-invariant metric on the symmetric space X D G=K. (4) The representation  is well-displacing.

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The concepts of welldisplacing representations and quasiisometric embeddings are discussed in more detail in § 8.2.1. We already mentioned that representations in higher Teichmüller spaces areAnosov representations. We now describe this in a little more detail using the existence of special boundary maps discussed in § 7.1. We want to emphasize that this is not the order in which results are proved. In many cases the proof of the existence of a continuous boundary map is intertwined with the proof of the Anosov property. For example one constructs first a not necessarily continuous boundary map, and then establishes the contraction properties for a (not continuous) section constructed out of this boundary map. Then using the contraction property one can conclude that the map is indeed continuous, hence defining a genuine Anosov section. (For an illustration of this strategy for maximal representations into the symplectic group, we refer the reader to [15].) The special “positivity conditions” for the map are usually derived from more specific properties of the representations. (1) Hitchin representations into PSL.n; R/, PSp.2n; R/, PO.n; n C 1/ are Anosov representations with P˙ being minimal parabolic subgroups [70]. Using the characterization of Hitchin representations via the existence of convex curves, we are given ' W S 1 ! RP n1 and we can take '  W S 1 ! .RP n1 / to be the dual curve, i.e. '  .t/ is the unique osculating hyperplane of the curve ' containing '.t/. We set  C D ',   D '  ; then . C ;   / W S 1 ! RP n1  .RP n1 / satisfies the hypothesis of Proposition 8.6, thus  is Anosov with respect to the parabolic subgroup stabilizing a line in Rn . In order to see that Hitchin representations are actually Anosov with respect to the minimal parabolic subgroup, note that for any point on the convex curve we can consider the osculating flag and obtain maps ˙ D  W S 1 ! G=Pmin . The convexity of ' implies the transversality condition on ˙ (see [70], Chapter 5) (2) Maximal representations are Anosov representation with P˙ being stabilizers of points in the Shilov boundary S{ of the Hermitian symmetric space [15], [16]. This means in particular that in this case the boundary maps ˙ D ' W S 1 ! S{ (Theorem 5.29) sending positively oriented triples to maximal triples are continuous. Since .C ;  / satisfies the transversality conditions required in Proposition 8.6, maximal representations are .PC ; P /-Anosov.

8.2 Quotients of higher Teichmüller spaces 8.2.1 Action of the mapping class group. The automorphism groups of 1 .S / and of G act naturally on Hom.1 .S/; G/,       Aut 1 .S/  Aut.G/  Hom 1 .S /; G ! Hom 1 .S /; G ; . ; ˛; / 7! ˛ B  B

1

:

When we consider the quotient of the representation variety Hom.1 .S /; G/=G, this action descends to an action of the outer automorphism group Out.1 .S // D

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Aut.1 .S //=Inn.1 .S// on Hom.1 .S/; G/=G. As discussed in § 2.1, if S is closed, the group of orientation-preserving outer automorphisms of 1 .S / is isomorphic to the mapping class group Map.S/, and we will refer to this action as the action of the mapping class group. The components of Hom.1 .S/; G/=G that form higher Teichmüller spaces are preserved by this action. In the case where G D PSU.1; 1/, the action of the mapping class group on Teichmüller space is properly discontinuous and the quotient M.S / is the moduli space of Riemann surfaces. Given a higher Teichmüller space, it is natural to consider its quotient by the action of the mapping class group, to study how it relates to the moduli space of Riemann surface, and to investigate its possible compactifications. The first question here is whether the action of the mapping class group is properly discontinuous on higher Teichmüller spaces. In order to answer this question the essential notion is that of a representation being well-displacing. For this let us introduce the translation lengths  and  of an element  2 1 .S /  f1g: ./ D inf d.p; p/; p2Sz

 ./ D inf dG .z; . /z/; z2X

where d is the lift of a hyperbolic metric on S and dG is a G-invariant Riemannian metric on the symmetric space X . Definition 8.7. A representation  W 1 .S / ! G is well-displacing if there exist constants A; B > 0 such that for all  2 1 .S /, A1  ./  B  . /  A . / C B: The translation length  depends on the choice of a hyperbolic metric on S . For any two choices the translations lengths are comparable, thus the definition of welldisplacing is independent of the chosen hyperbolic metric on S . It is shown in [73], [100], [59] that representations in higher Teichmüller spaces are well-displacing. Then a simple argument (see e.g. [73], [100]) shows that this implies that the action of the mapping class group on higher Teichmüller spaces is properly discontinuous. Remark 8.8. The notion of well-displacing is related to the notion of quasiisometric embeddings. A representation  W 1 .S/ ! G is a quasiisometric embedding if there exist constants A; B > 0 such that for all  2 1 .S /, A1 dG ../z; z/  B  d.p; p/  A dG .. /z; z/ C B; for some p 2 Sz and some z 2 X. Both notions can be defined more generally for representations of arbitrary finitely generated groups. The relation between the two notions is studied in [32]. Represen-

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tations in higher Teichmüller spaces are also quasiisometric embeddings [73], [100], [59], [57]. 8.2.2 Relation to moduli space. The notion of well-displacement also plays an important role when trying to obtain a mapping class group invariant projection from higher Teichmüller spaces to classical Teichmüller space. To describe this approach, recall that given a representation  W 1 .S/ ! G and a hyperbolic metric h 2 Hyp.S /, one can define the energy of a -equivariant smooth map f W Sz ! X into the symmetric space X D G=K as Z kdf k2 dvol; E .f; h/ WD S

where kdf k.p/, p 2 S , is the norm of the linear map dfp with respect to the hyperbolic metric on S and the G-invariant Riemannian metric on X. The map f is said to be harmonic if and only if it minimizes the energy in its -equivariant homotopy class. Setting E .h/ WD inff E .f; h/, where f ranges over all -equivariant smooth maps Sz ! X , we obtain a function E W F .S/ D Diff C 0 .S /nHyp.S / ! R; called the energy functional associated to the representation  W 1 .S / ! G. The energy functional is a smooth function on the Fricke space F .S /. In the case where G D PSU.1; 1/ and  is a discrete embedding, it is known that E has a unique minimum [36], [101], namely the hyperbolic structure determined by . In the general case, one would like to construct a mapping class group invariant projection from higher Teichmüller spaces to classical Teichmüller space by showing that the energy functional has a unique minimum. As a first step we have Theorem 8.9 ([73], Theorem 6.2.1). Let  W 1 .S / ! G be a well-displacing representation, then the energy functional E is a proper function on F .S /. In [73] Labourie describes an approach to realize the Hitchin component for PSL.n; R/ as a vector bundle over Teichmüller space in a equivariant way with respect to the mapping class group. Recall for this that the isomorphism (see (6.1))     hj W HomHit 1 .S/; PSL.n; R/ = PSL.n; R/ ! H 0 S; ˚nkD2 k.S;j / ; where H 0 .S; k.S;j / / is the vector space of holomorphic differentials (with respect to the fixed complex structure j on S) of degree k, is not mapping class group invariant. Consider the vector bundle E n over Teichmüller space T .S / realized as a space of complex structures on S, where the fiber over the complex structure j equals H 0 .S; ˚nkD3 k.S;j / /. Then E n has the same dimension as   HomHit 1 .S/; PSL.n; R/ = PSL.n; R/:

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Labourie defines the Hitchin map .j; !/ 7! hj .0; !/ ; where j 2 T .S/ is a complex structure and ! 2 H 0 .S; ˚nkD3 k.S;j / /. This map is equivariant with respect to the mapping class group action. Labourie proves that it is surjective [73], Theorem 2.2.1, and conjectures that H is a homeomorphism, which would imply Conjecture ([73], Conjecture 2.2.3). The quotient of the Hitchin component   HomHit 1 .S/; PSL.n; R/ = PSL.n; R/ by the mapping class group is a vector bundle over the moduli space of Riemann surfaces with fiber being the space of holomorphic k-differentials H 0 .S; ˚nkD3 kS /. In order to prove this conjecture it would be sufficient to show that for a Hitchin representation  2 HomHit .1 .S/; PSL.n; R//= PSL.n; R/ the energy functional E has a nondegenerate minimum. Conjecture 8.2.2 has been proved for n D 2 and n D 3. The proof for PSL.3; R/ is independently due to Labourie [72] and Loftin [76]. They rely on the description of HomHit .1 .S/; PSL.3; R// as deformation space of convex real projective structures due to Choi and Goldman, and use the theory of affine spheres developed in [26], [27] in order to prove Theorem 8.10 ([72], [76]). The quotient   Map.S/nHomHit 1 .S/; PSL.3; R/ = PSL.3; R/ is a vector bundle over the moduli space of Riemann surface with fiber being the space of cubic holomorphic differentials on the surface. For maximal representations, the quotients by the mapping class group are expected to look more complicated due to the fact that (1) the connected components consisting of maximal representations might have singularities, and (2) the space of maximal representations has several connected components, which need to be treated separately. (We will come back to this problem in § 8.4.) 8.2.3 Compactifications. Mapping class group equivariant compactifications of higher Teichmüller spaces are partially understood. A general construction to compactify the space of discrete, injective nonparabolic representations of a finitely generated group into G using the generalized marked length spectrum is given in [87]. This construction applies to give compactifications of higher Teichmüller spaces. Boundary points in this compactification can be interpreted as actions on R-buildings [88].

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For the Hitchin component HomHit .1 .S /; PSL.3; R//= PSL.3; R/, the identification with the deformation space of convex real projective structures allows to obtain a better understanding of this compactification, see e.g. [64], [77], [78], [30]. Through the study of degenerations of convex projective structures, Cooper et al. [30] obtain a description of boundary points as mixtures of measured laminations and special Finsler metrics (Hex metrics) on S . Fock and Goncharov construct tropicalizations of the spaces of positive representations which they expect to provide (partial) completions [38] when S is noncompact. But except for the case where G D PSL.2; R/ (treated in [39]), they do not define a topology of the union of the space of positive representations and its tropicalized counterpart. 8.2.4 Crossratios. Realizing @D  CP 1 , the restriction of the classical crossratio function on CP 1 , xy zt c.x; y; t; z/ D ; xt zy gives a continuous real valued PSU.1; 1/-invariant function on ˚  .@D/4 WD .x; y; z; t / 2 .@D/4 W x ¤ t y ¤ z : This crossratio and several generalizations (see e.g. [86], [75]) play an important role in the study of negatively curved manifolds and hyperbolic groups. Given a hyperbolic element  2 PSU.1; 1/, its period is defined as lc ./ D log c.  ; z;  C ; z/; where  C is the unique attracting fixed point and   the unique repelling fixed point of  in @D and z 2 @D  f ˙ g is arbitrary. The period of  equals the translation length  . / D infp2D dD .p; p/. Given a discrete embedding  W 1 .S/ ! PSU.1; 1/, let c WD '  c W .S 1 /4 ! R be the pullback of c by some -equivariant boundary map, be the associated crossratio function. Then c contains all information about the marked length spectrum of S with respect to the hyperbolic metric defined by . In particular, two discrete embeddings 1 , 2 are conjugate if and only if c1 D c2 . A generalized crossratio function is a 1 .S /-invariant continuous function ˚  .S 1 /4 D .x; y; z; t / 2 .S 1 /4 W x ¤ t y ¤ z ! R satisfying the following relations (see [71], Introduction): (1) (Symmetry) c.x; y; z; t / D c.z; t; x; y/. (2) (Normalization) c.x; y; z; t / D 0 if and only if x D y or z D t , c.x; y; z; t / D 1 if and only if x D z or y D t .

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(3) (Cocycle identity) c.x; y; z; t / D c.x; y; z; w/c.x; w; z; t /, c.x; y; z; t / D c.x; y; w; t /c.w; y; z; t /. Among such functions crossratios arising from a discrete embedding of the form 1 .S / ! PSU.1; 1/ are uniquely characterized by the functional equation 1  c.x; y; z; t / D c.t; y; z; x/: The study of generalized crossratio functions associated to higher Teichmüller spaces has been pioneered by Labourie. In particular, he associates a generalized crossratio function to any Hitchin representation into PSL.n; R/ and shows that crossratio functions arising from a Hitchin representation into PSL.n; R/ are characterized by explicit functional equations [71]. In [74] Labourie and McShane establish generalized McShane identities for the crossratios associated to Hitchin representations into PSL.n; R/. Remark 8.11. Related crossratio functions of four partial flags consisting of a line and a hyperplane are used in the work of Fock and Goncharov [38] in order to construct explicit coordinates for the space of positive representations into PSL.n; R/. In the context of maximal representations, crossratio functions have been defined and studied by Hartnick and Strubel [59]. They construct crossratio functions defined on a suitable subset S{4 of the fourfold product of the Shilov boundary of any Hermitian symmetric space of tube type. They show that there is a unique such crossratio function which satisfies some natural functorial properties. Given a maximal representation  W 1 .S/ ! G, a concrete implementation of the continuous boundary map ' W S 1 ! S{ (see Theorem 5.29) allows to pullback this crossratio function to a generalized crossratio function on .S 1 /4 . The well-displacing property of representations in higher Teichmüller spaces can be easily deduced from the existence of generalized crossratio functions. In all works investigating crossratio functions, the existence of boundary maps with special positivity properties (as discussed in § 7.1) plays an important role.

8.3 Geometric structures We already mentioned that Hitchin had asked in [62] about the geometric significance of Hitchin components, and one might raise the same question for maximal representations, even though the picture there seems to be more complicated due to the fact that the space of maximal representations has singularities and multiple components. Interpreting higher Teichmüller spaces as deformation spaces of geometric structures is not just of interest in itself. Any such interpretation gives an important tool to study these spaces, their quotients by the mapping class group, their relations to

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the moduli space of Riemann surfaces as well as their compactifications. This is illustrated by the fact that the deeper understanding of these questions for the Hitchin component of PSL.3; R/ relies on the Theorem by Choi and Goldman, which we already mentioned above: Theorem 8.12 ([49], [28]). The Hitchin component   HomHit 1 .S/; PSL.3; R/ = PSL.3; R/ parametrizes convex real projective structures on S. The original proof of this theorem relied on Goldman’s work on convex projective structures on surfaces [49], which implied that the deformation space of these structures is an open domain in HomHit .1 .S/; PSL.3; R//= PSL.3; R/. Goldman and Choi [28] then proved that this subset is furthermore closed, establishing the above theorem. In terms of the properties of Hitchin representations we have discussed so far, Theorem 8.12 is basically equivalent to the characterization of Hitchin representations into PSL.3; R/ by the existence of a convex map from S 1 into RP 2 (Theorem 7.1). We give a sketch of how Theorem 8.12 follows from Theorem 7.1 when n D 3. Sketch of a proof of Theorem 8.12 assuming Theorem 7.1. A convex real projective structure on S is a pair .N; f /, where N is the quotient =  of a strictly convex domain in RP 2 by a discrete subgroup  of PSL.3; R/, and f W S ! N is a diffeomorphism. Starting from a representation  W 1 .S/ ! PSL.3; R/ in the Hitchin component, let

 RP 2 be the strictly convex domain bounded by the convex curve .S 1 /  RP 2 . Then .1 .S// acts freely and properly discontinuously on

. The quotient

=.1 .S // is a real projective convex manifold, diffeomorphic to S. Conversely given a real projective structure on S , we can -equivariantly identify S 1 (identified with the boundary of 1 .S/) with the boundary of and get a convex curve  W S 1 ! @  RP 2 . Inspired by this proof and with Theorem 7.1 at hand for arbitrary n, one might try to follow a similar strategy in order to find geometric structures parametrized by the Hitchin component for PSL.n; R/. This works for n D 4, where we obtain the following Theorem 8.13 ([55]). The Hitchin component for PSL.4; R/ is naturally homeomorphic to the moduli space of properly convex foliated projective structures on T 1 S . A properly convex foliated projective structure is a locally homogeneous .PSL.4; R/; RP 3 /-structure on T 1 S satisfying the following additional conditions: • every orbit of the geodesic flow is locally a projective line,

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• every (weakly) stable leaf of the geodesic flow is locally a projective plane and the projective structure on the leaf obtained by restriction is convex. Using the convex curve provided by Theorem 7.1, one can consider the corresponding discriminant surface  RP 3 , i.e. the union of all its tangent lines. The complement RP 3  consists of two connected components, on both of which .1 .S // acts properly discontinuous. The quotient of one of the connected components by 1 .S / is homeomorphic to T 1 S, equipped with a properly convex foliated projective structure. The main work goes into establishing the converse direction, i.e. showing that the holonomy representation of a properly convex foliated projective structure on T 1 S lies in the Hitchin component – this is rather tedious. Remark 8.14. The above theorem implies that the Hitchin component for PSp.4; R/ is naturally homeomorphic to the moduli space of properly convex foliated projective contact structures on the unit tangent bundle of S. For n  5, the above strategy seems to fail in general. The first step in the strategy described above to find geometric structures parametrized by representations  W 1 .S/ ! G in higher Teichmüller spaces is to find domains of discontinuity for such representations in homogeneous spaces, more precisely in generalized flag varieties associated to G, on which 1 .S / is supposed to act with compact quotient. This problem becomes more difficult the bigger G gets, since 1 .S / is a group of cohomological dimension 2, whereas the dimension of the generalized flag varieties grows as G get bigger. So it comes a bit as a surprise that finding domains of discontinuity with compact quotient can be accomplished in the very general setting of Anosov representations. Theorem 8.15 ([56], [57]). Let G be a semisimple Lie group and assume that no simple factor of G is locally isomorphic to PSL.2; R/. Let  W 1 .S / ! G be a .PC ; P /Anosov representation. Let P D MAN be the minimal parabolic subgroup of G. Then there exists an open non-empty set   G=AN , on which 1 .S / acts freely, properly discontinuous and with compact quotient. Remark 8.16. The homogeneous space G=AN is the maximal compact quotient of G. In many cases the domain  descends to a domain of discontinuity in G=P . Remark 8.17. Theorem 8.15 holds more generally for Anosov representations of convex cocompact subgroups of Hadamard manifolds of strictly negative curvature or even of hyperbolic groups. The reader interested in the more general statement is referred to [57]. The main tool in order to define the domain of discontinuity  are the -equivariant continuous boundary maps ˙ W S 1 ! G=P˙ associated to the .PC ; P /-Anosov representation (see Proposition 8.5).

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There is some evidence that – at least in the case of higher Teichmüller spaces – the quotients  =.1 .S// are homeomorphic to the total spaces of bundles over S with compact fibers. This has been established for maximal representation into Sp.2n; R/ as well as for Hitchin representations into SL.2n; R/. Theorem 8.18 ([57]). (1) The Hitchin component for SL.2n; R/ parametrizes real projective structures on a compact manifold M , which is topologically an O.n/=O.n  2/-bundle over the surface S . (2) Maximal representations into Sp.2n; R/ parametrize real projective structures on a compact manifold M homeomorphic to an O.n/=O.n  2/-bundle over the surface S . Its isomorphism type depends on the connected component containing the representation.

8.4 Topological invariants The Hitchin component is by definition a single connected component, but the space of maximal representations is a priori only a union of connected components, and their might be more than one. In many cases, the exact number of connected components of the space of maximal representations has been computed using methods from the theory of Higgs bundles [52], [42], [41], [11], [12]. And the most interesting family in terms of the number of connected components are maximal representations into symplectic groups Sp.2n; R/: there are 3  22g connected components when n  3 [41] and .3  22g C 2g  4/ connected components when n D 2 [52]. Invariants to distinguish these connected components can be derived from the associated Higgs bundles, but topological invariants to distinguish the different connected components also arise from considering maximal representations as Anosov representations. Recall that in the definition of Anosov structures one considers the flat G-bundle G./ over T 1 S and the associated bundle O./. The first part of the data of an Anosov structure is a section  of O./. Since O./ is the G=H -bundle associated to G./ its sections are in one-to-one correspondence with reductions of the structure group of G./ from G to H . In general there is no canonical section, but in the case of Anosov structures we have Proposition 8.19 ([58]). If a section  of O./ with the properties required in Definition 8.1 exists, then it is unique. As a consequence, an Anosov representation  W 1 .S / ! G gives a canonical reduction of the G-principal bundle G./ to an H -principal bundle. This H -bundle is in general not flat; its characteristic classes give topological invariants of the Anosov representation  that live in H .T 1 S/. In the situation of maximal representations  W 1 .S / ! Sp.2n; R/, we have that H D GL.n; R/, embedded into Sp.2n; R/ as the stabilizer of two transverse

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Lagrangian subspaces. The topological invariants of significance are first and second Stiefel–Whitney classes, as well as an Euler class if n D 2. Theorem 8.20 ([58]). The topological invariants distinguish the connected components of Hommax .1 .S/; Sp.2n; R// n HomHit .1 .S /; Sp.2n; R//. opp Considering Hitchin representations as .Pmin ; Pmin /-Anosov representations, there is an additional first Stiefel–Whitney class, which distinguishes the connected components of HomHit .1 .S/; Sp.2n; R/. The invariants constructed using the Anosov property of maximal representations are in principle computable for a given representation  W 1 .S / ! Sp.2n; R/. Explicit computations for various representations allows us to describe model representations in any connected component. This is of particular interest for Sp.4; R/ as there are 2g  3 connected components in which every representation is Zariski dense (see also [10]). Besides the irreducible Fuchsian representation, which were introduced to define the Hitchin component, there are two other kinds of model representations: A twisted diagonal representation is a maximal representation  D . ˝ / W 1 .S/ ! SL.2; R/  O.n/  Sp.2n; R/; where  W 1 .S/ ! SL.2; R/ is a discrete embedding and  W 1 .S / ! O.n/ is an orthogonal representation; SL.2; R/  O.n/ sits in Sp.2n; R/ as the normalizer of the diagonal embedding SL.2; R/ ! SL.2; R/n  Sp.2n; R/: A hybrid representation is a maximal representation k D 1  2 W S D S1 [ S2 ! Sp.2n; R/; k D 3  2g; : : : ; 1, which is obtained by amalgamation of an irreducible Fuchsian representation on 1 .S1 / and a suitable deformation of an (untwisted) diagonal representation on 1 .S2 /. The subscript k indicates the Euler characteristic of S1 . The construction of hybrid representations relies on the additivity of the Toledo number and the Euler characteristic under gluing (see Proposition 5.10). Theorem 8.21 ([58]). For n  3, any maximal representation  W 1 .S / ! Sp.2n; R/ can be deformed either to an irreducible Fuchsian representation or to a twisted diagonal representation. When n D 2, there are 2g  3 connected components Hk , k D 1; : : : ; 2g  3 of Hommax .1 .S/; Sp.4; R// in which every representation has Zariski dense image. Representations in Hk can be deformed to k-hybrid representations. The information about model representations in each connected component can be used to obtain further information about the holonomies of maximal representations.

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For representations in the Hitchin components, Theorem 6.5 implies that for every  2 1 .S /  f1g the image ./ is diagonalizable over R with distinct eigenvalues. This does not hold for the other components of maximal representations. The Anosov property implies that for every  2 1 .S / n feg the image . / is conjugate to an element in GL.n; R/ < Sp.2n; R/. More precisely, we have: Theorem 8.22 ([58]). Let H be a connected component of     Hommax 1 .S/; Sp.2n; R/ n HomHit 1 .S /; Sp.2n; R/ ; and let  2 1 .S/  f1g be an element corresponding to a simple curve. Then there exist (1) a representation  2 H such that the Jordan decomposition of . / in GL.n; R/ has a nontrivial parabolic component; (2) a representation 0 2 H such that the Jordan decomposition of . / in GL.n; R/ has a nontrivial elliptic component. This result indicates that understanding the structure of the space of maximal representations is much more complicated than understanding the structure of Hitchin components, since already the conjugacy classes in which the holonomy of one element can lie in might differ from connected component to connected component.

9 Open questions and further directions In the previous sections we already mentioned some open questions regarding the quotients of higher Teichmüller spaces, their compactifications as well as their geometric significance. In this section we want to conclude our survey with mentioning some further directions in the study of higher Teichmüller spaces that to our knowledge have not yet been explored.

9.1 Positivity and other groups As we pointed out in § 7.1 an underlying common structure of higher Teichmüller spaces is that the homomorphisms in them admit equivariant boundary maps that satisfy some positivity or cyclic order preserving property. The relation between positive triples in the full flag variety and the weaker notion of maximal triples in the space of Lagrangians discussed in § 7.2 is very special. It would be very interesting to discover weaker notions of positivity of k-tuples in (partial) flag varieties for other groups that are neither split real forms nor of Hermitian type. Such notions of positivity might lead to discovering higher Teichmüller spaces for other Lie groups G, which are again characterized by the existence of special boundary maps.

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A first family of groups to look at could be G D PO.p; q/, which is of Hermitian type if .p; q/ D .2; q/ and a split real form if .p; q/ D .n; n C 1/ or .p; q/ D .n; n/. Every time there is a notion of positivity or cyclic ordering, the images of boundary maps tend to be more regular, namely rectifiable circles. This contrasts with the case of quasifuchsian deformations into PSL.2; C/ of compact surface groups in PSL.2; R/, where in fact the limit set, or – what is the same – the image of the boundary map, is a topological circle with Hausdorff dimension larger than 1, unless the deformed group is Fuchsian.3 This suggests to study the deformations of the homomorphism i B  W 1 .S/ ! G.C/; where i W G D G.R/B ! G.C/ is the natural inclusion and  W 1 .S / ! G is either a maximal representation into a group of Hermitian type or a Hitchin representation into a real split Lie group. Observe that i B  is Anosov for a suitable pair of parabolic subgroups and, as a result, small deformations of i B  are as well.

9.2 Coordinates and quantizations for maximal representations Fock and Goncharov describe explicit coordinates for the space of positive representations. For PSL.n; R/ these coordinates have a particular nice form. Based on the explicit coordinate system, they describe the cluster variety structure and quantizations of the space of positive representations. It would be interesting to construct similar explicit coordinate systems for the space of maximal representations, in particular when G D Sp.2n; R/. Theorem 8.22 gives a hint that constructing coordinates for the space of maximal representations is more involved. The structure of the coordinates also needs to be more complicated as they have to model the singularities of the space of maximal representations. The additivity of the Toledo number on the other hand implies that the space of maximal representations of a compact surface S can be built out of the space of maximal representations of a pair of pants. Having coordinates at hand, one might also ask for quantizations of the space of maximal representations or try to express the symplectic form on the space of maximal representations explicitly in coordinates.

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Chapter 14

The theory of quasiconformal mappings in higher dimensions, I Gaven J. Martin

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two geometric definitions . . . . . . . . . . . . . . . . . . . . 2.1 The linear distortion . . . . . . . . . . . . . . . . . . . . 2.2 Moduli of curve families . . . . . . . . . . . . . . . . . . 2.3 The geometric definition of quasiconformality . . . . . . 2.4 The modulus of some curve families . . . . . . . . . . . . 2.5 The Grötzsch and Teichmüller rings . . . . . . . . . . . . 2.6 Hölder continuity . . . . . . . . . . . . . . . . . . . . . . 2.7 Mori distortion theorem . . . . . . . . . . . . . . . . . . 3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Limits of quasiconformal mappings . . . . . . . . . . . . 4 The analytic definition of a quasiconformal mapping . . . . . . 5 The Liouville theorem . . . . . . . . . . . . . . . . . . . . . . 6 Gehring’s higher integrability . . . . . . . . . . . . . . . . . . 7 Further stability and rigidity phenomena . . . . . . . . . . . . . 8 Quasiconformal structures on manifolds . . . . . . . . . . . . . 8.1 The existence of quasiconformal structures . . . . . . . . 8.2 Quasiconformal 4-manifolds . . . . . . . . . . . . . . . . 8.3 The extension problem . . . . . . . . . . . . . . . . . . . 8.4 Boundary values of quasiconformal mappings . . . . . . 8.5 Generalised Beltrami systems . . . . . . . . . . . . . . . 9 Nevanlinna theory . . . . . . . . . . . . . . . . . . . . . . . . 10 Non-linear potential theory . . . . . . . . . . . . . . . . . . . . 10.1 A-harmonic functions . . . . . . . . . . . . . . . . . . . 10.2 Connections to quasiconformal mappings . . . . . . . . . 10.3 Removable singularities . . . . . . . . . . . . . . . . . . 11 Quasiregular dynamics in higher dimensions . . . . . . . . . . 11.1 Existence of equivariant measurable conformal structures 11.2 Fatou and Julia sets . . . . . . . . . . . . . . . . . . . . . 11.3 Dynamics of rational mappings . . . . . . . . . . . . . .

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11.4 Stoïlow factorisation . . . . . . . . . . . . . . . . . . . . 11.5 Smooth rational quasiregular mappings . . . . . . . . . . 11.6 The Lichnerowicz problem: rational maps of manifolds . 12 Quasiconformal group actions . . . . . . . . . . . . . . . . . . 12.1 Convergence properties . . . . . . . . . . . . . . . . . . 12.2 The elementary quasiconformal groups . . . . . . . . . . 12.3 Non-elementary quasiconformal groups and the conjugacy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Hilbert–Smith conjecture . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction Geometric function theory in higher dimensions is largely concerned with generalisations to Rn , n  3, of aspects of complex analysis, the theory of analytic functions and conformal mappings – particularly the geometric and function theoretic properties. In this sense it has been a successful theory with a variety of applications, many of which we will discuss below. The category of maps that one usually considers in the higher-dimensional theory are the quasiregular mappings, or, if injective, quasiconformal mappings. Both kinds of mappings have the characteristic property of bounded distortion. The higher-dimensional theory of quasiconformal mappings was initiated in earnest by Yu. G. Reshetnyak (USSR), F. W. Gehring (USA) and J. Väisälä (Finland) in 1960–62, see [23], [24], [81], [82]. There was earlier work, notably that of Ahlfors–Beurling (1950) on conformal invariants, Ahlfors (1954) and Callender (1959). While Ahlfors’ work was focussed on two-dimensions, the geometric ideas and techniques had clear generalities. Callender followed Finn & Serrin to establish Hölder continuity estimates for higher-dimensional quasiconformal mappings. We note that one of the most famous applications of the theory of higher-dimensional quasiconformal mappings, Mostow’s rigidity theorem (1967) [74], came just five years after the basic foundations were laid. The generalisations to non-injective mappings was initiated with Reshetnyak and the basic theory was comprehensively laid and significantly advanced in a sequence of papers from the Finnish school of O. Martio, S. Rickman and J. Väisälä in the late 1960s [62], [63], [64]. Both quasiconformal and quasiregular mappings solve natural partial differential equations (PDE) closely analogous to the familiar Cauchy–Riemann and Beltrami equations of the plane. The primary difference being that in higher dimensions these equations necessarily become nonlinear and overdetermined. Other desirable properties for a theory of the geometry of mappings are that they should preserve the natural Sobolev spaces which arise in consideration of the function theory and PDEs on subdomains of Rn , or more generally n-manifolds. Quasiconformal mappings do have these properties.

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In dimensions n  3, one needs to move away from the class of conformal mappings because of their remarkable rigidity properties. Perhaps most well known is the Liouville theorem, basically established in the 1970s independently and by different methods by Gehring and Reshetnyak as we discuss later. This rigidity is partly explained by the fact that the governing equations are overdetermined. This rigidity also has consequences for the pronounced differences between injective and non-injective mappings in higher dimensions. In two-dimensions, the celebrated Stoïlow factorisation theorem asserts that a quasiregular mapping f W  ! C admits a factorisation f D ' B g where g W  ! C is quasiconformal and ' is holomorphic. This factorisation, together with our more or less complete understanding of the structure of holomorphic functions in the plane, connects quasiregular & quasiconformal mappings strongly. In particular, if Bf is the branch set, Bf D fx 2  W f is not locally injective at xg;

(1.1)

then, in two-dimensions, the factorisation theorem quickly shows that Bf will be a y D C [ f1g  S2 , Bf discrete subset of . Thus if  is the Riemann sphere, C will be a finite point set. This is far from true in higher dimensions. First, wellknown results in geometric measure theory, see Federer [18], connect smoothness (in terms of differentiability) and local injectivity. Thus branched maps cannot be very smooth (C 1 is alright in three dimensions though). Second, the branch set Bf of a mapping of bounded distortion can have quite pathological topology, for instance it could be Antoine’s necklace – a Cantor set in S3 whose complement is not simply connected. This makes Rickman’s development of the higher-dimensional Nevanlinna theory all the more remarkable. As a consequence of the Nevanlinna theory, as in the classical case, one obtains best possible results concerning precompactness of families of mappings. Since quasiregular mappings are open and discrete at this point it is worth recalling Chernavskii’s theorem [14], [93] which asserts that if Bf is the branch set of a quasiregular mapping, then the topological dimension of both Bf and f .Bf / is less than or equal to n2 and therefore cannot separate. Further, fairly general topological results enable one to talk about the degree and topological index of such mappings. There are also second order equations related to the nonlinear governing equations for quasiconformal mappings. For example, the components of an analytic function are harmonic, while those of a quasiregular mapping are A-harmonic. These are basically the Euler–Lagrange equations for a conformally invariant integral for which the mapping in question is a minimiser. In this way such well-known non-linear differential operators as the p-Laplacian and the associated non-linear potential theory arise naturally in the theory of higher-dimensional quasiconformal mappings. This potential theory has significant topological implications for mappings of bounded distortion. These were first observed by Reshetnyak. Another fruitful idea when studying quasiconformal mappings and their properties is to view quasiregular mappings as conformal with respect to certain measurable Riemannian or measurable conformal structures. In two-dimensions this gives the direct connection with Teichmüller theory of course and this idea is greatly aided by

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the fact that one can solve the associated Beltrami equation, leading to the so-called measurable Riemann mapping theorem – or the existence theorem for quasiconformal mappings. In higher dimensions, unfortunately almost nothing useful is known about solving Beltrami systems. There are obvious reworkings of the classical results from the 1920s of Weyl and Schouten, which assume the vanishing of a second order tensor, when the conformal tensor G is sufficiently smooth. It is an extremely interesting problem to try and give reasonable conditions on G which guarantee local existence if G is perhaps only C 1C smooth let alone the most important case when G is only assumed measurable. The general higher-dimensional theory does provide good results about the regularity of solutions, really initiated by Gehring’s higher integrability and reverse Hölder inequalities from 1973 [26]. These generalise earlier results of Bojarski from 1955 in two-dimensions [9], but again totally different methods are needed to attack these non-linear equations in higher dimensions. I think it is fair to say that the Gehring’s higher integrability results revolutionised the theory and ultimately brought it closer to PDEs and nonlinear analysis as the techniques he developed had much wider application. We also understand, to a reasonable extent at least, both the uniqueness of solutions to higher-dimensional Beltrami systems as well as analytic continuation and so forth. Quasiconformal mappings provide a class of deformations which lie “between” homeomorphisms and diffeomorphisms but enjoy compactness properties neither do. The most recent developments in the theory concern mappings of finite distortion. Here the assumption concerning boundedness is removed and replaced with various control assumptions on the distortion or its associated tensors. Such mappings are even more flexible and to study them more refined techniques are necessary as the governing equations will be degenerate elliptic. However various compactness properties of families of mappings with finite distortion make them ideal tools for solving various problems in n-dimensional analysis. For instance in studying deformations of elastic bodies and the related extremals for variational integrals, mappings of finite distortion are often the natural candidates to consider. These ideas lead directly to the theory of non-linear elasticity developed by Antman and Ball, and many others. This theory of elasticity studies mappings (in certain Sobolev classes) which minimize various stored energy integrals. On seeks existence, regularity and so forth. The Jacobian determinant, in particular, has been subjected to a great deal of investigation. Of course there are many outstanding problems which are helping to drive the field, but which we won’t discuss here. These include determining precise geometric conditions on a domain to be quasiconformally equivalent to a ball (thus a generalised Riemann mapping theorem). As we will see in a moment, the Liouville theorem implies that in dimensions n  3 any domain conformally equivalent to the unit ball is a round ball or half-space. In two-dimensions Ahlfors gave a beautiful intrinsic characterisation of the quasiconformal images of the unit disk. While such a nice result is unlikely in higher dimensions, not a great deal is really known. In other directions, Iwaniec and his coauthors are advancing the connections between the higher-dimensional theory (largely as it pertains to the geometry of map-

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pings) and the calculus of variations. Particular advances concern generalising the theory to mappings of finite distortion [43], [5]. Here the distortion is no longer assumed uniformly bounded, but some additional regularity is necessary to get a viable theory. Most of the major results assume something close to the distortion function K.x; f / being of bounded mean oscillation. As the Jacobian of a mapping J.x; f / has automatically higher regularity as an H 1 function, one seeks to exploit the H 1 BMO duality discovered by Fefferman to gain information about the total differential from the distortion inequality jDf .x/jn  K.x; f /J.x; f / since meaning can be given to the right-hand side. There are very many interesting problems and deep connections to other areas here. There are still further generalisations and applications in the geometry and analysis of metric spaces. The connections with the higher-dimensional theory of quasiconformal mappings was pioneered by Heinonen and Koskela [34] and is a very active area of research today. Here is a sample of the successful and diverse applications of the higher-dimensional theory of quasiconformal mappings (some mentioned above and in no particular order of importance): • Compactness, equicontinuity and local to global distortion estimates. • The Liouville theorem and other stability and rigidity phenomena. • Gehring’s improved regularity and higher integrability. • Mostow rigidity – uniqueness of hyperbolic structures (n  3). • Sullivan’s uniformisation theorem – the existence of quasiconformal structures on topological n-manifolds (n ¤ 4). • Rickman’s versions of the Picard theorem and Nevanlinna theory. • Applications in nonlinear potential theory, A-harmonic functions and non-linear elasticity. • Tukia–Väisälä’s “quasiconformal geometric topology”. • Quasiconformal group actions and geometric group theory. • Donaldson and Sullivan’s “quasiconformal Yang–Mills theory”. • Painlevé type theorems and the structure of singularities. • Quasiconformal maps in metric spaces with controlled geometry. • Analysis and geometric measure theory in metric spaces. Mindful of the readership of a chapter such as this, we will not strive for maximum generality in the results we present. Also, we will seldom present complete proofs and discussions, but set the reader toward places where such discussions can be found. To that end there are a number of relatively recent books which the reader might consult for details of omissions here and which have a relatively broad focus. A reasonably

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complete account of the modern two-dimensional theory is given in Astala, Iwaniec & Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, 2009, [6]. For the analytic aspects of the theory in higher dimensions we have Iwaniec & Martin, Geometric function theory and non-linear analysis, 2001, [46]. For the nonlinear potential theory see Heinonen, Kilpeläinen & Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford, 1993, [36]. For analysis on metric spaces see Heinonen, Lectures on analysis on metric spaces, 2001 [33]. For the Nevanlinna and related theories of quasiregular mappings we have Rickman, Quasiregular mappings, 1993 [83]. Vuorinen, Conformal geometry and quasiregular mappings, 1988, [96] gives a detailed account of the distortion estimates and other geometric aspects of the theory which is further developed in by Anderson, Vamanamurthy and Vuorinen, [3]. Of course there are others, but these books should give a more or less complete overview. But not to forget the past, we cannot fail to mention the classics, Ahlfors, Lectures on quasiconformal mappings, 1966, [1] and Lehto & Virtanen, Quasiconformal mappings in the plane, 1973, [50], for the two-dimensional theory and of course Väisälä Lectures on n-dimensional quasiconformal mappings, 1971, [94], a book from which many of us learnt the basics of the higher-dimensional theory.

2 Two geometric definitions We will present the analytic definition of a quasiconformal mapping via Sobolev spaces and differential inequalities a little later. However we want to give a brief initial discussion to capture the idea of infinitesimal distortion. This is because the geometric definitions of quasiconformality are quite global in nature – asking us to test a Lipschitz condition against every family of curves in a given domain. It is this interplay between the local definition of quasiconformality and the global one that is a real strength of the theory. Once one has established an infinitesimal distortion condition (through properties of solutions to a PDE or some assumptions around differentiability), then one obtains large scale distortion estimates through considering various curve families and geometric estimates upon them.

2.1 The linear distortion Let  and 0 be domains in Rn and let f W  ! 0 be a homeomorphism. We will define quasiconformal mappings as mappings of “bounded distortion” and therefore we must discuss what distortion might mean. Suppose therefore, that x 2  and r < d.x; @/. We define the infinitesimal distortion H.x; f / of f at the point x as H.x; f / D lim sup jhj!0

max jf .x C h/  f .x/j : min jf .x C h/  f .x/j

(2.1)

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We further say that f is quasiconformal in  if H.x; f / is bounded throughout the domain : there exists an H < 1 such that H.x; f /  H for every x 2 . The essential supremum of this quantity is called the linear distortion of f , H.f / D kH.x; f /kL1 ./ :

(2.2)

Notice the pointwise everywhere assumption here in the definition of quasiconformality. It is necessary. If f is differentiable at x0 with non-singular differential, then we can multiply and divide by jhj in (2.1) and take limits. Quasiconformality quickly yields an inequality between the smallest and largest directional derivatives, max jDf .x0 /hj  H.x0 ; f / min jDf .x0 /hj:

jhjD1

jhjD1

A little linear algebra reveals that the left-hand side here is the square root of the largest eigenvalue of the positive definite matrix D t f .x0 /Df .x0 /, and the right-hand side is the smallest such. While geometrically appealing, unfortunately this quantity is not particularly useful in higher dimensions since it is not lower semicontinuous on the space of quasiconformal mappings [42]. Recall that a real-valued function h is lower semicontinuous if for all x0 , lim inf h.x/  h.x0 /: x!x0

A lower semicontinuous distortion function will guarantee distortion does not suddenly increase in the limit, a clearly desirable property. Interestingly, this failure is directly connected with the failure of rank-one convexity in the calculus of variations. However, there is a remarkable result here due to Heinonen and Koskela [35]. It turns out that the lim sup requirement of the definition at (2.1) is met once a lim inf condition holds. Theorem 2.1. Let f W  ! Rn be a homeomorphism. If there is H  < 1 such that for every x 2 , H  .x; f / D lim inf jhj!0

max jf .x C h/  f .x/j  H ; min jf .x C h/  f .x/j

(2.3)

then there is H D H.n; H  / < 1 such that for every x 2 , H.x; f /  H;

(2.4)

and consequently (2.3) is enough to guarantee the quasiconformality of f . Again, note the requirement of having a condition at every point of . The analytic definitions of quasiconformality will get around this problem by having a pointwise almost everywhere criteria. However, these conditions must of course give the boundedness of the linear distortion everywhere. Before we go in that direction we discuss the earliest natural definition of quasiconformality which is through the bounded distortion of a conformal invariant – the moduli of curve families.

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2.2 Moduli of curve families The most useful geometric definition of a quasiconformal mapping is through the modulus of a curve family. A curve family  is simply a collection of (rectifiable) curves; continuous maps  W Œ0; 1 ! . It is usual to identify a curve with its image as the quantities we wish to study must be invariant of parameterisation. Typically a curve family will be of the following sort, .E; F I / the set of all curves connecting E to F and lying in a domain  of Rn . Given a curve family , an admissible density is a non-negative Borel function  for which Z .s/ ds  1; for all  2 : (2.5) 

We see immediately that highly irregular curves (in particular those that are not rectifiable) in a family will not be relevant as their -length will most likely be infinite and so (2.5) will automatically be satisfied. The modulus of  is

Z M./ D inf

Rn

n .x/ dx

(2.6)

where the infimum is over all admissible densities for . There are more general discussions to be had here. We could, for instance, consider the p-modulus (or p-capacity) where we look at Z inf p .x/ dx Rn

over the same class of admissible functions (this was first considered by Fuglede). These quantities can be used to detect the size of sets in a similar fashion to Hausdorff dimension. As an example consider a set E  Bn and let  consist of all the curves in Bn connecting E to Sn . Then, depending on the size and structure of E, there may be a value p0 for which this quantity is zero – E has p0 -capacity zero. This has geometric and function theoretic consequences. For instance, sets of n-capacity (usually called conformal capacity) zero are typically negligible in the theory of quasiconformal mappings and so, for instance, removable for bounded mappings and so forth. However, these sets have Hausdorff dimension zero and so are very thin. The idea of the modulus of curve families is to develop the “length-area” method used by Ahlfors and Beurling to great effect in two-dimensions in their celebrated paper on conformal invariants [2] in 1950, although these ideas had been around and used in various ways in complex analysis since the 1920s. There are a few basic properties of the modulus of a curve family which fall out of the definition. Firstly M./ is increasing. If 1  2 , then M.1 /  M.2 /. If  contains a single “constant curve”, then M./ D 1. If 1 and 2 are curve families such that every curve in 2 has a subcurve in 1 , then M.2 /  M.1 /. Finally modulus is countably subadditive and additive on disjoint families.

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The most important fact is of course that modulus is a conformal invariant. For the moment a conformal mapping will be a diffeomorphism whose differential (the matrix Df ) is pointwise a scalar multiple of an orthogonal matrix. For a conformal map we therefore have the equality jDf .x/jn D J.x; f / D det Df .x/

(2.7)

(recall jDf .x/j D maxjhjD1 jDf .x/hj, the largest directional derivative). Next we establish that the modulus is conformally invariant. The proof is easy, but the reader should take note of how the differential inequalities between jDf .x/j and J.x; f / are used as this motivates the analytic definition. Theorem 2.2. Modulus is a conformal invariant. Proof. Let  be a curve family and set  0 D f ./. If 1 is admissible for , then .x/ D 1 .f .x//jDf .x/j is admissible for  since Z Z Z  ds D 1 .f .x//jDf .x/j ds D 1 ds  1: (2.8) 

0



Next

Z M./ 

Rn

n .x/ dx

Z D

Rn

Z D

Rn

Z D

Rn

1n .f .x//jDf .x/jn dx 1n .f .x//J.x; f / dx 1n .x/ dx:

Now taking the infimum over all 1 shows us that M./  M. 0 /. The converse inequality holds since f 1 is also a conformal mapping. We can now give an alternative definition of quasiconformality

2.3 The geometric definition of quasiconformality Let f W  ! 0 be a homeomorphism. Then f is K-quasiconformal if there exists a K, 1  K  1, such that 1 M./  M.f /  KM./ (2.9) K for every curve family  in . Of course in practise it is impossible to test the condition (2.9) against every curve family. That is why we seek equivalent conditions – either infinitesimal or by testing

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against only certain curve families – which guarantee quasiconformality. Once we have such things at hand (2.9) provides powerful global geometric information – provided we can find ways of computing, or at least estimating, the moduli of curve families. There are a couple of direct consequences from this definition that are not nearly so trivial when using the analytic definitions that follow. Primary among these are Theorem 2.3. Let f W  ! 0 D f ./ be K-quasiconformal and g W 0 ! Rn be K 0 -quasiconformal. Then • f 1 W 0 !  is K-quasiconformal, and • g B f W  ! Rn is KK 0 -quasiconformal. The general theory now develops by computing the modulus of special sorts of curve families. Then we estimate the modulus of more general curve families in terms of geometric data and define various special functions for the modulus of various curve families which are in some sense extremal for moduli problems (for instance the Grötzsch and Teichmüller curve families being the most common). We then obtain geometric information about quasiconformal mappings by studying what happens to special curve families under quasiconformal mappings using the Lipschitz estimates at (2.9) and comparing with the general estimates of various moduli. This approach quite quickly reveals that quasiconformal mappings are locally Hölder continuous, and establishes such things as an appropriate version of the Schwarz Lemma, and so forth. Of course the Lipschitz estimate at (2.9) must have further consequences for the differentiability and regularity of the homeomorphism f . A major part of the basic theory is in identifying these. This is typically done by connecting this geometric definition, with the analytic definition we give a bit later. To get further into the theory we must actually compute a couple of examples of the modulus of curve families.

2.4 The modulus of some curve families First, and most useful, is the calculation of the modulus of the curves in an annular ring.   Theorem 2.4. Let  D  Sn1 .a/; Sn1 .b/ W Bn .b/ n Bn .a/ , the set of curves connecting the boundary components of the annulus A.a; b/ D Bn .b/ n Bn .a/. Then, 

M./ D !n1

b log a

1n

where !n1 is the volume of the .n  1/-dimensional sphere.

(2.10)

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Proof. Choose an admissible density  for . The rays u .r/ D ru, u 2 Sn1 and a < r < b lie in  and so Hölder’s inequality gives us n Z b n Z b n Z  ds D .ru/ dr D .ru/ r .n1/=n r .1n/=n dr 1 u

Z

b

D

a

h

.ru/ r .n1/=n

Z

a b



 n

 .ru/ r

n1

a

in Z

dr

a

b

a

b log a

r 1 dr

n1

n1

:

Because this holds for every u in S n1 , we can integrate it over S n1 . Thus  Z Z b Z Z n dx  n dx D n .ru/ dn1 r n1 dr Rn

D

Sn1

Sn1

a

A

Z

Z

b

a

   b 1n n .ru/ r n1 dr d  !n1 log a

giving us the lower bound we want. Next, define  W Rn ! Œ0; 1 by ´ 1 1 jxj if x 2 A, log ab .x/ D 0 if x … A. Then  is an admissible function since Z b Z Z  ds  .r/ dr D 

a

b



log a

b a

1

r 1 dr D 1

for every rectifiable  in . Then Z n dx M./  Rn



Z b Z D

n

Sn1

a

D

1 log.b=a/



D

log

b a



D !n1

 .ru/ d r n1 dr  r n d r n1 dr

Z b Z a

n

Sn1 Z b

!n1

b log a

r 1 dr

a

1n

:

So we obtain the desired equality. Unfortunately, very few other moduli can be explicitly computed. An elementary estimate is the following

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Lemma 2.5. If E is an open set in Rn whose Lebesgue measure is finite, jEj < 1, and if  is a family of curves in E for which d D inff`. / W  2 g > 0, then M./ 

jEj < 1: dn

This is a direct consequence of the fact that  D d 1 E is an admissible density for . One now seeks ways to estimate, both from above and below, the modulus of certain curve families. With experience we quickly find that we are most often concerned with curve families that join two components of the boundary of a domain, and these are called “rings”, or sometimes condensers. Thus R.E; F I G/ is the family of curves joining E to F in the domain G and when G D Rn we simply write R.E; F /. The modulus of a ring R D R.E; F I G/ (or the capacity of the condenser) is 

Mod.R/ D

1 !n1

n1

M.R /

(2.11)

where R is the family of all curves joining E to F in G. In particular Lemma 2.6. The modulus of the annular ring A D fx W a < jxj < bg is Mod.A/ D log

b : a

Note now that bigger rings have smaller modulus. As the components get closer together we can expect the modulus to tend to 1. Now, with this notion the Lipschitz estimate of (2.9) gives rise to a new characterisation of quasiconformality. Suppose that f W  ! Rn is a mapping and suppose that E; F   are continua with R D R.E; F I /. We define f .R/ D R.f .E/; f .F /I f .//: Then we may say that f is quasiconformal if 1 Mod.R/  Mod.f .R//  KMod.R/ K for every ring in . Notice that this requires 

1 1 M.R / K !n1

n1





1 !n1

n1

M.f .R/ /



K

1 !n1

(2.12) n1

M.R /

and hence 1 K 1=.n1/

M.R /  M.f .R/ /  K 1=.n1/ M.R /

and the precise measure of distortion (namely K) differs from that at (2.9) unless n D 2.

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We note this as a warning, it is not really of relevance unless one is seeking optimal constants regarding various sorts of estimates – continuity and so forth. Nevertheless, there is room for confusion. Aspects of the higher-dimensional theory are concerned with the continuity of the modulus of rings in the Hausdorff topology. We recall that subcontinua of Rn , Ej , converge to E in the Hausdorff topology if sup fdistq .x; Ej /g C sup fdistq .y; E/g ! 0; x2E

as j ! 1.

y2Ej

Here distq refers to the spherical distance of Rn , jx  yj dist q .x; y/ D p : p 2 jxj C 1 jyj2 C 1

(2.13)

Theorem 2.7. Suppose that E and F are disjoint continua (distq .E; F / > 0) and that Ej ! E and Fj ! F in the Hausdorff topology. Then Mod.R.Ej ; Fj I Rn // ! Mod.R.E; F I Rn //;

as j ! 1.

Also as the modulus of curve families decreases under inclusion, the modulus of rings increases. Lemma 2.8. Suppose E0  E1 and F0  F1 in , then mod.R.E0 ; F0 I //  mod.R.E1 ; F1 I //:

(2.14)

2.5 The Grötzsch and Teichmüller rings We have computed the modulus of the annulus above. What we need now are some more general rings whose modulus we can estimate well and prove some extremal properties for. The first is the Grötzsch ring. We denote for t > 1, RG .t/ D R.Bn ; Œt; 1/ W Rn /; where by Œt; 1/ we mean f.s; 0; : : : ; 0/ W t  sg. Next is the Teichmüller ring. Here RT .t/ D R.Œ1; 0; Œt; 1/ W Rn / and then we set n .t/ D Mod.RG .t//;

n .t / D Mod.RT .t //:

These two quantities are functionally related, and the following properties are not difficult to establish.

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Lemma 2.9. For t > 1, n .t/ D 2n1 n .t 2  1/, both n and n are continuous, strictly monotone and lim n .t/ D C1;

t&1

lim n .t/ D C1;

t&0

lim n .t / D 0;

t %1

lim n .t / D C1:

t %1

In two-dimensions these quantities can be explicitly written down in terms of elliptic integrals, but there are no such formulas known in higher dimensions. As will become apparent in a moment, it is necessary to get fairly good estimates of these functions at these extreme values as it is from these that equicontinuity results can be deduced, although there are other ways of course. Theorem 2.7 establishes the continuity of these functions. Here are some important estimates on these moduli due to Gehring, [23]. They are asymptotically sharp as t ! 1, but slightly better, if somewhat more complicated, estimates are known [96]. The number n below is known as the Grötzsch ring constant. The value of n is unknown in any dimension other than two, however we do know 1=n ! e as n ! 1. n Theorem 2.10. For each n  2 there is a constant n 2 Œ4; 2e n1 /, 2 D 4, such that   1n 1n  n .t /  !n1 log.t / (2.15) !n1 log.n t / and

1n   1n  n .t  1/  !n1 log.t / : !n1 log.2n t /

(2.16)

What is most important about the Teichmüller and Grötzsch rings are the extremal properties. These are proved by a higher-dimensional generalisation, due to Gehring [25], of the classical technique of symmetrisation in the complex plane. This was first used by Teichmüller for these sorts of applications. Given x0 2 Rn , the spherical symmetrisation E v of E in direction v is defined as follows; for r 2 Œ0; 1, E C \ Sn1 .x0 ; r/ ¤ ; if and only if E \ Sn1 .x0 ; r/ ¤ ; and then E v \ Sn1 .x0 ; r/ is defined to be the closed spherical cap centred on x0 C rv with the same .n  1/-spherical measure as E \ Sn1 .x0 ; r/. Thus if E is connected, then so is E v and E v is rotationally symmetric about the ray x0 C rv, r > 0. We symmetrise a ring consisting of two components E and F by symmetrising E in the direction v to get E  and F in the direction v to get F  . Then E  and F  are disjoint as the spherical measure .E [ F / \ Sn1 .x0 ; r/ is strictly smaller than Sn1 .x0 ; r/ and so R D R.E  ; F  / is again a ring. Then we have the following very useful theorem: Theorem 2.11. Let x0 2 Rn and v 2 Sn1 and let R D R.E; F / be a ring and R D R.E  ; F  / be its symmetrisation. Then Mod.R /  Mod.R/:

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Next, a symmetrised ring contains a Möbius image of a Teichmüller ring. Namely, the two line segments E  \ Lv and F  \ Lv in the ray Lv D ft v W t 2 Rg. Note that only one of which may be of infinite length, however they might both be finite. In the latter case a Möbius transformation can be used to ensure one of the components is unbounded. This leads to the following important extremal property of Teichmüller rings. Theorem 2.12. Let R.E; F / be a ring with a; b 2 E and c; 1 2 F . Then 

Mod.R/  Mod RT



ja  cj : ja  bj

By conformal invariance we obtain the following corollary: Corollary 2.13. Let R.E; F / be a ring with a; b 2 E and c; d 2 F . Then    Mod.R/  Mod RT ja; b; c; d j/ ; where the cross ratio is defined by ja; b; c; d j D

ja  cjjb  d j : ja  bjjc  d j

(2.17)

2.6 Hölder continuity From the extremity of the Grötzsch and Teichmüller rings and estimates on their modulus we obtain modulus of continuity estimates for quasiconformal mappings. Ultimately these give Hölder continuity estimates once we estimate a certain distortion function which we now describe. Let f W   Rn ! Rn be K-quasiconformal, x 2  and put r D d.x; @/. For all y 2  with jxyj < r, the ring R D Bn .x; r/nŒx; y lies in  and is conformally equivalent to the Grötzsch ring n .r=jxyj/ by an obvious inversion. Next f .R/ is a ring with one finite component containing f .x/ and f .y/ and the other unbounded. By Theorem 2.12, the extremalilty of the Teichmüller ring, we have     d.f .x/; @/ d.x; @/

n  Kn : jf .x/  f .y/j jx  yj Thus



jx  yj jf .x/  f .y/j  'n;K d.f .x/; @/ d.x; @/



(2.18)

where 'n;K is the distortion function 'n;K .t/ D n1 .Kn .1=t //:

(2.19)

Because of (2.18) and its various generalisations to special situations, the distortion function is much studied. Gehring [23] showed that there was a constant Cn;K such that (2.20) 'n;K .t/  Cn;K t 1=K

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whenever t < 1=2 but there are much more refined estimates now, see [96]. Combining both (2.18) and (2.20) gives Hölder continuity, and in fact equicontinuity since the constants do not depend on the map in question, but only their distortion. Theorem 2.14. Let f W  ! 0 be a homeomorphism such that Mod.f .R//  KMod.R/;

(2.21)

for all rings R  . Then for all y < d.x; @/=2 we have the modulus of continuity estimate   jf .x/  f .y/j jx  yj 1=K  Cn;K : (2.22) d.f .x/; @0 / d.x; @/ In two-dimensions everything can be made rather more precise. For instance the following is well known. Theorem 2.15. Let f W D ! D, f .0/ D 0 be a K-quasiconformal mapping of the unit disk into itself. Then jf .z/j  411=K jzj1=K : This theorem has nice asymptotics as K ! 1 recovering the classical Schwarz inequality. Further, the K-quasiconformal map z 7! zjzj11=K shows the Hölder exponent to be optimal as well.

2.7 Mori distortion theorem Actually, if one follows the ideas above and estimates on a larger scale one achieves an important result of Mori [73], [96]. Theorem 2.16. There is a constant Cn;K such that if f W Rn ! Rn with f .0/ D 0 is K-quasiconformal with respect to rings, then jf .x/j  Cn;K jf .y/j

(2.23)

whenever jxj D jyj. We have the estimate

p Cn;K  Œn1 .n . 2/=K/2

where n is the Grötzsch ring modulus. When the normalisation f .0/ D 0 is removed we have Corollary 2.17. If f W Rn ! Rn is K-quasiconformal, then jf .x/  f .z/j  Cn;K jf .y/  f .z/j whenever jx  zj D jy  zj.

(2.24)

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Mori’s result is one of a class of results in the distortion theory of the geometry of mappings. Many other such can be found in Vuorinen’s book [96], including higherdimensional versions of the Schwarz lemma and so forth for quasiregular mappings. Two further interesting results for quasiconformal mappings measure the distortion of the cross ratio of the points x, y, z, 1. These in effect lead to the notion of quasisymmetry and when Möbius invariance is used to normalise away the behaviour at 1 we get the notion of quasi-Möbius mappings. The ideas are not particularly difficult and follow in much the same way as the distortion estimate of (2.18). Theorem 2.18 (Local quasisymmetry). For each K  1 and s 2 .0; 1/, there is a strictly increasing function s;K W Œ0; 1/ ! Œ0; 1/ with .0/ D 0 with the following properties. If x; y; z 2 Bn .0; s/ with x ¤ z, then 

jx  yj jf .x/  f .y/j  s;K jf .x/  f .z/j jx  zj



(2.25)

for every K-quasiconformal f W Bn ! Rn . Explicit (but a little complicated) formulas are easily obtained for the function s;K in terms of the Grötzsch and Teichmüller functions. Notice that from (2.18), and the obvious fact that s;K .1/  t;K .1/ if s  t , one immediately obtains the bound on the linear distortion H.x; f /  s;K .1/: After a rescaling argument, we also obtain a global version of this. Theorem 2.19 (Global quasisymmetry). For each K  1 there is a strictly increasing function K W Œ0; 1/ ! Œ0; 1/ with .0/ D 0 such that 

jx  yj jf .x/  f .y/j  K jf .x/  f .z/j jx  zj



(2.26)

for every K-quasiconformal f W Rn ! Rn .

3 Compactness An equally important aspect of quasiconformal mappings are their compactness properties. Usually these are couched in terms of normal family type results. Recall a family of mappings F D ff W   Rn ! Rn g is said to be normal if every sequence ffn g1 nD1  F contains a subsequence which converges uniformly on compact subsets of . The modulus of continuity estimates of Theorem 2.14 guarantee the equicontinuity, and therefore via the Arzela–Ascoli

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theorem, the normality of any suitably normalised family of K-quasiconformal mappings. Not only that of course, the bilipschitz estimate on the distortion of moduli also shows the family of inverses (restricted to a suitable domain of common definition) is also normal. These observations quickly lead to compactness results. The most elementary of these is the following. Theorem 3.1. Let   Rn and x0 ; y0 2 . Then the family FK D ff W  ! Rn ; f .x0 / D 0; f .y0 / D 1; and f is K-quasiconformalg is a normal family. Obviously something is necessary here as the family of conformal mappings of Rn , fx 7! nxg1 nD1 is not normal. If one wants to add the point at 1 to the discussion yn and consider families of K-quasiconformal maps defined on the Riemann sphere R normalisation at three points is all that is required to guarantee normality. Next, as convergence is given by Theorem 3.1 the issue is whether the limit map is actually quasiconformal. The next theorem establishes this.

3.1 Limits of quasiconformal mappings Theorem 3.2. Fix K. Let fj W  ! j be a sequence of K-quasiconformal mappings converging pointwise to f W  ! Rn . Then one of the following occurs. • f is a K-quasiconformal embedding and the convergence is uniform on compact subsets. • f ./ is a set with two points with one value attained only once. • f is constant. y n , and with the obvious interpretation of continuity at infinity and so When  D R forth, we have the following convergence properties of quasiconformal mappings of the Riemann sphere. yn ! R y n be a sequence of K-quasiconformal mapTheorem 3.3. Fix K. Let fj W R such that one of the following occurs. pings. Then there is a subsequence ffjk g1 kD1 yn ! R y n and both fj ! f • There is a K quasiconformal homeomorphism f W R k 1 y n , or ! f uniformly on R and fj1 k y n , possibly x0 D y0 , such that • There are constant x0 ; y0 2 R y n n fy0 g; fjk ! x0 locally uniformly in R and y n n fx0 g: ! y0 locally uniformly in R fj1 k

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Basically it is true that any sufficiently normalised family of quasiconformal mappings forms a normal family. However, there is a far reaching generalisation of these sorts of results. It is Rickman’s version of Montel’s Theorem which we discuss a bit below. There are also more general results about the normal family properties of families of mappings with finite distortion. Typically very little can be said but if, for instance, the distortion function jDf .x/jn K.x; f / D (3.1) J.x; f / has strong integrability properties such as being exponentially integrable, then there are very similar results to those of Theorems 3.1, 3.2 and 3.3 available [46].

4 The analytic definition of a quasiconformal mapping Examining the proof of Theorem 2.2, it becomes apparent that we should get the Lipschitz estimate of (2.9) if we were to have the pointwise estimate between the differential matrix Df and its determinant. jDf .x/jn  K J.x; f /

(4.1)

which of course is close to (2.7). We would need this for both f and its inverse of course, but at least where the differential is nonsingular if we write out the eigenvalues of the symmetric positive definite matrix Df t .x/Df .x/ as 1  2      n , then the inequality (4.1) reads as nn  K 2 .1  2      n / and this certainly implies n  K 2 1 and hence  K 2.n1/ n1 : 1  2      n  1 n1 n Therefore writing g D f 1 W 0 !  we would have jDg.y/jn  K n1 J.y; g/;

y D f .x/;

(4.2)

so g will also have a Lipschitz estimate, thus giving the bilipschitz estimate we want – albeit with different constants K. One of course needs some sort of Sobolev regularity to make all this work, and that leads us to the analytic definition of quasiconformal mappings. Let f W  ! 0 be a homeomorphism belonging to the Sobolev class 1;n ./ of functions whose first derivatives are locally Ln -integrable. Then f is Wloc K-quasiconformal if there exists a K, 1  K  1, such that jDf .x/jn  K J.x; f / at almost every point x 2 .

(4.3)

We again need to point out that the constant K here is not the same as that for rings. Further, it is not in general true that the composition of W 1;n mappings is again of

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Sobolev class W 1;n , nor is it true that the inverse of a W 1;n homeomorphism is W 1;n , so the fact that the composition of quasiconformal mappings and the inverse of a quasiconformal mapping are again quasiconformal, discussed in Theorem 2.3, are now not nearly so direct. There are advantages however; considering this definition, one sees that the hypothesis that f is a homeomorphism is largely redundant. We therefore say that f is 1;n Sobolev regularity. K-quasiregular if f satisfies (4.3) and has the appropriate Wloc 1;n In fact the hypothesis that f 2 Wloc ./ ensures that the Jacobian determinant of f is a locally integrable function and gives one a chance of establishing such things as the change of variable formula and so forth. From this purely analytic definition of quasiconformal mappings, Reshetnyak was able to establish important topological properties. Theorem 4.1 (Reshetnyak). A quasiregular mapping f W   Rn ! Rn is open and discrete. With this we now recall Chernavskii’s theorem [14]. Theorem 4.2. Let Bf denote the branch set of a quasiregular mapping f W  ! Rn , that is, Bf D fx 2  W f is not locally injective at xg: (4.4) Then the topological dimension of both Bf and f .Bf / is less than or equal to n  2. These two results, Theorems 4.1 and 4.2, together with fairly general topological degree theory and covering properties of branched open mappings established by Väisälä [93] and others give various quite general path lifting properties of these mappings [62], [63], [64], [83], and the well-known result of Poletsky [77]. With these properties at hand one can study the deformations of curve families in the more general setting of quasiregular mappings. The distortion bounds on the modulus enable the geometric methods of modulus to be used to great effect to build a theory analogous to that of analytic functions in the complex plane. There is a considerable body of research building around these topological questions for mappings of finite distortion. The questions become deep and subtle and beyond the scope of this chapter, but the interested reader can consult [46] and work forward to the many interesting current research directions. There are a few other consequences of the analytic definition that need to be recounted. These are key features for the analytic theory of these mappings showing sets of zero-measure are preserved. Theorem 4.3 (Condition N and N 1 ). Let f W  ! Rn be non-constant quasiregular mapping. • If A has measure 0, jAj D 0, then jf .A \ /j D 0.

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• If jBj D 0, then jf 1 .B/j D 0. • J.x; f / > 0 almost everywhere in . • jBf j D 0, and hence jf .Bf /j D 0.

5 The Liouville theorem In 1850, the celebrated French mathematician Joseph Liouville added a short note to a new edition of Gaspard Monge’s classic work Application de l’Analyse à la Géometrie, whose publication Liouville was overseeing. The note was prompted by a series of three letters that Liouville had received in 1845 and 1846 from the renowned British physicist William Thomson. Thomson, better known today as Lord Kelvin, had studied in Paris under Liouville’s in the mid-1840s, so these two giants of nineteenth century science were well acquainted. In his letters, Thomson asked Liouville a number of questions concerning inversions in spheres, questions that had arisen in conjunction with Thomson’s research in electrostatics, in particular, with the so-called principle of electrical images (we point out that the reflection in the unit sphere S2 of R3 is often referred to in physics as the “Kelvin transform.”) More about the interesting relationship between Thomson and Liouville can be found in Jesper Lützen’s magnificent biography of Liouville . The substance of Liouville’s note is conveyed by the following remarkable assertion: Theorem 5.1 (Liouville’s theorem). If  is a domain in Rn , n  3, then any conformal mapping f W  ! Rn is the restriction to  of a Möbius transformation y n. of R Exactly what is meant by a conformal mapping is a modern day issue around the regularity theory of solutions to PDEs such as the Cauchy–Riemann system below at (5.1). But in Liouville’s time he certainly understood such mappings to be many times differentiable and following his motivation for writing the article, Liouville couched his discussion in the language of differential forms rather than mappings. As a consequence, his original formulation bears little resemblance to the theorem above, although the relationship between the two formulations is quite clear via differential geometry. However Liouville’s title, “Extension au cas de trois dimensions de la question du tracé géographique” gives no hint whatsoever as to the results. It was only later that Liouville published his theorem in a form approximating the statement of it that we have given. As we hinted, the proof which Liouville outlined for his theorem makes use of certain implicit smoothness hypotheses which when unwound gives the added assumption that f is a mapping of class C 3 ./, that is, three times continuously differentiable, or yn better. In higher dimensions the group Möb.n/ of all Möbius transformations of R

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consists of all finite compositions of reflections in spheres and hyperplanes. It is easy to see that these mappings provide examples of conformal transformations. They are of course all C 1 ./. The smoothness assumption in Liouville’s theorem is not optimal. One would like to relax the injectivity assumption to allow the possibility of branching and also to relax the differentiability assumption as much as possible. The natural setting for Liouville’s theorem is a statement about the regularity of solutions to the Cauchy–Riemann system D t f .x/ Df .x/ D J.x; f /2=n Id;

almost everywhere in ;

(5.1)

where Id is the n  n identity matrix. If f W  ! R is a 1-quasiconformal mapping, then pointwise almost everywhere we must have the positive semidefinite matrix Df t .x/Df .x/ having a single eigenvalue with multiplicity n. Thus either Df .x/ D 0nn and J.x; f / D 0 or, as a little linear algebra will reveal, Df .x/ is a scalar multiple of an orthogonal transformation. In particular, with the analytic defi1;n ./ nition of quasiconformality we see that a 1-quasiconformal mapping f is a Wloc solution to the equation (5.1). With this formulation we have the following very strong version of the Liouville theorem established using the nonlinear Hodge Theory developed in [44], [41]. In two-dimensions it is analogous to the classical Looman–Menchoff Theorem. Note especially that there is no longer any assumption of injectivity – it is a consequence of the theorem. n

1;p ./ be a weak solution to the Theorem 5.2. Let   Rn , n  3 and let f 2 Wloc equation (5.1). If n is even, then any solution with p  n=2 is the restriction to  y n . If n is odd, then there is an D .n/ such that of a Möbius transformation of R any weak solution with p  n  is the restriction to  of a Möbius transformation y n. of R This is sharp in the following sense. In all dimensions n  2 and all p < n=2, there 1;p 1;n ./ which is not in Wloc ./ and so in particular is a weak solution to (5.1) in Wloc is not a Möbius transformation.

The discrepancy here between what is known in odd dimensions and even dimensions is one of the central unsolved problems in the theory. Further, although the results are very sharp in even dimensions, there remains the possibility of improvement. For instance it might be that the Liouville theorem remains true in Rn , n  3, 1;1 ./ solutions which are continuous. for weak Wloc We draw attention to one significant corollary of Liouville’s theorem: the only subdomains  in Rn with n  3 that are conformally equivalent to the unit ball Bn are Euclidean balls and half-spaces. This stands in stark contrast to the marvelous discovery by Riemann, announced in 1851 a year after Liouville’s note was published: any simply connected proper subdomain   C of the complex plane is conformally equivalent to the unit disk D. From the formulation of Liouville’s theorem in Theorem 5.2 we are naturally led to the basic connections between quasiregular mappings and non-linear PDEs through

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the Beltrami system. Let S.n/ denote the space of symmetric positive definite n  n matrices of determinant equal to 1. Geometrically S.n/ is a non positively curved complete symmetric space. Given a subdomain  of Rn and G W  ! S.n/ a bounded measurable mapping we define the Beltrami equation as D t f .x/ Df .x/ D J.x; f /2=n G.x/

for almost every x 2 .

(5.2)

To each non-constant quasiregular mapping, there corresponds a unique (tautological) Beltrami equation and we refer to G as the distortion tensor of the mapping f . A key approach to the modern theory is to examine properties and obtain geometric information about quasiregular mappings (and more general mappings of finite distortion) when they are viewed as solutions to this and related PDEs. These equations are studied from many points of view, as the Euler–Lagrange equations for the absolute minima of variational integrals, at the level of differential forms using exterior algebra and also as equations relating the Dirac operators of conformal and spin geometry, see [46] for results in these directions.

6 Gehring’s higher integrability In a remarkable paper in 1973, F. W. Gehring established that the Jacobian determinant of a K-quasiconformal mapping is integrable above the natural exponent. That is, the 1;n .; Rn / together with the bound on distortion implies that assumption f 2 Wloc 1;nC f 2 Wloc .; Rn / for some depending on n and K. Gehring gave explicit estimates on . While this result was already known in the plane due to the work of Bojarski [9], and perhaps anticipated in higher dimensions, it is impossible to overstate how important this result has proven to be in the theory of quasiconformal mappings and more generally Sobolev spaces and in the theory of non-linear PDEs. The techniques developed to solve this problem, for instance the well-known reverse Hölder inequalities, are still one of the main tools used in several disciplines, including non-linear potential theory, non-linear elasticity, PDEs and harmonic analysis. We state the following version of Gehring’s result as proved in [46] which also gives the result for quasiregular mappings. 1;q ./ satisfying Theorem 6.1. Let f W  ! Rn be a mapping of Sobolev class Wloc the differential inequality (6.1) jDf .x/jn  K J.x; f /: 1;p Then there are K ; K > 0 such that if q > n  K , then f 2 Wloc ./ for all p < n C K .

As an immediate corollary we have the following:

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Theorem 6.2. Let f W  ! Rn be a K quasiconformal mapping. Then there is 1;p pK > n such that f 2 Wloc K ./. The higher-dimensional integrability conjecture here would assert that if f satisfies 1;q ./ for some q > nK=.KC1/, then f actually lies in the Sobolev (6.1) and lies in Wloc 1;p space Wloc ./ for all p < nK=.K 1/. In dimension two this conjecture was proven by K. Astala [4]. In even dimensions rather more is known and the numbers K and

K > 0 can be related to the p-norms of certain singular integral operators which can be estimated. Indeed the conjecture would follow from the current conjectural identification of these norms. In odd dimensions rather less is known. In any case, Theorem 6.2 yields the following Corollary 6.3 (Reverse Hölder inequality). Let f W   Rn ! 0 be a K-quasiconformal mapping. Then there is p D p.n; K/ > 1 and C D C.n; K/ such that 

1 jQj

“

1=p p

J.x; f / dx Q

C  jQj

“ J.x; f / dx

(6.2)

Q

for all cubes Q such that 2Q  . Actually, our presentation here is a bit back to front as it is via the reverse Hölder inequality at (6.2) that the higher integrability Theorem 6.2 was first established. The restriction to cubes Q so that 2Q   is necessary but can be removed under assumptions about the regularity of 0 D f ./ – namely that it should be a John domain. There is another connection here to the nonlinear potential theory as the estimate shows the Jacobian J.x; f / to be an A1 Muckenhoupt weight on the cubes Q. Another interesting unsolved problem concerns the question of when a positive function can be the Jacobian of a quasiconformal mapping. Obviously the results above impose restrictions on such a function. Another is given by Reimann’s result (see [79]): Theorem 6.4 (Reimann’s theorem). Let f W Bn ! Rn be K-quasiconformal. Then log J.x; f / is a function of bounded mean oscillation.

7 Further stability and rigidity phenomena Along with the Liouville theorem there are other interesting phenomena which occur only in higher dimensions, n  3. For instance consider the following local to global homeomorphism property. In 1938 Lavrentiev [49] asserted that a locally homeomorphic quasiconformal mapping R3 ! R3 is a global homeomorphism onto. This assertion was proved correct by Zorich [98] in all dimensions n  3.

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Theorem 7.1. Let f W Rn ! Rn be a locally homeomorphic quasiregular mapping. If n  3, then f is a globally injective quasiconformal mapping onto Rn . The condition n  3 is essential as the exponential mapping e z in the plane demonstrates. Zorich’s theorem was generalised by Martio–Rickman–Väisälä in the following way (an earlier result of F. John proved the same result for locally bilipschitz mappings). Theorem 7.2. There is a positive constant r D r.n; K/ with the following property. If f W Bn ! Rn is a locally injective K-quasiregular mapping, then f jBn .0; r/ is injective. The number r.n; K/ in the above theorem is called the injectivity radius. Zorich’s result clearly follows from this result by scaling. There are also interesting local to global injectivity results for quasiregular mappings between Riemannian manifolds. In this vein the following result of Gromov is perhaps best known [31]. Theorem 7.3. If f W M ! N is a locally homeomorphic quasiregular mapping of a complete Riemannian n-manifold M of finite volume into a simply connected Riemannian manifold N with n  3, then f is injective and N nf .M / is of Hausdorff dimension zero. These results are very well presented in [83]. There are also stability results of a different nature. These are based on the compactness properties of mappings of finite distortion and the Liouville theorem. Roughly speaking one can show that in all dimensions as K ! 1, K-quasiregular mappings are uniformly well approximated by conformal mappings. Liouville’s theorem implies in dimension n  3 that conformal mappings are Möbius transformations. Thus in dimension n  3 for sufficiently small K we obtain local injectivity by virtue of the uniform approximation by a globally injective mapping, see [81] and for an interesting application [65]. For instance one has Theorem 7.4. For each n  3 there is a constant ı.n; / with the following properties. • ı.n; / ! 0 as ! 0C . • If f W Bn ! Rn is a K-quasiregular mapping with K  1 C , then there is a Möbius mapping  W Bn ! Rn such that sup j. 1 B f /.x/  xj < ı.n; /:

(7.1)

x2B

Also we mention the following connection between distortion and local injectivity. Again, this is a higher-dimensional phenomena. The map f W z 7! z 2 is a 1-quasiregular map of C with Bf D f0g ¤ ;. Contrast this with the following theorem.

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Theorem 7.5. There is a constant K0 > 1 with the following property. Let f W  ! Rn , n  3, be K-quasiregular. If K < K0 , then Bf D ;. That is, f is locally injective. The number K0 depends on the particular definition of the distortion K. But with the geometric and analytic definitions the number is expected to be equal to 2 (known as the Martio conjecture) . The best known bound is due to Rajala [78] and is only very slightly bigger than 1, but it is explicit and not derived from a compactness argument. It would be a major advance to establish the sharp result here. Among other consequences it is known that if the distortion tensor of a quasiregular mapping is close to continuous in the space of functions of bounded mean oscillation (BMO), or in particular continuous, then local injectivity follows. Closely related results can be found in [61]. Again, many of these sorts of results are based around compactness arguments and do not give effective information. These results explain why we really need to consider measurable conformal structures in the defining equation (5.2) as any degree of regularity of the distortion tensor forces local injectivity. Precisely what regularity is necessary is a study currently under intense investigation.

8 Quasiconformal structures on manifolds In this chapter we will not delve too deeply into the theory of quasiconformal mappings on manifolds. Of course the local theory, regularity, and compactness results pretty much follow from the Euclidean theory, but there are some quite subtle and interesting aspects to the theory that warrant deeper investigation. These investigations are far from complete at present. The starting point for questions concerning quasiconformal mappings and structures on manifolds is Sullivan’s uniformisation theorem which tells us that, apart from dimension n D 4, every topological manifold admits a unique quasiconformal structure (quasiconformal coordinates). This theorem is quite remarkable in that it allows analytical calculation on topological manifolds – manifolds which may not even admit a differentiable structure. Thus one may seek to calculate topological invariants analytically. Further, it is a consequence of uniqueness that two different smooth structures on the same compact manifold are quasiconformally equivalent. The classical uniformisation theorem in complex analysis states that every surface F admits a conformal structure. That is a set of local coordinates f.'˛ ; U˛ /g˛2A with S 1 ˛ U˛ D F and '˛ W U˛ ,! C in which the transition mappings '˛ 'ˇ W 'ˇ .U˛ \ Uˇ / ! C are conformal mappings for all ˛ and ˇ between planar subdomains. Every surface has a simply connected covering space which inherits this conformal structure. y C or the The monodromy theorem then implies that this covering space is one of C, y C, Cnfz0 g unit disk D D fz 2 C W jzj < 1g. For every (orientable) surface except C,

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and the torus, the universal cover is the unit disk and the group of cover translations is a subgroup of the group of conformal automorphisms of D, that is, a group of linear fractional transformations, called a Fuchsian group. This result is of course one of the most profound results in complex analysis. The theory of Fuchsian groups developed by Poincaré laid the foundations for the study of discrete groups of transformations of more general spaces and geometries. Quasiconformal mappings play an essential rôle in the study of Fuchsian groups and their orbit spaces, Riemann surfaces. The theory of Teichmüller spaces uses quasiconformal mappings to study the various conformal structures on a given Riemann surface. This is amply demonstrated in the contents of this book. However, what we want to consider here is the extent to which the uniformisation theorem might be true in higher dimensions. Because of the rigidity of conformal mappings in space it is not to be expected that every n-manifold admits a conformal structure. Although Perelman’s recent proof of Thurston’s geometrisation theorem, building on earlier work of Hamilton, suggests that this is nearly the case in dimension 3.

8.1 The existence of quasiconformal structures In general, given any pseudo-group of homeomorphisms of Euclidean space one can define the associated category of manifolds using the pseudo-group to provide local coordinates. The two most familiar pseudo-groups are of course the pseudo-group of homeomorphisms, giving rise to topological manifolds, and the pseudo-group of diffeomorphisms, giving rise to C 1 or smooth manifolds. Other examples of possible structures would be piecewise linear, real analytic, complex analytic and so forth. One of the fundamental problems of topology has been to determine when a topological manifold admits a “nicer” structure than that given a priori and how many different sorts of similar structures exist on a particular manifold. For instance one might ask: when does a topological manifold admit a smooth structure? Given a manifold with two potentially different smooth structures, are they the same by a smooth change of coordinates? Notice that the possibility of admitting a smooth structure is a topological invariant. That is, if M and N are homeomorphic and M is smooth, then N admits a smooth structure, obtained by simply declaring that the homeomorphism is a smooth map. Due to the work of Moise and others in three dimensions, the differences between smooth and topological structures first shows up in dimension 4. Because of the work of Freedman [20] and Donaldson [15] we know that there are plenty of 4 manifolds which do not admit any smooth structure and, quite surprisingly, topological manifolds as simple as 4-dimensional Euclidean space which admit many different smooth structures. A similar situation persists in higher dimensions. Notice that in order to do calculus or study function theory on a manifold some smoothness assumptions are necessary on the coordinate charts. From the geometric point of view quasiconformal manifolds would seem a natural starting point. We say

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a manifold M admits Sa quasiconformal structure if there is a set of local coordinates f.'˛ ; U˛ /g˛2A with ˛ U˛ D M and '˛ W U˛ ,! Rn in which the transition mappings '˛ 'ˇ1 W 'ˇ .U˛ \ Uˇ / ! Rn

(8.1)

are quasiconformal mappings of subdomains of Rn for all ˛ and ˇ. Notice that there is no assumption on the distortion of the transition charts other than boundedness. Since quasiconformal mappings of subdomains of Rn have Ln -integrable first derivatives they admit enough structure so as to be able to define differentiation, speak of differential forms and exterior derivatives, define a de Rham type cohomology theory and discuss the index theory of certain differential operators. We can speak of conformal and quasiregular mappings between quasiconformal manifolds and study conformal invariants of such manifolds. The reader should be aware of the complexity and some nuances of the very definition of the various Sobolev classes of mappings between manifolds with measurable metric tensors, see Bethuel [7] and Hajtasz [32]. The circle of ideas concerning the question of regularity of topological manifolds from the point of view of analysis is interesting and important. There are two principal properties of a pseudo-group of transformations in a given category, denoted CAT, (for instance smooth, piecewise linear or quasiconformal) to imply that a topological manifold admits such a structure, and if it does so, then it is unique. These are • Deformation. Two CAT homeomorphisms which are uniformly close in the C 0 topology can be deformed one to the other through CAT homeomorphisms (a suitable relative version of this statement is also necessary). • Approximation. Any homeomorphism Bn ,! Rn can be uniformly approximated in the C 0 -topology by a CAT homeomorphism. In a remarkable piece of work D. Sullivan established the deformation property in all dimensions for the category of quasiconformal mappings [87] and he also laid the foundations for much of the recent work in geometric topology in the quasiconformal category, notably the work of Tukia and Väisälä [92]. Sullivan also established the approximation property for n ¤ 4. The basic tool was a hyperbolic version of the Edwards–Kirby furling technique of geometric topology. As a consequence of Sullivan’s work we have the following remarkable result which one might regard as an analogue of the 2-dimensional uniformisation theorem. Theorem 8.1. Every topological n-manifold, n ¤ 4, admits a unique quasiconformal structure.

8.2 Quasiconformal 4-manifolds The revolution in our understanding of the theory of 4-manifolds initiated by Donaldson and Freedman has not left the theory of quasiconformal mappings untouched. As

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we have discussed above Sullivan’s uniformisation theorem implies that every topological n-manifold, n ¤ 4, admits a quasiconformal structure. This leaves open the question of what possible structures can exist on an arbitrary topological 4-manifold. Donaldson and Sullivan attacked this problem in a beautiful paper in 1990 [16] which heralded many new ideas into the theory of quasiconformal mappings. Their approach was to take a quasiconformal 4-manifold and develop the associated globalYang–Mills theory on such a manifold and thereby produce the same sorts of invariants associated to intersection forms that are used to distinguish the topological manifolds which admit smooth structures from those that do not. Recall that in 1982 Freedman gave a complete classification of compact simply connected 4-manifolds [20] by establishing the 4-dimensional h-cobordism theorem. Thus there is exactly one simply connected topological 4-manifold for each given unimodular intersection form. In 1983 Donaldson [15] showed the only negative definite forms which are realised as the intersection forms of smooth compact simply connected 4-manifolds are the standard diagonalisable forms (the hypothesis on simple connectivity was later removed). These results then provide a mechanism for finding topological 4-manifolds which do not admit smooth structures. If one could develop the necessary Yang–Mills theory for manifolds with less smoothness assumptions, then one could similarly provide examples of topological 4-manifolds which do not admit quasiconformal structures. The development of this theory in the quasiconformal category is highly nontrivial and significant technical obstructions need to be overcome. Notice that for instance quasiconformal manifolds do not admit Riemannian metrics and the smooth construction depends on splitting the curvature into self-dual and anti-self-dual parts to define the anti-self-dual moduli space of connections modulo gauge equivalence. These are the objects from which the invariants are computed. In the quasiconformal category, Donaldson and Sullivan set up some differential geometric invariants on a quasiconformal manifold M based around the existence of a measurable conformal structure. Principally these were the anti-self dual Yang–Mills equations. Since the Yang–Mills equations are conformally invariant, the measurable conformal structure can be used to define the anti-self-dual connections. The analysis of these connections requires the non-linear Hodge theory and the improved regularity properties of quasiregular mappings. The fact that quasiconformal mappings preserve the “correct” Sobolev spaces plays no small part in this development. Their main results are as follows. Theorem 8.2. There are topological 4-manifolds which do not admit any quasiconformal structure. Theorem 8.3. There are smooth compact (and therefore quasiconformal) 4-manifolds which are homeomorphic but not quasiconformally homeomorphic. As far as we are aware the question of whether there are quasiconformal 4-manifolds which do not admit smooth structures remains open. Also as a consequence of the

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deformation and approximation theory properties we discussed above with regard to Sullivan’s uniformisation theorem we obtain the following curiosity in dimension 4. Corollary 8.4. There is an embedding ' of the 4-ball, ' W B4 ,! R4 , which cannot be approximated uniformly in the spherical metric by a quasiconformal homeomorphism. This means that there is > 0 such that if sup q.f .x/; '.x// < ;

(8.2)

x2B

then f is not K-quasiconformal for any K < 1. The methods developed by Donaldson and Sullivan perhaps allow one to extend the Atiyah–Singer index theory of the first order elliptic differential operators to quasiconformal 4-manifolds (and to other even dimensions), and to study the de Rham cohomology. Earlier results along these lines had been developed and studied by Teleman using the Lipschitz structures on topological n-manifolds guaranteed by Sullivan’s results. Recent results of Sullivan and others seem to suggest that the Seiberg–Witten equations cannot be used so effectively in the quasiconformal category. Thus there is perhaps a distinction between the topological, quasiconformal and smooth categories in dimension 4.

8.3 The extension problem The deformation and approximation theory developed by Sullivan for quasiconformal mappings has other important applications. One of these is the extension or boundary value problem: can a quasiconformal homeomorphism f W Rn ! Rn be lifted to a quasiconformal mapping of RnC1 ? Actually, since f .1/ D 1 defines a quasiconformal homeomorphism of the Riemann sphere Sn  Rn [ f1g the problem is usually formulated as asking if given a quasiconformal homeomorphism f W Sn ! Sn , is there a quasiconformal F W BnC1 ! BnC1 such that F jSn D f ? If the answer is “yes” we would also like it to be quantitative. The answer to this rather elementary question took rather a long time to find. In one dimension it is a well-known theorem ofAhlfors and Beurling concerning the boundary values of quasiconformal mappings of the disk (and quasisymmetric mappings). In dimension three Carleson gave a proof which relied on some combinatorial/piecewise linear topology which is not available in higher dimensions [12]. Tukia and Väisälä developed and applied Sullivan’s ideas to solve this problem, [92]. We remark that it is not at all obvious that this should be the case though. For instance, it follows from Milnor’s work that there is a diffeomorphism f W S6 ! S6 which cannot be extended to a diffeomorphism of B7 . The solution to the lifting problem shows there is however a quasiconformal extension (since a diffeomorphism of the sphere is quasiconformal).

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Theorem 8.5. Let K  1. There is K  D K  .n; K/ such that if f W Sn ! Sn is K-quasiconformal, then there exits a K  -quasiconformal F W BnC1 ! BnC1 such that F jSn D f . This theorem is quite nontrivial to establish. Basically one constructs an obvious extension which will not in general be a homeomorphism but is “almost quasiconformal” at large scales in the hyperbolic metric of the ball Bn . Such things are called quasi-isometries in the literature. The approximation results of Sullivan & Tukia– Väisälä show that such mappings can be approximated in the C 0 topology of the hyperbolic metric of the ball by quasiconformal maps. The technical condition they require is called '-solid. Finally, any two maps which are a bounded distance apart in the hyperbolic metric agree on the sphere – an elementary consequence of hyperbolic geometry – and so the constructed quasiconformal approximation is an extension of the given boundary values.

8.4 Boundary values of quasiconformal mappings The converse problem to the problem discussed above is well known and rather easier: Theorem 8.6. Let F W Bn ! Bn be a K-quasiconformal homeomorphism. Then F yn ! R y n , F  jBn D F . Further, if f D F  jSn1 , extends quasiconformally to F  W R n  3, then f W Sn1 ! Sn1 is K-quasiconformal. Actually for this theorem we may as well assume F .0/ D 0 and the extension can be effected by reflection; for jxj > 1 define F  .x/ D

F .x=jxj2 / : jF .x=jxj2 /j2

The difficulty now lies in establishing that F has a continuous extension to the boundary, but this follows from quite direct modulus estimates. Since F 1 also satisfies the same hypothesis, it also has a continuous extension to Sn1 , so F has a homeomorphic extension and it directly follows that this extension is quasiconformal. This leads one directly to consider the Carathéodory problem for the boundary values of quasiconformal mappings. Recall that Carathéodory proved that a conformal map of the unit disk ' W D !  extends homeomorphically to the boundary if and only if @ is a Jordan curve. The topological obstructions to such a result in higher dimensions are manifest, especially considering that the quasiconformal image of a ball could have as boundary a wildly knotted sphere. However, it is quite clear the extension result will remain valid if locally the boundary is quasiconformally equivalent to the boundary of the ball (a notion referred to as collaring). However not much beyond this is known.

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Finally here, the reader familiar with complex analysis will be well aware of the substantial theory around the structure and properties of bounded analytic functions. There is an analogous theory for quasiregular mappings and while there are some interesting results there remains some substantial issues to be resolved. A major question concerns the existence almost everywhere of radial limits, a well-known and useful result for analytic mappings. (f has radial limits at  2 @Bn if whenever Bn 3 xn !  such that 1  jxn j  jxn  j, then f .xn / has a limit). Thus one might ask the following: given a bounded quasiregular mapping f W Bn ! Rn is it true that f has radial limits almost everywhere? At this point I believe it is not known even if f has a single radial limit. These sorts of results are known with additional assumptions such as finite Dirichlet energy, see e.g. [96].

8.5 Generalised Beltrami systems Recall that a measurable conformal structure on a domain   Rn is a measurable map G W  ! S.n/, the non-positively curved symmetric space of positive definite symmetric n  n matrices of determinant equal to 1. We will always assume that such a map is bounded and such an assertion is equivalent to the assumption that there is a constant K < 1 such that max jG.x/j  K min jG.x/j

j jD1

(8.3)

jjD1

for almost every x 2 . The number K plays the role of an ellipticity constant in the associated nonlinear PDE we shall encounter. We can use these ideas to study Beltrami systems on manifolds. In what follows we avoid technicalities by simply discussing what happens locally in Rn – the tangent space to a smooth manifold. The bounded measurable conformal structure G can be used to define an innerproduct on the tangent spaces to  by the rule hu; viG D hu; G.x/vi;

u; v 2 T x :

(8.4)

Thus (8.3) implies that the unit balls in the metric h; iG on the tangent space have uniform eccentricity when viewed in the Euclidean metric h; i. For this reason a measurable conformal structure is often referred to as a bounded ellipse field. z is another domain and H W  z ! S.n/ is a measurable conSuppose now that  z formal structure defined on . The generalised Beltrami system is the PDE D t f .x/ H.f .x// Df .x/ D J.x; f /2=n G.x/

for almost every x 2 

(8.5)

z is assumed to be a mapping of Sobolev class where a solution f W  !  We wish to place an ellipticity condition on equation (8.5) to link this with the theory of quasiconformal and quasiregular mappings. This takes the form

1;n Wloc ./.

kdS .G; In /k1 C kdS .H; In /k1  M < 1;

(8.6)

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where dS is the metric of S.n/ and In is the n  n identity matrix. This metric is discussed in Wolf’s book [97]. The assumption at (8.6) bounds from above and below the ratio of the largest to the smallest eigenvalues of G and H and applying norms shows that (8.5) together with (8.6) gives the existence of a constant K D K.M; n/ such that jDf .x/jn  KJ.x; f /; that is, f is quasiconformal. Next, the following calculation is very informative. If u; v 2 T x , then almost everywhere hf u; f viH D hDf .x/u; Df .x/viH D hDf .x/u; H.f .x//Df .x/viH D hu; D t f .x/H.f .x//Df .x/vi D hu; J.x; f /2=n G.x/vi D J.x; f /2=n hu; viG : This shows that f preserves the inner-product between tangent vectors up to a scalar multiple. Therefore Df preserves angles between tangent vectors and f the angle between curves (almost everywhere). Thus f can be viewed as a conformal mapping z H /. between the spaces .; G/ and .; It is fair to say the theory of the equation (8.5) is complete and about as good as one could wish for in two-dimensions. This is because when written in complex notation and with a bit of simplification it reduces to the linear first order equation @f @f @f D .z/ C .z/ @zN @z @z

(8.7)

with the ellipticity bounds j.z/j C j.z/j  k < 1 for almost all z. The measurable functions  and  can be explicitly determined from G and H . When H is the identity, we have  0 and the usual Beltrami equation, @f @f D .z/ ; @zN @z

(8.8)

with kk1 D k < 1 which readers familiar with Teichmüller theory will no doubt recognise. A thorough modern account of the theory of Beltrami equations is given in [6]. In higher-dimensions we have already commented above on the various forms of topological rigidity that occur for solutions to Beltrami systems. It is basically the following result which assures us we are going to have to deal with discontinuous conformal structures if there is to be a viable theory of branched mappings preserving a conformal structure. We will call these things “rational mappings” later.

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1;n Theorem 8.7. Let f W  ! f ./ be a Wloc .; f .//, n  3, solution to the equation (8.5) where both G and H are continuous. Then f is a local homeomorphism.

This result can be strengthened in various ways – continuity is really too strong here, see [46]. As an easily seen consequence of Theorem 7.5 we further note that if the ellipticity constants are close enough to 1, that is, if G and H are sufficiently and uniformly close to the identity, then f is also a local homeomorphism.

9 Nevanlinna theory The classical theorem of Picard of 1879 initiated the value distribution theory of holomorphic functions in the complex plane. It simply states that an entire function which omits two values is constant. Nevanlinna theory is a far reaching extension of Picard’s theorem and concerns the distribution of the values of an entire function. It was developed around 1925. Ahlfors subsequently brought many new geometric ideas, including the use of quasiconformal mappings, into Nevanlinna theory. Given y we define for any Borel set  and any y 2 C a meromorphic function f W C ! C the counting function (9.1) n.; y/ D #ff 1 .y/ \ g where the number of points is counted according to multiplicity. The function A.r/ is defined to be the average of n.r; y/ D n.B.r/; y/ with respect to the spherical measure y An important result in the area is Ahlfors’ theorem concerning the so-called on C. defect relation. Given a nonconstant meromorphic function there is a set E  Œ1; 1/ of finite logarithmic measure Z dr 0. If F is a family of K-quasiregular mappings y n such that each f 2 F omits q values af ; af ; : : : ; aqf yn ! R for fW  R n;K 1 2 which the spherical distances .aif ; ajf / > ; then F is a normal family.

i ¤ j;

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Note here that qn;K does not depend on . Then an elementary topological argument gives a sharper version in the quasiconformal case. Theorem 9.5. Let > 0. If F is a family of K-quasiconformal mappings f W   yn ! R y n such that each f 2 F omits two values af and af for which the spherical R 1

distance  .a1f ; a2f / > , then F is a normal family.

2

Another useful normal families criterion is through Zalcman’s lemma in higher dimensions due to Miniowitz [71]. We say a family of mappings F is normal at a point x0 if there exists an open neighbourhood U of x0 on which the family F jU D ff jU W f 2 F g is normal. Theorem 9.6 (Zalcman’s lemma). Let K  1 and F a family of K-quasiregular y n . Then F is not normal at x0 2 Bn if and only if there is a mappings f W Bn ! R sequence of positive numbers rj & 0, a sequence of points xj ! x0 and a sequence of mappings ffj gj1D1  F such that if we define 'j .x/ D fj .xj C rj x/;

(9.6)

then 'j converges uniformly on compact subsets of Rn to a non-constant K-quasiy n. regular mapping ' W Rn ! R The term quasimeromorphic is sometimes used for quasiregular mappings which assume the value 1 continuously in the spherical metric. Again, these theorems and their near relatives remain true in much more general settings. These have been worked out by Rickman, his students and others, see for instance [40] and the references therein.

10 Non-linear potential theory The nonlinear potential theory as it pertains to the theory of higher-dimensional quasiconformal and quasiregular mappings is quite comprehensively covered in the book of Heinonen, Kilpaläinen and Martio [36]. One of the central results is the fact that quasiconformal and quasiregular mappings are morphisms for the class of p-harmonic functions. Typically the modern theory deals with structurally nice measures d defined on Rn and giving weighted Sobolev space. These measures, called admissible, are almost always of the form d D !.x/ dx, so “ !.x/ dx .E/ D E

and ! is an admissible weight satisfying the four conditions (1) Doubling. There is a constant c1 such that .2B/  c1 .B/;

B D Bn .x; r/:

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(2) Testing. If  is open and 'i 2 C 1 ./ with “ “ j'i jp d ! 0; and j'i  vjp d ! 0 



Lp ./,

then v D 0. for vector valued v in (3) Sobolev embedding. There are constants ˛ > 1 and c2 such that for all balls B D B.x0 ; r/  Rn and all ' 2 C01 .B/ we have “ “ 1=.˛p/  1=p  1 1 ˛p p j'j d  c2 r j'j d: j.B/j B j.B/j B (4) Poincaré inequality. There is a constant c3 such that for all balls B D B.x0 ; r/  Rn and all bounded ' 2 C 1 .B/ we have “ “ p p j'  'B j d  c3 r jr'jp d B

B

’ where 'B is the -average of ', 'B D j.B/j B ' d. The constants c1 , c2 and c3 do not matter so much, but the constant ˛ plays an important role in regularity. From these one directly gets the weighted Poincaré inequality 1

Theorem 10.1 (Poincaré inequality). If  is a bounded domain, then for all ' 2 C01 ./ we have “ “ p p p j'j d  c2 diam ./ jr'jp d: 



The first direct connection with the theory of higher-dimensional quasiconformal mappings is the following. Theorem 10.2. If f W Rn ! Rn is quasiconformal and J.x; f / denotes its Jacobian determinant, then !.x/ D J.x; f /1p=n is an admissible weight whenever 1 < p < n. There is now a substantial literature on admissible weights and their role in generalising many of the basic results of analysis to more general settings (including metric spaces).

10.1 A-harmonic functions One of the key tasks of nonlinear potential theory is to develop techniques to study the quasilinear elliptic equation div A.x; ru/ D 0;

(10.1)

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generalising the Laplace equation A.x; / D . To get any viable theory one needs structural conditions on A and these are usually of the form A.x; /    !.x/jjp for an admissible weight !, together with various (less important) technical assumptions. The weighted p-Laplace equation is div.!.x/jrujp2 ru/ D 0;

(10.2)

the solutions of which are local minimisers of the weighted energy integral “ jrujp !.x/ dx: In a similar manner, the general A-harmonic equation is connected with the local extrema, satisfying the Euler–Lagrange equations, of variational integrals of the form “ F .x; ru/ dx: The precise assumptions on A take the following form. Suppose that 0 < ˛  ˇ < 1 and suppose A W Rn  Rn ! Rn is a mapping such that x 7! A.x; / is measurable for all  2 Rn , and  7! A.x; / is continuous for almost every x 2 Rn and finally we require that for all  2 Rn and almost all x 2 Rn we have the estimates: • A.x; /    ˛!.x/jjp , • jA.x; /j  ˇ!.x/jjp1 , • .A.x; 1 /  A.x; 2 //  .1  2 / > 0, 1 ¤ 2 , • A.x; / D jjp2 A.x; /,  2 R n f0g. These conditions describe the allowable degenerate behaviour of the equation 1;p ./ is a weak solution to the equation (10.1) in  (10.1). A function u 2 Wloc if for every ' 2 C01 ./ we have “ A.x; ru.x//  r'.x/ dx D 0; (10.3) 

and a super-solution if the left-hand side is non-negative. A real-valued function h W  ! R is called A-harmonic if it is a continuous weak solution to (10.1). These functions are the main object of study in the theory in as much as the harmonic functions are for classical potential theory. We then have the following theorem telling us the Dirichlet problem has a solution for Sobolev boundary values. Theorem 10.3. Let   Rn be a bounded domain and 2 W1;p ./. Then there is 1;p a unique solution u 2 W1;p ./ of the equation (10.1) with u  2 W0; ./.

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The notation should be self explanatory, but the last space here consists of those functions whose p th power of their absolute value is integrable with respect to d and which vanish at the boundary in the Sobolev sense. The theory then develops by studying regularity (giving continuity) and compactness properties of solutions (locally uniformly bounded families are equicontinuous), firmly establishing the connection with the variational formulation and then the Harnack principle and maximum principle. For instance: Theorem 10.4 (Strong maximum principle). A non-constant A-harmonic function defined in a domain  cannot achieve its maximum or minimum value. Of course once connected with quasiconformal mappings the maximum principle will imply that quasiregular mappings are open. The capacity and A-harmonic measure theory are deep and interesting and have important consequences, but it would lead us too far astray to develop them here.

10.2 Connections to quasiconformal mappings If f D .f 1 ; f 2 ; : : : ; f n / is a solution to the Beltrami system (5.2), then u D f i satisfies the following equation of elliptic type: div.hG 1 ru; rui.n2/=2 G 1 ru/ D 0; 1

(10.4)

1

where G D G .x/ is the inverse of the distortion tensor. This is easily seen by unwinding the tautological Beltrami equation that f satisfies. Therefore u is an A-harmonic function. With the choice G D Idnn (the Cauchy– Riemann system and conformal mappings) we see that u is n-harmonic. These ideas lead to an alternative proof of the Liouville theorem which was developed by Reshetnyak. Actually, if f is a solution of the Beltrami system, then remarkably u D log jf j satisfies an A-harmonic equation as well. Theorem 10.5. Let f W  ! Rn be a non constant quasiregular mapping and let b 2 Rn . Then the function u.x/ D log jf .x/  bj is A-harmonic in the open set  n f 1 .b/. Here A satisfies the structure equations with p D n, !.x/ D 1, ˛ D 1=K and ˇ D K. Now the theory of A-harmonic functions implies directly that quasiregular mappings are open and discrete (we discussed open mapping property above). Roughly, discreteness follows from the omitted discussion of A-harmonic measure. Here we need the fact that the polar sets (fx W log jf .x/  bj D 1g) have conformal capacity zero. This implies they have Hausdorff dimension zero and hence are totally disconnected. A topological degree argument then completes the proof of Reshetnyak’s theorem – discussed earlier as Theorem 4.1.

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10.3 Removable singularities There is a classical theorem of Painlevé concerning removable sets for analytic functions. It states that if   C is a planar domain, E   is a closed subset and f W nE ! C is a bounded analytic function, then f has an analytic extension to . This result has also found generalisations in higher dimensions and we give a brief account of that here. Details are to be found in [46] A closed set E  Rn is removable under bounded K-quasiregular mappings if for every open set   Rn any bounded K-quasiregular mapping f W nE ! Rn extends to a K-quasiregular mapping of . We stress here that f need not even be locally injective, nor even of bounded topological degree. Theorem 10.6. There is D .K/ > 0 such that closed sets of Hausdorff dimension

are removable under bounded K-quasiregular mappings. In particular, sets of Hausdorff dimension 0 are always removable for bounded quasiregular mappings. In light of conjectures regarding the p-norms of the Hilbert transform on forms and the relationship between s-capacity and Hausdorff dimension we formulate the following conjecture regarding the optimal result, see [46]. Conjecture. Sets of Hausdorff d -measure zero, d D n=.KC1/  n=2, are removable under bounded K-quasiregular mappings. In dimension two, Astala [4] has verified this conjecture for all d < 2=.K C 1/ and the borderline cases are well in hand. In response to these questions Rickman [85] has constructed examples to show that the results are, in some sense, best possible. Theorem 10.7. There are Cantor sets E of arbitrarily small Hausdorff dimension and bounded quasiregular mappings R3 nE ! R3 . For such mappings E is necessarily non-removable. Here we must have K ! 1 as the Hausdorff dimension tends to zero.

11 Quasiregular dynamics in higher dimensions There are a number of recent developments in the theory of higher-dimensional quasiconformal and quasiregular mappings which link these areas to questions of dynamics and different types of rigidity phenomena in dimension n  3 than those we have seen earlier. A related survey can be found in [57]. A self-mapping of an n-manifold is rational or uniformly quasiregular if it preserves some bounded measurable conformal structure (see below at (11.1)). In what follows we will assume that the manifold in question is Riemannian for simplicity, but Sullivan’s theorem shows that the existence

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of a bounded measurable Riemannian structure is a purely topological notion, at least when n ¤ 4. The bounded measurable structure which is preserved will typically not be the underlying Riemann structure – indeed most often this structure will necessarily be discontinuous. There is a close analogy between the dynamics of rational maps of closed manifolds y The theory and the classical Fatou–Julia theory of iteration of rational mappings of C. n y where many classical results is particularly interesting on the Riemann n-sphere R find their analogue, some of which we will discuss below. In higher dimensions other interesting aspects come into play. An analytic mapping of a closed surface is a homeomorphism unless the surface is S2 or the 2-torus – where of course the map is covered by multiplication on the complex plane. Thus there cannot be any interesting Fatou/Julia type theories on surfaces other than S2 where it is very highly developed. We cannot expect this situation to persist exactly in higher dimensions, but informed by the two-dimensional case we might expect an interaction between the curvature of the manifold and the existence or otherwise of rational mappings. Thus we introduce the Lichnerowicz problem of classifying those manifolds admitting rational endomorphisms. Once we have examples of nontrivial rational maps of such spaces (for instance the n-spheres) we can ask to what extent the classical Fatou/Julia theory remains true. What is the structure of the Julia set and can we classify dynamics on the Fatou set? yn ! y n is a measurable map G W R Recall that a measurable conformal structure on R S.n/, the non-positively curved symmetric space of positive definite symmetric n  n y n / map matrices of determinant equal to 1. A W 1;n .R yn ! R yn fWR will preserve this conformal structure if it satisfies the generalised Beltrami system D t f .x/ G.f .x// Df .x/ D J.x; f /2=n G.x/

y n: for almost every x 2 R

(11.1)

With an ellipticity assumption on G as before, there is a K < 1 such that any y n / solution to (11.1) is K-quasiregular. The composition of quasiregular W 1;n .R maps is certainly quasiregular, and so the set of solutions to the equation (11.1) forms a semigroup under composition. Let us denote the semigroup of solutions to (11.1) by Rat.G/, the maps rational with respect to the measurable conformal structure G.

11.1 Existence of equivariant measurable conformal structures The nonpositive curvature of the space S.n/ allows one to make various averaging constructions, first noted by Sullivan in two-dimensions for group actions and develop by Tukia in higher-dimensions. Slightly refining these arguments gives us the following theorem, see [46], Chapter 21. Theorem 11.1 (Semigroup). Let F be an abelian semigroup of quasiregular mappings of a manifold M n such that each f 2 F is K-quasiregular. Then there is a measurable

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conformal structure GF W M n ! S.n/ and for each f 2 F , D t f .x/ GF .f .x// Df .x/ D J.x; f /2=n GF .x/ for almost every x:

(11.2)

The condition that the semigroup be abelian is too strong, but some condition is necessary even in two-dimensions, see [38]. In [46] a “left-right” coset condition is given which is automatically true in the abelian case. In the case of quasiconformal groups, the result is more general, but because of Lelong’s Theorem 11.12 (described y n. below) the result is really only of interest in Bn , Rn and R Theorem 11.2 (Group). Let  be a group of quasiconformal self-homeomorphisms y n such that each g 2  is K-quasiconformal. Then there is a of a domain   R measurable conformal structure G W  ! S.n/ and for each g 2 , D t g.x/ G .g.x// Dg.x/ D J.x; g/2=n G .x/ for almost every x 2 :

(11.3)

11.2 Fatou and Julia sets The semigroup of solutions to (11.1) has the property that the composition of its elements cannot increase the distortion beyond a uniform bound. This is surprising inasmuch as if a solution is branched, then its iterates f , f B f , …, f B f B    B f have ever increasing degree. We will see soon that there can even be fixed points of a rational mapping which are also branch points. It would seem that there should be little or no distortion at a fixed point in order for it not to grow under iteration, yet we know from higher-dimensional topological rigidity that there must be some distortion at branch points. It is very interesting to see how these observations reconcile. First we note the following obvious fact. Theorem 11.3. Rat.G/, the space of quasiregular solutions to (11.1), is closed under composition and there is a K < 1 such that each f 2 Rat.G/ is K-quasiregular. Because of Rickman’s version of Montel’s Theorem (Theorem 9.4) there is a reasonably complete Fatou–Julia theory associated with the iteration of rational mappings. But first we need to state that there are examples. yn yn ! R Theorem 11.4. For each n  2 there is a K-quasiregular mapping f W R B2 Bn n with the property that all the iterates f D f B f , …, f D f B f B    B f are also K-quasiregular. Thus the family ff Bn W n  1g is a quasiregular semigroup. As a corollary from the existence invariant conformal structures, Theorem 11.1, we have the following. Corollary 11.5. For each n  2 there is a bounded measurable conformal structure G defined on Sn which admits non-injective rational mappings. In these cases Rat.G/ is infinite, contains mappings of arbitrary high degree and is not precompact.

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y n was initiated in The study of these quasiregular rational mappings for Sn  R joint work with Iwaniec [45], but have since been developed by V. Mayer, K. Peltonen and others, see [58], [59], [67], [68]. There are strong restrictions on the geometry and topology of closed manifolds admitting nontrivial rational mappings, for instance they cannot be negatively curved. The Fatou set F .f / of a rational mapping f is the open set where the iterates form a normal family (that is, have locally uniformly convergent subsequences). The Julia set J.f / is the complement of the Fatou set yn n F : JDR If the degree of f  2, the only interesting case for us, then the Julia set is nonempty, closed and a completely invariant set, f 1 .J/ D J: y n itself (the Lattès type examKnown examples of Julia sets include Cantors sets, R ples), codimension one spheres and somewhat more complicated sets which separate R3 into infinitely many components. There are very many interesting and unanswered questions about what sets could be Julia sets. For instance it is known in three dimensions, that only very simple knots (torus knots in S3 ) can possibly be Julia sets.

11.3 Dynamics of rational mappings We first consider the classification of fixed points. In [39] it was shown that uniformly quasiregular mappings are locally Lipschitz near a fixed point x0 which is not a branch point. This is then used to show that the family F D ff W  > 1g is a normal family, where f .z/ D f .z=/. This is relatively straightforward as f B f D .f B f / and so fB2 is Lipschitz and linearizes again. Moreover, this more or less implies that all limits of convergent subsequences of F are uniformly quasiconformal mappings – that is, the cyclic group hgi D fg n W n 2 Zg for g 2 F is a uniformly quasiconformal group as discussed in the next section. The set of all such limit mappings is called the generalized derivative of f at x0 . Uniformly quasiconformal mappings have been classified as either loxodromic, elliptic or parabolic. It follows that the elements of the generalized derivative are either all constant, all elliptic, or all loxodromic, and this allows for a classification of the fixed points of a uniformly quasiregular mapping as attracting/repelling (generalized derivative loxodromic), neutral (generalized derivative elliptic), or super-attracting (fixed point is a branch point). All these types of fixed points can occur. Perhaps the most surprising is Mayer’s construction of the Lattès example. This example is derived from a functional equation in a similar manner as the classical example, building on a classification of Martio and Srebro for automorphic y n is the unit sphere and both the quasimeromorphic mappings [66]. The Julia set in R

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origin and 1 are super attracting fixed points – the branch set being a family of lines passing through the origin. We record this in the following theorem: y n have Theorem 11.6. The Fatou set of a rational quasiregular self-mapping f of R y precisely the same types of stable components U as rational functions of R2 . They are either (1) attracting: there is x0 2 U with f .x0 / D x0 and as n ! 1, f Bn .x/ ! x0 locally uniformly in U ;

(11.4)

(2) super-attracting: there is x0 2 U with f .x0 / D x0 and x0 2 Bf . Necessarily then (11.4) holds; (3) parabolic: there is x0 2 @U with f .x0 / D x0 and (11.4) holds; (4) Siegel: f W U ! U is quasiconformal and ff Bn W n 2 Zg is a compact Lie group. Further, there are examples of types .1/, .2/ and .3/. For attracting and repelling fixed points we know the following: Theorem 11.7. A rational quasiregular mapping is locally quasiconformally conjugate to the map x 7! 2x near a repelling fixed point and is quasiconformally conjugate to x 7! 12 x near an attracting fixed point. Here, by quasiconformal conjugacy we mean that there is a quasiconformal homeomorphism ' defined in a neighbourhood V of the origin such that ' B f B ' 1 .x/ D 2x: Roughly, after a quasiconformal change of coordinates we get standard dynamics. There is no standard conformal model for a super-attractive fixed point – a consequence of the Liouville theorem. This sort of dynamics is quite novel. There are examples with parabolic dynamics. It can be shown that a map with a parabolic fixed point can be constructed in such a way that it does not admit a quasiconformal linearization in its attracting parabolic petal (unlike the rational case) due to the existence of wild translation arcs. This builds on Mayer’s work in [69]. Question. An interesting problem is to decide whether or not it is possible to have a “Siegel disk” of type .4/ described above for a non-injective rational quasiregular mapping. Such a domain would presumably be a ball or solid torus with irrational rotational dynamics. That the map is injective on a Siegel domain, and that it would generate a compact Lie group as described does follow. A classical result is the density of repellors, that is, repelling fixed points, in the Julia set. This is not known in complete generality yet for higher-dimensional rational mappings, but we do know the following.

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Theorem 11.8. The set of repelling and neutral fixed points is dense in the Julia set. In certain cases, with assumptions on the topological structure of the Julia set, for instance separating, we do know repellors are dense.

11.4 Stoïlow factorisation The following factorisation theorem shows that in fact rational quasiregular mappings are quite common. This is a variant of the well-known theorem of Stoïlow’s, see [60]. yn ! R y n is a non-constant quasiregular mapping, Theorem 11.9. Suppose g W R n  2. Then there exists a rational quasiregular mapping f whose Julia set is a y n such that g D f B h. yn ! R Cantor set, and a quasiconformal mapping h W R y D S2 ) is unique up to Classically the factorization (for quasiregular maps of C Möbius transformation. If ' B f D B g, then there is a Möbius transformation ˆ such that ' B ˆ D . Clearly this statement cannot hold in higher dimensions if ' and are merely assumed rational. However if we fix the invariant conformal structure, then we can make uniqueness statements up to a finite-dimensional Lie group. Notice the following easy implication showing that no distinction can be made between the branch sets of rational quasiregular mappings and completely general quasiregular mappings, [54]. y3 ! Theorem 11.10. Let X D Bf be the branch set of a quasiregular mapping f W R y 3 . Then X is the branch set of a rational quasiregular mapping. R This is a little surprising given how complicated these branch sets can be. We recall from [37] that the branch set of a quasiregular mapping could be as wild as Antoine’s necklace, a Cantor set in S3 whose complement is not simply connected.

11.5 Smooth rational quasiregular mappings The technique used in the construction of the factorisation is sufficiently robust that if y n /, then the quasiconformal homeomorphism h, the map f is smooth of class C k .R and consequently the rational mapping ', can be chosen to be smooth of the same class. Typically one does not expect branched (not locally injective) quasiregular mappings to be smooth, however there are examples of M. Bonk and J. Heinonen [10] of a quasiregular map f W S3 ! S3 which is C 3 .S3 / for every > 0. Kaufman, Tyson and Wu extended these results to higher dimensions, [48]. The following theorem (which was certainly known to Bonk and Heinonen) is a consequence. y n with nonTheorem 11.11. There are smooth rational quasiregular mappings of R empty branch set, Bf ¤ ;. Indeed,

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• for each > 0, there is a C 3 .S3 / rational quasiregular mapping ' whose Julia set is a Cantor set; • for each > 0, there is a C 2 .S4 / rational quasiregular mapping ' whose Julia set is a Cantor set; • for each n  5 there is an D .n/ > 0 and a C 1C .Sn / rational quasiregular mapping ' whose Julia set is a Cantor set. Note that although these maps are smooth, any invariant conformal structure must be quite irregular (at least discontinuous) near the branch set. Indeed it is known that the rational mappings described here are structurally stable (or generic); there is a single attracting fixed point, no relations between critical points and the Julia set is ambiently quasiconformally equivalent to the middle thirds Cantor set.

11.6 The Lichnerowicz problem: rational maps of manifolds A natural question now is to ask what sort of manifolds support rational quasiregular endomorphisms. As we have noted, in two-dimensions it is an easy application of the signature formula for branched coverings to see that only the sphere and torus admit branched self-maps. In higher dimensions the question is more complicated – though a complete answer was given by Kangaslampi in three dimensions [47]. Here we shall consider such mappings acting on closed manifolds M of dimension at least two and our problem is to determine what kind of rational mappings can act on a given manifold. Recall that a rational map f W M ! M is surjective since the continuity and openness of a quasiregular map implies that the image f M is both compact and open; hence f M D M . The first part of our problem is a non-injective version of the answer given by Ferrand [19] to a conjecture of Lichnerowicz. Theorem 11.12 (Lichnerowicz conjecture). Let K < 1. Up to quasiconformal equivalence, the only compact n-manifold which admits a non-compact family of Kquasiconformal mappings is the standard n-sphere Sn A noninjective rational map f of a closed manifold M will have the semi-group ff n g1 nD1 non-compact – the Julia set of f is always non-empty. Lelong’s result suggests the existence of such a map should imply severe restrictions on the manifold M . The first of these is the following obstruction for the existence, [59]. Theorem 11.13. If M n is a closed n-manifold and f W M n ! M n is a non-injective quasiregular rational mapping, then there exists a non-constant quasiregular mapping g W Rn ! M n .

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This theorem is proved using a version of Zalcman’s lemma. The semigroup ff Bn g generated by f cannot be normal as the degree increases. Therefore there is a point x0 2 M n where the iterates fail to be normal. Choose a quasiconformal chart ' W Bn ! M n with '.0/ D x0 . The proof is then in showing that one can balance the dynamics in M n and the scaling in Rn through a judicious choice of sequence n ! 0 such that f Bn .'.n x// is normal and converges to g as n ! 1. The limit is obviously defined on Rn . Manifolds admitting such a map g are called qr-elliptic, and answering a question of Gromov, Varopoulos, Saloff-Coste and Coulhon [95] showed that such manifolds must in turn have a fundamental group of at most polynomial growth. Corollary 11.14. If M n is a closed n-manifold and f W M n ! M n is a non-injective rational quasiregular mapping, then 1 .M n / has polynomial growth. Hence M n cannot admit a metric of negative curvature. In fact there are further important consequences of qr-ellipticity. Bonk and Heinonen established an upper bound on the dimension of the de Rham coholomogy ring H  .M / for any closed oriented qr-elliptic n-manifold [11] and this was generalised by Pankka in other directions [75] to consider mappings whose distortion function is bounded in Lp , [75]. This has obvious implications in generalising this corollary. Rickman has shown that the 4-manifold M D .S2  S2 / # .S2  S2 / is qr-elliptic, this gives an example which is nontrivial, simply connected and closed [86]. We do not know if this manifold admits a rational quasiregular mapping though. The generalized Lichnerowicz problem seeks to determine all closed manifolds which admit non-injective rational quasiregular mappings: From [59] we have the following. Theorem 11.15. Let f be a non-injective rational quasiregular map of the closed manifold M and suppose that f is locally homeomorphic, so the branch set Bf D ;. Then M is the quasiconformal image of a Euclidean space form. By a Euclidean space form we mean the quotient of Rn under a Bieberbach group (co-finite volume lattice)   IsomC .Rn /. The two other types of space forms are the quotients by torsion free co-finite volume lattices of isometries of the n-sphere and of hyperbolic n-space. As a sort of converse we also have the following. Theorem 11.16. If M is quasiconformally equivalent to a Euclidean space form, then M admits no branched quasiregular (and in particular no branched quasiregular rational) mappings. In the case of the sphere, lens spaces and other spherical manifolds the existence of rational quasiregular maps is due to [44] and Peltonen [76]. These results suggest

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that there are few such mappings in three or more dimensions as compared with the y space of rational functions of the Riemann sphere S2 D C. Theorem 11.17. Any non-injective rational quasiregular map of a closed Euclidean space form M is the quasiconformal conjugate of a conformal map. We remark that, in this second result, we no longer suppose that the map has to be locally injective. This result is surprising because it is false for globally injective mappings. Indeed, Mayer shows that there are uniformly quasiconformal (even bi-Lipschitz) maps of three (or higher) dimensional tori which cannot be quasiconformally conjugate to a conformal map [68]. Next, we can distinguish space forms according to the type of rational quasiregular maps they support: Theorem 11.18. If M is a closed space form, then we have the following characterization: (1) M admits a branched quasiregular rational map if and only if M is a spherical space form. (2) M admits a non-injective, locally injective quasiregular rational map if and only if M is a Euclidean space form. (3) M admits no non-injective quasiregular rational map if and only if M is a hyperbolic space form.

12 Quasiconformal group actions This final section is intended as an introduction to the theory of quasiconformal groups. The theory is modelled on the classical theory of discrete groups of Möbius transformations and the connections with hyperbolic geometry and low-dimensional topology are well known. y n is called a quasiconformal A group  of self-homeomorphism of a domain   R group if there some K < 1 such that each element of  is K-quasiconformal. As a consequence of Theorem 11.2 we know that every quasiconformal group admits an invariant conformal structure. Thus quasiconformal groups are really the groups of conformal transformations of (bounded measurable) conformal structures. Quasiconformal groups were introduced by Gehring and Palka [28] in their study of quasiconformally homogeneous domains. They asked whether in fact every quasiconformal group is the quasiconformal conjugate of a Möbius group. Sullivan and Tukia established this in two dimensions, see [89]. In higher dimensions the first example of a quasiconformal group not conjugate to a Möbius group was given by Tukia [90]. He gave an example of a connected Lie group acting quasiconformally on Rn , n  3, which was not isomorphic to a Möbius group. Tukia’s group was

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in fact constructed as the topological conjugate of a Lie group. The obstruction to quasiconformal conjugacy to a Möbius group lies in the fact that the orbit of a point under the group was constructed to be the product of an infinite Von-Kock snowflake and Rn2 . This orbit is certainly not quasiconformally equivalent to a hyperplane, the orbit of a point in an isomorphic Möbius group. Certain discrete subgroups of Tukia’s group were also shown not to be quasiconformally conjugate to Möbius groups [52]. Generalising these examples, McKemie [70] has shown that similar examples both in the discrete and non-discrete case can be found with K arbitrarily close to 1. Examples from topology of “exotic” smooth involutions also give finite quasiconformal groups not conjugate to Möbius groups. For example Giffen [30] shows how to construct a smooth periodic transformations of the n-sphere, n  4, whose fixed point set is a knotted co-dimension 2-sphere. The fixed point set of any Möbius transformation, or its topological conjugate, must be unknotted. Clearly any finite group of diffeomorphisms is a quasiconformal group. A further important example of an infinite quasiconformal group not topologically conjugate to a Möbius group was given in dimension three by Freedman and Skora [21]. This example differs from the others in that the topological fact used concerns the linking of the fixed point sets of elliptic elements. Such linking can be more complicated for quasiconformal groups than for Möbius groups. There are important applications of quasiconformal groups. Even in 1-dimension (where the term quasisymmetric group is used). Here Gabai [22] and Casson–Jungreis [13] proved, building on important earlier work of Tukia [91], Zieschang and others, that discrete quasisymmetric groups are topologically conjugate to Möbius groups. This is a far reaching generalisation of the Nielsen realisation problem. It was already known from work of Mess and Scott, that this would also imply an important result in low-dimensional topology, namely the Seifert fibered space conjecture: a compact 3-manifold with a normal infinite cyclic subgroup of its fundamental group is a Seifert fibered space. In this brief survey we will only outline the basic facts such as the classical trichotomy classification of elements into elliptic, parabolic and loxodromic. We then recall two fundamental results in the area. The first asserts that a “sufficiently large” discrete quasiconformal group is the quasiconformal conjugate of a Möbius group. Secondly, we show that quasiconformal groups are Lie groups; the quasiconformal version of the Hilbert–Smith conjecture.

12.1 Convergence properties Suppose that  is a quasiconformal group of self-homeomorphisms of a domain   Rn . Then  is discrete if it contains no infinite sequence of elements converging locally uniformly in  to the identity. Since the identity is isolated in a discrete group  it follows that each element of  is isolated in  in the compact open topology. The group  is said to be discontinuous at a point x 2  if there is a neighbourhood U

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of x such that for all but finitely many g 2  we have g.U / \ U D ;. We denote by O./ the set of all x 2  such that  is discontinuous at x. O./ is called the ordinary set of . The set L./ D  n O./ (12.1) is called the limit set of . Clearly O./ is open and L./ is closed. Both sets are -invariant. That is, g.O.// D O./

and

g.L.// D L./;

(12.2)

for each g 2 . A discontinuous group (one for which O./ ¤ ;) is discrete, the converse is false. The following theorem, a consequence of the compactness results we discussed earlier, is central to what follows. Theorem 12.1. If  is a discrete quasiconformal group and fgj g is an infinite sequence in , then there are points x0 and y0 and a subsequence fgjk g  fgj g for which we y n n fy0 g and gj ! y0 locally uniformly in have gjk ! x0 locally uniformly in R k y n n fx0 g. R It is the above compactness property, dubbed the convergence property, which led to the theory of convergence groups, see [27], [53], [91] which are closely related to Gromov’s theory of hyperbolic groups.

12.2 The elementary quasiconformal groups A discrete quasiconformal group  is said to be elementary if L./ consists of fewer than three points. For each g 2 , a quasiconformal group, we set ord.g/ D inffm > 0 W g m D identityg; y n W g.x/ D xg: fix.g/ D fx 2 R Let  be a discrete quasiconformal group. It is an easy consequence of the convergence properties that if g 2  and ord.g/ D 1, then 1  #fix.g/  2 . We define three types of elements in a discrete quasiconformal group as • g is elliptic, if ord.g/ < 1; • g is parabolic, if ord.g/ D 1 and #fix.g/ D 1; • g is loxodromic, if ord.g/ D 1 and #fix.g/ D 2. In a discrete quasiconformal group this list of elements is exhaustive. However, in the non-discrete case there is one other type of element which needs to be considered. If  is a quasiconformal group and g 2  has the property that there is a sequence of integers kj ! 1 for which g kj ! identity

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y n , then we call g a quasirotation. In this case hgi has a nice structure, uniformly in R we shall see in a moment that its closure in the space of homeomorphisms is a compact abelian Lie group. Theorem 12.2. Let  be a discrete quasiconformal group. • The limit set L./ D ; if and only if  is a finite group of elliptic elements. • The limit set consists of one point, L./ D fx0 g, if and only if  is an infinite group of elliptic or parabolic elements. • The limit set consists of two points, L./ D fx0 ; y0 g, if and only if  is an infinite group of loxodromic elements which fix x0 and y0 and elliptic elements which either fix or interchange these points. In addition  must contain at least one loxodromic element and at most finitely many elliptic elements which fix x0 and y0 . This leads to the classification. Theorem 12.3. If  is a discrete quasiconformal group, then each g 2  is either elliptic, parabolic or loxodromic. Moreover, g and g k are always elements of the same type for each integer k ¤ 0.

12.3 Non-elementary quasiconformal groups and the conjugacy problem In this section we record a few observations about non-elementary groups. We first note the following. • If g is a parabolic element with fixed point x0 in a quasiconformal group , then lim g j D x0

j !1

and

lim g j D x0

j !1

(12.3)

y n n fx0 g locally uniformly in R • If g be a loxodromic element with fixed points x0 , y0 of a quasiconformal group . Then these points can be labelled in such a way that lim g j D x0

j !1

and

lim g j D y0

j !1

(12.4)

y n n fx0 g respectively y n n fy0 g and R locally uniformly in R y n has the very useful property that it 12.3.1 The triple space. A Möbius group of R extends to the upper-half space HnC1 D fx 2 RnC1 W x D .x1 ; x2 ; : : : ; xnC1 /; xnC1 > 0g

(12.5)

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via the Poincaré extension as a Möbius group. The existence of such an extension is unknown for quasiconformal groups and would have important topological consequences. Next we introduce an alternative for the upper-half space for which any y n naturally extends to. This substitute is the triple group of homeomorphisms of R n space T , a 3n-manifold defined by y n and u, v, w are distinct g: T n D f.u; v; w/ W u; v; w 2 R

(12.6)

There is a natural projection p W T ! H defined by the property that p.u; v; w/ is the orthogonal projection of w (in hyperbolic geometry) onto the hyperbolic line joining u and v. This map has the property that if X  HnC1 is compact, then p 1 .X /  T n is compact. y n , there is a natural action of f on T n , which Given a self-homeomorphism f of R for notational simplicity we continue to call f , by the rule n

nC1

f .u; v; w/ D .f .u/; f .v/; f .w//;

.u; v; w/ 2 T n :

(12.7)

If f is in fact a Möbius transformation of HnC1 we find that the projection p commutes with the action of f on T n , that is, f B p D p B f W T n ! HnC1 . Using the convergence properties of quasiconformal groups it is easy to see that a y n is discrete if and only if it acts discontinuously on T n . quasiconformal group of R 12.3.2 Conjugacy results. In order to establish the best known result on the quasiconformal conjugacy of a quasiconformal group to a Möbius groups we need to discuss a special type of limit point. The following definition is easiest to use in the quasiconformal setting, though it has useful purely topological counterparts. A point x0 2 L./ n f1g is called a conical limit point if there is a sequence of numbers f˛j g, ˛j ! 0, and a sequence fgj g   such that the sequence hj .x/ D gj .˛j x C x0 / y n n fy0 g. converges locally uniformly in Rn to a quasiconformal mapping h W Rn ! R We extend the definition to include 1 in the usual manner. The term radial limit point is also common in the literature. We should also compare the definition of conical limit point with the conclusion of the Zalcman Lemma 9.6. The next result is elementary. Lemma 12.4. Let x0 be a loxodromic fixed point of a K-quasiconformal group . Then x0 is a conical limit point. The loxodromic elements of a quasiconformal group are always quasiconformally conjugate to Möbius transformations [27]. However, this fact relies on some quite deep topology. Parabolic elements are not always conjugate to Möbius transformations [69] and as observed above there are examples of elliptic elements which are not conjugate to Möbius transformations. We now relate the regularity of the conformal structure at a conical limit point and the question of conjugacy. The proof is basically through a linearisation procedure which enables one to move to an invariant conformal

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structure which is constant after a quasiconformal conjugacy. This quickly gives us the quasiconformal conjugacy to a Möbius group. Theorem 12.5. Let  be a discrete K-quasiconformal group and let G be a invariant conformal structure. Suppose that G is continuous in measure at a conical y n such that point of . Then there is a quasiconformal homeomorphism h of R 1 h B  B h is a Möbius group. Since a measurable map is continuous in measure almost everywhere, Theorem 12.5 implies that a quasiconformal group whose conical limit set has positive measure is the quasiconformal conjugate of a Möbius group. There are a number of results asserting that the set of conical limit points is large, here are two. Theorem 12.6. Let  de a discrete quasiconformal group. Suppose that either • the action of  on the triple space is cocompact, that is, T n =  is compact, or z of HnC1 in such a • the group  can be extended to a quasiconformal group  nC1 z = is compact. way that H n y and every limit point is a conical point. Thus  is the quasiconformal Then L./ D R conjugate of a Möbius group.

12.4 Hilbert–Smith conjecture Hilbert’s fifth problem was formulated following Lie’s development of the theory of continuous groups. It has been interpreted to ask if every finite-dimensional locally Euclidean topological group is necessarily a Lie group. This problem was solved by von Neumann in 1933 for compact groups and by Gleason and Montgomery and Zippin in 1952 for locally compact groups, see [72] and the references therein. A more general version of the fifth problem asserts that among all locally compact groups  only Lie groups can act effectively on finite-dimensional manifolds. This problem has come to be called the Hilbert–Smith Conjecture. It follows from the work of Newman and of Smith together with the structure theory of infinite abelian groups that the conjecture reduces to the special case when the group  is isomorphic to the p-adic integers. In 1943 Bochner and Montgomery solved this problem for actions by diffeomorphisms. The Lipschitz case was established by Repovš and Šˇcepin [80]. In the quasiconformal case we have the following result [55]. Theorem 12.7. Let  be a locally compact group acting effectively by quasiconformal homeomorphisms on a Riemannian manifold. Then  is a Lie group. Here we wish to make the point that there is no a priori distortion bounds assumed for elements of . If one assumes a priori bounds on the distortion of elements , then precompactness of the family of all K-quasiconformal mappings enables the

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local compactness hypothesis in Theorem 12.7 to be dropped. The hypothesis of effectiveness (that is, the hypothesis that the representation of  in the appropriate homeomorphism group is faithful) is redundant if we give  the topology it inherits from the compact open topology of maps. We usually view  simply as a quasiconformal transformation group. Corollary 12.8. Let  be a quasiconformal group acting on a Riemannian manifold. Then  is a Lie group. This result has an important consequence in the property of analytic continuation and also uniqueness statements for solutions of quite general Beltrami systems, see [56]. Acknowledgement. This work was partially supported by the New Zealand Marsden Fund.

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Part C

Dynamics

Chapter 15

Infinite-dimensional Teichmüller spaces and modular groups Katsuhiko Matsuzaki

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics of Teichmüller modular groups . . . . . . . . . . . . 2.1 Teichmüller spaces and modular groups . . . . . . . . . . 2.2 Orbits of Teichmüller modular groups . . . . . . . . . . . 2.3 Subgroups of Teichmüller modular groups . . . . . . . . 2.4 Application to the infinite-dimensional Teichmüller theory 3 The action on the asymptotic Teichmüller space . . . . . . . . . 3.1 Asymptotic Teichmüller spaces and modular groups . . . 3.2 Asymptotically elliptic subgroups . . . . . . . . . . . . . 3.3 The asymptotically trivial mapping class group . . . . . . 4 Quotient spaces by Teichmüller modular groups . . . . . . . . 4.1 Geometric moduli spaces . . . . . . . . . . . . . . . . . 4.2 Several Teichmüller spaces . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction The theory of Teichmüller space is studied in various fields of mathematics but the complex-analytic approach has one advantage in a sense that it can deal with Teichmüller spaces of finite and infinite type Riemann surfaces in parallel and simultaneously. However that may be, Teichmüller spaces of analytically infinite Riemann surfaces are infinite-dimensional and they display several aspects and phenomena different from those of the finite-dimensional cases, and some results involve much more difficult and complicated arguments for their proofs. On the other hand, through these generalization and unification of theories, arguments given for finite-dimensional cases become clearer and more transparent in some occasions. One of the recent developments of the infinite-dimensional Teichmüller theory is brought by the fact that the biholomorphic automorphism group of the Teichmüller space is completely determined. Namely, it is proved that every biholomorphic auto-

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morphism is induced by a quasiconformal mapping class of the base Riemann surface, which is called a Teichmüller modular transformation. This result was first proved by Royden [44] for Teichmüller spaces of compact Riemann surfaces, and through succeeding researches due to Kra, Earle, Gardiner and Lakic (see [3] and [9]), it has been proved in full generality by Markovic [31]. See the report in Volume II of this Handbook [12]. After this, it should be natural that we investigate the moduli space of an infinite type Riemann surface, which is the quotient space of the Teichmüller space by the Teichmüller modular group (now known to be the biholomorphic automorphism group). In fact, compared with the finite-dimensional theory, the study of moduli spaces has not been developed yet in the infinite-dimensional case. In this chapter, we survey recent results concerning the dynamics of modular groups of infinite-dimensional Teichmüller spaces and their quotient spaces. As we mentioned above, the Teichmüller modular group is the automorphism group of the Teichmüller space induced geometrically by the quasiconformal mapping class group. Although they can be identified with each other in almost all cases, we prefer to use “Teichmüller modular group” since we consider its dynamics on the Teichmüller space. Occasionally “mapping class group” is still used when its property as a surface automorphism is a matter in question. Unlike in the finite-dimensional case, the action of the Teichmüller modular group is not necessarily discontinuous in our case. In general, we say that a group  acts on a Hausdorff space X discontinuously if, for every point x 2 X , there is a neighborhood U of x such that the number of elements  2  satisfying .U / \ U ¤ ; is finite. If X is locally compact, this is equivalent to saying that  acts on X properly discontinuously. However, since our Teichmüller spaces are not locally compact, we use the term “discontinuous” instead of “properly discontinuous”. Because of such non-discontinuous action, the topological moduli space obtained simply by taking the quotient of the Teichmüller space by the modular group does not have a nice geometric structure, which might be a reason why this subject matter has not been so attractive. However, assuming these facts in a positive way, conversely, we can specify the set of points where the Teichmüller modular group does not act discontinuously, and observe some properties of this set, called the limit set. As is well known, limit sets play an important role in the theory of Kleinian groups and the iteration of rational maps. We import this concept for the study of dynamics of Teichmüller modular groups. Then, the non-homogeneity of Teichmüller space appears to be tangible and in particular it provides an interesting research subject, which is to understand the interaction between the hyperbolic structure of an infinite type Riemann surface and the behavior of the orbit of the corresponding point in the infinite-dimensional Teichmüller space. The region of discontinuity is the complement of the limit set and the quotient restricted to this set inherits a geometric structure from the Teichmüller space. However, another problem is caused by the fact that this region is not always dense in the Teichmüller space. To overcome this difficulty, we introduce the concept of region of stability, which is the set of points where the Teichmüller modular group acts in a stable way. Stability is defined by closedness of the orbit. Then the region of stability

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sits in the Teichmüller space as an open dense subset and the metric completion of the quotient of the region of stability defines the stable moduli space. As a generalization of the moduli space for finite type Riemann surfaces, we expect that the stable moduli space should be an object we have to work with. In the actual arguments developed in this theory, a comparison between countability and uncountability, such as the Baire category theorem, appears at several places. Uncountability is expressed in the non-separability of the Teichmüller space of an infinite type Riemann surface and by the cardinality of its Teichmüller modular group. In various situations, ideas of our arguments lie in how to pick out countability in the presence of these uncountable circumstances and how to use it in each specific case. Typically, countability comes from  -compactness of Riemann surfaces and from compactness of a family of normalized quasiconformal homeomorphisms with bounded dilatation in the compact-open topology. Further, by considering a fiber of the projection of the Teichmüller space onto the asymptotic Teichmüller space, we are able to extract countability in an implicit manner. The asymptotic Teichmüller space is a new concept for infinite-dimensional Teichmüller spaces introduced by Gardiner and Sullivan [26] and developed by Earle, Gardiner and Lakic [4], [5], [6]. It parametrizes the deformation of complex structures on arbitrarily small neighborhoods of the topological ends of an infinite type Riemann surface. Therefore, the projection from the Teichmüller space means ignoring the deformation of the complex structures on any compact regions, and hence, in each fiber of this projection, all these ignored deformations constitute a separable closed subspace in the Teichmüller space. The Teichmüller modular group acts preserving the fiber structure on the Teichmüller space and it induces a group of biholomorphic automorphisms of the asymptotic Teichmüller space, which is called the asymptotic Teichmüller modular group. In this way, we can divide the action of the Teichmüller modular group into that on the fibers and that on the asymptotic Teichmüller space. Between these actions, a study of the action of a stabilizer subgroup of a fiber, which is called an asymptotically elliptic subgroup, has already been developed to some extent. In this chapter we review the dynamics of asymptotically elliptic subgroups. The dynamics of the asymptotic Teichmüller modular group will be an interesting future research project. In the next section (Section 2), we survey fundamental results on the dynamics of the Teichmüller modular group, without considering asymptotic Teichmüller spaces. We summarize results contained in a series of papers by Fujikawa [13], [14] and [15], in particular the concept of limit set of Teichmüller modular groups and the bounded geometry condition on hyperbolic Riemann surfaces. Concerning the stability of Teichmüller modular groups and several criteria for stable actions of particular subgroups, especially those for closed subgroups in the compact-open topology, we extract the arguments from [42] and [41] and edit them in a new way. Then, in Section 3, we add the consideration of asymptotic Teichmüller spaces. In particular, the action on a fiber over the asymptotic Teichmüller space is discussed in detail. We survey several results on asymptotically elliptic subgroups obtained in

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[36], [37], [38], [39] and [40]. The topological characterization of a quasiconformal mapping class that acts trivially on the asymptotic Teichmüller space is excerpted from [16] and [19]. As an application of this result, we explain a version of the Nielsen realization theorem for the asymptotic Teichmüller modular group, which is obtained in [20]. Finally, in Section 4, we construct moduli spaces and several quotient spaces by subgroups of Teichmüller modular groups. The stable moduli space is introduced here, which is one of the aims persued in [42]. As a quotient space by the subgroup consisting of all mapping classes acting trivially on the asymptotic Teichmüller space, we obtain an intermediate Teichmüller space. When R satisfies the bounded geometry condition, this coincides with the enlarged moduli space, which is the quotient by the stable quasiconformal mapping class group. This is a subgroup of the mapping class group given by the exhaustion of mapping class groups of topologically finite subsurfaces. Then the asymptotic Teichmüller modular group is canonically realized as the automorphism group of the intermediate Teichmüller space. These arguments are demonstrated in [19]. Throughout this chapter, our original work [19] and [42] is frequently cited as basic references. The research announcement of [42] appeared in [33].

2 Dynamics of Teichmüller modular groups In this section, we develop the theory of dynamics of Teichmüller modular groups acting on infinite-dimensional Teichmüller spaces. For an analytically finite Riemann surface, which is a Riemann surface obtained from a compact Riemann surface by removing at most a finite number of points, the mapping class group and its action on the Teichmüller space are well known and broadly studied. For an analytically infinite Riemann surface whose Teichmüller space is infinite-dimensional, we also consider mapping classes in the quasiconformal category. Their action on the infinite-dimensional Teichmüller space induces Teichmüller modular transformations just like in the finitedimensional cases. Especially in this case, non-homogeneity of the Teichmüller space indicates an interesting interaction between the dynamics of orbits and hyperbolic structures on the base Riemann surface. In the first part of this section, we give basic concepts on the dynamics of Teichmüller modular groups. Then we discuss fundamental techniques for treating various kinds of subgroups of these groups. We also show some application of these theories to infinite-dimensional Teichmüller spaces.

2.1 Teichmüller spaces and modular groups Throughout this chapter, we assume that a Riemann surface R is hyperbolic, that is, it is represented as a quotient space D=H of the unit disk D endowed with the hyperbolic metric by a torsion-free Fuchsian group H . Without specific mention, we always re-

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gard R to have the hyperbolic structure, but when a hyperbolic geometrical aspect of R is a matter in question, we sometimes call R a hyperbolic surface. If the limit set ƒ.H / of the Fuchsian group H is a proper subset of the unit circle @D, then H acts properly x  ƒ.H / and a bordered Riemann surface .D x  ƒ.H //=H is discontinuously on D obtained, which contains R as its interior. In this case, .@D  ƒ.H //=H is called the boundary at infinity of R and denoted by @1 R. We are mainly interested in the case where the fundamental group 1 .R/ Š H is infinitely generated, namely, R is of infinite topological type. (Conversely, if 1 .R/ is finitely generated, then R is said to be of finite topological type. Furthermore, if 1 .R/ is cyclic, we call R elementary. ) We now define the Teichmüller space for R and its Teichmüller modular group. Teichmüller spaces in general The Teichmüller space T .R/ of an arbitrary Riemann surface R is the set of all equivalence classes of quasiconformal homeomorphisms f of R onto another Riemann surface. Two quasiconformal homeomorphisms f1 and f2 are defined to be equivalent if there is a conformal homeomorphism h W f1 .R/ ! f2 .R/ such that f21 BhBf1 is homotopic to the identity on R. Here the homotopy is considered to be relative to the boundary at infinity @1 R when @1 R is not empty. It is proved in Earle and McMullen [10] that the existence of a homotopy is equivalent to saying that there is an isotopy to the identity of R through uniformly quasiconformal automorphisms (relative to @1 R if @1 R ¤ ;). The equivalence class of f is called its Teichmüller class and denoted by Œf . The Teichmüller space T .R/ has a complex Banach manifold structure. When R is analytically finite, T .R/ is finite-dimensional, and otherwise T .R/ is infinitedimensional. A distance between p1 D Œf1  and p2 D Œf2  in T .R/ is defined by dT .p1 ; p2 / D 12 log K.f /, where f is an extremal quasiconformal homeomorphism in the sense that its maximal dilatation K.f / is minimal in the homotopy class of f2 B f11 (relative to the boundary at infinity if it is not empty). This is called the Teichmüller distance. In virtue of a compactness property of quasiconformal maps, the Teichmüller distance dT is complete on T .R/. This coincides with the Kobayashi distance on T .R/ with respect to the complex Banach manifold structure. Consult [12], [24], [25], [28], [30] and [43] for the theory of Teichmüller space. Quasiconformal mapping class groups For an arbitrary Riemann surface R, the quasiconformal mapping class group MCG.R/ is the group of all homotopy classes Œg of quasiconformal automorphisms g of R (relative to @1 R if @1 R ¤ ;). Each element Œg is called a mapping class and it acts on T .R/ from the left in such a way that Œg W Œf  7! Œf B g 1 . It is evident from the definition that MCG.R/ acts on T .R/ isometrically with respect to the Teichmüller distance. It also acts biholomorphically on T .R/. Definition 2.1. Let  W MCG.R/ ! Aut.T .R// be the homomorphism defined by Œg 7! Œg , where Aut.T .R// denotes the group of all isometric biholomorphic

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automorphisms of T .R/. The image Im   Aut.T .R// is called the Teichmüller modular group and is denoted by Mod.R/. Except for a few low-dimensional cases,  is injective. In particular, if R is analytically infinite, then  is always injective. This was first proved by Earle, Gardiner and Lakic [4] and another proof was given by Epstein [11]. We will discuss again this proof in Section 2.4 later. The map  is also surjective except in the one-dimensional case. This was finally proved by Markovic [31] after a series of pioneering works. We refer to the account [12] in Volume II of this Handbook. Hence, when there is no risk of confusion, we sometimes identify MCG.R/ with Mod.R/. Bounded geometry condition We often put some moderate assumptions concerning the geometry of hyperbolic Riemann surfaces which make the analysis of Teichmüller modular groups easier. Definition 2.2. We say that a hyperbolic surface R satisfies the lower boundedness condition if the injectivity radius at every point of R is uniformly bounded away from zero except in horocyclic cusp neighborhoods of area 1. We say that R satisfies the upper boundedness condition if the injectivity radius at every point of R is uniformly bounded from above, where R is some connected subsurface of R such that the inclusion map R ! R induces a surjective homomorphism 1 .R / ! 1 .R/. Then R satisfies the bounded geometry condition if both the lower and upper boundedness conditions are satisfied and if the boundary at infinity @1 R is empty. These conditions are quasiconformally invariant and hence we may regard them as conditions for the Teichmüller space T .R/. For example, an arbitrary non-universal normal cover of an analytically finite Riemann surface satisfies the bounded geometry condition (see [13]). A pair of pants is a hyperbolic surface with three geodesic boundary components and zero genus, where geodesic boundaries can degenerate to punctures. When a hyperbolic surface R can be decomposed into the union of pairs of pants, we say that R has a pants decomposition. If R has a pants decomposition such that all the lengths of boundary geodesics of the pairs of pants are uniformly bounded from above and from below, then R satisfies the bounded geometry condition. However, the converse is not true. Counter-examples can be easily obtained by considering a planar non-universal normal cover of an analytically finite Riemann surface with a puncture (see [17]).

2.2 Orbits of Teichmüller modular groups Except for the universal Teichmüller space T .D/, which is the Teichmüller space of the unit disk D, and for the Teichmüller spaces T .R/ of the punctured disk or the threepunctured sphere R, no Teichmüller space T .R/ is homogeneous in a sense that the

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Teichmüller modular group Mod.R/ acts transitively on T .R/. Actually, these are the only hyperbolic Riemann surfaces which have no moduli. In the non-homogeneous case, the aspects of the dynamics of Mod.R/ are different depending on the points p 2 T .R/, and the geometric structure of the Riemann surface corresponding to p reflects the action of Mod.R/ at p. Limit sets For an analytically finite Riemann surface R, the Teichmüller modular group Mod.R/ acts on T .R/ properly discontinuously. Although Mod.R/ has fixed points on T .R/, each orbit is discrete and each stabilizer subgroup is finite. Hence an orbifold structure on the moduli space M.R/ is induced from T .R/ as the quotient space by Mod.R/. However, this is not always satisfied for analytically infinite Riemann surfaces. Hence, we introduce the concept of limit set for the Teichmüller modular group Mod.R/. For a subgroup   Mod.R/, the orbit of p 2 T .R/ under  is denoted by .p/ and the stabilizer subgroup of p in  is denoted by Stab .p/. In the case where  D Mod.R/, Stab .p/ is denoted by Stab.p/. For an element  2 , the set of all fixed points of  is denoted by Fix./. The set of all common fixed points of the elements of  is denoted by Fix./. Definition 2.3. For a subgroup   Mod.R/ and for a point p 2 T .R/, we say that q 2 T .R/ is a limit point of p if there exists a sequence fn g of distinct elements of  all limit points of p is denoted by ƒ.; p/. such that n .p/ converge to q. The set ofS The limit set for  is defined by ƒ./ D p2T .R/ ƒ.; p/ and the elements of ƒ./ are called the limit points of . A point p 2 T .R/ is a recurrent point of  if p 2 ƒ.; p/ and the set of all recurrent points of  is denoted by Rec./. It follows from the definition that Rec./  ƒ./ and these sets are -invariant. In fact, we have the following fact. Proposition 2.4 ([13], [18]). For a subgroup   Mod.R/, the limit set ƒ./ coincides with Rec./ and it is a closed set. Moreover, p 2 T .R/ is a limit point of  if and only if either the orbit .p/ is not a discrete set or the stabilizer subgroup Stab .p/ consists of infinitely many elements. The notion of limit set was originally defined for a Kleinian group and it was also defined for the iteration of a holomorphic function as the Julia set. Some properties of our limit set are common to the original settings but some are not. For instance, the limit set is the smallest invariant closed subset in the original setting, but this is not true for the case of Teichmüller modular groups. Discontinuity and stability The complement of the limit set should be defined as the region of discontinuity. Hence we define discontinuity of the action at a point p 2 T .R/ as the complementary

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condition for p to be a limit point. By weakening the property of discreteness of the orbit, we also introduce another concept of manageable action, stability, which will be important for our arguments later. Definition 2.5. Let  be a subgroup of Mod.R/. We say that  acts at p 2 T .R/ • discontinuously if .p/ is discrete and Stab .p/ is finite; • weakly discontinuously if .p/ is discrete; • stably if .p/ is closed and Stab .p/ is finite; • weakly stably if .p/ is closed. If  acts at every point p in T .R/ (weakly) discontinuously or stably, then we say that  acts on T .R/ (weakly) discontinuously or stably, respectively. The set of points p 2 T .R/ where  acts discontinuously is denoted by ./ and called the region of discontinuity for . The set of points p 2 T .R/ where  acts stably is denoted by ˆ./ and called the region of stability for . Note that   Mod.R/ acts at p 2 T .R/ discontinuously if and only if there exists a neighborhood U of p such that the number of elements  2  satisfying .U / \ U ¤ ; is finite. When T .R/ is locally compact (i.e., finite-dimensional), this condition is the same as proper discontinuity. The discontinuity and stability criteria mentioned above have obvious inclusion relations that immediately follow from their definitions. The following theorem says that the converse assertion holds under a certain countability assumption. This fact is based on the Baire category theorem and the uncountability of perfect closed sets in a complete metric space. Lemma 2.6 ([42]). Assume that   Mod.R/ contains a subgroup 0 of countable index (that is, the cardinality of the cosets = 0 is countable) such that 0 acts at p 2 T .R/ weakly discontinuously. If  acts at p (weakly) stably, then  acts at p (weakly, resp.) discontinuously. The region of discontinuity ./ is always an open set because it is the complement of the limit set ƒ./ as it follows from Proposition 2.4. However, we only see that the region of stability ˆ./ becomes an open set under a certain condition upon . This is also obtained by an argument based on the Baire category theorem. Lemma 2.7 ([42]). If   Mod.R/ contains a subgroup 0 of countable index such that 0 acts on T .R/ stably, then the region of stability ˆ./ is open. We regard these two lemmata as fundamental principles of our arguments on the dynamics of Teichmüller modular groups and we utilize them in later discussion. If  acts discontinuously, then every subgroup of  acts discontinuously. However, this property is not necessarily satisfied for stability. This is because any subset of a discrete set is always discrete whereas any subset of a closed set is not always closed. By this evidence, we have the following claim.

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Proposition 2.8. Let fi gi2I be a family of subgroups T of Mod.R/ such that each i acts stably at p 2 T .R/. Then the intersection  D i2I i acts stably at p 2 T .R/. Boundedness and divergence Now we consider another aspect of the dynamics of Mod.R/. We classify the action of subgroups of Mod.R/ according to the global behavior of their orbits. Definition 2.9. Let  be a subgroup of Mod.R/. If the orbit .p/ of some p 2 T .R/ is a bounded set in T .R/, we say that  is of bounded type. On the other hand, if the orbit .p/ is divergent to the infinity of T .R/, meaning that .p/ is infinite and each bounded subset of T .R/ contains only finitely many of .p/, we say that  is of divergent type. Note that the notions of bounded type and of divergent type are well-defined for  since these properties of the orbit are independent from the choice of p 2 T .R/. When R is analytically finite, T .R/ is locally compact and Mod.R/ acts properly discontinuously on T .R/. Hence every infinite subgroup of Mod.R/ is of divergent type and of course every finite subgroup of Mod.R/ is of bounded type. However, for a general R, there are subgroups of Mod.R/ neither of bounded type nor of divergent type even for infinite cyclic subgroups. See [36], where we have tried to give a classification of the Teichmüller modular transformations. If we restrict subgroups of Mod.R/ to certain classes, then they have the dichotomy of boundedness and divergence. We will discuss later these classes having a nature similar to the finitedimensional cases. Example 2.10. Here we give an example of a Teichmüller modular transformation  2 Mod.R/ such that hi acts discontinuously on T .R/ and it is neither of bounded type nor of divergent type. Let S be a closed hyperbolic surface of genus 3 and take three mutually disjoint non-dividing simple closed geodesics a, b and c on S. Cut S along a and b to make a totally geodesic surface S 0 of genus 1 with four boundary components, and give a pants decomposition for S 0 having c and the copies of a and b as boundary geodesics. We prepare copies of S 0 and paste them to make an abelian covering surface R0 of S with the covering transformation group isomorphic to Z2 . Then we index all the lifts of c to R0 in such a way that cnk is the image of some fixed lift c00 under the covering transformation corresponding to .n; k/ 2 Z2 . We extend the pants decomposition of S 0 to R0 such that the action of the covering transformation group Z2 preserves this decomposition. By assigning the geodesic lengths `.cnk / to each cnk and keeping the lengths of the other boundary curves of the pants decomposition invariant, we can obtain various hyperbolic Riemann surfaces R. This is performed by a locally quasiconformal deformation but it is not necessarily globally quasiconformal. A suitable choice of f`.cnk /g gives an interesting example of a mapping class Œg corresponding to the

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element of the covering transformation n 7! n C 1, k 7! k in Z2 . For our purpose, we define ˚  `.cnk / D exp  2jkj h.2jkj.jkjC1/=2 n/ ; where h is a periodic function of period one defined on R such that h.x/ D x for 0  x  1=2 and h.x/ D 1  x for 1=2  x  1. Then, we see that the mapping class Œg gives a Teichmüller modular transformation  2 Mod.R/ such that h i acts discontinuously on T .R/. Furthermore, we can find subsequences fni g and fnj g such that f ni .p/g is bounded and f nj .p/g is divergent for any p 2 T .R/. See [21]. The classification by divergence and boundedness is more restrictive than that by discontinuity and instability. Proposition 2.11. If a subgroup   Mod.R/ is of divergent type, then  acts discontinuously on T .R/. On the contrary, if an infinite subgroup  is of bounded type, then  does not act stably on T .R/. Proof. The first statement is obvious from the definition. The second statement will be seen later on by the arguments on elliptic subgroups (Theorem 2.34).

2.3 Subgroups of Teichmüller modular groups In this subsection, we list up several subgroups of the Teichmüller modular group, which have special properties with respect to their action on Teichmüller space. We intend to summarize a glossary of basic facts on their dynamics. Countable index subgroups The following subgroup of countable index in Mod.R/ plays an important role in our arguments for the application of Lemmata 2.6 and 2.7. Definition 2.12. For a homotopically non-trivial simple closed curve c, we define MCGc .R/ to be a subgroup of MCG.R/ consisting of all mapping classes that preserve c: MCGc .R/ D fŒg 2 MCG.R/ j g.c/  cg; where  means free homotopy equivalence. We denote the image of MCGc .R/ under the representation  W MCG.R/ ! Mod.R/ by Modc .R/. Proposition 2.13. For any non-trivial simple closed curve c in R, the subgroup MCGc .R/ is of countable index in MCG.R/. For an arbitrary subgroup G  MCG.R/, there is a subgroup G 0  MCGc .R/ of countable index in G. Proof. The countability of the indices comes from the fact that the number of free homotopy classes of non-trivial simple closed curves on R is countable. The latter statement is obtained by taking the intersection of G with MCGc .R/.

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As this proposition shows, we can say that  -compactness of Riemann surfaces is at the basis of the countability involved in the dynamics of Teichmüller modular groups. Countable subgroups It is well known that the mapping class group of an analytically finite Riemann surface is finitely generated and in particular countable. However, for almost all analytically infinite Riemann surfaces, the quasiconformal mapping class groups are uncountable. We consider countable subgroups of Mod.R/. The fundamental lemma 2.6 includes the following claims in particular if   Mod.R/ itself is countable. Theorem 2.14. Let  be a countable subgroup of Mod.R/. Then  acts (weakly) discontinuously at p 2 T .R/ if and only if  acts (weakly, resp.) stably at p. In particular ./ D ˆ./. The following subgroup of MCG.R/ can be regarded as the exhaustion of mapping class groups of topologically finite subsurfaces of R. Definition 2.15. A mapping class Œg 2 MCG.R/ is called trivial near infinity (or essentially trivial) if some representative g W R ! R is the identity outside some topologically finite bordered subsurface V  R possibly having punctures such that V is closed in R [ @1 R. Let MCG1 .R/ be the subgroup of MCG.R/ consisting of all mapping classes trivial near infinity. Then we call it the stable mapping class group. The image of MCG1 .R/ under the representation  W MCG.R/ ! Mod.R/ is denoted by Mod1 .R/. Since MCG1 .R/ admits an exhaustion by countable groups, it is countable. Moreover, MCG1 .R/ is a normal subgroup of MCG.R/. This group plays an important role when we consider the action of MCG.R/ on the asymptotic Teichmüller space later on. Under the assumption that R satisfies the bounded geometry condition, we see that MCG1 .R/ acts nicely on T .R/ as the following theorem asserts. Theorem 2.16 ([16]). If R satisfies the bounded geometry condition, then Mod1 .R/ acts discontinuously on T .R/. Moreover, whenever R is of infinite topological type, this action is fixed-point free. If MCG.R/ itself is countable for a hyperbolic Riemann surface R of infinite topological type, then the geometry of R is much more restricted (in the opposite direction to the boundedness) by this assumption and we have the following stronger result. Note that the existence of such a Riemann surface R is also known. To all appearances, this theorem is a generalization of the case where R is analytically finite. Theorem 2.17 ([35]). If Mod.R/ is countable, then it acts discontinuously on T .R/.

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In [37], an example of a Riemann surface R of infinite topological type satisfying MCG1 .R/ D MCG.R/ is given. Closed subgroups The compact-open topology on the space of all homeomorphic automorphisms of R induces a topology on the quasiconformal mapping class group MCG.R/. More precisely, we say that a sequence of mapping classes Œgn  2 MCG.R/ converges to a mapping class Œg 2 MCG.R/ in the compact-open topology if we can choose representatives gn 2 Œgn  and g 2 Œg such that gn converge to g locally uniformly on R. When R has boundary at infinity @1 R, assuming that the quasiconformal automorphisms gn and g extend to @1 R, we further require that these extensions converge locally uniformly on R [ @1 R. If Œgn  converge to Œg in the compact-open topology, then there are quasisymmetric automorphisms gQ n and gQ of the unit circle @D corresponding to Œgn  and Œg respectively such that gQ n converge uniformly to g. Q Definition 2.18. We say that a subgroup G of MCG.R/ is discrete if it is discrete in the compact-open topology on MCG.R/, and closed if it is closed. For a subgroup G, x the closure of G in MCG.R/. We also use the same terminology for we denote by G x by .G/. x the corresponding subgroup  D .G/ of Mod.R/ and define the closure  T It is clear that the intersection i2I Gi of closed subgroups Gi of MCG.R/ is also closed. For a point p 2 T .R/ such that the stabilizer Stab.p/ in Mod.R/ is trivial (see Theorem 2.37 in Section 2.4), the orbit of p defines a topology on Mod.R/ by using the Teichmüller distance on T .R/. However, this topology does not coincide with the x compact-open topology introduced above. In fact, the orbit .p/ for p 2 T .R/ does not necessarily coincide with the closure of the orbit .p/ in the topology on T .R/. Closed subgroups have preferable properties. The following theorem provides an algebraic condition on G for being closed. Theorem 2.19 ([41]). Assume that @1 R D ;. If G  MCG.R/ is a finitely generated abelian group, then G is discrete, and in particular closed. This result is no more valid for a countable group in general. The stable mapping class group MCG1 .R/, which is countable, is not closed in almost all cases. In what follows, we will give a closed subgroup that contains MCG1 .R/. We consider the end compactification R of a Riemann surface R by adding all the ends of R and by providing this union with the canonical topology. Here an end means a topological end if the boundary at infinity @1 R is empty. However, if y of R with respect to @1 R and then take @1 R ¤ ;, we first consider the double R y which we define to be R . y of R, the closure of R in the end compactification R This has been introduced in [22]. Every quasiconformal automorphism of R extends to a homeomorphic automorphism of R . Furthermore this extension preserves the

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cuspidal ends. The extension restricted to the ideal boundary R  R is determined by the mapping class of the quasiconformal automorphism. Clearly, every mapping class trivial near infinity fixes all the ends except cuspidal ends. Definition 2.20. The subgroup of MCG.R/ consisting of all mapping classes that fix all the ends except the cuspidal ends is called the pure mapping class group and denoted by MCG@ .R/. Proposition 2.21. The pure mapping class group MCG@ .R/ is a closed normal subgroup of MCG.G/ which contains MCG1 .R/. Proof. The property of fixing the ends is preserved under convergence in the compactopen topology. Stationary subgroups I: with closedness To generalize certain properties shared by the mapping class group of an analytically finite Riemann surface, we will consider a subgroup of MCG.R/ that keeps the images of some compact bordered subsurface bounded. Definition 2.22. We call a subgroup G of MCG.R/ stationary if there exists a compact bordered subsurface V of R such that every representative g of every mapping class Œg 2 G satisfies g.V / \ V ¤ ;. The corresponding subgroup  D .G/ of Mod.R/ is also called stationary. x in the It is clear from the definition that if G is stationary, then so is the closure G compact-open topology. A basic feature of stationary subgroups in connection with their closedness and discreteness can be summarized as the following theorem. Theorem 2.23. Let  be a stationary subgroup of Mod.R/. If  is closed then it acts stably on T .R/. If  is infinite and discrete then it is of divergent type, and in particular it acts discontinuously on T .R/. Proof. Compactness of a family of stationary quasiconformal automorphisms with uniformly bounded dilatations yields that if there is a sequence Œgn  in MCG.R/ such that n .p/ is bounded in T .R/ for n D Œgn  and for p 2 T .R/, then a subsequence of Œgn  converges to some Œg 2 MCG.R/ in the compact-open topology. Suppose that a stationary subgroup  is closed. If n .p/ converges to q 2 T .R/ for some sequence n 2  and for p 2 T .R/, then we see that  D Œg belongs to  and .p/ D q. This implies that the orbit .p/ is closed for every p 2 T .R/. Since the stabilizer is finite for any stationary subgroup,  acts stably on T .R/. Suppose that  is infinite and discrete. In this case, we see that there is no subsequence n 2  such that n .p/ is bounded in T .R/. This implies that  is of divergent type.

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x acts stably on T .R/ Corollary 2.24. If a subgroup  of Mod.R/ is stationary, then  x and .p/ D .p/ for every p 2 T .R/. x is stationary and closed, it acts stably on T .R/ by Theorem 2.23. Then Proof. Since  x x .p/ is closed for every p 2 T .R/, which gives .p/  .p/. To prove the converse inclusion, we take an arbitrary point q 2 .p/ and consider a sequence n 2  such that n .p/ ! q as n ! 1. As in the proof of Theorem 2.23, since  is stationary, x such that .p/ D q. This implies q 2 .p/. x we have  2  By imposing an algebraic condition on  as before, we have another corollary obtained from Theorems 2.19 and 2.23. Corollary 2.25. Assume that @1 R D ;. If a finitely generated infinite abelian group   Mod.R/ is stationary, then  is of divergent type. Note that for an infinite cyclic group  this has been proved in [36] without imposing the condition @1 R D ;. We expect that the statement of the corollary is always valid without this assumption. As an example of a stationary subgroup, we have the pure mapping class group MCG@ .R/ in many cases as the following proposition states. See [16] for details. Recall that MCG@ .R/ is also closed by Proposition 2.21. Proposition 2.26. Assume that R has at least three non-cuspidal topological ends. Then the pure mapping class group MCG@ .R/ is stationary. In this case, Mod@ .R/ D .MCG@ .R// acts stably on T .R/. Proof. This is because a pair of pants that divides three non-cuspidal topological ends has nonempty intersection with its image under every element of MCG@ .R/. The latter statement is a consequence of Theorem 2.23. The subgroup Modc .R/ defined before is closed and it is stationary if R is nonelementary. Hence Modc .R/ acts stably on T .R/ by Theorem 2.23. More generally, we have the following. Lemma 2.27. Assume that R is non-elementary. Each subgroup  of Mod.R/ contains a stationary subgroup  0 of countable index in . In addition, if  is closed, then  0 can be taken to be closed and hence acting stably on T .R/. Proof. Set  0 D  \ Modc .R/. Then this is stationary since so is Modc .R/, and it is of countable index in  by Proposition 2.13. Furthermore if  is closed, then  0 is closed since Modc .R/ is closed. We mentioned that the region of stability ˆ./ for a subgroup   Mod.R/ is not necessarily open. However, by applying Lemmata 2.7 and 2.27, we can now recognize a sufficient condition for the region of stability to be open.

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Theorem 2.28. For a closed subgroup  of Mod.R/, the region of stability ˆ./ is an open subset of T .R/. In particular, ˆ.Mod.R// is open. Stationary subgroups II: with bounded geometry Another feature of a stationary subgroup is that under the bounded geometry condition it acts discontinuously on T .R/. Note that we cannot drop any of the three assumptions in the bounded geometry condition (lower boundedness, upper boundedness and @1 R D ;) for the validity of this claim. Theorem 2.29 ([23], [14]). Let  be a stationary subgroup of Mod.R/. If R satisfies the bounded geometry condition, then  acts discontinuously on T .R/. We apply this theorem to Modc .R/. Then Lemma 2.6 implies the following result. Theorem 2.30. Assume that R satisfies the bounded geometry condition. Then a subgroup  of Mod.R/ acts (weakly) discontinuously at p 2 T .R/ if and only if  acts (weakly, resp.) stably at p. In particular ./ D ˆ./. We also see in the following theorem that ˆ./ D ./ is non-empty in this case, which has been proved in [13]. Later we will see a stronger assertion that ˆ.Mod.R// is always dense in T .R/ without any assumption on the geometry of R. Theorem 2.31. If R satisfies the bounded geometry condition, then, for any subgroup  of Mod.R/, ./ is non-empty. Proof. We only have to show the statement for  D Mod.R/. Since R satisfies the lower boundedness condition, we choose an arbitrary simple closed geodesic c on R and give a deformation of the hyperbolic structure by pinching c such that it becomes the unique shortest simple closed geodesic with respect to the new hyperbolic structure. Let p be the corresponding point on T .R/. For a neighborhood U of p, we consider the smallest subgroup 0 of Mod.R/ that contains f 2 Mod.R/ j .p/ 2 U g. If U is sufficiently small, 0 is contained in Modc .R/. Since Modc .R/ acts discontinuously on T .R/ by Theorem 2.29, so does 0 and hence Mod.R/ acts discontinuously at p. If we do not assume a geometric condition on R, this result is not satisfied any more. For instance, if R does not satisfy the lower boundedness condition or if the boundary at infinity @1 R is not empty, then .Mod.R// D ;. As a conjecture, we expect that the converse of this claim is also true. Conjecture 2.32. The region of discontinuity .Mod.R// is not empty if and only if R satisfies the lower boundedness condition together with @1 R D ;.

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Elliptic subgroups If a Teichmüller modular transformations Œg 2 Mod.R/ has a fixed point on T .R/, then it is called elliptic according to Bers [1]. This is equivalent to a condition that the mapping class Œg 2 MCG.R/ is realized as a conformal automorphism of the Riemann surface f .R/ corresponding to p D Œf , that is, fgf 1 is homotopic to a conformal automorphism of f .R/ (relative to the boundary at infinity if it is not empty). Such a mapping class Œg is called a conformal mapping class. When R is an analytically finite Riemann surface, Œg 2 Mod.R/ is elliptic if and only if it is periodic (of finite order). We extend the concept of ellipticity to the case where R is not necessarily analytically finite. In this case, elliptic modular transformations can be of infinite order. Definition 2.33. A subgroup  of Mod.R/ is called elliptic if it has a common fixed point on T .R/. If  D .G/ is an elliptic subgroup of Mod.R/ fixing p D Œf  2 T .R/, then the subgroup G of MCG.R/ is realized as a group of conformal automorphisms of f .R/. Hence G is a countable group. Furthermore G is discrete in the compact-open topology, and if G is an infinite group, then it is not stationary. We characterize elliptic subgroups by their orbits. It is clear that any orbit .q/ of an elliptic subgroup  is bounded since dT ..q/; p/ D dT .q; p/ for a common fixed point p and for all  2 . The following theorem says that the converse is also true. Theorem 2.34. A subgroup  of Mod.R/ is elliptic if and only if  is of bounded type, that is, the orbit .p/ for some p 2 T .R/ is bounded. In the case where R is analytically finite, the Nielsen realization theorem, which was finally proved by Kerckhoff [29], is equivalent to saying that every finite subgroup of Mod.R/ is elliptic. The realization theorem says that every finite subgroup of MCG.R/ is realized as a group of conformal automorphisms of a Riemann surface corresponding to the fixed point. Since a finite subgroup has a bounded orbit, Theorem 2.34 can be regarded as a generalization of the realization theorem. The proof is essentially based on a theorem due to Markovic [32], which asserts that a uniformly quasisymmetric group on the unit circle @D is conjugate to a Fuchsian group by a quasisymmetric homeomorphism. Next, we see that most infinite elliptic subgroups  have an indiscrete orbit in T .R/. Since  is countable, Theorem 2.14 implies that this is equivalent to the statement that the orbit is not closed. Theorem 2.35. Assume that an elliptic subgroup   Mod.R/ has an infinite descending sequence fn g1 nD1 of proper subgroups  ¥ 1 ¥ 2 ¥    . Then the S union X D n1 Fix.n / is not closed in T .R/ and, at each p 2 Xx  X,  does not act weakly discontinuously, in other words, the orbit .p/ is not a discrete set. In

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particular, if an elliptic subgroup  contains an element of infinite order, the above assumption is always satisfied and the conclusion is valid. Proof. The strict inclusion relation n ¥ nC1 gives the strict inclusion relation Fix.n / ¤ Fix.nC1 / for every n by passing to a subsequence if necessary. This has been proved in [34]. Then the Baire category theorem implies that X is not closed and x Take any point p 2 Xx  X and consider a sequence fn .p/g1 Xx  X is dense in X. nD1 for n 2 n  nC1 . Then we see that n .p/ ¤ p and limn!1 n .p/ D p. This shows that the orbit .p/ is not a discrete set. Note that an arbitrary countable group can be realized as a group of conformal automorphisms of some Riemann surface (cf. [27]). Hence there is an example of an infinite elliptic subgroup  that does not contain an infinite descending sequence of proper subgroups. For these groups, we do not know whether their orbits are discrete or not. Finally, we show that each element of the countable group Mod1 .R/, which comes from a mapping class trivial near infinity, can be represented by the composition of elliptic modular transformations of infinite order if they exist. Theorem 2.36. Assume that Mod.R/ contains an elliptic modular transformation of infinite order. Then any element of the countable subgroup Mod1 .R/ can be written as a composition of some elliptic elements of infinite order of Mod.R/. Proof. For an elliptic modular transformation  2 Mod.R/ of infinite order, we may assume that it fixes the base point o 2 T .R/ and hence the corresponding mapping class is realized as a conformal automorphism g of R. Take an arbitrary simple closed geodesic c in R. Since hgi acts on R properly discontinuously, there is an integer k ¤ 0 such that g k n .c/ \ c D ; for every integer n ¤ 0. Then, for gO D g k , a family of simple closed geodesics fgO n .c/gn2Z are mutually disjoint. Moreover in this case, there is a collar neighborhood Ac of c such that fgO n .Ac /gn2Z are mutually disjoint. We rename gO as g and reset  D Œg . For each n 2 Z, let tg n .c/ be a quasiconformal automorphism of R that is obtained by a Dehn twist supported on g n .Ac / D Ag n .c/ . We set gc D g B tc and define c D Œgc  . Since g 1 B tc B g D tg 1 .c/ , we have gcn D g n B tg .n1/ .c/ B    B tg 1 .c/ B tc for every integer n  1. From this expression, we see that the maximal dilatation of gcn is equal to that of tc because the support of the quasiconformal automorphisms tg i .c/ .0  i  n  1/ are mutually disjoint. This implies that the orbit fcn .o/gn2Z is bounded. Hence, by Theorem 2.34, c is also an elliptic modular transformation. It is easy to see that the order of c is infinite. Since tc D g 1 B gc , the corresponding modular transformation c D Œtc  is written as a composition of elliptic elements. Every element of MCG1 .R/ can be written as a composition of mapping classes obtained by Dehn twists along simple

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closed geodesics because this is true for the pure mapping class group of any compact bordered surface. Since every modular transformation c corresponding to a Dehn twist is the composition of elliptic elements of infinite order, we see that every element of Mod1 .R/ is also written as a composition of such elements. Note that no non-trivial elliptic modular transformations belong to Mod1 .R/ if R is analytically infinite.

2.4 Application to the infinite-dimensional Teichmüller theory We gave some basic concepts in the dynamics of the Teichmüller modular group. They have a general theoretical nature and will be developed by finding interesting applications to Teichmüller theory. Here we review some of such applications. Fixed point loci of Teichmüller modular groups For a finite-dimensional Teichmüller space T .R/, it is well known that the union of the fixed point loci of all non-trivial elements of the mapping class group MCG.R/ is nowhere dense in T .R/ except in a few cases of low dimensions. For instance, if R is a closed Riemann surface of genus 2, there exists an involution Œg in MCG.R/ that fixes all the points of T .R/. A Riemann surface having such a symmetry is called an exceptional surface. The representation  W MCG.R/ ! Aut.T .R// is injective for a non-exceptional Riemann surface R. For an infinite-dimensional Teichmüller space, a claim analogous to the above statement says that the union of the fixed point loci of MCG.R/ is contained in a countable union of nowhere dense subsets. This has been proved by Epstein [11]. The complement of this countable union is called a residual set which is dense in T .R/. The existence of a point in the residual set where the isotropy subgroup of MCG.R/ is trivial in particular shows that the representation  W MCG.R/ ! Aut.T .R// is injective. On the other hand, from a viewpoint of dynamical systems, we can consider a problem of whether the set of points p 2 T .R/ such that Stab.p/ has an element of infinite order is dense in the limit set of Mod.R/, in analogy with the same question in the dynamics of Kleinian groups. However, it is proved in [42] that this is not true for the dynamics of Teichmüller modular groups. This means that the fixed point loci of Mod.R/ is a thinner set even in the limit set and similar arguments for proving this fact give the following extension of the aforementioned result. Theorem 2.37 ([42]). The interior of the set of points p 2 T .R/ for which Stab.p/ is trivial is dense in T .R/. Biholomorphic automorphisms of Teichmüller spaces Now it is known that every biholomorphic automorphism of the Teichmüller space of dimension greater than one is a Teichmüller modular transformation. For finitedimensional Teichmüller spaces, this result was first proved by Royden [44]. In the

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general case, the proof is carried out by the combination of two theorems. Earle and Gardiner [3] proved the automorphism theorem, which asserts that the above claim is true if a Riemann surface satisfies a so-called isometry property. Then Markovic [31] finally proved that every non-exceptional Riemann surface satisfies the isometry property. See the exposition in Volume II of this Handbook [12]. See also [9] for an adaptation of this idea to the proof in the finite-dimensional case. If we assume the isometry property against the chronological order, then we can state an essential part of the automorphism theorem as follows. Theorem 2.38. Assume that the Teichmüller space T .R/ has dimension greater than one. Then, for every biholomorphic automorphism of T .R/ and for every point p 2 T .R/, there exists an element p 2 Mod.R/ such that .p/ D p .p/. For an analytically finite Riemann surface, once this theorem is proved (actually Royden’s arguments imply this statement), then it is easy to obtain the result that any biholomorphic automorphism is a Teichmüller modular transformation. This is due to the fact that the Teichmüller modular group acts properly discontinuously in this case. However, in the general case, we still need an extra argument to reach the desired result. This step was included in the proof of the automorphism theorem in [3]. Fujikawa [15] found that there is a certain case where the original argument of Royden can be applied without change. The assumption for this case is described by using the region of discontinuity of the Teichmüller modular group. Theorem 2.39. For a biholomorphic automorphism of T .R/, assume that there exists a subgroup  of Mod.R/ with ./ ¤ ; such that, for every point p 2 ./, there is an element p 2  satisfying .p/ D p .p/. Then coincides with an element of . Proof. Choose a point p 2 ./. The stabilizer of p is a finite group in general and this does not make a trouble in the proof as is shown in [15], but here we assume that the stabilizer is trivial for the sake of simplicity. Actually, the choice of such a p is always possible because the set of all such p is dense in T .R/ as is seen before. Take a disk Up .r/ with center at p and radius r > 0 such that .Up .r//\Up .r/ D ; for every non-trivial  2 . Then consider a smaller disk Up .r=2/ of radius r=2 and choose an arbitrary point q 2 Up .r=2/. It is clear that dT ..q/; q/ > r for every nontrivial  2 . We consider p1 B q 2  and, from the fact that the biholomorphic automorphism preserves the Kobayashi distance on T .R/, we have dT .p1 B q .q/; q/ D dT .q .q/; p .q//  dT .q .q/; p .p// C dT .p .p/; p .q// D dT . .q/; .p// C dT .p; q/ < r: This estimate implies that p1 B q should be trivial and hence q D p for every q 2 Up .r=2/. Therefore D p restricted to Up .r=2/. Then, by the rigidity of holomorphic functions, we conclude that coincides with p on T .R/.

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In order to apply this theorem, it is necessary to find geometric or algebraic conditions under which ./ is not empty. Theorem 2.31 says that this is true if R satisfies the bounded geometry condition. Here we give the following condition on  to guarantee ./ ¤ ;. The assumption ./ D ˆ./ is satisfied, for example, when  is countable by Theorem 2.14. Lemma 2.40. Assume that R is non-elementary. If   Mod.R/ is a closed subgroup such that ./ D ˆ./, then ./ ¤ ;. Proof. As in the proof of Theorem 2.31, we can choose a neighborhood U of some p 2 T .R/ and a simple closed geodesic c on R so that the smallest subgroup 0 that contains f 2  j .p/ 2 U g is contained in Modc .R/. Note that this is possible even if R does not satisfy the bounded geometry condition. See [42]. x0 is contained in x0 of 0 . Since Modc .R/ and  are closed,  Take the closure  x0 . Hence  x0 is stationary and both of them. Since Modc .R/ is stationary, so is  x0 acts stably on T .R/. This implies that  acts stably closed, and by Theorem 2.23,  at p and thus ˆ./ ¤ ;.

3 The action on the asymptotic Teichmüller space We regard Teichmüller space as a fiber space over a certain base space called the asymptotic Teichmüller space. An asymptotically conformal homeomorphism of a Riemann surface is a quasiconformal map that is close to a conformal map as we go to the infinity of the surface. The asymptotic Teichmüller space is defined by replacing the roles of conformal homeomorphisms with asymptotically conformal ones in the definition of the Teichmüller space. Since the quasiconformal mapping class group acts on the Teichmüller space preserving the fibers, its action can be divided into that on each fiber and that on the asymptotic Teichmüller space. In this section, we are mainly concerned with the former action, which is given by a group of asymptotically conformal mapping classes. It acts on the Teichmüller space as an asymptotically elliptic subgroup of the Teichmüller modular group, having certain similarity to Teichmüller modular groups of analytically finite Riemann surfaces.

3.1 Asymptotic Teichmüller spaces and modular groups We introduce the asymptotic Teichmüller space and define the action of the quasiconformal mapping class group on this space. Asymptotic Teichmüller spaces The asymptotic Teichmüller space has been introduced by Gardiner and Sullivan [26] for the unit disk and by Earle, Gardiner and Lakic [4], [5], [6] for an arbitrary Riemann surface.

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Definition 3.1. We say that a quasiconformal homeomorphism f of a Riemann surface R is asymptotically conformal if, for every  > 0, there exists a compact bordered subsurface V of R such that the maximal dilatation K.f jRV / of the restriction of f to R  V is less than 1 C . We say that two quasiconformal homeomorphisms f1 and f2 of R are asymptotically equivalent if there exists an asymptotically conformal homeomorphism h W f1 .R/ ! f2 .R/ such that f21 BhBf1 is homotopic to the identity (relative to @1 R if @1 R ¤ ;). The asymptotic Teichmüller space AT .R/ of R is the set of all asymptotic equivalence classes ŒŒf  of quasiconformal homeomorphisms f of R. Since a conformal homeomorphism is asymptotically conformal, there is a natural projection ˛ W T .R/ ! AT .R/ that maps each Teichmüller equivalence class Œf  2 T .R/ to the asymptotic equivalence class ŒŒf  2 AT .R/. The asymptotic Teichmüller space AT .R/ has a complex manifold structure such that ˛ is holomorphic. Each fiber of the projection ˛ is a separable closed subspace of T .R/. Moreover ˛ induces a quotient distance dAT on AT .R/ from the Teichmüller distance, which is called the asymptotic Teichmüller distance. We do not know yet whether this distance coincides with the Kobayashi distance on AT .R/ or not. See [5], [6] and [8]. The asymptotic Teichmüller space AT .R/ is of interest only when R is analytically infinite. Otherwise AT .R/ is trivial, that is, it consists of just one point. Conversely, if R is analytically infinite, then AT .R/ is not trivial. In fact, it is infinite-dimensional and non-separable. Asymptotic Teichmüller modular groups Like in the case of Teichmüller space, every mapping class Œg 2 MCG.R/ induces a biholomorphic automorphism Œg of AT .R/ by ŒŒf  7! ŒŒf B g 1 , which is also isometric with respect to dAT . Note that since the projection ˛ W T .R/ ! AT .R/ is not known to be a holomorphic split submersion, the fact that Œg is holomorphic is not so trivial. See [6] and [7]. Definition 3.2. Let Aut.AT .R// be the group of all biholomorphic isometric automorphisms of AT .R/. For a homomorphism AT W MCG.R/ ! Aut.AT .R// given by Œg 7! Œg , we define the asymptotic Teichmüller modular group ModAT .R/ of R to be the image AT .MCG.R//. Unlike the representation  W MCG.R/ ! Aut.T .R//, the homomorphism AT is not injective, namely, Ker AT ¤ fŒidg unless R is either the unit disc or the once-punctured disc. See [4]. Definition 3.3. We call an element of Ker AT asymptotically trivial and call Ker AT the asymptotically trivial mapping class group. We also call an element of the corresponding subgroup .Ker AT / of Mod.R/ asymptotically trivial.

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The action of ModAT .R/ on AT .R/ has been studied by Fujikawa [17]. In particular, the limit set of ModAT .R/ in AT .R/ is investigated.

3.2 Asymptotically elliptic subgroups In order to investigate the action of the quasiconformal mapping class group on a fiber over the asymptotic Teichmüller space, we consider the stabilizer subgroup of the fiber in the Teichmüller modular group. The projection of this subgroup to the asymptotic Teichmüller modular group fixes the base point of the fiber on the asymptotic Teichmüller space. Asymptotically elliptic modular transformations Now we define asymptotic conformality for quasiconformal mapping classes and asymptotic ellipticity for Teichmüller modular transformations. Definition 3.4. A mapping class Œg 2 MCG.R/ is called asymptotically conformal if there is a quasiconformal homeomorphism f of R such that fgf 1 is homotopic to an asymptotically conformal automorphism of f .R/ (relative to the boundary at infinity if it is not empty). A Teichmüller modular transformation Œg 2 Mod.R/ is called asymptotically elliptic if Œg 2 ModAT .R/ has a fixed point ŒŒf  on AT .R/. It is clear that a mapping class Œg 2 MCG.R/ is asymptotically conformal if and only if the corresponding Teichmüller modular transformation Œg 2 Mod.R/ is asymptotically elliptic. An elliptic modular transformation is of course asymptotically elliptic. However the converse is not true. A trivial example is a mapping class caused by a single Dehn twist. This is not a conformal mapping class, but it acts trivially on AT .R/. In particular, it has a fixed point on AT .R/. Petrovic [45] first dealt with an asymptotically conformal mapping class that acts on AT .R/ non-trivially (in fact non-periodically) and that has no fixed point on T .R/. See also [40] for another example. When R is analytically finite, every Teichmüller modular transformation is asymptotically elliptic since AT .R/ consists of a single point. Asymptotically elliptic modular transformations are generalization of the Teichmüller modular transformations of analytically finite Riemann surfaces in this sense. Similarly, we define asymptotic ellipticity for subgroups of Mod.R/. Definition 3.5. A subgroup  D .G/ of Mod.R/ is called asymptotically elliptic if AT .G/  ModAT .R/ has a common fixed point on AT .R/. It is clear from the definition that a subgroup consisting of asymptotically trivial modular transformations is asymptotically elliptic. Elliptic subgroups are always countable. For asymptotically elliptic subgroups, this is not valid in general, but if we impose the bounded geometry condition on R, this is true as is shown in [38].

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Theorem 3.6. Assume that R satisfies the bounded geometry condition. Then every asymptotically elliptic subgroup  of Mod.R/ is countable. Proof. By Lemma 2.27, we can take a stationary subgroup  0 of countable index in . If  is uncountable, then so is  0 . On the other hand,  0 acts discontinuously on T .R/ by Theorem 2.29. In particular, the uncountable group  0 acts discontinuously on the fiber over the fixed point on AT .R/, which is separable. However, this is impossible. Like in the case where Mod.R/ is countable, if the entire Mod.R/ is asymptotically elliptic, this restrictive condition gives us a stronger consequence. Theorem 3.7 ([38]). If Mod.R/ itself is asymptotically elliptic, then Mod.R/ is countable and acts discontinuously on T .R/. There is an example of R such that Mod.R/ is asymptotically elliptic. Furthermore, the entire Mod.R/ can be asymptotically trivial. See [35] and [37] for these examples. The action on the fiber We consider the action of an asymptotically elliptic subgroup   Mod.R/ restricted to the fiber over the fixed point on AT .R/. For any point p 2 T .R/, we denote the fiber of the projection ˛ W T .R/ ! AT .R/ containing p by Tp , that is, Tp D ˛ 1 .˛.p//. If   Mod.R/ is asymptotically elliptic having a common fixed point ˛.p/ 2 AT .R/, then  preserves the fiber Tp . We investigate an abelian action of such a subgroup and obtain the following. Theorem 3.8. Assume that @1 R D ;. Let  be an asymptotically elliptic subgroup of Mod.R/ that is finitely generated infinite abelian. Then, for every point p 2 T .R/ over the fixed point of  on AT .R/, one of the following alternative conditions is satisfied: (1)  fixes p; (2)  acts discontinuously at p and the orbit .p/ is bounded; (3) .p/ is divergent, that is,  is of divergent type. In any case, .p/ is a discrete set. Before the proof of Theorem 3.8, we extend the definition of the stationary property for a subgroup of MCG.R/ to any sequence of mapping classes. A sequence fŒgi g1 iD1 in MCG.R/ is called stationary if there exists a compact bordered subsurface V of R such that every representative gi of each mapping class Œgi  satisfies gi .V / \ V ¤ ;. On the contrary, a sequence fŒgi g1 iD1 is called escaping if, for every compact bordered subsurface V of R, there exists some representative gi of each mapping class Œgi 

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such that fgi .V /g diverges to the infinity of R as i ! 1. Note that a sequence fŒgi g  MCG.R/ can be neither stationary nor escaping, but we can always choose a subsequence that is either stationary or escaping. A sequence fi g in Mod.R/ is also called stationary or escaping if so is fŒgi g  MCG.R/ for i D Œgi  . Proof. If  is stationary, then Corollary 2.25 says that  is of divergent type. This is also true for any stationary subsequence fi g in  and we see that fi .p/g diverges to the infinity of T .R/ for such a sequence. Suppose that there is a subsequence fi g in  such that fi .p/g has an accumulation point in Tp . By replacing the subsequence if necessary, we may assume that fi .p/g converges to p. Moreover, by the previous paragraph, we see that this subsequence fi g is escaping. Then we use Lemma 3.9 below to show that the whole group  fixes the point p. This is the situation of Condition (1). Next, suppose that there is a subsequence fi g in  such that fi .p/g is bounded in Tp . Then fi g should be an escaping subsequence as before, and in this case, we see by Lemma 3.9 that the whole orbit .p/ is bounded. This is the situation of either Conditions (1) or (2). By excluding the case discussed in the previous paragraph, we have Condition (2). Finally, if there is no subsequence fi g in  such that fi .p/g is bounded, then this means that the orbit .p/ is divergent. This is the situation of Condition (3). Lemma 3.9 ([36], [40]). Assume that @1 R D ;. Let  be an asymptotically elliptic abelian subgroup of Mod.R/. Let fi g be an escaping sequence of . Then the following are satisfied for any point p 2 T .R/ over the fixed point of  on AT .R/. • If fi .p/g converges to p, then  fixes p. • If fi .p/g is bounded, then  is of bounded type. In both cases,  is elliptic. By Theorem 2.34, we see that Conditions (1) or (2) of Theorem 3.8 occur if and only if  is elliptic. Note that there is a case where  satisfies (2) but has no fixed point in Tp , which is shown in [39]. Condition (3) occurs if and only if  is asymptotically elliptic but not elliptic. In this case,  acts discontinuously on T .R/. This gives the following corollary. Corollary 3.10. Assume that @1 R D ;. Let  be an asymptotically elliptic subgroup of Mod.R/ that is finitely generated infinite abelian. Then either  is elliptic or  acts discontinuously on T .R/. Note that if R satisfies the bounded geometry condition in Theorem 3.8 and Corollary 3.10, then we can weaken the assumption on  for the same claim. Namely, we have only to assume that  is an infinite abelian group. This is based on Theorem 2.29. As an application of the previous facts, we have the following result, which has been obtained in [39] and [19]. We believe that this should be proved without any assumption on R.

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Proposition 3.11. Assume that R satisfies the bounded geometry condition. Then no non-trivial elliptic modular transformation of Mod.R/ is asymptotically trivial. Proof. Let  be an elliptic modular transformation of Mod.R/. If  is of infinite order, then by Theorem 2.35, there is an orbit of p 2 T .R/ under h i that is not a discrete set. On the other hand, if  is asymptotically trivial, then in particular h i preserves the fiber Tp , and the orbit should be a discrete set by Theorem 3.8. This is a contradiction. In the case where  is of finite order, we see that  cannot be asymptotically trivial by a certain geometric argument.

3.3 The asymptotically trivial mapping class group The asymptotically trivial mapping class group contains the stable mapping class group. They do not necessarily coincide, but when R satisfies the bounded geometry condition, they coincide. We explain the relationship between these groups and then discuss certain results obtained from their coincidence. Relation to the stable mapping class group It is evident from the definition that the stable mapping class group is contained in the asymptotically trivial mapping class group and the pure mapping class group. Moreover, there is an inclusion relation between the latter two groups. Theorem 3.12 ([16], [20]). The following inclusion relations are satisfied in general: MCG1 .R/  Ker AT  MCG@ .R/: We expect that the closure MCG1 .R/ of the stable mapping class group in the compact-open topology should contain Ker AT . Since MCG@ .R/ is closed, the inclusion MCG1 .R/  MCG@ .R/ is clear. If R has a sequence of mutually disjoint simple closed geodesics whose lengths tend to zero, then a mapping class given by the simultaneous Dehn twists along all these curves belongs to Ker AT but not to MCG1 .R/. However, if R satisfies the bounded geometry condition, then there is no such sequence of curves, and in fact there is no such mapping class. Theorem 3.13 ([19], [20]). Assume that R satisfies the bounded geometry condition. Then MCG1 .R/ D Ker AT is satisfied. An application of this theorem will be given in the next section. Finite subgroups in the asymptotic Teichmüller modular group We deal with periodic elements, and more generally, finite subgroups of the asymptotic Teichmüller modular group. Recall that every finite subgroup of the Teichmüller

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modular group has a fixed point on the Teichmüller space, which is a special case of Theorem 2.34. We consider a similar property on the asymptotic Teichmüller space. We assume that R satisfies the bounded geometry condition. Let Œg 2 MCG.R/ be a mapping class such that Œg 2 ModAT .R/ is periodic of order n. This means that Œg n  2 Ker AT , and since Ker AT D MCG1 .R/ by Theorem 3.13, we have Œg n  2 MCG1 .R/. Then we see that, outside some topologically finite bordered subsurface, Œg is a periodic mapping class. By standard arguments, we can find a complex structure such that Œg can be realized as a conformal automorphism off the subsurface, that is, Œg is asymptotically conformal. This is equivalent to saying that this complex structure gives a fixed point of Œg on AT .R/. Therefore, every periodic element of ModAT .R/ has a fixed point on AT .R/. This has been proved in [19]. The Nielsen realization theorem for the mapping class group MCG.R/ is solved by finding a fixed point on T .R/. Analogously, we formulate the following fixed point theorem for ModAT .R/, the asymptotic version of the realization theorem. The proof is also carried out by a similar argument as above relying on the fact that Ker AT D MCG1 .R/. Theorem 3.14 ([20]). Assume that R satisfies the bounded geometry condition. Then every finite subgroup of ModAT .R/ has a common fixed point on AT .R/. In the light of Theorem 2.34, we further propose the following. Problem 3.15. Find a common fixed point on AT .R/ when the orbit of a subgroup of ModAT .R/ is bounded. Realization in asymptotic Teichmüller modular groups Every countable group can be realized as a group of conformal automorphisms of some Riemann surface. Actually, a stronger result has been periodically proved since the first proof was given by Greenberg [27], which asserts that we can find a Riemann surface R whose conformal automorphism group is precisely isomorphic to the given countable group. This fact says that every countable group can be obtained as the stabilizer of some point in some Teichmüller modular group. Then we may ask the same question for the asymptotic Teichmüller modular group. If we see that the kernel of the representation AT W MCG.R/ ! Aut.AT .R// contains no conformal mapping classes besides the trivial one, then every countable group can also be realized as the stabilizer subgroup of ModAT .R/. In fact, Proposition 3.11 gives the following theorem. Note that it is easy to make a hyperbolic Riemann surface R to satisfy the bounded geometry condition as well as to avoid extra asymptotically conformal automorphisms of R other than the conformal ones.

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Theorem 3.16. For any countable group H , there exists a hyperbolic Riemann surface R satisfying the bounded geometry condition such that the stabilizer subgroup for some point of AT .R/ in ModAT .R/ is isomorphic to H . Here we will give a concrete construction of a Riemann surface R such that the Thompson group is realized in some stabilizer subgroup of ModAT .R/, according to de Faria, Gardiner and Harvey [2]. Let E be the middle-third Cantor set in the unit interval and set R D C  E, which has one puncture at 1. Given a hyperbolic metric, R satisfies the bounded geometry condition. Indeed, each step for the construction of the Cantor set by removing the middle-third interval defines a pair of pants, and this procedure induces a canonical pants decomposition of R such that all the lengths of boundary geodesics of the pairs of pants are uniformly bounded from above and from below. Then there is a quasiconformal homeomorphism f of R D C  E preserving the upper and lower half-planes respectively such that for any non-cuspidal topological ends e and e 0 of f .R/, there are neighborhoods U and U 0 of e and e 0 respectively that are conformally equivalent. Set p D Œf  2 T .R/. Let G be the subgroup of MCG.R/ consisting of all mapping classes that have representatives preserving the upper and lower half-planes. Then, by the choice of p, we see that each mapping class of G is realized as an asymptotically conformal automorphism of the Riemann surface f .R/ corresponding to p. This means that  D .G/ is an asymptotically elliptic subgroup of Mod.R/. Since R satisfies the bounded geometry condition, Theorem 3.6 tells us that  is a countable group. Also note that MCG.R/ itself is stationary because every representative of each mapping class maps any neighborhood of the puncture in such a way that it has non-empty intersection with its image. Hence by Theorem 2.29,  acts discontinuously on T .R/. The Thompson group F is the group of all piecewise-linear automorphisms of the unit interval Œ0; 1 fixing 0 and 1 having the following property. For some integer n  0, the domain and the range are divided into nC1 subintervals. These subintervals are obtained by n time half-division of intervals such that at each step we choose one of the intervals made by the previous steps and divide it into two half intervals. (The subdivision in the domain and in the range is not the same.) Then such a division of the domain and the range intervals gives a unique piecewise-linear homeomorphism by the correspondence of the subintervals in order. The Thompson group F is an infinitely generated group without torsion. It has been proved in [2] that AT .G/ is isomorphic to F . This means that F can be realized as a subgroup of the stabilizer for ˛.p/ 2 AT .R/ in ModAT .R/.

4 Quotient spaces by Teichmüller modular groups If a Riemann surface R is analytically finite, the moduli space M.R/ of all complex structures on R is obtained as a quotient space of the Teichmüller space T .R/ by

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the Teichmüller modular group Mod.R/. In this case, Mod.R/ acts properly discontinuously on T .R/ and hence M.R/ inherits complex and geometric structures from T .R/. However, this is not always the case where R is a general Riemann surface. We have to consider other quotients which inherit certain structures of T .R/. Especially, we introduce the stable moduli space and the enlarged moduli space. The former is obtained as the completion of the quotient of the region of stability by Mod.R/ whereas the latter is the quotient of T .R/ by the stable mapping class group. In this section, we assume that a Riemann surface R is non-elementary.

4.1 Geometric moduli spaces We introduce a new moduli space, which has a complete distance induced from the Teichmüller distance. We give two different ways for its construction and show that the resulting spaces coincide. The moduli space of stable points No matter how the action of Mod.R/ is far from discontinuity, we can define the moduli space M.R/ D T .R/= Mod.R/ which is a topological space for the quotient topology. We call this M.R/ the topological moduli space. Moreover a pseudodistance dM on M.R/ is induced from the Teichmüller distance dT on T .R/. Namely, letting  W T .R/ ! M.R/ the projection, we define the pseudo-distance by dM .; / D inffdT .p; q/ j .p/ D ; .q/ D g for any  and in M.R/. However, this is not always a distance because the infimum is not necessarily attained. Hence we want to consider the following smaller subset in M.R/. Definition 4.1. The moduli space of stable points is defined by Mˆ .R/ D ˆ.Mod.R//= Mod.R/; where ˆ.Mod.R// is the region of stability for Mod.R/. For the region of discontinuity .Mod.R//, the quotient space M .R/ D .Mod.R//= Mod.R/ inherits complex and geometric structures from T .R/. In particular, M .R/ is a complex Banach orbifold. On the other hand, Mˆ .R/ is an open subset of M.R/ including M .R/ where the restriction of the pseudo-distance dM becomes a distance. If R satisfies the bounded geometry condition, then Mˆ .R/ D M .R/ by Theorem 2.30. The distance dM on Mˆ .R/ defines the length of a path in Mˆ .R/. For any two points in Mˆ .R/, consider all paths in Mˆ .R/ connecting these points and take the i on Mˆ .R/, which infimum over their lengths. This defines an intrinsic distance dM i is called the inner distance with respect to dM . Clearly dM  dM .

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Definition 4.2. The metric completion of Mˆ .R/ with respect to the inner distance i is denoted by Mˆ .R/i and called the stable moduli space. dM Closure equivalence We use a stronger equivalence relation than the usual orbit equivalence under Mod.R/. This makes the quotient space a metric space. Definition 4.3. For a subgroup  of Mod.R/, we define two points p and q in T .R/ to be equivalent if q 2 .p/. This gives an equivalence relation and the equivalence class containing p is .p/. This is called closure equivalence. The quotient space by the closure equivalence is denoted by T .R/== . Let N W T .R/= ! T .R/== be the canonical projection. The inverse image N 1 .s/ for s 2 T .R/== coincides with the closure f g of a single point set f g in T .R/= , where  is an arbitrary point in N 1 .s/. This corresponds to the fact that the equivalence classes containing p 2 T .R/ are .p/ and .p/ for the orbit equivalence and for the closure equivalence, respectively. Clearly, f g D fg if and only if the corresponding orbit .p/ is closed, namely,  acts at p weakly stably. The Teichmüller distance dT induces a quotient distance d on T .R/== ; it satisfies a property that d .s; s 0 / D 0 implies s D s 0 . This is because the equivalence classes are closed in T .R/. Hence T .R/== is a complete metric space. Now, by setting  D Mod.R/, we have our definition of the moduli space. Definition 4.4. The complete metric space T .R/== Mod.R/ with the distance d is called the geometric moduli space and denoted by M .R/. If Mod.R/ acts on T .R/ weakly stably, then the geometric moduli space M .R/ is nothing but the topological moduli space M.R/ and the pseudo-distance dM coincides with the distance d . However, if it does not act weakly stably, the projection N W M.R/ ! M .R/ is not injective and dM is not a distance on M.R/. In fact, M.R/ does not satisfy the first separability axiom (T1 -axiom) in this case. Proposition 4.5 ([42]). The following conditions are equivalent: (1) the Teichmüller modular group Mod.R/ acts weakly stably on T .R/; (2) the projection N W M.R/ ! M .R/ is injective; (3) the topological moduli space M.R/ is a T1 -space, in other words, every single point constitutes a closed set; (4) the pseudo-distance dM on M.R/ is a distance. A sufficient condition for R and Mod.R/ not to satisfy the conditions in Proposition 4.5 is also given in [42] as follows.

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Theorem 4.6. Assume that R satisfies the bounded geometry condition and Mod.R/ contains an elliptic element of infinite order. Then the topological moduli space M.R/ is not a T1 -space. In particular, for an infinite cyclic cover R of an analytically finite Riemann surface, M.R/ is not a T1 -space. Proof. Since Mod.R/ contains an elliptic element of infinite order, it does not act weakly discontinuously by Theorem 2.35. Since R satisfies the bounded geometry condition, this implies that Mod.R/ does not act weakly stably by Theorem 2.30. Then Proposition 4.5 asserts that M.R/ is not a T1 -space. Genericity of stable points We give several properties of the stable points which show that they are generic in T .R/ in the following sense. We apply these properties to the investigation of the structure of moduli spaces. Theorem 4.7 ([42]). Assume that R is non-elementary. The region of stability ˆ.Mod.R// is open, connected and dense in T .R/. Note that we have seen that ˆ.Mod.R// is open by Theorem 2.28. The following corollary is an easy consequence of the density of ˆ.Mod.R//. Corollary 4.8. The geometric moduli space M .R/ is isometric to the completion Mˆ .R/ of the moduli space of the stable points with respect to the distance dM . Concerning the connectivity, ˆ.Mod.R// has a stronger property than just a topological one. Namely, the distance between two points in ˆ.Mod.R// measured by dM is comparable in some sense, with the length of a path connecting them in ˆ.Mod.R//, i . This in particular gives the folwhich approximates the distance measured by dM lowing. Theorem 4.9 ([42]). The geometric moduli space M .R/ is locally bi-Lipschitz equivalent to the stable moduli space Mˆ .R/i . If R satisfies the bounded geometry condition, ˆ.Mod.R// D .Mod.R//. This implies that our moduli space M .R/ has an open dense connected subregion .Mod.R//= Mod.R/ which has the complex Banach orbifold structure induced from T .R/. One of the problems we are interested in is to give a characterization of each point in M .R/ explaining this equivalence class geometrically.

4.2 Several Teichmüller spaces In general, we can define the quotient T .R/=  by a subgroup  of Mod.R/ as a certain reduction of the Teichmüller space or a certain extension of the moduli space in some appropriate sense.

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An example: the reduced Teichmüller space As an example, we present a familiar Teichmüller space, which can be defined as the quotient of the following subgroup of the Teichmüller modular group. Let MCG# .R/ be the subgroup of MCG.R/ consisting of all elements Œg such that g is freely homotopic to the identity of R, where R is assumed to have the boundary at infinity @1 R but the homotopy is not assumed to be relative to @1 R. It is clear that MCG# .R/ is normal in MCG.R/. As usual, we set Mod# .R/ D .MCG# .R//. Proposition 4.10. Assume that R is non-elementary. The subgroup Mod# .R/ is the intersection of the subgroups Modc .R/ taken over all non-trivial simple closed curves c on R. Hence Mod# .R/ acts stably on T .R/. Proof. The first statement is well-known. See for instance [11]. The second statement is a consequence of Proposition 2.8. The space T .R/= Mod# .R/ D T .R/== Mod# .R/ is called the reduced Teichmüller space T # .R/ with the quotient distance d # , and Mod.R/= Mod# .R/ is the reduced Teichmüller modular group Mod# .R/. It acts on .T # .R/; d # / isometrically. Relative Teichmüller spaces We have already seen the important roles of the subgroup Modc .R/. Here we consider the quotient space of T .R/ by this group. In Proposition 2.13, we have seen that Modc .R/ is of countable index in Mod.R/. And, since Modc .R/ is stationary and closed if R is non-elementary, it acts stably on T .R/ by Theorem 2.23. Moreover, if R satisfies the bounded geometry condition, then it acts discontinuously on T .R/. Definition 4.11. The quotient space T c .R/ D T .R/= Modc .R/ is called the relative Teichmüller space with respect to c. Since Modc .R/ acts stably on T .R/, the relative Teichmüller space T c .R/ is a complete metric space with the quotient distance dO . This divides the action of Mod.R/ on T .R/ into the stable part under Modc .R/ on T .R/ and the countable part under Mod.R/= Modc .R/ on T c .R/. More precisely, let P  T .R/ be the orbit of p under Mod.R/ and Py  T c .R/ the image of P under the projection T .R/ ! T c .R/. Assume that Modc .R/ acts discontinuously on T .R/. In this case, if Py is closed in T c .R/, then Py is discrete and, as a consequence, we see that Mod.R/ acts discontinuously on T .R/. This yields a result similar to Lemma 2.6. Another feature of T c .R/ is the fact that T c .R/ is not separable if R is of infinite topological type, which is obtained in [42]. If we impose the extra assumption that R satisfies the bounded geometry condition, this fact can be easily seen as is shown below. From the non-separability of T c .R/, we can prove that the topological moduli space M.R/ is not separable either. In fact, every countable subset is nowhere dense in M.R/, and this is also true for the geometric moduli space M .R/.

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Theorem 4.12. Assume that R satisfies the bounded geometry condition. Then the geometric moduli space M .R/ is not separable if R is of infinite topological type. Proof. If R satisfies the bounded geometry condition, then Modc .R/ acts discontinuously on T .R/ by Theorem 2.29. On the other hand, T .R/ is not separable when R is of infinite topological type. Then T c .R/ D T .R/= Modc .R/ is not separable. Since Modc .R/ is of countable index in Mod.R/, M.R/ D T .R/= Mod.R/ is not separable either. By considering the moduli space of stable points Mˆ .R/, which is open and dense in M.R/, we also see that M .R/ is not separable by Corollary 4.8. The intermediate Teichmüller space We consider quotient spaces of T .R/ by the stable mapping class group and the asymptotically trivial mapping class group. When R satisfies the bounded geometry condition, by Theorem 3.13, they coincide. Definition 4.13. For the subgroup Mod1 .R/ of Mod.R/ corresponding to the stable z .R/ D T .R/= Mod1 .R/ is called the mapping class group, the quotient space M enlarged moduli space. If R is of infinite topological type and satisfies the bounded geometry condition, then Mod1 .R/ acts on T .R/ discontinuously and freely by Theorem 2.16. Then z .R/ is a complex Banach manifold which has complex the enlarged moduli space M and metric structures induced from T .R/. Since Mod1 .R/ is a normal subgroup of z .R/ as a Mod.R/, the quotient group Mod1 .R/ D Mod.R/= Mod1 .R/ acts on M biholomorphic and isometric automorphism group that induces a quotient map onto the topological moduli space M.R/. This will be a way of considering a geometric structure on M.R/. Definition 4.14. For the subgroup .Ker AT / of Mod.R/ corresponding to the asymptotically trivial mapping class group, the quotient space T .R/ D T .R/=.Ker AT / is called the intermediate Teichmüller space . Since Ker AT acts on AT .R/ trivially, the definition of T .R/ immediately gives the following. Proposition 4.15 ([19]). The projection T .R/ ! T .R/ factorizes the projection ˛ W T .R/ ! AT .R/. Hence there are natural projections from T .R/ onto both AT .R/ and M.R/. In fact, T .R/ is the smallest quotient space of T .R/ by a subgroup of Mod.R/ having this property. z .R/ Since Mod1 .R/  Ker AT by Theorem 3.12, the enlarged moduli space M z .R/ D lies always between T .R/ and T .R/. If R is analytically finite, then M T .R/ D M.R/, and the asymptotic Teichmüller space AT .R/ is just one point.

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z .D/ D T .D/. Indeed, On the other hand, if R is the unit disk D, then T .D/ D M Ker AT is trivial for the unit disk D, and thus T .D/ D T .D/=.Ker AT / D T .D/. Now we assume that R satisfies the bounded geometry condition. Then we have z .R/ D T .R/. In this case, MCG1 .R/ D Ker AT by Theorem 3.13 and hence M 1 we have the relationship between Mod .R/ and ModAT .R/. Theorem 4.16 ([19]). Assume that R is of infinite topological type and satisfies the bounded geometry condition. Then the asymptotic Teichmüller modular group ModAT .R/ is geometrically isomorphic to the automorphism group Mod1 .R/ of z .R/ D T .R/. M For the representation of the quasiconformal mapping class group MCG.R/ in the automorphism group Aut.T .R// of Teichmüller space, it has been proved that the kernel is trivial and the image is the entire group in almost all cases. In contrast to these facts, for the representation of MCG.R/ in the automorphism group Aut.AT .R// of the asymptotic Teichmüller space, we obtain that the kernel is characterized topologically as the stable mapping class group MCG1 .R/ and the image can be represented as the automorphism group of the intermediate Teichmüller space T .R/ in the case where R satisfies the bounded geometry condition. Acknowledgement. This work was partially supported by JSPS Grant B #20340030.

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[10] C. Earle and C. McMullen, Quasiconformal isotopies. In Holomorphic functions and moduli, Vol. I (D. Drasin et al., eds.), Math. Sci. Res. Inst. Publ. 10, Springer, New York 1988, 143–154. [11] A. L. Epstein, Effectiveness of Teichmüller modular groups. In In the tradition of Ahlfors and Bers (I. Kra and B. Maskit, eds.), Contemp. Math. 256, Amer. Math. Soc., Providence, RI, 2000, 69–74. [12] A. Fletcher and V. Markovic, Infinite dimensional Teichmüller spaces. In Handbook of Teichmüller space (A. Papadopoulos, ed.), Volume II, EMS Publishing House, Zürich 2009, 65–91. [13] E. Fujikawa, Limit sets and regions of discontinuity of Teichmüller modular groups. Proc. Amer. Math. Soc. 132 (2004), 117–126. [14] E. Fujikawa, Modular groups acting on infinite dimensional Teichmüller spaces. In In the tradition of Ahlfors and Bers, III (W. Abikoff and A. Haas, eds.), Contemp. Math. 355, Amer. Math. Soc., Providence, RI, 2004, 239–253. [15] E. Fujikawa, Another approach to the automorphism theorem for Teichmüller spaces. In In the tradition of Ahlfors-Bers, IV (R. Canary et al., eds.), Contemp. Math. 432, Amer. Math. Soc., Providence, RI, 2007, 39–44. [16] E. Fujikawa, Pure mapping class group acting on Teichmüller space. Conform. Geom. Dyn. 12 (2008), 227–239. [17] E. Fujikawa, Limit set of quasiconformal mapping class group on asymptotic Teichmüller space. In Teichmüller theory and moduli problem (I. Biswas, R. Kulkarni and S. Mitra, eds.), Ramanujan Math. Soc. Lect. Notes Ser. 10, Ramanujan Mathematical Society, Mysore 2010, 167–178. [18] E. Fujikawa and K. Matsuzaki, Recurrent and periodic points for isometries of L1 spaces. Indiana Univ. Math. J. 55 (2006), 975–997. [19] E. Fujikawa and K. Matsuzaki, Stable quasiconformal mapping class groups and asymptotic Teichmüller spaces. Amer. J. Math. 133 (2011), 637–675. [20] E. Fujikawa and K. Matsuzaki, The Nielsen realization problem for asymptotic Teichmüller modular groups. Trans. Amer. Math. Soc. 365 (2013), 3309–3327. [21] E. Fujikawa and K. Matsuzaki, Non-divergent infinitely discrete Teichmüller modular transformation. In Topics in finite or infinite dimensional complex analysis (K. Matsuzaki and T. Sugawa, eds.), Tohoku University Press, Sendai 2013, 97–102. [22] E. Fujikawa, K. Matsuzaki, and M. Taniguchi, Structure theorem for holomorphic selfcovers and its applications. In Infinite dimensional Teichmüller space and moduli space, RIMS Kokyuroku Bessatsu B17, Res. Inst. Math. Sci. (RIMS), Kyoto 2010, 21–36. [23] E. Fujikawa, H. Shiga, and M. Taniguchi, On the action of the mapping class group for Riemann surfaces of infinite type. J. Math. Soc. Japan 56 (2004), 1069–1086.

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[24] F. P. Gardiner, Teichmüller theory and quadratic differentials. Pure and Applied Mathematics, Wiley-Interscience Publication, John Wiley & Sons, Inc. 1987. [25] F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller theory. Math. Surveys Monogr. 76, Amer. Math. Soc., Providence, RI, 2000. [26] F. P. Gardiner and D. P. Sullivan, Symmetric structure on a closed curve. Amer. J. Math. 114 (1992), 683–736. [27] L. Greenberg, Conformal transformations of Riemann surfaces. Amer. J. Math. 82 (1960), 749–760. [28] Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces. Springer 1992. [29] S. P. Kerckhoff, The Nielsen realization problem. Ann. of Math. 117 (1983), 235–265. [30] O. Lehto, Univalent functions and Teichmüller spaces. Grad. Text in Math. 109, Springer, New York 1986. [31] V. Markovic, Biholomorphic maps between Teichmüller spaces. Duke Math. J. 120 (2003), 405–431. [32] V. Markovic, Quasisymmetric groups. J. Amer. Math. Soc. 19 (2006), 673–715. [33] K. Matsuzaki, Dynamics of Teichmüller modular groups and general topology of moduli spaces: Announcement. In Perspectives of hyperbolic spaces II, RIMS Kokyuroku 1387, Res. Inst. Math. Sci. (RIMS), Kyoto 2004, 81–94. [34] K. Matsuzaki, Inclusion relations between the Bers embeddings of Teichmüller spaces. Israel J. Math. 140 (2004), 113–124. [35] K. Matsuzaki, A countable Teichmüller modular group. Trans. Amer. Math. Soc. 357 (2005), 3119–3131. [36] K. Matsuzaki, A classification of the modular transformations on infinite dimensional Teichmüller spaces. In In the tradition of Ahlfors-Bers, IV (R. Canary et al., eds.), Contemp. Math. 432, Amer. Math. Soc., Providence, RI, 2007, 167–178. [37] K. Matsuzaki, A quasiconformal mapping class group acting trivially on the asymptotic Teichmüller space. Proc. Amer. Math. Soc. 135 (2007), 2573–2579. [38] K. Matsuzaki, Quasiconformal mapping class groups having common fixed points on the asymptotic Teichmüller spaces. J. d’Analyse Math. 102 (2007), 1-28. [39] K. Matsuzaki, The action of elliptic modular transformations on asymptotic Teichmüller spaces. In Teichmüller theory and moduli problem (I. Biswas, R. Kulkarni and S. Mitra, eds.), Ramanujan Math. Soc. Lect. Notes Ser. 10, Ramanujan Mathematical Society, Mysore 2010, 481–488. [40] K. Matsuzaki, Properties of asymptotically elliptic modular transformations of Teichmüller spaces. In Infinite dimensional Teichmüller space and moduli space (E. Fujikawa, ed.), RIMS Kokyuroku Bessatsu B17, Res. Inst. Math. Sci. (RIMS), Kyoto 2010, 73–84. [41] K. Matsuzaki, Polycyclic quasiconformal mapping class subgroups. Pacific J. Math. 251 (2011), 361–374. [42] K. Matsuzaki, Dynamics of Teichmüller modular groups and topology of moduli spaces of Riemann surfaces of infinite type. Preprint. [43] S. Nag, The complex analytic theory of Teichmüller spaces. Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York 1988.

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Chapter 16

Teichmüller spaces and holomorphic dynamics Xavier Buff, Guizhen Cui, and Lei Tan

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Teichmüller spaces for rational maps . . . . . . . . . . 1.1 The Teichmüller space of a marked sphere . . . . 1.2 The Teichmüller space of a rational map . . . . . . 1.3 Thurston’s pullback map . . . . . . . . . . . . . . 1.4 Epstein’s deformation space . . . . . . . . . . . . 2 Thurston’s theorem with marked points . . . . . . . . . 2.1 Thurston obstructions . . . . . . . . . . . . . . . 2.2 Main results . . . . . . . . . . . . . . . . . . . . 2.3 Classical results from hyperbolic geometry . . . . 2.4 From TZ to Ratd . . . . . . . . . . . . . . . . . . 2.5 Contraction of Thurston’s pullback maps . . . . . 2.6 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . 2.7 Proof of Proposition 2.12. . . . . . . . . . . . . . 3 Applications of Thurston’s theorem and related results . 3.1 Geyer’s sharpness result for harmonic polynomials 3.2 Applications of Thurston’s theorem . . . . . . . . 4 Epstein’s transversality results . . . . . . . . . . . . . . 4.1 Formal invariants of a cycle . . . . . . . . . . . . 4.2 Quadratic differentials with higher order poles . . 4.3 The Fatou–Shishikura inequality . . . . . . . . . . 4.4 Transversality for multiplier loci . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction Let f .z/ D p.z/=q.z/ be a rational map with p and q relatively prime polynomials. The degree d D deg.f / of f is defined to be the maximum of the degrees of p and q. In the following we will always assume that deg.f / > 1. The iteration of f generates a holomorphic dynamical system on the Riemann y and partitions the sphere into two dynamically natural subsets C y D Jf tFf , sphere C,

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where by definition  ˚ y j ff Bn jU gn0 is a normal family on some neighborhood U of z : Ff D z 2 C The set Jf (resp. Ff ) is called the Julia set (resp. the Fatou set) of f . Roughly speaking, Ff consists of the set of initial values z such that the long term behavior of the iterated orbit .f Bn .z//n0 is insensitive to small perturbations of z. The simplest example is given by f .z/ D z 2 , for which Ff D fjzj ¤ 1g and Jf D fjzj D 1g. With a little effort one can also show that for f .z/ D z 2  2, Jf D Œ2; 2. There are however very few rational maps for which the Julia set can be described by smooth equations, as Jf often presents a fractal shape. The orbit of a point z is simply ff Bn .z/; n  0g. We say that z is periodic if there is p such that f p .z/ D z. By a classical result of Fatou and Julia, there are at most finitely many periodic points outside the Julia set Jf (more precisely, all repelling periodic points are in the Julia set and there are finitely many non-repelling periodic points; see Theorem 4.3 below), and Jf is compact containing uncountably many points, in which the periodic points form a countable dense subset. The rational map f is proper and the Julia and Fatou sets are completely invariant: f 1 .Jf / D f .Jf / D Jf and f 1 .Ff / D f .Ff / D Ff . As a consequence, f maps each Julia (resp. Fatou) component onto another Julia (resp. Fatou) component as a proper map. y With finitely many exceptions, every We consider f as a branched covering of C. y has exactly d preimages. More precisely, denote by Cf the set of points value w 2 C y where f is not locally injective. These points are called the critical points of f . z2C y X f 1 .Vf / ! C y X Vf Let Vf D f .Cf / be the set of critical values of f . Then f W C is an (unramified) covering of degree d . The postcritical set Pf of f is defined to be  [ ˚  f Bn .z/ : Pf D closure z2Cf ; n1

In a certain sense, this set captures the essence of the dynamical system generated by f . We say that f is postcritically finite if Pf is finite. This is equivalent to the fact that all critical points of f are eventually periodic under iteration. A rational map f is hyperbolic if it is uniformly expanding near its Julia set. These are the natural analogues of Smale’s Axiom A maps in this setting. If in addition the Julia set is connected, the dynamics of f on Jf is equivalent to the dynamics of a map f0 which is postcritically finite. We may also forget the analytic nature of a rational map and consider it as a topological (orientation preserving) branched covering of the two-sphere S 2 . As the notions of degree, critical points, postcritical set and postcritical finiteness are topological, they are naturally defined for a branched covering as well. In the early eighties, Thurston gave a complete topological characterization of postcritically finite rational maps (see [72], [18]), which can be stated roughly as follows: The set of postcritically finite rational maps (except for the Lattès examples)

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are in one-to-one correspondence with the homotopy classes of postcritically finite branched self-coverings of S 2 with no Thurston obstructions (see Section 2.1 for a more precise statement). This result has then become a fundamental theorem in the theory of holomorphic dynamics, together with some surprising applications outside the field. Teichmüller theory plays an essential role in Thurston’s proof of his theorem. An outline goes as follows: To a postcritically finite branched covering F of S 2 one can associate the Teichmüller space T of the punctured sphere S 2 X PF . The pullback of complex structures by F induces a weakly contracting operator  on T . The main point is to prove that in the absence of obstructions,  has a unique fixed point in T . This fixed point represents a complex structure that is invariant (up to isotopy) by F , thus turns F into an analytic branched covering, i.e. a rational map. Therefore in order to build a rational map with desired combinatorial properties one may first construct a branched covering F as a topological model (this is a lot more flexible than building holomorphic objects, for example one may freely cut, paste and interpolate various holomorphic objects), and then check whether F has Thurston obstructions (this is not always easy). If not then Thurston’s theorem ensures the existence of a rational map with the same combinatorial properties. In practice, one sometimes needs a slightly generalized version of Thurston’s theorem, namely one with a larger marked set than the mere postcritical set. We will call it ‘marked Thurston’s theorem’. The main purpose of writing up this chapter is to provide a self-contained proof of this theorem. As one can see below, the proof follows essentially the same line as that presented by Douady and Hubbard ([18]), except for some refinements in the estimates. For instance to get a strong contraction of the pullback operator on the appropriate Teichmüller space, we had to raise the operator to a large power (instead of just to its second power). Just to illustrate the power of Thurston’s characterization theorem we will mention some of its applications. There are many such applications. These include Rees’ descriptions of parameter spaces [55], Kiwi’s characterization of polynomial laminations (using previous work of Bielefield–Fisher–Hubbard [4] and Poirier [53]), Rees, Shishikura and Tan’s studies on matings of polynomials ([54], [66], [69], [70]), Pilgrim and Tan’s cut-and-paste surgery along arcs ([52]), and Timorin’s topological regluing of rational maps ([75]), among many others. Furthermore, one of the two main outstanding questions in the field, namely, the density of hyperbolicity in the quadratic polynomial family, can be reduced to the assertion that every (infinitely renormalizable) quadratic polynomial p is a limit of certain postcritically finite ones pn obtained via Thurston’s theorem and McMullen’s quotienting process ([38]). The detailed knowledge of the combinatorics of the parameter space of quadratic polynomials (which follows from a special case of Thurston’s theorem) was used by Sørensen ([67]) to construct highly non-hyperbolic quadratic polynomials with non-locally connected Julia sets, and this in turn was used by Henriksen ([28]) to show that McMullen’s combinatorial rigidity property fails for cubic polynomials.

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We will give a more complete, but by no means exhaustive, list of applications and related results. We mention in particular an interesting result of L. Geyer be´ atek ([34]) proved that yond the field of complex dynamics. Khavinson and Swi¸ harmonic polynomials z  p.z/, where p is a holomorphic polynomial of degree n > 1, have at most 3n  2 roots, and the bound is sharp for n D 2; 3. Bshouty and Lyzzaik ([8]) extended the sharpness of the bound to the cases n D 4; 5; 6 and 8, using purely algebraic methods. Finally L. Geyer ([21]) settled the sharpness for all n at once, by constructing ‘à la Thurston’ a polynomial p of degree n with real coefficients and with mutually distinct critical points z1 ; z2 ; : : : ; zn1 such that p.zj / D zj . We will also present the notion of deformation space of a rational map introduced by Adam Epstein in his PhD thesis (in fact, the construction applies to finite type transcendental maps on compact Riemann surfaces, which was his original motivation). Those are smooth sub-manifolds of appropriate Teichmüller spaces of spheres with marked points. In the dynamical setting, the relation between Epstein’s deformation spaces and spaces of rational maps is somewhat comparable to the relation between Teichmüller spaces and moduli spaces in the classical theory of Riemann surfaces. Interesting transversality properties are more easily expressed and proved in those deformation spaces, and we believe they will attract an increasing amount of interest in the coming years.

1 Teichmüller spaces for rational maps In this section we will recall the classical theory of the Teichmüller space of a marked sphere, define the Teichmüller space associated to a rational map, the Thurston’s pullback map and Epstein’s deformation space.

1.1 The Teichmüller space of a marked sphere y All homeomorphisms S 2 ! C y Let S 2 be an oriented surface homeomorphic to C. we will consider are orientation preserving. Let Z  S 2 be finite with #Z  4. Then, y an injection and • MZ is the space of equivalence classes Œi , with i W Z ,! C i1  i2 if there is a Möbius transformation M such that M B i1 D i2 . • TZ D Teich.S 2 ; Z/, the Teichmüller space of the marked sphere .S 2 ; Z/, is y defined to be the space of equivalence classes of homeomorphisms  W S 2 ! C with   ', if there is a Möbius transformation M such that M B 'jZ D jZ and M B ' D  B h, with h a homeomorphism isotopic to the identity rel Z. Here

Chapter 16. Teichmüller spaces and holomorphic dynamics

is the diagram .S 2 ; Z/ h

'



.S 2 ; Z/



/ C; y '.Z/







721



M



/ C; y .Z/ .

y containing at least three points, we denote by Q.X / the • For a finite set X  C y which are holomorphic outside space of integrable quadratic differentials on C X. Equivalently, Q.X/ is the space of meromorphic quadratic differentials on y holomorphic outside X with at worst simple poles in X . By the Riemann– C, Roch theorem, the number of poles minus the number of zeros of a meromorphic y is equal to 4, taking into account multiplicities. It quadratic differential on C follows that Q.X/ is a C-linear space of dimension dim Q.X / D #X  3: • The space Q.X/ is equipped with the norm Z Z ˇ ˇ jqj D ˇq.x C iy/ˇ dxdy: kqk D y C

C

y represents a point  2 TZ , the cotangent space to TZ at  may If W S 2 ! C   be canonically identified to Q .Z/ . • We equip T TZ with the dual norm ˇ ˇ kk D sup ˇhq; iˇ for all  2 T TZ : q2Q. .Z// kqk1

• The induced Teichmüller metric on TZ is given by   1 dTZ Œ1 ; Œ2  D inf log K.h/ 2 where the infimum is taken over all the quasiconformal homeomorphisms y !C y such that  1 B h B 2 is homotopic to the identity rel Z and where hW C 1 K.h/ is the quasiconformal distortion K.h/ D

N 1 C k@h=@hk 1 : N 1  k@h=@hk1

1.2 The Teichmüller space of a rational map y !C y be a rational map with deg.f /  2. Let f W C The grand orbit of a point z is defined to be ˚ 0  y j f n .z 0 / D f m .z/ for some n; m 2 N : z 2C

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The extended Julia set, denoted by Jyf , is the closure of the grand orbits of all periodic points and all critical points. We always have Pf [ Jf  Jyf : • M.f /, the moduli space of f , denotes the space of conformal equivalence classes of rational maps quasiconformally conjugate to f , that is ² ˇ ³. ˇ there is a quasiconformal map h M.f / D g ˇ  such that h B f D g B h where g  G if there is a Möbius transformation H such that g B H D H B G. y which commute • QC.f / is the group of quasiconformal automorphisms of C with f . • QC0 .f /  QC.f / is the normal subgroup consisting of those quasiconformal automorphisms which are isotopic to the identity in an appropriate sense: there is a family .h t /, t 2 Œ0; 1, with h0 D id, h1 D h such that each h t is quasiconformal, h t B f D f B h t , and .t; z/ 7! .t; h t .z// is a homeomorphism from y onto itself. Note that h t must be the identity on the set of periodic Œ0; 1  C points as well as on the postcritical set for all t 2 Œ0; 1. Consequently h t is the identity on Jyf . • Mod.f / D QC.f /=QC0 .f /, the modular group of f , denotes the group of isotopy classes of quasiconformal automorphisms of f up to isotopy, that is, y !C y is a quasiconformal the group of equivalence classes Œ, such that  W C homeomorphism,  B f D f B , and   ' if ' D  B h with h 2 QC0 .f /. This group contains as a subgroup the set of Möbius transformations commuting with f , denoted by Aut.f /. • T .f /, the Teichmüller space of f , is the set of equivalence classes of pairs .g; / such that g is a rational map, is a quasiconformal conjugacy between f and g (i.e. B f D g B ), and .g1 ; 1 /  .g2 ; 2 / if there is a Möbius transformation M such that g1 D M B g2 B M 1 and M B 2 D 1 B h with h 2 QC0 .f /: M

y Jyg / o .C; 2 g2

2





y Jyg / o .C; 2

y Jyf / .C;

2

h

f

y Jyf / .C;

/ .C; y Jyf / 

h

1

f

/ .C; y Jyf /

) / .C; y Jyg / 1 

1

g1

/ .C; y Jyg / 1 5

M

For example when M 2 Aut.f /, then .f; M /  .f; id/. Note that M.f / D T .f /=Mod.f /:

Chapter 16. Teichmüller spaces and holomorphic dynamics

723

y !C y of degree d . This space Let Rat d denote the space of all rational maps f W C can be realized as the complement of a hyper-surface in the projective space P 2d C1 .C/ by considering f .z/ D p.z/=q.z/ where p and q are relatively prime polynomials in y acts on z with d D maxfdeg p; deg qg. The group of Möbius transformations Aut.C/ 1 y Rat d by conjugacy: if  2 Aut.C/ and f 2 Ratd , then   f D  B f B  2 Rat d . Theorem 1.1 (McMullen and Sullivan, [39]). The group Mod.f / acts properly discontinuously by holomorphic automorphisms on T .f /. There is a natural holoy parameterizing the morphic injection of complex orbifolds M.f / ! Ratd =Aut.C/ rational maps g quasiconformally conjugate to f . Each connected component of the Fatou set F of a rational map f of degree d  2 properly maps to a connected component of F . Such a Fatou component U is periodic if there is a p  1 such that f p .U / D U and is preperiodic if f k .U / is periodic for some k  0. If U is not preperiodic, then it is called a wandering Fatou component. Sullivan, using the Measurable Riemann Mapping Theorem in Teichmüller theory, proved that if f had a wandering Fatou component, then the Teichmüller space T .f / y would be infinite dimensional, contradicting the previous theorem since Rat d =Aut.C/ has dimension 2d  2. Thus, Theorem 1.2 (Sullivan, [68]). Every Fatou component of a rational map is preperiodic. Since Sullivan, the Measurable Riemann Mapping Theorem has been applied in almost every domain of holomorphic dynamics. We recommend the monograph of Branner–Fagella, [6], for a detailed account of relative results and references. The following classification of periodic Fatou components goes back to Fatou and is rather elementary. Assume U is a periodic Fatou component of period p. Then U is either • a superattracting basin: there is a point z0 in U , fixed by f p , with .f p /0 .z0 / D 0, attracting all points of U under iteration of f p ; • an attracting basin: there is a point z0 in U , fixed by f p , with 0 < j.f p /0 .z0 /j < 1, attracting all points of U under iteration of f p ; • a parabolic basin: there is a point z0 in @U with .f p /0 .z0 / D 1, attracting all points of U ; • a Siegel disk: U is conformally isomorphic to the unit disk, and f p jU is conformally conjugate to an irrational rotation; • a Herman ring: U is conformally isomorphic to an annulus fr < jzj < Rg with 0 < r < R < 1, and f p jU is conformally conjugate to an irrational rotation. If U is an attracting basin, then f acts properly discontinuously on U X Jyf and the quotient .U X Jyf /=f is isomorphic to a punctured torus. If U is a parabolic basin, then f acts properly discontinuously on U X Jyf and the quotient .U X Jyf /=f is isomorphic to a punctured sphere.

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Theorem 1.3 (McMullen and Sullivan, [39]). The space T .f / is canonically isomorphic to a connected finite-dimensional complex manifold, which is the product of a polydisk and the traditional Teichmüller spaces associated to punctured tori and punctured spheres. In particular, the obstruction to deforming a quasiconformal conjugacy between two rational maps to a conformal conjugacy is measured by finitely many complex moduli.

1.3 Thurston’s pullback map Let F W S 2 ! S 2 be an orientation preserving branched covering of degree d  2. The set CF of critical points, the set VF of critical values and the postcritical set PF are defined in the same way as for a rational map. Assume Y  S 2 is a finite set containing at least three points with VF  Y . Then there is a Thurston’s pullback map &F W TY ! TF 1 .Y / which may be defined y This as follows. Let  2 TY be represented by a homeomorphism  W S 2 ! C. 2 homeomorphism  defines a complex structure c on S which can be pulled-back via F W S 2 ! S 2 to a complex structure F  c on S 2 (one has to use the removable singularity theorem to define the complex structure near the critical points of F ). The Uniy formization Theorem guarantees the existence of an isomorphism W .S 2 ; F  c/ ! C. Then, &F is defined by &F

TY 3 Π! Π 2 TF 1 .Y / : It is not obvious that this definition is independent on the choice of  and show this now. First, note that  B F B 1 is analytic (thus a rational map): / y C

S2

f 2Ratd

F

 S2

. We will



(1.1)

 / y C.

y and 1 W S 2 ! Assume  2 TY is represented by the homeomorphisms 0 W S 2 ! C 2 2 y Let M W C y !C y be a Möbius transformation and let h W S ! S be a homeomorC. y phism isotopic to the identity rel Y , such that M D 0 B h B 11 . Let 0 W S 2 ! C 2 y !C y (resp. 1 W S ! C y and f1 W C y ! C) y satisfy diagram (1.1). Since and f0 W C Y VF , there is a lift k W S 2 ! S 2 which is a homeomorphism isotopic to the identity rel F 1 .Y / such that h B F D F B k. We therefore have a commutative

Chapter 16. Teichmüller spaces and holomorphic dynamics

725

diagram: N

y o C f1 2Ratd



y o C

1

1

S2 

k

/ S2

h



F

S2

0

/* y C

F

/ S2

0



f0 2Ratd

/4 y C.

M

Since M , f0 and f1 are analytic, the homeomorphism N D 0 B k B 11 is analytic, thus a Möbius transformation. As a consequence, 0 and 1 represent the same point in TF 1 .Y / . Proposition 1.4. The map &F W TY ! TF 1 .Y / is analytic. y and  W S 2 ! C y be such that f D  B F B 1 2 Ratd . The Proof. Let W S 2 ! C Teichmüller spaces TY and TF 1 .Y / are canonically identified to quotients of the unit y and the map &F W TY ! TF 1 .Y / is ball of the space of Beltrami differentials on C induced by the C-linear (thus analytic) map  7! f  . Assume now X  F 1 .Y / contains at least three points. Then, there is an analytic submersion $ W TF 1 .Y / ! TX which consists in forgetting points in F 1 .Y / X X. We shall use the notation F for the Thurston’s pullback map F D $ B &F W TY ! TX : As a composition of analytic maps, this map is itself analytic. We will be particularly interested in the case that F is postcritically finite (i.e., PF is finite) and X D Y D PF . y !C y is a rational map and q is a meromorphic quadratic differential Now, if f W C y the pullback f  q and the push forward f q may be defined in coordinates as on C, follows:   2 • if q D b.y/dy 2 , then f  q D a.x/dx 2 with a.x/ D b f .x/  .f 0 .x/ . P • if q D b.y/dy 2 , then f q D c.z/dz 2 with c.f .y// D y2f 1 .z/ .fb.y/ 0 .y//2 . It follows that f .fpoles.f  q/g/  fpoles.q/g and f 1 .fpoles.q/g/  fpoles.f  q/g [ Cf I on the other hand, fpoles.f q/g  f .Cf / [ f .fpoles.q/g/: y Let W S 2 ! C y represent F . / 2 TX Let  2 TY be represented by  W S 2 ! C. 1 2 Ratd . Then, the cotangent space to TY at  is canonically with f D  B F B

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identified to Q..Y // and the cotangent space to TX at F . / is canonically identified to Q. .X //. By means of those identifications, the adjoint map of the derivative D F W T TY ! TF ./ TX is the push forward operator f W Q. .X // ! Q..Y //.

1.4 Epstein’s deformation space In his Ph.D. thesis, generalizing a construction due to Thurston which will be described below, Adam Epstein introduced the following deformation space. Definition. Let F W S 2 ! S 2 be an orientation preserving branched covering of y containing at least three points such degree d  2. Let X and Y be finite subsets of C 1 that X  Y \ F .Y / and VF  Y . Define Y .F / D f 2 TY j ./ D F . /g; DefX

where  W TY ! TX is the submersion which consists in forgetting points in Y X X and F W TY ! TX is the Thurston’s pullback map induced by F . Y .f / is an analytic subset of TY . We will see that Given its definition, the set DefX in most cases, it is either empty or a smooth submanifold of TY (Theorem 1.5 below). We may first briefly discuss why this space is interesting from a dynamical point of view. y represents a point  2 Def Y .F /, then, there is a unique Note that if  W S 2 ! C X y representing ./ D F ./ and coinciding with  on X. In that case, W S2 ! C the map f D  B F B 1 is a rational map of degree d and we have the following commutative diagram:



/ C; y

.S 2 ; X / F



.S 2 ; Y /





.X/ 

 with

f

/ C; y .Y /



jX D jX and  isotopic to relative to X:

(1.2)

Y Any point of DefX .F / is represented by a triple .; ; f / as in this diagram. If Y .F /, .1 ; 1 ; f1 / and .2 ; 2 ; f2 / are two triples representing the same point  2 DefX then the rational maps f1 and f2 are Möbius conjugate by the Möbius transformation sending 1 .X/ to 2 .X/. In particular, there is a natural map Y y .F / ! Ratd =Aut.C/: ˆ W DefX

In addition, for x 2 X we have

    f .x/ D  F .x/ :

In particular,  sends cycles of F contained in X to cycles of f .

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Chapter 16. Teichmüller spaces and holomorphic dynamics

If F is postcritically finite, there exists a smallest function F W S 2 ! N [ f1g such that .x/ D 1 if x … PF and .x/ is a multiple of .y/  degy F for each y 2 F 1 .x/. The function F is called the orbifold signature of F . We say that an orientation-preserving branched covering F is a .2; 2; 2; 2/-map if F is postcritically finite and its orbifold signature takes the value 2 exactly at 4 points and the value 1 otherwise. This happens exactly when #PF D 4, CF \ PF D ; and all critical points of F are simple. Theorem 1.5 (Epstein, [20]). If F is not a .2; 2; 2; 2/-map or if X does not contain Y the entire postcritical set of F , then the deformation space DefX .F / is either empty or a smooth manifold of TY of dimension #.Y  X /. Y Proof. Let  be a point of DefX .F / represented by a triple .; ; f /. By the Implicit Function Theorem, it is enough to show that the linear map

D   D F W T TY ! T. / TX is surjective. The cotangent space to TY at  is canonically identified to Q..Y // and the cotangent space to TX at ./ D F ./ is canonically identified to Q..X //. The adjoint map of D   D F is the linear map rf D id  f W Q..X // ! Q..Y //. It is enough to prove that this linear map is injective. If there were a q 2 Q..X // such that q D f q, according to Lemma 1.6 below, f would be a .2; 2; 2; 2/-map and the set of poles of q would be Pf . As a consequence, we would have Pf  .X /. The restriction of F to X is conjugate to the restriction of f to .X /. Thus, F would be a .2; 2; 2; 2/-map with PF  X, contradicting our assumptions. y ! C y be a Lemma 1.6 (Thurston’s contraction principle, [72], [18]). Let f W C rational map of degree d  2. Then for any integrable meromorphic quadratic y we have kf qk kqk, with equality if and only if f  f q D d  q. differential q on C, Furthermore, if q D f q for some q ¤ 0 then f is a .2; 2; 2; 2/-map and the set of poles of q is Pf . Proof. The inequality kf qk kqk follows easily from the triangle inequality: if y X Vf is a simply connected open set of full measure and if fgi gi2f1;:::;d g are U C the inverse branches of f on U , then Z Z ˇX Z Z Z ˇ Z X ˇ ˇ jf qj D jf qj D gi q ˇ

jgi qj D jqj

jqj: ˇ y C

U

U

gi

U g i

f 1 .U /

y C

The case of equality follows easily. As a consequence, if q D f q, we have f  q D d  q. In particular, the set Z of poles of q satisfies f .Z/  Z and f 1 .Z/  Z [ Cf . Thus, #Z C .2d  2/  #Z C #Cf  #f 1 .Z/  d  #Z  .2d  2/:

(1.3)

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Xavier Buff, Guizhen Cui, and Lei Tan

This implies 4.d  1/  #Z.d  1/. As d > 1, we have #Z 4. Assume q ¤ 0. Then, q has at least 4 poles, thus #Z D 4 and all inequalities in (1.3) become equalities. The leftmost equality in (1.3) implies Z \ Cf D ; and #Cf D 2d  2, which means that all critical points of f are simple. The middle equality means that f 1 .Z/ D Z t Cf so Vf  SZ. But f .Z/  Z (if q has a pole at z, then f q has a pole at f .z/). So Pf D n0 f n .Vf /  Z. It remains to show Z  Pf . Note that f 1 .Z X Pf / is contained in Z [ Cf and is disjoint from Cf [ Pf . So f 1 .Z X Pf /  Z X Pf and hence f n .Z X Pf /  Z X Pf for any n. But f 1 .z/ consists of d distinct points for any z which is not a critical value. This proves that Z X Pf , as a set with at most 4 points, must be empty. Therefore f is a .2; 2; 2; 2/-map and the set of poles of q is Pf . Corollary 1.7. Let f be a rational map of degree d > 2 that is not a .2; 2; 2; 2)-map. Then the operator rf D id  f is injective on the space of integrable meromorphic y quadratic differentials on C. Y Characterizing the cases for which the deformation space DefX .f / is not empty is not an easy task. Thurston’s theorem below gives precise conditions under which this space is not empty (actually is a single point) when F is postcritically finite and X D Y D Z is a finite forward invariant set containing PF .

2 Thurston’s theorem with marked points Let us define an equivalence relation on the set of pairs .F; Z/ such that F W S 2 ! S 2 is an orientation-preserving branched covering of a topological sphere S 2 of degree deg.F /  2 and Z  S 2 is a finite set satisfying PF  Z and F .Z/  Z. An equivalence .; / between two pairs .F0 ; Z0 / and .F1 ; Z1 / is a pair of homeomorphisms ; W S 2 ! S 2 such that .Z0 / D .Z0 / D Z1 , jZ0 D jZ0 , the two maps  and are isotopic rel Z0 and F1 B D  B F0 . In this situation, we say that .F0 ; Z0 / is combinatorially equivalent to .F1 ; Z1 /. In the case that Z D PF and #Z < 1, Thurston’s characterization theorem ([72], [18]) provides a necessary and sufficient condition for .F; Z/ to be combinatorially y !C y a rational map (we say that .f; X / is a rational equivalent to .f; X/ with f W C representative). We will now present the condition.

2.1 Thurston obstructions A Jordan curve disjoint from Z is said null-homotopic (resp. peripheral) rel Z if one of its complementary component contains zero (resp. one) point of Z. A Jordan curve that is disjoint from Z, such that each of its two complementary components contains at least two points of Z, is said non-peripheral rel Z.

Chapter 16. Teichmüller spaces and holomorphic dynamics

729

We say that D f 1 ; : : : ; k g is a multicurve of .F; Z/, if each i is a Jordan curve disjoint from Z and is non-peripheral rel Z, and the j ’s are mutually disjoint and mutually non-homotopic rel Z. We say that is .F; Z/-stable if every curve of F 1 . / is either homotopic rel Z to a curve of or null-homotopic or peripheral rel Z. This implies that for any m > 0, every curve of F m . / is either homotopic rel Z to a curve of or null-homotopic or peripheral rel Z. Each such induces an .F; Z/-transition matrix F together with its leading eigenvalue  as follows: Let . i;j;ı /ı denote the components of F 1 . j / homotopic to i rel Z (there might be no such components). Then F W i;j;ı ! j is a topological covering of a certain degree di;j;ı . The transition matrix is defined to be F D P 1=d . This is a non-negative matrix. By the Perron–Frobenius Theorem i;j;ı ı there is a non-negative eigenvalue  that coincides with the spectral radius of F . We say that an .F; Z/-stable multicurve is a Thurston obstruction for .F; Z/ if   1. In the particular case Z D Pf , we simply say that is a Thurston obstruction for F .

2.2 Main results Theorem 2.1 (Marked Thurston’s theorem). Let F W S 2 ! S 2 be a postcritically finite branched covering which is not a .2; 2; 2; 2/-map. Let Z  S 2 be finite with PF  Z and F .Z/  Z. If .F; Z/ has no Thurston obstructions, then the combinatorial equivalence class of .F; Z/ contains a rational representative which is unique up to Möbius conjugacy. More precisely, if .; / is an equivalence between two rational representatives .f1 ; X1 / and .f2 ; X2 /, then there is a (unique) Möbius transformation M which is isotopic to both  and rel X1 and satisfies M B f1 D f2 B M . Remark. Our statement is slightly more general than Thurston’s original theorem (see [72], [18]), where Z D PF . We actually prove more. Theorem 2.2. Let F W S 2 ! S 2 be a branched covering and Z  S 2 be a finite set containing at least three points x0 , x1 , x2 with PF  Z and F .Z/  Z. y be any given orientation preserving homeomorphism. Define Let 0 W S 2 ! C y is a homeomorphism agreeing with 0 on .n ; fn / recursively such that n W S 2 ! C fx0 ; x1 ; x2 g and such that the map fn D n1 B F B n1 is a rational map. If F is not a .2; 2; 2; 2/-map and .F; Z/ has no Thurston obstructions, then • the Thurston pullback map F W TZ ! TZ has a unique fixed point  ; • the sequence Œn  converges to  in the Teichmüller space TZ ; y and • ffn g converges uniformly to a rational map f on C;

y • n .Z/ converges pointwise to a set X  C. Moreover, there is an equivalence .; / between .F; Z/ and .f; X / with ; y both representing the fixed point  2 TZ . C

W S2 !

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An easy corollary of the above theorem is that if Z contains more than deg.F / C 1 fixed points then .F; Z/ is necessarily obstructed. There might be a direct proof of this fact without using Thurston’s theorem. It is easy to see that Theorem 2.2 implies Theorem 2.1. The sequence .n ; fn / appearing in the previous theorem is called Thurston’s algorithm for the pair .F; 0 /. Its definition is sketched on the commutative diagram below.

 S2

2

 /C y2 f2

F



S2

1

F



S2

0



(2.1)

/C y1 

f1

/C y0

Let us now state without proof a result of McMullen ([38], Theorem B4) which is closely related to the previous discussion. Again this form is slightly stronger than McMullen’s original version but the proof goes through without any trouble. y !C y be a rational map (not necessarily postcritically finite), Theorem 2.3. Let f W C y y y and f .Z/ y  Z. y Let

and let Z  C be closed (not necessarily finite) with Pf  Z y y is finite). be a .f; Z/-stable multicurve (defined in a similar way as in the case that Z Then  1. If  D 1, then either f is a .2; 2; 2; 2/-map; or f is not postcritically finite, and includes a curve that is contained in a Siegel disk or a Herman ring.

2.3 Classical results from hyperbolic geometry In this chapter we will make the following convention on the choice of the multiplicative constant in a hyperbolic metric. 1) The hyperbolic metric on the unit disc D is H it is

jdzj . =z

2jdzj , 1jzj2

and on the upper half plane

2) The modulus of an open annulus A is denoted by mod.A/, and   log r : mod f1 < jzj < rg D 2 3) For S a hyperbolic Riemann surface and a closed geodesic on S , we use `S . / (or `. / if there is no confusion) to denote the hyperbolic length of . Set w. / D  log `. / (one should consider it as a kind of logarithmic width).

Chapter 16. Teichmüller spaces and holomorphic dynamics

731

4) For any non-peripheral simple closed curve on S 2 X Z and any point  2 TZ y we denote by `. ;  / the length of the unique simple represented by  W S 2 ! C, y closed geodesic in C X .Z/ homotopic to . / and w. ;  / D  log `. ;  /. Denote w./ D sup w. ; / where the supremum is taken over all simple closed geodesics in S 2 X Z. 5) For any constant C > 0, set ˚  TZ .C / D  2 TZ j w./ C ² ³ ˇ ˇ  log `. ;  / C for every non-peripheral D  2 TZ ˇ : simple closed curve in S 2 X Z The following result is a version of Wolpert’s Lemma which gives an upper bound for ratios of hyperbolic lengths in terms of Teichmüller distances. Lemma 2.4. Let 1 ; 2 2 TZ . Assume dT .1 ; 2 / D. Then for any non peripheral simple closed curve in S 2 X Z, ˇ ˇ ˇw. ; 1 /  w. ; 2 /ˇ 2D: If in addition 1 2 TZ .C /, then 2 2 TZ .C C 2D/. Proof. Let D 0 > D be arbitrary. Let 1 , 2 be representatives of 1 ; 2 respectively. There is a quasi-conformal homeomorphism h W 1 .S 2 XZ/ ! 2 .S 2 XZ/ homotopic to 2 B 11 with 12 log K.h/ D 0 . Set S1 D 1 .S 2 X Z/ and S2 D 2 .S 2 X Z/. Let 1 be a closed geodesic on S1 and 2 the closed geodesic on S2 homotopic to h. 1 /. Let A1 ! S1 be an annular cover associated to 1 and A2 ! S2 be an annular cover associated to 2 . Then   mod.A1 / D and mod.A2 / D : `S1 . 1 / `S2 . 2 / In addition, h W S1 ! S2 lifts to a K.h/-quasiconformal homeomorphism between A1 and A2 , and according to Grötzsch’s inequality, mod.A1 / K.h/  mod.A2 / This yields

and

mod.A2 / K.h/  mod.A1 /:

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ log `S1 . 1 / ˇ D ˇ log mod.A1 / ˇ log K.h/: ˇ ˇ ˇ ˇ

`S2 . 2 /

mod.A2 /

Therefore for any non peripheral simple closed curve in S 2 X Z, jw. ; 1 /  w. ; 2 /j D j log `. ; 1 /  log `. ; 2 /j log K.h/ 2D 0 : As D 0 > D is arbitrary, we may replace D 0 by D in the inequality. Lemma 2.5. Let S be a hyperbolic Riemann surface.

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Xavier Buff, Guizhen Cui, and Lei Tan

(1) (Short geodesics are simple and disjoint) Let 1 , 2 be distinct closed geodesics on S . Then p `. i / < 2 log.1 C 2/; i D 1; 2 (2.2) H) 1 \ 2 D ; and 1 , 2 are simple. (2) Let A  S be an open annulus whose equator is homotopic to a simple closed geodesic on S. Then  modA

: (2.3) `. / (3) (Collar) For any simple closed geodesic on S , there is a canonical annulus CS . /  S whose equator coincides with , with   1: (2.4) modCS . / > `. / Moreover if two simple closed geodesics , are disjoint, then CS . / and CS . / are disjoint. Proof. This is a classical result in hyperbolic geometry. Part (3) is attributed to Buser and to Bers (“the collar lemma”). See e.g. Hubbard, [29]. Lemma 2.6 (Short geodesics under a forgetful map). Let S be a hyperbolic Riemann p surface and S 0 D S X Q with Q  S a finite set. Choose L < 2 log.1 C 2/. Set q D #Q. Let be a simple closed geodesic on S . Denote by f i0 gi2I the set of simple closed geodesics on S 0 homotopic to in S such that the hyperbolic length `0i WD `S 0 . i0 / satisfies `0i < L. Set ` D `S . /. Then (1) For every i 2 I , `0i  `, and #I q C 1 (in particular it is finite). (2)

qC1 1 1 qC1 X 1 1   < C ; 0 < `  L `i ` 

(2.5)

i2I

in particular if I D ; then

1 `



1 



qC1 L

< 0.

Proof. The fact `0i  ` follows from Schwarz’ Lemma. Apply (2.2) to S 0 . We know that the i0 ’s are pairwise disjoint. Also, any pair i0 , 0 j enclose an annulus in S (since they are homotopic in S and disjoint) containing at least one point of Q (since they are not homotopic in S 0 ). It follows that there are at most q C 1 such curves. 0 0 pairwise disjoint. There is therefore It follows from (2.4) that the collars S CS . i / are an open annulus A  S containing i2I CS 0 . i0 / with equator homotopic to on S . The right-hand side of (2.5) is trivial if I D ;, otherwise,   X #I qC1 X  (2.3)  Grötzsch (2.4) X  .q C 1/

 1 < modCS 0 . i0 / modA

: 0 0 `i `i ` i 2I

i2I

i2I

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Chapter 16. Teichmüller spaces and holomorphic dynamics

We now prove the left-hand side inequality of (2.5). We Pfirst decompose CS . / into t (1 t q C 1) pairwise disjoint annuli Cj such that jt D1 modCj D modCS . /, Cj  S 0 , and the core curves of Cj are pairwise non-homotopic in S 0 . For each j , let ıj be the geodesic on S 0 homotopic in S 0 to the core curve Cj . We have then t X (2.4)   1 < modCS . / D modCj ` j D1  X X  D C modCj  modCj  L

(2.3)



 modCj > L

.q C 1/ C L

X

modCj

 modCj > L

.q C 1/ C L

X

 : `S 0 .ıj /

 modCj > L

Assume that the index set of the rightmost term is non empty. Then `S 0 .ıj / < L so ıj D i0 for some i 2 I , in particular I ¤ ;. In this case .q C 1/ X   : 1< C ` L `0i i2I

If I D ;, then necessarily no Cj satisfies modCj >

 L

and we have

.q C 1/  1< : ` L The left-hand inequality of (2.5) is now proved.

2.4 From TZ to Ratd From now on, we fix three points x0 , x1 , x2 in Z  F 1 .Z/. A point  2 TZ may be y sending x0 , x1 , x2 to respectively 0, 1, represented by a homeomorphism  W S 2 ! C y 1. Its restriction  W Z ! C only depends on . Similarly, &F . / 2 TF 1 .Z/ may y sending x0 , x1 , x2 to respectively be represented by a homeomorphism W S 2 ! C 1 y is a rational map. The restriction  W F 1 .Z/ ! C 0, 1, 1, so that f D  B F B of and the rational map f only depend on . Indeed, if dx stands for the local degree of F at x 2 S 2 , then f D f D   P =Q where P .z/ D

Y x2F 1 .x0 / x¤x2



z

 .x/

dx

;

Q .z/ D

Y x2F 1 .x2 / x¤x2



z

dx

 .x/

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Xavier Buff, Guizhen Cui, and Lei Tan

and  is the value taken by Q =P at any point of TF 1 .Z/ is analytic, the map TZ 3  7! .f ;  ;

/

 

 F 1 .x1 / . Since &F W TZ !

y Z  .C/ y F 1 .Z/ 2 Ratd  .C/

is analytic. It is true, although not elementary, that the image of TZ under the map  7! f is closed in Ratd . We shall circumvent the difficulties by introducing the following space. We shall denote by RZ;F the set of triples 1 .Z/

y Z  .C/ y F .f; ; / 2 Ratd  .C/

such that •  and are injections sending x0 , x1 , x2 to respectively 0, 1, 1, •  B F D f B on F 1 .Z/ and • the local degree of F at x is equal to that of f at .x/ for all x 2 F 1 .Z/.  1  In particular, setting Y D .Z/ and X D F .Z/ D f 1 .Y /, we have the following commutative diagram: / .C; y Cf  X /

.S 2 ; CF  F 1 .Z// F





.S 2 ; VF  Z/



f

/ .C; y Vf  Y /:

y Given c > 0, we shall denote by RZ;F .c/ Let dCy stand for the spherical distance in C.   the subset of RZ;F consisting of those triples .f; ; / for which dCy z1 ; z2  c for   any pair of distinct points z1 ¤ z2 in F 1 .Z/ . Lemma 2.7. For all c > 0, the set RZ;F .c/ is a compact subset of RZ;F . For all C > 0 there exists c > 0 such that  2 TZ .C / H) .f ;  ;

/

2 RZ;F .c/:

Proof. Let .fn ; n ; n / be a sequence of triples in RZ;F .c/. Set Yn D n .Z/ and   y is compact, extracting a subsequence if Xn D n F 1 .Z/ D fn1 .Yn /. Since C  necessary, we may assume that the sequences .n W Z ! Yn / and n W F 1 .Z/ !  Xn converge respectively to maps  W Z ! Y and W F 1 .Z/ ! X for some finite y Since the spherical distance between distinct points in Xn is at least sets X; Y  C. c > 0, the limit W Z ! X is a bijection and the spherical distance between distinct points in X is at least c. The sequence of rational maps fn converges to f D   P =Q where Y  Y  d d P .z/ D z  .x/ x ; Q.z/ D z  .x/ x x2F 1 .x0 / x¤x2

x2F 1 .x2 / x¤x2

Chapter 16. Teichmüller spaces and holomorphic dynamics

and  is the value taken by Q=P at any point of f B

.x/ D lim fn B

n .x/



735

 F 1 .x1 / . If x 2 F 1 .Z/, then

D lim n B F .x/ D  B F .x/:

The local degree of f at a point .x/ is at least dx , and for all y 2 Y , the number of preimages of y by f , counting multiplicities, is X   deg f; .x/ dD x2.BF /1 .y/



X

dx

x2.BF /1 .y/



X

X

dx D d  # 1 .y/:

z2 1 .y/ x2F 1 .z/

Thus, # 1 .y/ D 1, i.e.  is injective, and the local degree of f at .x/ is dx . All this shows that .f; ; / 2 RZ;F .c/. y Z  .C/ y F 1 .Z/ . This proves that RZ;F .c/ is a compact subset of Rat d  .C/ in RZ;F .c/ for some c > 0. Let us now prove that the image of TZ .C  / is contained  1 1 Set Y D  .Z/ and X D  F .Z/ D f .Y /. By definition of TZ .C /, the y X Y is bounded from below by e C . length of any simple closed geodesic 2 C y y Since Y contains the critical values of f , the map f W CXX  ! CXY is a covering. y It follows that the length of any simple closed geodesic ı 2 C X X is bounded from below by e C . As a consequence, as  ranges in TZ .C / and x; y range in F 1 .Z/ with x ¤ y, the spherical distance between  .x/ and  .y/ is uniformly bounded away from 0 as required.

2.5 Contraction of Thurston’s pullback maps Let F W S 2 ! S 2 be a branched covering of degree d  2 with a finite postcritical y be a finite set with #Z  4, PF  Z and F .Z/  Z. Setting set PF . Let Z  C X D Y D Z, the conditions in Section 1.4 are satisfied and thus, Thurston’s pullback map F W TZ ! TZ is well defined. From now on, we set k D #Z

and

G D F Bk :

Recall that the tangent space to TZ at some point  represented by equipped with the dual norm: ˇ ˇ kk D sup ˇhq; iˇ for all  2 T TZ :

y is W S2 ! C

q2Q. .Z// kqk1

We will now show that FBk is contracting, and even uniformly contracting on TZ .C / for C > 0. It will be useful to notice that FBk D G . Indeed, assume  is a point in y be a homeomorphism representing  Bi . / so TZ , and for i 2 Œ0; k, let i W S 2 ! C F 1 y !C y is a rational map for i 2 Œ0; k  1. Then, we have that fi D i B F B i C1 W C

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Xavier Buff, Guizhen Cui, and Lei Tan

the following commutative diagram with Zi D i .Z/: k

.S 2 ; Z/ F



k1

.S 2 ; Z/  .S 2 ; Z/ F

2



1

.S 2 ; Z/ F



0

.S 2 ; Z/ Set  D 0 ,

fk1



/ .C; y Zk1 /  / .C; y Z2 / f1



/ .C; y Z1 / 

f0

/ .C; y Z0 /:

D k and g D f0 B f1 B    B fk1 . Then, the commutative diagram / .C; y Zk /

.S 2 ; Z/ G

g





.S 2 ; Z/ shows that

/ .C; y Zk /



/ .C; y Z0 /

  FBk ./ D Π D G ΠD G . /:

Lemma 2.8. If there is a set X  Z such that #X  4 and G 1 .X /  Z [ CG , then F W S 2 ! S 2 is a .2; 2; 2; 2/-map. Proof. Define recursively X0 D X

and XiC1 D F 1 .Xi / X CF ;

so that Xk D G 1 .X/ X CG  Z. In particular, #Xk #Z D k. Since F 1 .Xi /  XiC1 [ CF , we have the following inequalities (compare with (1.3)): #Xi C1 C .2d  2/  #XiC1 C #CF  #F 1 .Xi /  d  #Xi  .2d  2/: This implies #XiC1  4  d  .#Xi  4/. In particular, d k4 > k  4  #Xk  4  d k4  .#X4  4/         d k1  .#X1  4/  d k  .#X0  4/  0:

(2.6)

Chapter 16. Teichmüller spaces and holomorphic dynamics

737

As a consequence, #Xi D 4 for i D 0; 1; 2; 3; 4 and inequalities (2.6) must be equalities for i D 0; 1; 2; 3: • #CF D 2d  2, thus the critical points of F are simple. • #Xi C1 C #CF D #F 1 .Xi /, thus CF \ XiC1 D ;. In particular, CF \ X1 D ;. • #F 1 .Xi / D d  #Xi  .2d  2/, thus VF  Xi . In particular, VF B3 D VF [F .VF /[F B2 .VF /  X1 and VF B4 D F .VF B3 /[F B3 .VF /  X0 : We now claim that X1 D PF . Indeed • X1  PF since otherwise a point in X1 X PF would have d 3 > 4 preimages in X4 whereas #X4 D 4. • PF  X1 since VF  VF B2  VF B3  VF B4  X0 ; so that 2 #VF #VF B2 #VF B3 #VF B4 4 which forces the non-decreasing sequence VF Bi to stabilize: there exists i0 3 such that VF Bi D VF Bi0 for i  i0 . We then have PF D VF Bi0 D VF B3  X1 . Summarizing, we see that PF D X1 has cardinality 4, CF \ PF D CF \ X1 D ; and all the critical points of F are simple. Thus, F is a .2; 2; 2; 2/-map. Lemma 2.9 (Contraction). Suppose that F W S 2 ! S 2 is not a .2; 2; 2; 2/-map. Then kD FBk k < 1 (where k D #Z) for any  2 TZ .1 Proof. A point  2 TZ yields a triple .g ;  ;  / 2 RZ;G such that g B  D  B G Bk on G 1 .Z/. Thenorm of the linear map   D F D D G is equal to the norm of its adjoint .g / W Q  .Z/ ! Q  .Z/ . The result is a consequence of the following more general lemma. 2 2 Lemma 2.10.  Assume  F WS ! S is not a .2; 2; 2; 2/-map and .g; ; / 2 RZ;G . Then g W Q .Z/ ! Q .Z/ has norm strictly less than 1.

Proof. Due to Lemma 1.6 we  already  know that kg k 1, with equality if and only if there is a non-zero q 2 Q .Z/ such that q D d k g  .g q/: Assume by contradiction that kg k D 1. Let Y  .Z/ be the set of poles of g q. Then, every point of g 1 .Y / is either a pole of q, or a critical point of g. So, g 1 .Y /  .Z/ [ Cg . As a consequence, X D  1 .Y / satisfies X  Z and G 1 .X /  Z [ CG . According to Lemma 2.8, this contradicts the fact that F is not a .2; 2; 2; 2/-map. 1 It is known that in the classical version of Thurston’s theorem where Z D P , one may choose k D 2. In F the general version, it is possible to prove that we may choose k  2 such that d k2 > #.Z X PF /.

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Xavier Buff, Guizhen Cui, and Lei Tan

Lemma 2.11 (Uniform contraction on TZ .C /). If F is not a .2; 2; 2; 2/-map, then for each C > 0, there is  < 1 such that kD FBk k  for all  2 TZ .C /. Proof. We proceed by contradiction and assume we can find a sequence n 2 TZ .C / such that kDn FBk k tends to 1 as n tends to 1. Consider the corresponding sequence of triples .gn ; n ; n / 2 RZ;G . Set Xn D n .Z/ and Yn D n .Z/. The norm kDn FBk k is equal to the norm of .gn / W Q.Xn / ! Q.Yn /. Thus, we can find a sequence of quadratic differentials qn 2 Q.Xn / of norm 1 so that k.gn / qn k tends to 1 as n tends to 1. According to Lemma 2.7, this sequence belongs to a compact subset of RZ;G . So, extracting a subsequence if necessary, we may assume that the triple .gn ; n ; n / converges to .g; ; / 2 RZ;G . According to the previous lemma, the norm of g W Q.X / ! Q.Y / is less than 1. The poles of the quadratic differentials qn are simple and stay uniformly away from each other for the spherical distance. It follows that we may extract a further subsequence such that qn converges locally uniformly outside X to some q 2 Q.X / of norm 1. The sequence of quadratic differentials .gn / qn then converges locally uniformly to g q 2 Q.Y / outside Y . Since the poles of .gn / qn are in Yn , they remain uniformly away from each other for the spherical distance. As a consequence, kg qk D lim .gn / qn D 1 D kqk. This contradicts the previous observation that kg k < 1.

2.6 Proof of Theorem 2.2 Proposition 2.12 (Short geodesics do not become shorter). Assume that .F; Z/ has no Thurston obstructions. Given 0 2 TZ , set n D FBn .0 /. Then there is a positive integer m depending only on deg.F / and #Z, a positive constant C depending only on deg.F /, #Z and dT .0 ; 1 /, such that for all n  0, w.n / > C H) w.nCm / < w.n /: We will postpone the proof of this proposition to Section 2.7. Proof of Theorem 2.2 assuming Proposition 2.12. Given 0 2 TZ , set n D FBn .0 / and D D dT .0 ; 1 /. Let ı be a geodesic of TZ connecting 0 and 1 and for n  0, set ın D FBn .ı/. According to Lemma 2.4, F W TZ ! TZ is contracting and so, for all n  0, we have length.ınC1 / length.ın /. It then follows from Lemma 2.9 that for all n  0, dT .n ; nC1 / dT .0 ; 1 / D

w.nC1 / w.n / C 2D: (2.7)   Let m and C be given by Proposition 2.12. Set C1 D max C; w.0 / . We claim that the sequence .n /n0 remains in TZ .C1 C2mD/. Indeed, for n  0, let jn 2 Œ0; n be the largest integer j such that j 2 TZ .C1 /. If jn D n, then we are done. Otherwise, and

Chapter 16. Teichmüller spaces and holomorphic dynamics

739

let us write n D .jn C 1/ C q m C r with 0 r < m. For j 2 Œjn C 1; n, we have w.j / > C1  C . It follows from Proposition 2.12 and Lemma 2.4 that w.n / w.jn C1Cr / w.jn / C .r C 1/  2D C1 C m  2D: Set C2 D C1 C.mC1/2D. According to Lemma 2.4, ın  TZ .C2 / for all n  0. Set k D #Z. By Lemma 2.11, there is a constant  < 1 such that kD FBk k <  for any  2 TZ .C /. It follows that, for any n  0 and any 1 j k, dT .nk ; nkCj / n dT .0 ; j / jDn kDn : Therefore .n /n0 is a Cauchy sequence in TZ and hence converges to a fixed point  in TZ . We now prove that the fixed point is unique, independent of the choice of 0 . Indeed, let  and  0 be two fixed points of F . Let ı be the geodesic joining  and  0 . Recall that k D #Z. Then FBk .ı/ is a curve joining  and  0 and its length is less than that of ı, which contradicts the fact that ı is the shortest curve joining  and  0 . Finally, as n 2 TZ tends to  2 TZ , the sequence .fn ; n ; n / 2 RZ;F y is an orientation preserving tends to .f ;  ;  /. This shows that if 0 W S 2 ! C homeomorphism sending x0 , x1 , x2 to 0, 1, 1 and if .n ; fn / is defined recursively y is a homeomorphism sending x0 , x1 , x2 to 0, 1, 1 and such such that n W S 2 ! C that the map fn D n1 B F B n1 is a rational map, then • fn D fn converges to f and y • n .Z/ D n .Z/ converges pointwise to a set X  C. Since F . / D , the bijection  W Z ! X coincides with the bijection  W Z ! X. It follows that f .X/  X and that f is postcritically finite with Pf  X . y be the homeomorphism representing  sending x0 , x1 , x2 Finally, let  W S 2 ! C y be the homeomorphism representing F . / D  sending to 0, 1, 1. Let W S 2 ! C x0 , x1 , x2 to 0, 1, 1 with f B D  B F . Then, .; / is an equivalence between .F; Z/ and .f; X/.

2.7 Proof of Proposition 2.12. Notice that from the definition of the transition matrix, given a degree d and an integer p, there are only finitely many possible transition matrices F with F of degree d and of size at most p  3. Therefore there are finitely many such matrices with leading eigenvalue  < 1. The integer m, depending only on d and p, is chosen such that every such matrix F with  < 1 satisfies kFm k < 1=2, where k  k is relative to the sup-norm of R (this is possible due to the spectral radius formula kFn k1=n !  ). n!1 p Set A D  log.2 log. 2 C 1// and D D dT .0 ; 1 /. We choose at first any J > m.log d C 2D/, and set B D .p  3/J C A. For the moment choose any C > B and assume w.n / > C for some n  0. We want to show that w.nCm / < w.n /, up to a further adjustment of C .

740

Xavier Buff, Guizhen Cui, and Lei Tan

y represent n . Set P D .Z/ and Let  W S 2 ! C Ln D fw. ; n / j a non-peripheral Jordan curve on S 2 X Zg: Now let a; bŒ be the leftmost gap in ŒA; C1ŒXLn of length J . Set ² ˇ ³ ˇ a non-peripheral Jordan curve on S 2 X Z :

D ˇ with w. ; n / 2  a; C1 Œ Then w. ; n /  b for 2 . By Lemma 2.5, the set of with w. ; n / > A consists of pairwise disjoint non-peripheral simple closed curves in S 2 X Z. But Z consists of exactly p points. It follows that there are at most p  3 elements of Ln greater than A. By assumption w.n / > C > B D .p  3/J C A. So at least one element of Ln is greater than .p  3/J C A. It follows that b < B and ¤ ;. Claim (a). The multicurve is .F; Z/-stable. Proof. For this we will only use the fact that J > log d C 2D. 1 is a rational Let ' be a representative of nC1 D F .n /such that f D  B F B '  1 0 00 1 map. Set T D '.Z/ and T D ' F .Z/ D f .P /. Given 2 , let be a non-peripheral Jordan curve in F 1 . /. Let 0 (respectively 00 ) be the geodesic y X T 0 (respectively in C y X T 00 ). Since f W C y X T 00 ! C y XP homotopic to '. / in C 0 00 is a holomorphic covering, and since T  T , we have   00 `CXT f . 00 / D deg.F W ! /  `. ; n / 00 . / D deg.F W ! /  `CXP y y and 00 00 0 `CXT 00 . /  `CXT 0 . /  `CXT 0 . / D `. ; nC1 /: y y y

Thus w. ; nC1 /  w. ; n /  log deg.F W ! /  w. ; n /  log d  b  log d: By Lemma 2.4, we have jw. ; nC1 /  w. ; n /j 2D. Thus w. ; n /  b  log d  2D > a since b  a D J > log d C 2D. This shows that is homotopic rel Z to a curve in

. That is, is an .F; Z/-stable multicurve. This ends the proof of Claim (a). 1 Set G D F Bm . Let be a representative of nCm such that  g1D  B G B 1 is m 0 00 G .Z/ D g .P /. a rational map of degree d . Set P D .Z/ and P D Then P 0  P 00 .

y X P 00 of length less than d m  e b is Claim (b). Every simple closed geodesic in C 00 1 homotopic (rel P ) to a component of g B . / for some unique choice of 2 . y X P 00 of length less than d m  e b . Proof. Let ˇ be a simple closed geodesic in C b y Then  g.ˇ/  is a simple closed geodesic in C X P with length less than e , that is, w g.ˇ/  b. Thus g.ˇ/ is homotopic, rel P , to . / for some unique choice of

741

Chapter 16. Teichmüller spaces and holomorphic dynamics

2 . The critical values of g are contained in P . We may then lift the homotopy by g to get Claim (b). Set L D d m e B . Note that L depends only on p, d and D. Let D f 1 ; : : : ; s g be the non-empty .F; Z/-stable multicurve defined above. Define v; v 0 2 R by vi D

1 `. i ; n /

and

vi0 D

1 : `. i ; nCm /

y X P 0 and Q D P 00 X P 0 . Set q D #Q D #P 00  #P 0 D #P 00  p. We have Set S D C 00 #P D #g 1 .P / < d m  #P  1 D d m  p  1 as P 00 contains at least two critical points. It follows that q C 1 .d m  1/p. Furthermore L D d m  e B D d m  e .p3/J e A m

< d m  e .p3/ log d e A

p m

d m  e  log d e A D 2 log. 2 C 1/: By the left inequality of (2.5), we have, for any i , X 1 qC1 1 1 vi0 D < C C `. i ; nCm / `CXP  L 00 .ˇ/ y ˇ 2Wi

D

X

ˇ 2Wi

1 .d m  1/p C C ; `CXP  L 00 .ˇ/ y 1

y X P 00 homotopic to where Wi is the set of all simple closed geodesics on C 0 00 m B y X P ) less than L D d  e . P , and of length (in C

(2.8)

. i / rel

Claim (c). Each curve ˇ of Wi is homotopic rel P 00 to some . /, for a component

of G 1 . / of a unique choice 2 . Furthermore is homotopic rel Z to i , and 1 1 1 D : `CXP .ˇ/ deg.G W

! / `. ; n / 00 y Also the map ˇ 7! is injective. y X P 00 less than d m  e B which is less than Proof. Let ˇ 2 Wi . It has length in C   b m Claim (b), it is homotopic rel P 00 to a component . / of g 1 . / D d   e . By  G 1 . / for a unique choice of 2 . But being non-peripheral rel Z, the curves in G 1 . / are pairwise non-homotopic rel G 1 .Z/. Thus the curves in G 1 . / are pairwise non-homotopic rel P 00 . This shows that is unique. As ˇ and . / are homotopic rel P 00 , they are also homotopic rel P 0 . But ˇ is homotopic rel P 0 to . i / by the definition of Wi . We conclude that . / is also homotopic rel P 0 to . i /. y X P is a holomorphic covering, the curve g.ˇ/ is the simple y X P 00 ! C As g W C y closed geodesic of C X P homotopic to . / rel P . So .g.ˇ// D deg.G W ! /  `. ; n /: `CXP 00 .ˇ/ D deg.g W ˇ ! g.ˇ//  `CXP y y

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 1  The injectivity of ˇ 7! follows from the fact that every curve in G . / y X P 00 . This proves the is homotopic rel P 00 to a unique simple closed geodesic of C claim. It follows from this claim that  1 X X X 1 1

D .G v/i `CXP deg.G W ! / `. ; n / 00 .ˇ/ y   ˇ 2Wi

2

Z i

where the sum is taken over all curves in G 1 . / homotopic to i rel Z, and the right equality is due to the definition of the transition matrix. It follows from (2.8) that for any i , .d m  1/p 1 vi0 .G v/i C C :  L Therefore .d m  1/p .d m  1/p 1 1

kG k  jvj C C ; jv 0 j jG vj C C  L  L where jvj denotes the sup norm of R . As the multicurve is .F; Z/-stable, we have G D .F /m . By the choice of m, we have kG k 12 . Thus jv 0 j

1 1 .d m  1/p jvj C C : 2  L

If





.d m  1/p 1 C ; jvj > 2  L then jv 0 j < jvj, that is, w.nCm / < w.n /. Now we see that if we choose ²

C D max log





³

.d m  1/p 1 C C log 2 ; B ;  L

then the proposition is proved.

3 Applications of Thurston’s theorem and related results 3.1 Geyer’s sharpness result for harmonic polynomials The power of Thurston’s theorem is beautifully illustrated by a result of L. Geyer. We present this result here. Let  denote the map z 7! zN . We say that P .z/ D ad z d C    C a1 z C a0 is a Geyer polynomial if P has all coefficients real, all critical points simple, at most one critical point real, and if P maps each critical point c to its complex conjugate c. N Theorem 3.1 (Geyer, [21]). For every d  2, there is a Geyer polynomial P of degree d .

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This result solved a sharpness problem in the study of harmonic polynomials. It has been conjectured by Wilmshurst ([77]) that for any polynomial P of degree d  2, the equation P .z/ D zN ´ atek ([34]) proved the conjecture and has at most 3d  2 solutions. Khavinson and Swi¸ showed that for d D 2; 3 there are polynomials realizing the bound. Then Crofoot and Sarason noticed that the bound 3d  2 would be realized by a Geyer polynomial of degree d if it exists. Later on Bshouty and Lyzzaik proved that such polynomials exist for d D 4; 5; 6 and 8 ([8]). But their method seems to be difficult to reach the remaining degrees. Proof of Theorem 3.1. The idea is to first construct a topological model, and then prove the existence using Thurston’s theorem. y of degree d Fix any d  2. Assume that there exists a branched covering of C 1 satisfying G .1/ D 1, G B  D  B G, all critical points are simple, at most one critical point is real, and each critical point c is mapped to its complex conjugate c. N (Cf. Geyer, [21] for a construction). The postcritical set of G coincides with the set of critical points CG . Set Z D CG . Let  W z 7! zN . Notice that all critical points of G are periodic (of period 1 or 2). A theorem of S. Levy proves that in this case .G; Z/ has no Thurston obstructions. Furthermore, fix a non-real critical point c of G. Let .n ; fn / be the sequence in Thurston’s algorithm (2.1) so that every n fixes pointwise 1, c, c, N and 0 D  B 0 B  1 . It follows from Theorem 2.2 that fn converges uniformly to a polynomial P combinatorially equivalent to G. We want to prove that P is real. For this we will show that fn is real for every n. N is again a Set 10 D  B 1 B  1 and F D  B f1 B  1 . Then F .z/ D f1 .z/ polynomial and we have the following chains of commutative diagrams: 10

y o C G



G



y o C

y C



1



y C

/ y C 

0



f1

/ y C

# / y C 



F

/ y ; C.

0

Due to the uniqueness of the normalized making 0 B G B 1 holomorphic, we conclude that  B 1 B  D 10 D 1 . So 1 is real. This in turn implies that f1 is real. Thus, 0 real implies that 1 , f1 are real, and so n and fn are real. But fn ! P , so P is real.

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3.2 Applications of Thurston’s theorem There are many applications of Thurston’s theorem in holomorphic dynamics. In most cases, there is no need to work directly with Teichmüller spaces. One just needs to study Thurston obstructions. As illustrated by Geyer’s result above, the general procedure of an application goes as follows: a. Construct a postcritically finite branched covering F with some specific dynamical properties (if possible). b. Check whether F has Thurston obstructions. c. In the case of absence of obstructions use Thurston’s theorem to get a (unique up to Möbius conjugation) rational map f combinatorially equivalent to F , therefore having the same dynamical properties. Here is a case where there is an obstruction of topological nature: there is no branched covering of degree 4 having one double critical point c and four simple critical points sharing two critical values v and w. To prove it by contradiction, draw a segment linking v to w through the critical value coming from c, and pullback this segment. One runs easily into trouble due to the Jordan curve theorem. Another interesting case is that although it is easy to construct a cubic branched covering F with 4 distinct and fixed critical points, no cubic rational map has this property. So such a F must have a Thurston obstruction. It is in general difficult to apply Thurston’s theorem effectively, namely to check whether a specific branched covering has Thurston obstructions or not. Each successful application is usually a theorem in its own right. Here is brief account of some related results: • Topological polynomials. These are the branched coverings of S 2 with one backward invariant point. S. Levy ([37], [22]) reduced Thurston’s obstructions to some specific type of obstructions (called the Levy cycles). An easy consequence is that if every critical point eventually lands in a periodic cycle containing a critical point, then the map is unobstructed. In this case the map is combinatorially equivalent to a polynomial. • Matings of two polynomials. This is a surgery procedure in order to obtain rational maps whose Julia set is the gluing of two postcritically finite polynomial (therefore simpler) Julia sets. Obstructions often occur. Via the works of Milnor, Rees, Sharland, Shishikura, Tan, among others, some families of maps have been well understood. They include quadratic rational maps and Newton’s method of cubic polynomials. See for example [42], [54], [62], [64], [69], [70], [66]. One may consult the beautiful animations in the webpage of Chéritat [14], as well as the popularization article [71]. It has been known that two pairs of polynomials may lead to the same rational map. An amazing recent work of Rees shows that the number of pairs giving the same rational map can be arbitrarily large [58]. There are also results on matings of postcritically infinite polynomials (see for example [1], [27], [79]).

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• Captures. This is a surgery procedure to deform a polynomial so that the point at 1 glides along a certain path and gets ‘captured’ by a bounded orbit. Again, obstructions may occur and the procedure is highly non-injective. See the works of Wittner, Rees ([78], [55]–[59]), among others. • Blowing up an arc surgery. This consists of cutting open an invariant arc of a rational map in order to create a rational map of higher degree. This has been used in the works of Pilgrim and Tan [52], [49] to construct a variety of rational maps with interesting dynamical properties – Fatou component boundaries which are homeomorphic to a figure-8, symmetries, Sierpinski carpet Julia sets, maps with cylinders, etc. • Classifications of a family of rational maps. This consists of studying a full set of combinatorics that arises in a given family. Such combinatorics may take the form of Hubbard trees, external rays, spiders, kneading sequences, laminations, graphs, etc. See for example the works of Bielefield, Geyer, Hubbard, Kiwi, Mikulich, Poirier, Rees, Rückert, Schleicher ([4], [21], [30], [35], [44], [45], [53], [55]–[59]), among others. See also Douady–Hubbard–Sullivan’s proof of the monotonicity of the topological entropy in the logistic family presented by Milnor and Thurston in [43]. • Criteria of absence of Thurston’s obstructions. Several techniques have been developed in various situations. See for example work of Bonnot, Braverman, Pilgrim, Shishikura, Tan and Yampolsky [5], [51], [52], [65], [66]. • Perron algebraic number as the exponential of the topological entropy. Thurston ([74]) proved recently that any positive algebraic number greater than the modulus of its Galois conjugates can be realized as the leading eigenvalue of a transition matrix associated to a polynomial action on its Hubbard tree. • Folding surgery. This is a new type of surgery providing examples of postcritically finite rational maps whose Julia set contains wandering separating continua, see [16]. It is known, due to works of Thurston, Kiwi and Levin, [74], [35], [36], that such continua do not exist for polynomials with locally connected Julia sets (in particular for postcritically finite polynomials). • Selinger studies compactifications of rational map Teichmüller spaces, [61]. Work of Bonk, Haïssinsky, Meyer and Pilgrim, [9], [23], [24], [25], [26], [40], [41], [50], [51] study postcritically finite branched coverings of S 2 , in particular those with Thurston obstructions. Rivera-Letelier, [60], studies some weakly hyperbolic rational maps with the help of the convergence of Thurston’s algorithm. • Bisets as algebraic invariant of combinatorial equivalent classes. Let f be a y X Pf ; t /. postcritically finite rational map. Let t … Pf . Set G D 1 .C y X Pf , linking t to a point in Define Mf to be the set of homotopic paths in C f 1 .t /. This set is equipped with a right action of G by amending a curve ı 2 G first before taking 2 Mf to get :ı, and with a left action of G by taking 2 Mf

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first and then by following the corresponding lift by f of ı 2 G. These two actions commute and make Mf into a G-biset. Nekrachevych introduced this notion and proved that Mf is a complete invariant of the combinatorial equivalence class of f ([46]). Bartholdi and Nekrachevych then used this invariant to solve the so-called twisted rabbit problem of Hubbard, [2]. See also their related works as well as that of K. Bux, G. Kelsey and R. Perez [3], [33], [47], [48], among others. • Extensions of Thurston’s theorem beyond postcritically finite maps. Thurston’s original theorem can only be applied to postcritically finite rational maps. On the one hand, all these maps have a connected Julia set; on the other hand, they form a totally disconnected subset in the parameter space (except for the Lattès examples). Therefore the theorem alone cannot characterize the combinatorics of disconnected Julia sets, nor the dynamical bifurcations through continuous parameter perturbations. Up to now there are several extensions of Thurston’s theory to postcritically infinite rational maps. David Brown [7], supported by previous work of Hubbard and Schleicher [30], extended it to uni-critical polynomials with an infinite postcritical set (but always with a connected Julia set), and pushed it even further to the infinite degree case, namely the exponential maps. Hubbard–Schleicher–Shishikura [31] extended Thurston’s theorem to postcritically finite exponential maps. Zhang announced a corresponding result for maps that have a fixed Siegel disc with bounded type rotation number and are postcritically finite elsewhere. Jiang–Zhang [32], in parallel with Cui–Tan [15] solved the characterization problem for sub-hyperbolic rational maps with possibly disconnected Julia set. The proof of the former uses similar ideas as Thurston’s. Whereas that of the latter reduces the situation to a postcritically finite setting and applies the marked Thurston theorem (the unmarked one is not enough for this purpose), and at the same time provides a combination result together with a detailed description of the structure of disconnected Julia sets, alongside a Thurston-like theory for maps that are only partially defined. Zhang, [80], generalized Thurston’s theorem to maps with a fixed Siegel disc of bounded rotation number (and postcritically finite elsewhere). A generalization to maps with parabolic periodic points is also under preparation ([17]). Wang, [76], developed a Thurston-like theory for rational maps with Herman rings and Siegel disks, by combining the work of [15] and [80] together with a surgery technique of Shishikura [63]. y !C y be a postcrit• Covering properties of Thurston’s pullback maps. Let f W C ically finite rational map with postcritical set Pf . It induces a Thurston pullback map f W TPf ! TPf which has a unique fixed point ~ D Œid 2 TPf . For any  2 TPf , the sequence .fn .//n0 converges to ~ as n ! C1. We mention here a result about the covering properties of f . Theorem 3.2 (Buff–Epstein–Koch–Pilgrim, [11]). (1) Assume f is a polynomial of degree  2 whose critical points are all periodic. Then f .TPf / is open and dense in

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TPf and f W TPf ! f .TPf / is a covering map. In particular the derivative of f at ~ is invertible. 2 (2) The rational map f .z/ D 2z3z3 C1 is postcritically finite. The associated Thurston pullback map f W TPf ! TPf is a ramified covering whose group of deck transformations acts transitively on the fibers, and the derivative of f at ~ is not invertible. (3) There are explicit postcritically finite polynomials and rational maps f for which f W TPf ! TPf is constant. For example, this is the case for the polynomial 

f .z/ D 2i z 2 

1Ci 2

2

:

4 Epstein’s transversality results From now on, we assume that y !C y is a rational map, • fWC y containing at least three points with Vf  Y and • X and Y are finite subsets of C 1 X  Y \ f Y , and • either f is not a .2; 2; 2; 2/-map or X does not contain the entire postcritical set Pf . In Section 1.4, we used Thurston’s contraction principle, i.e., the injectivity of the operator rf D id  f acting on the space of meromorphic quadratic differentials y having at most simple poles, to show the smoothness of the deformation space on C Y Y DefX .f /. In addition, let ~ stand for the basepoint in DefX .f / represented by the Y .f / triple .id; id; f / as in (1.2). Then, the proof shows that the cotangent space to DefX at ~ is canonically identified with the quotient space Q.Y /=rf Q.X /. Right after his Ph.D. thesis, Epstein observed that he could deduce corresponding results for appropriate loci of maps with given multipliers, parabolic degeneracies, and holomorphic indices, from the injectivity of rf on appropriate spaces of meromorphic quadratic differentials with higher order poles. The reader who is not a dynamicist is invited to focus on the statements related to the multipliers, since we think those are the most easily accessible ones.

4.1 Formal invariants of a cycle y is a periodic point of Let us recall the following classical definitions. A point x 2 C Bp f of period p if f .x/ D x for some least integer p  1. The multiplier  of the cycle ˚  hxi D x; f .x/; : : : ; f B.p1/ .x/

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Xavier Buff, Guizhen Cui, and Lei Tan

y ! Tx C. y The cycle is is the eigenvalue of the derivative Dx .f Bp / W Tx C • superattracting if  D 0, • attracting if 0 < jj < 1, • repelling if jj > 1, • irrationally indifferent if jj D 1 and  is not root of unity, and • parabolic if  is a root of unity. The holomorphic index of f Bp at x is the residue  D Resx

d    B f Bp

where  is a local coordinate at x. It is remarkable that this residue does not depend on the choice of local coordinate . If  ¤ 1, then D

1 : 1

When  D e2ir=s is an s-th root of unity with r co-prime to s, there are • a unique integer m  1 called the parabolic multiplicity of f Bp at x, • a unique complex number ˇ 2 C called the résidu itératif of f Bp at x and • a (non unique) local coordinate  vanishing at x such that the expression of f Bp is 







ms C 1  ˇ  2ms C O. 2msC2 /: 2 Such a coordinate  is called a preferred coordinate for f Bp at x. The résidu itératif ˇ of f Bp at x is related to the holomorphic index  of f Bps at x by  7!  1 C  ms C

D

ms C 1 ˇ  2 s

(see for example Buff–Epstein, [10]). Let us now assume that x 2 U is a periodic point of f of period p and let hxi be the cycle containing x. The formal invariants of the cycle are by definition the formal invariant of f Bp at any point of the cycle (they do not depend on the point of the cycle).

4.2 Quadratic differentials with higher order poles We shall say that two quadratic differentials q1 and q2 which are defined and meromory represent the same divergence at z if q1  q2 phic in a neighborhood of a point z 2 C has at most a simple pole z. We shall denote by Dz the vector space of divergences Œqz at z.

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For .f; X; Y / as above, let C  X be a union of cycles of f contained in X. Denote by DC the direct sum M Dz : DC D z2C

In other words, a divergence at z is a polar part of degree 2 of meromorphic quadratic differentials at z. y C .Y /) the set of meromorphic quadratic y C .X/ (respectively Q We shall denote by Q y which are holomorphic outside X (respectively Y ) and have at differentials on C y C .X / and Q.Y /  Q y C .Y / and most simple poles outside C . Note that Q.X /  Q moreover, we have the canonical identifications y C .Y /=Q.Y / ' DC : y C .X/=Q.X/ ' Q Q In addition, the linear operator rf descends to the quotient space (we keep the notation rf for the induced map) and we have the following commutative diagram with exact columns and rows: Thus, the following diagram commutes: 0

0

0

 / Q.X/

rf

 / Q.Y /

K.f /

 /Q y C .X/

rf

 /Q y C .Y /

DC .f /

 / DC

rf

 / DC

 0

/ Q.Y /=rf Q.X /

 0

y C .X / ! Q y C .Y / and DC .f / is where K.f / is the kernel of the linear map rf W Q the kernel of the linear map rf W DC ! DC .

4.3 The Fatou–Shishikura inequality According to the Snake Lemma, there is a linear map Hf W DC .f / ! Q.Y /=rf Q.X / such that the following sequence is exact: Hf

0 ! K.f / ! DC .f / ! Q.Y /=rf Q.X /:

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Xavier Buff, Guizhen Cui, and Lei Tan

Adam Epstein then gave a complete description of DC .f /. And, by analyzing K.f /, he proved that Hf is injective on a certain subspace of DC .f / (the space DC[ .f / defined below). Proposition 4.1 (Epstein [19]). The space DC .f / is computed cycle by cycle: M DC .f / D Dhxi .f /: hxiC

y be a periodic point of f of period p. Let x 2 C (1) The projection Dhxi ! Dx restricts to an isomorphism Dhxi .f / ! Dx .f Bp / whose inverse is Gx W Dx .f

Bp



/ ! Dhxi .f /;

Œqx 7!

p1 M

fBk q

 f Bk .x/

:

kD0

(2) If hxi is superattracting, then Dx .f Bp / D 0. Bp (3) If hxi is attracting, repelling or irrationally h i indifferent, then Dx .f / is the one-

dimensional vector space spanned by at x.

d 2 2 x

for any local coordinate  vanishing

(4) If hxi is parabolic with multiplier e2ir=s , parabolic multiplicity m and résidu itératif ˇ, then Dx .f Bp / is the direct sum of the m-dimensional vector space Dxm .f Bp / spanned by 2

d d 2 d 2 ; : : : ; skC2 ; : : : ; .m1/sC2 2 x   x x together with the one-dimensional vector space spanned by

d 2 . msC1  ˇ 2msC1 /2 x for any preferred coordinate  for f Bp at x. Let us now introduce the subspace DC[ .f /  DC .f / defined by M [ DC[ .f / D Dhxi .f / hxiC

where [ • Dhxi .f / D f0g if hxi is superattracting or repelling, [ .f / D Dhxi .f / if hxi is attracting or rationally indifferent or parabolic with • Dhxi 0; S has an ideal triangulation. Any ideal triangulation of S has 2m ideal triangles and 3m edges. The edges of an ideal triangulation  of S are enumerated as f1 ; : : : ; 3m g. Let ƒ.S / denote the set of isotopy classes of ideal triangulations of S. The set ƒ.S / admits a natural action of the symmetric group on the set f1; 2; : : : ; 3mg, S3m ,

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acting by permuting the indices of the edges of . Namely 0 D ˛./ for ˛ 2 S3m if i D 0˛.i / . Another important transformation of ƒ.S/ is provided by the i -th diagonal exchange map i W ƒ.S/ ! ƒ.S/ defined as follows. Suppose that the i -th edge i of an ideal triangulation  2 ƒ.S/ is adjacent to two triangles. Then i ./ is obtained from  by replacing the edge i by the other diagonal 0i of the square formed by the two triangles, as illustrated in Figure 1.

0i

i !

i

Figure 1

Lemma 2.1. The reindexings and diagonal exchanges satisfy the following relations: (1) .˛ˇ/./ D ˛.ˇ.// for every ˛, ˇ 2 S3m ; (2) .i /2 D Id; (3) ˛ B i D ˛.i/ B ˛ for every ˛ 2 S3m ; (4) If i and j do not belong to the same triangle of  2 ƒ.S/, then i B j ./ D j B i ./; (5) If three triangles of an ideal triangulation  2 ƒ.S/ form a pentagon with diagonals i , j as in Figure 2, then i B j B i B j B i ./ D ˛i$j ./; where ˛i$j 2 S3m denotes the transposition exchanging i and j .

i

j

Figure 2

To construct the quantum Teichmüller space, we need the following two results of R. C. Penner [24] (see also J. L. Harer [14]).

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Theorem 2.2. Given two ideal triangulations ; 0 2 ƒ.S/, there exists a finite sequence of ideal triangulations  D .0/ , .1/ , …, .n/ D 0 such that each .kC1/ is obtained from .k/ by a diagonal exchange or by a reindexing of its edges. Theorem 2.3. Given two ideal triangulations ; 0 2 ƒ.S/ and given two sequences 0  D .0/ , .1/ , …, .n/ D 0 and  D N .0/ , N .1/ , …, N .n/ N D  of diagonal exchanges and reindexings connecting them as in Theorem 2.2, these two sequences can be related to each other by successive application of the following moves and of their inverses. These moves correspond to the relations in Lemma 2.1. (1) Replace …, .k/ , ˇ..k/ /, ˛.ˇ..k/ //, … by …, .k/ , .˛ˇ/..k/ /, …where ˛, ˇ 2 S3m . (2) Replace …, .k/ , i ..k/ /, .k/ , … by …, .k/ , … . (3) Replace …, .k/ , i ..k/ /, ˛ B i ..k/ /, … by …, .k/ , ˛..k/ /, ˛.i/ B ˛..k/ /, … where ˛ 2 S3m . (4) Replace …, .k/ , i ..k/ /, j B i ..k/ /, … by …, .k/ , j ..k/ /, i B j ..k/ /, … where i , j are two edges which do not belong to a same triangle of .k/ . (5) Replace …, .k/ , i ..k/ /, j B i ..k/ /, i B j B i ..k/ /, j B i B j B i ..k/ /, i B j B i B j B i ..k/ /, … by …, .k/ , ˛i$j ..k/ /, … where i , j are two diagonals of a pentagon of .k/ as in Figure 2.

2.2 Shear coordinates for the Teichmüller space If the Euler characteristic of S is negative, i.e., m WD 2g  2 C p > 0; S admits complete hyperbolic metrics. The Teichmüller space T .S / of S consists of all isotopy classes of complete hyperbolic metrics on S . W. Thurston [27] associated to each ideal triangulation a global coordinate system which is called shear coordinates for the Teichmüller space T .S/ (see also [4], [11]). An end of a surface S with a complete hyperbolic metric d 2 T .S / can be of two types: a cusp with finite area bounded on one side by a horocycle; and a funnel with infinite area bounded on one side by a simple closed geodesic. The convex core Conv.S; d / of .S; d / is the smallest non-empty closed convex subset of .S; d /, and is bounded in S by a family of disjoint simple closed geodesics. Each cusp end of .S; d / is also a cusp end of Conv.S; d /, while each funnel end of S faces a boundary component of Conv.S; d /. The enhanced Teichmüller space Tz .S/ consists of all isotopy classes of complete hyperbolic metrics d 2 T .S/ enhanced with an orientation of each boundary component of Conv.S; d /.

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Under an enhanced hyperbolic metric d 2 Tz .S /, each edge i of an ideal triangulation  is realized by a unique d -geodesic gi such that each end of gi either converges towards a cusp end of S, or spirals around a boundary component of Conv.S; d / in the orientation specified by d 2 Tz .S/. The enhanced hyperbolic metric d 2 Tz .S / associates to the edge i of  a positive number xi defined as follows. The geodesic gi separates two triangle components Ti1 and Ti2 of Conv.S; d /  fgi g. The hyperbolic plane H2 is the universal covering of S endowed with the metric d . Lift gi , Ti1 and Ti2 to a geodesic gQ i and two triangles Tzi1 z in H2 . See Figure 3. and Tzi2 in H2 so that the union gQ i [ Tzi1 [ Tzi2 forms a square Q z in such a In the upper half-space model for H2 , let z , zC , zr , zl be the vertices of Q H2

z Q Tzi1

gQ i Tzi2

zl

z

zr

zC

Figure 3

way that gQ i goes from z to zC and, for this orientation of gQ i , zr , zl are respectively z given by the orientation of S. to the right and to the left of gQ i for the orientation of Q Then, .zr  z /.zl  zC / : xi WD  cross-ratio .zr ; zl ; z ; zC / D  .zr  zC /.zl  z / The real numbers fxi g are the exponential shear coordinates of the enhanced hyperbolic metric d 2 Tz .S/. The shear coordinates are ln xi . It turns out that fxi g defines a homeomorphism  W Tz .S / ! R3m C . Therefore the exponential shear coordinates associates a parametrization  W Tz .S / ! R3m C to each ideal triangulation  2 ƒ.S/ (endowed with an indexing of its edges). We now investigate the coordinate changes 0 B 1 associated to two ideal triangulations. If 0 D ˛./ is obtained by reindexing the edges of  by ˛ 2 S3m , then 0 B 1 is the permutation of the coordinates by ˛. For a diagonal exchange, we have the following result. Proposition 2.4 (Liu [22]). Suppose that the ideal triangulations , 0 are obtained from each other by a diagonal exchange, namely that 0 D i ./. Label the edges of  involved in this diagonal exchange as i , j , k , l , m as in Figure 4. If 0 .x1 ; x2 ; : : : ; x3m / and .x10 ; x20 ; : : : ; x3m / are the exponential shear coordinates asso0 ciated to  and  of the same enhanced hyperbolic metric, then xh0 D xh for every h 62 fi; j; k; l; mg, xi0 D xi1 and the following cases occur:

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Ren Guo j0

j

m

k

i

0i

0m

i !

0k

0l

l Figure 4

Case 1. If the edges j , k , l , m are distinct, then xj0 D .1 C xi /xj ;

xk0 D .1 C xi1 /1 xk ;

xl0 D .1 C xi /xl ;

0 xm D .1 C xi1 /1 xm :

Case 2. If j is identified with k , and l is distinct from m , then xj0 D xi xj ;

xl0 D .1 C xi /xl ;

0 xm D .1 C xi1 /1 xm :

Case 3 (the inverse of Case 2). If j is identified with m , and k is distinct from l , then xj0 D xi xj ; xk0 D .1 C xi1 /1 xk ; xl0 D .1 C xi /xl : Case 4. If j is identified with l , and k is distinct from m , then xj0 D .1 C xi /2 xj ;

xk0 D .1 C xi1 /1 xk ;

0 xm D .1 C xi1 /1 xm :

Case 5 (the inverse of Case 4). If k is identified with m , and j is distinct from l , then xj0 D .1 C xi /xj ;

xk0 D .1 C xi1 /2 xk ;

xl0 D .1 C xi /xl :

Case 6. If j is identified with k , and l is identified with m (in which case S is a 3-times punctured sphere), then xj0 D xi xj ;

xl0 D xi xl :

Case 7 (the inverse of Case 6). If j is identified with m , and k is identified with l (in which case S is a 3-times punctured sphere), then xj0 D xi xj ;

xk0 D xi xk :

Case 8. If j is identified with l , and k is identified with m (in which case S is a once-punctured torus), then xj0 D .1 C xi /2 xj ;

xk0 D .1 C xi1 /2 xk :

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2.3 The Chekhov–Fock algebra Fix an ideal triangulation  2 ƒ.S/. The complement S  has 6m spikes converging towards the punctures, and each spike is delimited by one i on one side and one j  on the other side, with possibly i D j . For i , j 2 f1; : : : ; 3mg, let aij denote the number of spikes of S   which are delimited on the left by i and on the right by j , and set  ij D aij  aji :

Note that ij 2 f2; 1; 0; 1; 2g, and that ji D ij . Let q be an arbitrary complex number. The Chekhov–Fock algebra associated to the ideal triangulation  is the algebra Tq defined by generators X1 , X11 , X2 , X21 , 1 …, X3m , X3m , with each pair Xi˙1 associated to an edge i of , and by the relations 

Xi Xj D q 2ij Xj Xi : This algebra has a well-defined fraction division algebra Tyq which consists of all formal fractions PQ1 with P , Q 2 Tq and Q ¤ 0, and two such fractions P1 Q11 and P2 Q21 are identified if there exists S1 , S2 2 Tq  f0g such that P1 S1 D P2 S2 and Q1 S1 D Q2 S2 . The algebras Tq and Tyq strongly depend on the ideal triangulation . As one moves from one ideal triangulation  to another 0 , Chekhov and Fock [11], [9], [10] (see also [22]) introduce coordinate change isomorphisms ˆq0 W Tyq0 ! Tyq . We denote by X10 , X20 , …, Xn0 the generators of Tyq0 associated to the edges 01 , 02 , …, 0n of 0 , and by X1 , X2 , …, Xn the generators of Tyq associated to the edges 1 , 2 , …, n of . Definition 2.5. Suppose that the ideal triangulations , 0 2 ƒ.S/ are obtained from each other by an edge reindexing, namely that 0i D ˛.i/ for some permutation ˛ 2 S3m . Then we define a map ˛O from the set of generators of the algebra Tyq0 to Tyq by ˛.X O i0 / D X˛.i/ ;

for any i D 1; : : : ; 3m:

Suppose that the ideal triangulations , 0 are obtained from each other by a diagonal exchange, namely that 0 D i ./. Label the edges of  involved in this diagonal exchange as i , j , k , l , m as in Figure 4. Then we define a map y i on the set of generators of the algebra Ty q0 to Ty q such that X 0 7! Xh for every  h   h 62 fi; j; k; l; mg, Xi0 7! Xi1 and the following cases occur: Case 1. If the edges j , k , l , m are distinct, then Xj0 7! .1 C qXi /Xj ;

Xk0 7! .1 C qXi1 /1 Xk ;

Xl0 7! .1 C qXi /Xl ;

0 Xm 7! .1 C qXi1 /1 Xm :

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Case 2. If j is identified with k , and l is distinct from m , then Xj0 7! Xi Xj ;

Xl0 7! .1 C qXi /Xl ;

0 Xm 7! .1 C qXi1 /1 Xm :

Case 3 (the inverse of Case 2). If j is identified with m , and k is distinct from l , then Xj0 7! Xi Xj ;

Xk0 7! .1 C qXi1 /1 Xk ;

Xl0 7! .1 C qXi /Xl :

Case 4. If j is identified with l , and k is distinct from m , then Xj0 7! .1 C qXi /.1 C q 3 Xi /Xj ; Xk0 7! .1 C qXi1 /1 Xk ;

0 Xm 7! .1 C qXi1 /1 Xm :

Case 5 (the inverse of Case 4). If k is identified with m , and j is distinct from l , then Xj0 7! .1 C qXi /Xj ;

Xl0 7! .1 C qXi /Xl ;

Xk0 7! .1 C qXi1 /1 .1 C q 3 Xi1 /1 Xk : Case 6. If j is identified with k , and l is identified with m (in which case S is a 3-times punctured sphere), then Xj0 7! Xi Xj ;

Xl0 7! Xi Xl :

Case 7 (the inverse of Case 6). If j is identified with m , and k is identified with l (in which case S is a 3-times punctured sphere), then Xj0 7! Xi Xj ;

Xk0 7! Xi Xk :

Case 8. If j is identified with l , and k is identified with m (in which case S is a once-punctured torus), then Xj0 7! .1 C qXi /.1 C q 3 Xi /Xj ; Xk0 7! .1 C qXi1 /1 .1 C q 3 Xi1 /1 Xk : y i can be extended to the whole algebra Ty q0 as It turns out that the maps ˛O and   algebra homomorphisms from Tyq0 to Tyq . y i is that they are reduced to the correThe motivation of the definition of ˛O and  sponding shear coordinate changes (Proposition 2.4) when q D 1. Proposition 2.6 (Liu [22]). If an ideal triangulation 0 is obtained from another one  by an operation ; where  D ˛ for some ˛ 2 S3m ; or  D i for some i , then O W Tyq0 ! Tyq as in Definition 2.5 is an isomorphism between the two algebras. y i satisfy the following relations which Proposition 2.7 (Liu [22]). The map ˛O and  correspond to the relations in Lemma 2.1:

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c D ˛O B ˇO for every ˛, ˇ 2 S3m ; (1) ˛ˇ yi B  y i D Id; (2)  yi D  y ˛.i/ B ˛O for every ˛ 2 S3m ; (3) ˛O B  yi B  yj D (4) If i and j do not belong to the same triangle of  2 ƒ.S/, then  y y j B i ; (5) If three triangles of an ideal triangulation  2 ƒ.S/ form a pentagon with diagonals i , j as in Figure 2, then yj B  yi B  yj B  y i D ˛O i$j : yi B  

(2.1)

2.4 The quantum Teichmüller space Theorem 2.8 (Liu [22]). There is a family of algebra isomorphisms ˆq0 W Tyq0 ! Tyq defined as , 0 2 ƒ.S/ ranges over all pairs of ideal triangulations, such that: (1) ˆq00 D ˆq0 B ˆq0 00 for every , 0 , 00 2 ƒ.S/; (2) ˆq0 is the isomorphism defined in Definition 2.5 when 0 is obtained from  by a reindexing or a diagonal exchange. (3) ˆq0 depends only on  and 0 . The quantum (enhanced) Teichmüller space of S can now be defined as the algebra  G  Tyq =  TySq D 2ƒ.S/

where the relation  is defined by the property that, for X 2 Tyq and X 0 2 Tyq0 , X  X 0 () X D ˆq;0 .X 0 /: The quantum Teichmüller space TySq is a noncommutative deformation of the algebra of rational functions on the enhanced Teichmüller space Tz .S /.

3 The quantum Teichmüller space: properties In this section we survey some interesting properties and applications of the quantum Teichmüller space. The uniqueness of the construction of the quantum Teichmüller space is established by H. Bai [1]. In [3], [5], [6], [23], it is shown that the quantum Teichmüller space TySq has a rich representation theory which also produces an invariant of hyperbolic 3-manifolds.

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We would like to mention the following related important works without providing more details. H. Bai [2] shows that Kashaev’s 6j -symbols [16], [17] are intertwining operators of local representations of quantum Teichmüller spaces introduced in [3]. Note that appearance of Kashaev’s 6j -symbols in quantum Teichmüller theory at roots of unity is already explicit in [18] (see the operator Th;x;y in Proposition 10). C. Hiatt [15] proves that for the torus with one hole and p  1 punctures and the sphere with four holes there is a family of quantum trace functions in the quantum Teichmüller space, analogous to the non-quantum trace functions in Teichmüller space, satisfying the properties proposed by Chekhov and Fock in [10]. For a punctured surface S , a point of its Teichmüller space T .S / determines an irreducible representation of its quantization TSq . J. Roger [25] analyzes the behavior of these representations as one goes to infinity in TS . He shows that an irreducible representation of TSq limits to a direct sum of representations of TSq , where S is obtained from S by pinching a multicurve  to a set of nodes. The result is analogous to the factorization rule found in conformal field theory. The skein algebra and the quantum Teichmüller space are considered as two different quantizations of the character variety consisting of all representations of surface groups in PSL2 .C/. F. Bonahon and H. Wong [7], [8] construct a homomorphism from the skein algebra to the quantum Teichmüller space which, when restricted to the classical case, corresponds to the equivalence between these two algebras through trace functions.

3.1 Uniqueness The original definition of the quantum Teichmüller space was motivated by geometry. However, H. Bai [1] shows that it is intrinsically tied to the combinatorics of the set ƒ.S /. Indeed, H. Bai proves that the coordinate change isomorphisms considered in Definition 2.5 are the only ones which satisfy a certain number of natural conditions. The discrepancy span D.; 0 / of two ideal triangulations , 0 is the closure of the union of those connected components of S   which are not isotopic to a component of S  0 . The coordinate change isomorphisms ˆq0 are said to satisfy the Locality Condition if the following holds. Let  and 0 be two ideal triangulations indexed in such a way that i  D.; 0 / when i  k, and 0i D i when i > k. Then (1) ˆq0 .Xi0 / D Xi for every i > k; (2) ˆq0 .Xi0 / D fi .X1 ; X2 ;    ; Xk / for every i  k, where fi is a multi-variable rational function depending only on the combinatorics of  and 0 in D.; 0 / in the following sense: For any two pairs of ideal triangulations .; 0 /, .00 ; 000 / for which there exists a diffeomorphism W D.; 0 / ! D.00 ; 000 / sending i

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to j00 and j0 to j000 for every 1  j  k, then ˆq0 .Xi0 / D fi .X1 ; X2 ;    ; Xk / and ˆq00 000 .Xi000 / D fi .X100 ; X200 ;    ; Xk00 / for the same rational function fi . Proposition 3.1 (Bai [1]). The algebra isomorphisms ˆq0 W Tyq0 ! Tyq in Theorem 2.8 satisfies the Locality Condition. Theorem 3.2 (Bai [1]). Assume that the surface S satisfies .S / < 2. Then the family of coordinate change isomorphisms ˆq0 in Theorem 2.8 is unique up to a uniform rescaling and/or inversion of the Xi . Namely, if q q 0 0 0 q ‰ 0 W C.X1 ; X2 ; : : : ; Xn /0 ! C.X1 ; X2 ; : : : ; Xn /

is another family of isomorphisms satisfying the conditions of Theorem 2.8 and the Locality Condition, then there exists a non-zero constant 2 C.q/ and a sign " D ˙1 q q 1 0 such that ‰ 0 D ‚ B ˆ0 B ‚0 for any pair of ideal triangulations ,  , where q q ‚ W C.X1 ; X2 ; : : : ; Xn / ! C.X1 ; X2 ; : : : ; Xn / is the isomorphism defined by ‚ .Xi / D Xi" for every i. Theorem 3.2 is false when S is the once-punctured torus or the 3-times punctured sphere. The uniqueness property for the twice-punctured torus and the 4-times punctured sphere has not been established.

3.2 Representations In this subsection, our exposition follows [6] closely. A standard method to move from abstract algebraic constructions to more concrete applications is to consider finite-dimensional representations. In the case of algebras, this means algebra homomorphisms valued in the algebra End.V / of endomorphisms of a finite-dimensional vector space V over C. Elementary considerations show that these can exist only when q is a root of unity. Theorem 3.3 (Bonahon–Liu [6]). Suppose that q 2 is a primitive N -th root of unity. For any ideal triangulation  of a surface S, every irreducible finite-dimensional representation of the Chekhov–Fock algebra Tq has dimension N 3gCp3 if N is odd, and N 3gCp3 =2g if N is even. Up to isomorphism, such a representation is classified by (1) a non-zero complex number xi 2 C  associated to each edge of ; (2) a choice of an N -th root for each of p explicit monomials in the numbers xi ;

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(3) when N is even, a choice of a square root for each of 2g explicit monomials in the numbers xi . Conversely, any such data can be realized by an irreducible finite-dimensional representation of Tq . Theorem 3.3 shows that the Chekhov–Fock algebra has a rich representation theory. Unfortunately, for dimension reasons, its fraction algebra Tyq and, consequently, the quantum Teichmüller space TySq cannot have any finite-dimensional representation. This leads us to introduce the polynomial core TSq of the quantum Teichmüller space TySq , defined as the family fTq g2ƒ.S/ of all Chekhov–Fock algebras Tq , considered as subalgebras of TySq , as  ranges over the set ƒ.S/ of all isotopy classes of ideal triangulations of the surface S. Theorem 3.3 says that, up to a finite number of choices, an irreducible representation of Tq is classified by certain numbers xi 2 C  associated to the edges of the ideal triangulation  of S. There is a classical geometric object which is also associated to  with the same edge weights xi . Namely, we can consider in the hyperbolic 3-space H3 the pleated surface that has pleating locus , that has shear parameter along the i -th edge of  equal to the real part of ln xi , and that has bending angle along this edge equal to the imaginary part of ln xi . In turn, this pleated surface has a monodromy representation, namely a group homomorphism from the fundamental group 1 .S / to the group IsomC .H3 / Š PSL2 .C/ of orientation-preserving isometries of H3 . This construction associates to a representation of the Chekhov–Fock algebra Tq a group homomorphism r W 1 .S/ ! PSL2 .C/, well-defined up to conjugation by an element of PSL2 .C/. Theorem 3.4 (Bonahon–Liu [6]). Let q be a primitive N -th root of .1/N C1 , for instance q D e2i=N . If D f  W Tq ! End.V /g2ƒ.S/ is a finite-dimensional irreducible representation of the polynomial core TSq of the quantum Teichmüller space TySq , the representations  induce the same monodromy homomorphism r W 1 .S / ! PSL2 .C/. Theorem 3.4 is essentially equivalent to the property that, for the choice of q indicated, the pleated surfaces respectively associated to the representations  W Tq ! End.V / and  B ˆq0 W Tq0 ! End.V / have (different pleating loci but) the same monodromy representation r W 1 .S/ ! PSL2 .C/. The homomorphism r is the hyperbolic shadow of the representation . Not every homomorphism r W 1 .S/ ! PSL2 .C/ is the hyperbolic shadow of a representation of the polynomial core, but many of them are: Theorem 3.5 (Bonahon–Liu [6]). An injective homomorphism r W 1 .S / ! PSL2 .C/ is the hyperbolic shadow of a finite number of irreducible finite-dimensional representations of the polynomial core TSq , up to isomorphism. More precisely, this number

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of representations is equal to 2l N p if N is odd, and 22gCl N p if N is even, where l is the number of ends of S whose image under r is loxodromic. Let ' be a diffeomorphism of the surface S . Suppose in addition that ' is homotopically aperiodic (also called homotopically pseudo-Anosov), so that its (3-dimensional) mapping torus M' admits a complete hyperbolic metric. The hyperbolic metric of M' gives an injective homomorphism r' W 1 .S / ! PSL2 .C/ such that r' B '  is conjugate to r' , where '  is the isomorphism of 1 .S / induced by '. The diffeomorphism ' also acts on the quantum Teichmüller space and on its polynomial core TSq . In particular, it acts on the set of representations of TSq and, because r' B '  is conjugate to r' , it sends a representation with hyperbolic shadow r' to another representation with shadow r' . Actually, when N is odd, there is a preferred representation ' of TSq which is fixed by the action of ', up to isomorphism. This statement means that, for every ideal triangulation , we have a representation

 W Tq ! End.V / of dimension N 3gCp3 and an isomorphism Lq' of V such that

'./ B ˆ'./ .X/ D Lq'   .X /  .Lq' /1 in End.V / for every X 2 Tq , for a suitable interpretation of the left hand side of the equation. Theorem 3.6 (Bonahon–Liu [6]). Let N be odd. Up to conjugation and up to multiplication by a constant, the isomorphism Lq' depends only on the homotopically aperiodic diffeomorphism ' W S ! S and on the primitive N -th root q of 1. Explicit computations of these invariants for the once-punctured torus or the 4times punctured sphere are provided in X. Liu [23]. H. Bai, F. Bonahon and X. Liu [3] investigate another type of representations of the quantum Teichmüller space, called local representations, which are somewhat simpler to analyze and more closely connected to the combinatorics of ideal triangulations.

4 Kashaev algebra In this section, we establish the theory of Kashaev algebra which is parallel to the theory of the quantum Teichmüller space. The exposition follows [13] closely.

4.1 Decorated ideal triangulations Let S be an oriented surface of genus g with p  1 punctures and negative Euler characteristic, i.e., m D 2g  2 C p > 0: A decorated ideal triangulation  of S in the sense of Kashaev [18] is an ideal triangulation where the ideal triangles are enumerated as f1 ; 2 ; : : : ; 2m g and there

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is a mark (a star symbol) at a corner of each ideal triangle. Denote by 4.S / the set of isotopy classes of decorated ideal triangulations of the surface S. The set 4.S/ admits a natural action of the symmetric group S2m on the set f1; 2; : : : ; 2mg, acting by permuting the indexes of the ideal triangles of . Namely 0 .  0 D ˛. / for ˛ 2 S2m if i D ˛.i/ Another important transformation of 4.S / is provided by the diagonal exchange 'ij W 4.S / ! 4.S/ defined as follows. Suppose that two ideal triangles i , j share an edge e such that the marked corners are opposite to the edge e: Then 'ij . / is obtained by rotating the interior of the union i [ j 90B in the clockwise order, as illustrated in Figure 5 (2). The last transformation of 4.S/ is the mark rotation i W 4.S / ! 4.S /. i . / is obtained by relocating the mark of the ideal triangle i from one corner to the next corner in the counterclockwise order, as illustrated in Figure 5 (1). 

i !

i



i0

.1/  

i

j



'ij !

i0 j0

.2/



Figure 5

Lemma 4.1. The reindexings, diagonal exchanges and mark rotations satisfy the following relations: (1) .˛ˇ/. / D ˛.ˇ.// for every ˛, ˇ 2 S2m ; (2) 'ij B 'ij D ˛i$j ; where ˛i$j denotes the transposition exchanging i and j ; (3) ˛ B 'ij D '˛.i/˛.j / B ˛ for every ˛ 2 S2m ; (4) 'ij B 'kl ./ D 'kl B 'ij ./, for fi; j g ¤ fk; lg; (5) If three triangles i , j , k of an ideal triangulation  2 4.S / form a pentagon and their marked corners are located as in Figure 6, then the Pentagon Relation holds: !j k B !ik B !ij . / D !ij B !j k . /; where ! D  B ' B  ;

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(6) i B i B i D Id; (7) i B j D j B i ; (8) ˛ B i D ˛.i/ B ˛ for every ˛ 2 S2m .

  i

k j 

 Figure 6

Lemma 4.1 is essentially contained in Kashaev [19] where !ij is used as the diagonal exchange. The following two results about decorated ideal triangulations can be easily proved using Penner’s result about ideal triangulations [24]. Theorem 4.2. Given two decorated ideal triangulations ;  0 2 4.S /, there exists a finite sequence of decorated ideal triangulations  D .0/ , .1/ , …, .n/ D  0 such that each .kC1/ is obtained from .k/ by a diagonal exchange or by a mark rotation or by a reindexing of its ideal triangles. Theorem 4.3. Given two decorated ideal triangulations ;  0 2 4.S / and given two 0 sequences  D .0/ , .1/ , …, .n/ D  0 and  D N.0/ , N.1/ , …, N.n/ N D  of diagonal exchanges, mark rotations and reindexings connecting them as in Theorem 4.2, these two sequences can be related to each other by successive applications of the following moves and of their inverses. These moves correspond to the relations in Lemma 4.1. (1) Replace …, .k/ , ˇ..k/ /, ˛ B ˇ..k/ /, … by …, .k/ , .˛ˇ/..k/ /, …where ˛, ˇ 2 Sn . (2) Replace …, .k/ , 'ij ..k/ /, 'ij B 'ij ..k/ / … by …, .k/ , ˛i$j ..k/ /, …. (3) Replace …, .k/ , 'ij ..k/ /, ˛ B 'ij ..k/ /, … by …, .k/ , ˛..k/ /, '˛.i/˛.j / B ˛..k/ /, …where ˛ 2 Sn . (4) Replace …, .k/ , 'kl ..k/ /, 'ij B 'kl ..k/ /, … by …, .k/ , 'ij ..k/ /, 'kl B 'ij ..k/ /, …where fi; j g ¤ fk; lg. (5) Replace …, .k/ , !ij ..k/ /, !ik B !ij ..k/ /, !j k B !ik B !ij ..k/ /, …, by …, .k/ , !j k ..k/ /, !ij B !j k ..k/ /, …where ! D  B ' B  .

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(6) Replace …, .k/ , i ..k/ /, i B i ..k/ /, .k/ … by …, .k/ , …. (7) Replace …, .k/ , i ..k/ /, j B i ..k/ /, … by …, .k/ , j ..k/ /, i B j ..k/ /, …. (8) Replace …, .k/ , i ..k/ /, ˛ B i ..k/ /, … by …, .k/ , ˛..k/ /, ˛.i/ B ˛..k/ /, ….

4.2 Kashaev coordinates For a decorated ideal triangulation  of a punctured surface S, Kashaev [18] associated to each ideal triangle i two numbers ln yi , ln zi . A Kashaev coordinate is a vector .ln y1 ; ln z1 ; : : : ; ln y2m ; ln z2m / 2 R4m : Denote by .y1 ; z1 ; : : : ; y2m ; z2m / the exponential Kashaev coordinate for the deco0 0 ; z2m / the exponential Kashaev rated ideal triangulation  . Denote by .y10 ; z10 ; : : : ; y2m 0 coordinate for the decorated ideal triangulation  . Kashaev [18] introduces the change of coordinates as follows. Definition 4.4 (Kashaev [18]). Suppose that a decorated ideal triangulation  0 is obtained from another one  by reindexing the ideal triangles, i.e.,  0 D ˛. / for some ˛ 2 S2m ; then we define .yi0 ; zi0 / D .y˛.i/ ; z˛.i/ / for any i D 1; : : : ; 2m: Suppose that a decorated ideal triangulation  0 is obtained from another one  by a mark rotation, i.e.,  0 D i ./ for some i; then we define .yj0 ; zj0 / D .yj ; zj / for any j ¤ i while   zi 1 ; : .yi0 ; zi0 / D yi yi Suppose a decorated ideal triangulation  0 is obtained from another one  by a diagonal exchange, i.e.,  0 D 'ij ./ for some i; j; then we define .yk0 ; zk0 / D .yk ; zk / for any k … fi; j g while .yi0 ; zi0 ; yj0 ; zj0 /





yi zi yj zj D ; ; ; : yi yj C zi zj yi yj C zi zj yi yj C zi zj yi yj C zi zj

Kashaev [18] considered !ij instead of 'ij . There is a natural relationship between Kashaev coordinates and Penner coordinates for the decorated Teichmüller space which is established in [18]. For an exposition, see also Teschner [26].

4.3 Generalized Kashaev algebra: triangulation-dependent For a decorated ideal triangulation  of a punctured surface S, Kashaev [18] introduced ˙ ˙ ; Z2m ; with Yi˙ , Zi˙ an algebra K q on C generated by Y1˙ ; Z1˙ ; Y2˙ ; Z2˙ ; : : : ; Y2m

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associated to an ideal triangle i ; and by the relations: Yi Yj D Yj Yi ; Zi Zj D Zj Zi ; Yi Zj D Zj Yi if i ¤ j; Zi Yi D q 2 Yi Zi : Kashaev’s original definition is Yi Zi D q 2 Zi Yi : We adopt our convention to make it compatible with the quantum Teichmüller space [22]. Kashaev’s parameter q corresponds to our q 1 . y q is the fraction division algebra of K q . The algebra K y q respectively coincide with the LauIn particular, when q D 1; K q and K ˙ ˙ ˙ ˙ rent polynomial algebra CŒY1 ; Z1 ; : : : ; Y2m ; Z2m and the rational fraction algebra q y C.Y1 ; Z1 ; : : : ; Y2m ; Z2m /. The general K and K q can be considered as deformay 1. tions of K 1 and K y q depends on the decorated ideal triangulation  . We introduce The algebra K algebra isomorphisms in the following. Definition 4.5. Let a, b be two arbitrary nonzero complex numbers. Suppose that a decorated ideal triangulation  0 is obtained from another one  by reindexing the ideal triangles, i.e.,  0 D ˛. / for some ˛ 2 S2m ; then we define a y q0 to K y q by map ˛O on the set of generators of K ˛.Y O i0 / D Y˛.i/ ; for any i D 1; : : : ; 2m; ˛.Z O i0 / D Z˛.i/ ; for any i D 1; : : : ; 2m: Suppose that a decorated ideal triangulation  0 is obtained from another one  by a mark rotation, i.e.,  0 D i ./ for some i; then we define a map Oi on the set of y q0 to K y q by generators of K

Oi .Yj0 / D Yj

if j ¤ i;

Oi .Zj0 / D Zj

if j ¤ i;

Oi .Yi0 /

Oi .Zi0 /

D D

aYi1 Zi ; Yi1 :

Suppose a decorated ideal triangulation  0 is obtained from another one  by a diagonal exchange, i.e.,  0 D 'ij ./ for some i; j; then we define a map 'Oij on the set y q0 to K y q by of generators of K 'Oij .Yi0 / D .bYi Yj C Zi Zj /1 Zj ; 'Oij .Zi0 / D b.bYi Yj C Zi Zj /1 Yi ; 'Oij .Yj0 / D .bYi Yj C Zi Zj /1 Zi ; 'Oij .Zj0 / D b.bYi Yj C Zi Zj /1 Yj :

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y q0 It turns out that the maps ˛; O Oi and 'Oij can be extended to the whole algebra K y q0 to K y q : as algebra homomorphisms between from K Kashaev [18] considered a special case of these maps when a D q 1 ; b D q: From the definition, when q D 1; we get the coordinate change formula in Definition 4.4. Proposition 4.6 (Guo–Liu [13]). If a decorated ideal triangulation  0 is obtained from another one  by an operation ; where  D ˛ for some ˛ 2 S2m ; or  D i y q0 ! K y q as in Definition 4.5 is an for some i , or  D 'ij for some i, j , then O W K isomorphism between the two algebras. Proposition 4.7 (Guo–Liu [13]). The maps ˛; O Oi and 'Oij satisfy the following relations which correspond to the relations in Lemma 4.1: c D ˛O B ˇO for every ˛, ˇ 2 S2m ; (1) ˛ˇ (2) 'Oij B 'Oij D ˛O i$j ; (3) ˛O B 'Oij D 'O˛.i/˛.j / B ˛O for every ˛ 2 S2m ; (4) 'Oij B 'Okl D 'Okl B 'Oij for fi; j g ¤ fk; lg; (5) If three triangles i ; j ; k of an ideal triangulation  2 4.S / form a pentagon and their marked corners are located as in Figure 7, then the Pentagon Relation holds: !Oj k B !O ik B !O ij D !O ij B !Oj k ; where !O  D O B 'O B O ; (6) Oi B Oi B Oi D Id; (7) Oi B Oj D Oj B Oi ; (8) ˛O B Oi D O˛.i/ B ˛O for every ˛ 2 S2m .

  i

k j





Figure 7 (same as Figure 6)

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4.4 Generalized Kashaev algebra: triangulation-independent Theorem 4.8 (Guo–Liu [13]). For two arbitrary complex numbers a, b, there is a family of algebra isomorphisms y q0 ! K yq ‰ q 0 .a; b/ W K defined as  ,  0 2 4.S/ ranges over all pairs of decorated ideal triangulations, such that the following holds: q q q 0 00 (1) ‰ 2 4.S /; 00 .a; b/ D ‰ 0 .a; b/ B ‰ 0 00 .a; b/ for every  ,  ,  q 0 (2) ‰ 0 .a; b/ is the isomorphism of Definition 4.5 when  is obtained from  by a reindexing or a mark rotation or a diagonal exchange. q 0 (3) ‰ 0 .a; b/ depends only on  and  .

y q .a; b/ associated to a surface S is defined as The generalized Kashaev algebra K S the algebra  G  y q .a; b/ D y q .a; b/ =  K K S 24.S /

y q .a; b/ and X 0 2 where the relation  is defined by the property that, for X 2 K y q0 .a; b/, K q 0 X  X 0 () X D ‰ ; 0 .a; b/.X /:

5 Kashaev coordinates and shear coordinates To understand the quantization using shear coordinates and the quantization using Kashaev coordinates, we first need to understand the relationship between these two coordinates.

5.1 Decorated ideal triangulations Given a decorated ideal triangulation ; by forgetting the mark at each corner, we obtain an ideal triangulation : We call  the underlying ideal triangulation of . Let 1 ; 2 ; : : : ; 3m be the components of the ideal triangulation . Denote by 1 ; : : : ; 2m the ideal triangles. For the ideal triangulation , we may consider its dual graph. Each ideal triangle  corresponds to a vertex  of the dual graph. Denote by 1 ; 2 ; : : : ; 3m the dual edges. If an edge i bounds one side of the ideal triangles  and one side of  , then the dual edge i connects the vertices  and  . In a decorated ideal triangulation , each ideal triangle  (embedded or not) has three sides which correspond to the three half-edges incident to the vertex  of the

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dual graph. The three sides are numbered 0, 1, 2 in counterclockwise order such that the 0-side is opposite to the marked corner.

5.2 Space of Kashaev coordinates Let us recall that a Kashaev coordinate associated to a decorated ideal triangulation  is a vector .ln y1 ; ln z1 ; : : : ; ln y2m ; ln z2m / 2 R4m , where ln y and ln z are associated to the ideal triangle  . Denote by K the space of Kahsaev coordinates associated to  . We see that K D R4m : Given a vector .ln y1 ; ln z1 ; : : : ; ln y2m ; ln z2m / 2 K , we associate a number to each side of each ideal triangle as follows. For the ideal triangle  , we associate • ln h0 WD ln y  ln z to the 0-side; • ln h1 WD ln z to the 1-side; • ln h2 WD  ln y to the 2-side. Therefore ln h0 C ln h1 C ln h2 D 0. We can identify the space K D R4m with a subspace of R6m D f.: : : ; ln h0 ; ln h1 ; ln h2 ; : : : /g satisfying ln h0 Cln h1 Cln h2 D 0 for each ideal triangle  .

5.3 Exact sequence The enhanced Teichmüller space parametrized by shear coordinates is Tz D R3m D f.ln x1 ; ln x2 ; : : : ; ln x3m /g, where ln xi is the shear coordinate at edge i . We dez fine P3ma map f1 W T ! R by sending .ln x1 ; ln x2 ; : : : ; ln x3m / to the sum of entries i D1 ln xi : Suppose  is the underlying ideal triangulation of the decorated ideal triangulation  . We define a map f2 W K ! Tz as a linear function by setting ln xi D ln hs C ln ht whenever i bounds the s-side of  and the t -side of  ( may equal ), where s; t 2 f0; 1; 2g. class in Another map f3 W H1 .S; R/ ! K is defined as follows. A homology P  H1 .S; R/ is represented by a linear combination of oriented dual edges: 3m c iD1 i i . If  s the orientation of i is from the s-side of  to the t -side of  , by setting ln h D ci and ln ht D ci , we obtain a vector .: : : ; ln h0 ; ln h1 ; ln h2 ; : : : / 2 R6m : The boundary P  map of chain complexes sends 3m iD1 ci i to a linear combination of vertices. In this combination, the term involving the vertex  is .ci i C cj j C ck k / where i , j , k (two of them may coincide) bound three sides of  and  t D 1 if t starts at the side of  bounded by  t while  t D 1 if t ends at the side of  bounded by  t . Therefore .ci i C cj j C ck k / D .ln h0 C ln h1 C ln h2 / :

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P  0 1 2 Since the chain 3m iD1 ci i is a cycle, we must have ln h C ln h C ln h D 0. Therefore this vector .: : : ; ln h0 ; ln h1 ; ln h2 ; : : : / is in the subspace K . Combining the three maps, we obtain Theorem 5.1 (Guo–Liu [13]). The following sequence is exact: f3 f2 f1 0 ! H1 .S; R/ ! K ! Tz ! R ! 0:

From the above theorem, we see that K is a fiber bundle over the space Ker.f1 / whose fiber is an affine space modeled on H1 .S; R/: To be precise, given a vector s 2 Ker.f1 /; let v 2 f21 .s/: Then f21 .s/ D v C H1 .S; R/.

5.4 Relation to bivecotrs Consider the linear isomorphism M W K ! K ; .ln y1 ; ln z1 ; : : : ; ln y2m ; ln z2m / 7!

(5.1)

.: : : ; ln h0 ; ln h1 ; ln h2 ; : : : /:

Proposition 5.2 (Guo–Liu [13]). If .ln x1 ; ln x2 ; : : : ; ln x3m / D f2 B M.ln y1 ; ln z1 ; : : : ; ln y2m ; ln z2m /; then

3m X i;j D1

X @ @  @ @ ^ D .f2 / B M ^ ; @ ln xi @ ln xj @ ln y @ ln z D1 2m

ij

  D aij  aji and aij is the number of corners of the ideal triangulation  where which is delimited in the left by i and on the right by j .

ij

The left hand side of the equality is the Weil–Petersson Poisson structure on the enhanced Teichmüller space [9].

5.5 Compatibility of coordinate changes Proposition 5.3 (Guo–Liu [13]). Suppose that the decorated ideal triangulations  and  0 have underlying ideal triangulations  and 0 respectively. The following diagram is commutative: Tz  o  Tz0 o

f2

f2

K  K 0

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where the two vertical maps are corresponding coordinate changes. The coordinate changes of Kashaev coordinates are given in Definition 4.4. The coordinate changes of shear coordinates are given in Proposition 2.4.

6 Relationship between quantum Teichmüller space and Kashaev algebra In this section, we establish a natural relationship between the quantum Teichmüller y q .a; b/. space TySq and the generalized Kashaev algebra K S

6.1 Homomorphism For a ideal triangle  , we associate three elements in K q to the three sides of  as follows: • H0 WD Y Z1 to the 0-side; • H1 WD Z to the 1-side; • H2 WD Y1 to the 2-side. Lemma 6.1. For any s; t 2 f0; 1; 2g and 2 1; 2; : : : ; 3m; Hs Ht D q 2st Ht Hs ; where st C  ts D 0 and 10 D 02 D 21 D 1: Suppose  is the underlying ideal triangulation of  , the Chekhov–Fock algebra Tq is the algebra over C defined by generators X1˙1 , X2˙1 , …, Xn˙1 associated to the 

components of  and by relations Xi Xj D q 2ij Xj Xi . We define a map F from the set of generators of Tq to K q . Suppose that the edge i bounds the s-side of  and the t -side of  . We define F .Xi / D q ı  ts Hs Ht 2 K q ;

(6.1)

where  t s is defined in Lemma 6.1 and ı is the Kronecker delta, i.e., ı D 1 and ı D 0 if ¤ . When D ; Xi is well-defined, since q  t s Hs Ht D q st Ht Hs due to Lemma 6.1. This definition is natural since when q D 1 we get the relationship between the Kashaev coordinates and the shear coordinates which is given by the map M and f2 . In fact when q D 1, the generators Y ; Z are commutative. They reduce to the geometric quantities y , z associate to  : Hs and Xi are reduced to hs and xi .

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Lemma 6.2. F .Xi /F .Xj / D q 2ij F .Xj /F .Xi / for any generators Xi and Xj . It turns out that F can be extended to the whole algebra Tq as an algebra homomorphism from Tq to K q .

6.2 Compatibility y q is the fraction division algebra of K q . The algebraic isomorphism Recall that K q y and K y q0 is defined in Definition 4.5. between K Lemma 6.3. Suppose that a decorated ideal triangulation  0 is obtained from  by a mark rotation  for some 2 f1; 2; : : : ; 2mg. Let  be the common underlying ideal triangulation of  and  0 . The following diagram is commutative if and only if a D q 2 . Tyq O

F

/K y q O O

Id

Tyq

F 0

/K y q0

Lemma 6.4. Suppose that a decorated ideal triangulation  0 is obtained from  by a diagonal exchange ' : Let  and 0 be the underlying ideal triangulation of  and  0 respectively. Then 0 is obtained from  by a diagonal exchange with respect to the edge i which is the common edge of  and  . The following diagram is commutative if and only if b D q 3 . Tyq O

F

/K y q O

yi

Tyq0

'O 

F 0

/K y q0

Theorem 6.5 (Guo–Liu [13]). Suppose the decorated ideal triangulations  and  0 have underlying ideal triangulations  and 0 respectively. The following diagram is commutative if and only if a D q 2 ; b D q 3 . Tyq O

F

q

q

‰; 0 .a;b/

ˆ;0

Tyq0

/K y q O

F 0

/K y q0

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Recall that the quantum Teichmüller space of S is defined as the algebra  G  TySq D Tyq =  2ƒ.S/

where the relation  is defined by the property that, for X 2 Tyq and X 0 2 Tyq0 , X  X 0 () X D ˆq;0 .X 0 /: y q .a; b/ associated to a surface S is defined as The generalized Kashaev algebra K S the algebra  G  y q .a; b/ D y q .a; b/ =  K K

S

24.S /

y q .a; b/ and X 0 2 where the relation  is defined by the property that, for X 2 K q y 0 .a; b/, K q 0 X  X 0 () X D ‰ ; 0 .a; b/.X /: Corollary 6.6. The homomorphism F induces a homomorphism y q .a; b/ TySq ! K S if and only if a D q 2 , b D q 3 .

6.3 Quotient algebra Furthermore, consider the element H D q

P

i 0 certainly has the cardinality of the continuum; if in addition to that one wants to calculate the dimension of R then at first one has to turn R into a space in some way. Initially it has been tried to describe the elements of R with  coordinates or also with  >  coordinates satisfying    relations1 . But the different calculations of  on these grounds stood in no relationship to each other, and it was not a priori clear but rather a miracle that the calculations always lead to the same value of  . I think that one should not primarily ask for an explicit representation of the points of R via numbers in a coordinate system, but study the inner structure of the “space” R. One ought to start introducing a notion of neighbourhood in R. But up to now there is only one approach2 which is not suitable for a foundation. And as long as R is not a space with a notion of neighbourhood, it does not have a dimension in the sense of analysis or set theory. Then v. d. Waerden3 has proved within the framework of algebraic geometry that an algebraic function field depends on  parameters if certain pathologies are excluded. I 1 [The footnotes are Teichmüller’s except when the contrary is specified.] Compare e.g. Enzyklopädie der mathematischen Wissenschaften, Band II, 2. Teil, p. 147 f. (W. Wirtinger). 2 O. Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale, Abhandl. Preuß. Akad. Wiss. 1939, Nr. 18 oder 54. 3 B. L. van der Waerden, Zur algebraischen Geometrie. XI. Projektive Äquivalenz und Moduln von ebenen Kurven. Math. Ann. 114.

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will not go into this since in this approach the problem is replaced by a purely algebraic one whose solution does not provide a function theorist with the insight he is seeking. I will now study the problem not for its own sake, but to obtain some knowledge about R for the investigation of quasiconformal mappings etc. Therefore the applicability in the theory of Riemann surfaces is crucial for me. One does not only need R to have the structure of a neighbourhood space or an algebraic variety but also of an analytic manifold, i.e. in every neighbourhood of a point of R, a coordinate system with  coordinates is wanted, and all the coordinate transformations should be analytic. Then the notion of neighbourhood is provided automatically, and in our case it is not difficult to choose a class of coordinate systems transforming algebraically into one another. But since the set R can be made into an analytic manifold in different ways, we have to make sure that the choice is uniquely determined by certain properties. And these determining properties must ensure the applicability in function theory. It turns out that R contains certain singular manifolds. But we will construct a covering space R without singularities. In the near future I will not be able to publish all my extensive considerations in detail. Therefore at this point I just give a short overview of the methods and results. And I will leave out the calculation of first sections in R completely. As I am realizing now, my solution of the problem is based mainly on three newly introduced notions, namely topological determination, analytic family of Riemann surfaces, winding piece coordinates. The topological determination4 of closed surfaces of genus g is done as follows: Let H0 be a fixed and H an arbitrary closed Riemann surface of genus g. Let H be a topological map from H0 onto H. (The adjective “topological” will be left out from now on.) We will have to deal with pairs .H; H / of a surface H and a map of the fixed H0 onto H. Two such pairs are called equal .H; H / D .H0 ; H 0 /, if firstly H D H0 and secondly the map H 01 H from H0 onto itself can be deformed into the identity (i.e. if to every point p of H0 and every t with 0  t  1 one can associate continuously an image point q.p; t / on H0 such that q.p; 0/ D p and q.p; 1/ D H 01 H p for all p). The pairs .H; H / form classes of “equal” pairs. We denote such a class by H and we call it a topologically determined Riemann surface of genus g. So by H a surface H is given and moreover, a map H from H0 to H determined up to deformation. For a map A from H to H0 we set AH D H0 $ A.H; H / D .H0 ; AH /; i.e. A transforms the topologically determined Riemann surface H into the topologically determined Riemann surface H0 D .H0 ; H 0 / corresponding to the surface H0 D AH and the composed map H 0 D AH from H0 to H0 . If for given H and 4 Compare

No. 49–51 of the article mentioned in footnote 2.

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H0 a conformal map A from H to H0 with AH D H0 exists then we call H and H0 conformally equivalent. We denote a class of conformally equivalent topologically determined surfaces H of genus g by h. These classes are the points of the “space” R. In contrast to this the classes of conformally equivalent surfaces H of genus g denoted by h, form the “space” R. The moduli problem consists of asking for the properties of the space R. But it turns out that it is better to study the space R at first. We now come to the most important notion that we have to introduce here, namely the notion of an analytic family of Riemann surfaces. – As a start, let us recall the notion of an analytic manifold. Let M be any neighbourhood space, i.e. a set where to every point p is associated a collection of neighbourhoods U.p/ such that the Hausdorff axioms for open neighbourhoods are satisfied. Moreover, to every point p of M let there be associated at least one topological map A from a neighbourhood U of p to a domain G in the n-dimensional complex .z1 ; : : : ; zn /-number space. And if p0 is a point in U and if to that point is associated a topological map A0 from a neighbourhood U0 of p0 to a domain G 0 in the .z10 ; : : : ; zn0 /-space then the map A0 A1 .z1 ; : : : ; zn / D .z10 ; : : : ; zn0 / shall be given by n analytic functions zi0 D @fi fi .z1 ; : : : ; zn / with nonvanishing functional determinant j @z j. In this way M is k provided with the structure of an n-dimensional complex analytic manifold. Of course one can impose even more conditions on the “coordinate transformations” z ! z 0 (e.g. that the fi should be algebraic functions that are regularly analytic at the chosen point). – The simplest example (n D 1) is the Riemann surface in the most general sense of the word. – For the time being we do not require M to be connected. Let P be such an r-dimensional complex analytic manifold whose point p is described by r coordinates p1 ; : : : ; pr . Suppose that to every point p in P is associated a closed Riemann surface H D H.p/ of fixed genus g. The point t of such a H.p/ is given by a coordinate t (a “uniformizing local parameter”). Now we consider pairs .p; t/: p is a point of P and t is a point of the associated surface H.p/. These pairs .p; t/ shall be the points of a new .r C 1/-dimensional complex analytic manifold M. Namely, in M there shall be a distinguished analytic coordinate system .p1 ; : : : ; pr ; t / consisting of the coordinates .p1 ; : : : ; pr / of p and a last coordinate5 t that for fixed p1 ; : : : ; pr is a uniformizing local parameter of the associated surface H.p/ D H.p1 ; : : : ; pr /. The transformation to a second “distinguished” coordinate system p10 ; : : : ; pr0 ; t 0 of M is then given by the formulae p0 D f .p1 ; : : : ; pr /

. D 1; : : : ; r/I

t 0 D g.p1 ; : : : ; pr ; t /;

where @p0 @t

D 0;

5 Better “subordinate”; compare O. Teichmüller, Über Extremalprobleme der konformen Geometrie. Deutsche Mathematik 6 (1941).

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ˇ 0ˇ ˇ @p ˇ ˇ ˇ ˇ @p ˇ ¤ 0 

and

@t 0 ¤ 0: @t

Clearly, the coordinate transformations of this type form a group. A family H.p/ that has been made into an .r C 1/-dimensional manifold M in the way just described, is called an analytic family of Riemann surfaces, and P is called its parameter manifold. The important point is that in M the last coordinate t being a local parameter on H.p/ may not be chosen independently for every p but in such a way that a certain analytic dependence of t on p is ensured. Such a t , forming an analytic coordinate system in M together with p1 ; : : : ; pr , I will call a permanent uniformizing local parameter in contrast to an arbitrary uniformizing local parameter that is only useful for a certain surface H. [Example: The two-sheeted covering surface of the z-plane ramified over z D 0; 1; 1; p. Here we have one parameter p. P is the p-plane punctured in the points q0, p p 1, 1. Let t D z in the neighbourhood of all nonramified points, t D z; z  1; z1 p in the ramification points 0, 1, 1 and t D z  p in the ramification point z D p where the signs of the square roots have to be chosen in an appropriate continuous way. Then to the neighbourhood of every point of M, i.e. every pair .p; t/ consisting of a p of P and a point t of the surface corresponding to p, is associated a coordinate system .p; t / and all the requirements are satisfied.] There is a connection between the notion of an analytic family of Riemann surfaces with the notion of topological determination. One can prove the following: For every point p.0/ of P there is a connected neighbourhood U and a map Ap associating to every p in U and every point t.0/ of H.0/ D H.p.0/ / a point Ap t.0/ D t of H.p/ continuously such that the map t0 ! t for fixed p is a topological map from H.0/ onto H.p/. If now Bp is a map with the same properties then Bp1 Ap can be deformed into the identity (by letting p go to p.0/ ), therefore Ap is uniquely determined up to deformation (in a neighbourhood U of p.0/ !). Now if H.0/ has been determined topologically by choosing a map H .0/ from H0 onto H.0/ to obtain H.0/ D .H.0/ ; H .0/ /, we set H.p/ D Ap H.0/ D .H.p/; Ap H .0/ /: In this way we obtain a topological determination H.p/ for all H.p/ for a neighbourhood of p.0/ . By well-known principles of analytic continuation there exists a relatively unramified covering P of P such that to every point p of P corresponds a topologically determined surface H.p/. Of course then the underlying surface H is precisely the surface H.p/ associated to the point p of P underlying p, and in a neighbourhood of every p.0/ the topological determination of H coincides with H.p/ D Ap H.0/ :

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Then we simply say the topological determination depends continuously on p. In this way we obtain the analytic manifold of topologically determined Riemann surfaces. Often P will coincide with P. From now we just write P and p instead of P and p. Now we have to give correct formulations for several notions that are usually just defined for a fixed Riemann surface, for our Riemann surfaces H.p/ depending on parameters p1 ; : : : ; pr . These will be based on the permanent local parameter t . Let prC1 ; : : : ; ps be extra parameters in addition to p1 ; : : : ; pr (the coordinates in P) that may or may not occur. A point t of H.p/ depends analytically on p1 ; : : : ; ps if the coordinate t of t depends analytically on p1 ; : : : ; ps in every neighbourhood, where p1 ; : : : ; pr ; t is a coordinate system of M (or where t is a permanent local parameter). An integral divisor a D t1    tn is a product of n equal or different “prime divisors” t of the same surface H. The prime divisors are in one-to-one correspondence with the points and will be denoted with the same letters. To begin with, we assume that for all these n-points the same permanent local parameter t applies. Then if t belongs to t D t , let s1 ; : : : ; sn be the elementary symmetric functions of t1 ; : : : ; tn . We say, that a depends analytically on p1 ; : : : ; ps if s1 ; : : : ; sn are analytic functions of p1 ; : : : ; ps . One proves that this does not depend on the choice of the permanent local parameter t . If one cannot or does not want to use the same t for all t , one can partition the t (in a neighbourhood of .p1.0/ ; : : : ; ps.0/ /) in classes such that for every class there is one permanent local parameter and the t of different classes are always different. If then for every class the product of their prime divisors depends analytically on p1 ; : : : ; ps , we say that a depends analytically on p1 ; : : : ; ps . One proves that this does not depend on the choice of the partition. If ab D c and a and c depend analytically on p1 ; : : : ; ps , then the same is true for b. Because of the last theorem we define: a fractional divisor d depends analytically on p1 ; : : : ; ps , if d D a , where a und b are integral divisors depending analytically b on p1 ; : : : ; ps . Now we come to the important notion of a principal part. A principal part at the place t of the surface H is an additive residue class of the additive group of all functions that are regular up to poles in a neighbourhood of t modulo the normal subgroup of all the functions that are regular at t. If to finitely many points t we have associated principal parts, we call this a system of principal parts. When does such a system depend analytically on p1 ; : : : ; ps ? Let t be a permanent local parameter and let Q.t/ D t n C ˇn1 t n1 C    C ˇ0 be a polynomial of degree n with leading coefficient 1 whose zeros are all contained in the domain to which t is restricted. Let P .t / D ˛0 C    C ˛n1 t n1

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be a polynomial of degree n1. If ˛0 ; : : : ; ˛n1 ; ˇ0 ; : : : ; ˇn1 depend analytically on P .t/ p1 ; : : : ; ps , then we say that the system of principal parts of Q.t depends analytically / on p1 ; : : : ; ps . A system of principal parts depends analytically on p1 ; : : : ; ps if it is P .t/ equal to the system of principal parts of such a quotient Q.t under the condition that / all its poles are contained in the part of H uniformized by t . Otherwise one splits the system of principal parts into separate parts (just as above for the integral divisor) and requires the parts to depend analytically on p1 ; : : : ; ps . In order to justify this definition as well as for further applications one needs the following proposition: If f .p1 ; : : : ; pr ; t / for .p1 ; : : : ; pr / in a domain P, 0 <  < jt j < R, is regularly analytic and if f for fixed p as a function of t alone in jt j < R has at most n poles but no other singularities, then there are polynomials P .t / D ˛0 .p1 ; : : : ; ps / C    C ˛n1 .p1 ; : : : ; ps /t n1 ; Q.t/ D t n C ˇn1 .p1 ; : : : ; ps /t n1 C    C ˇ0 .p1 ; : : : ; ps /; whose coefficients depend analytically on p1 ; : : : ; ps , and there is a function r.p1 ; : : : ; ps ; t / that is regular for .p1 ; : : : ; pr / in P, jt j < R, such that f D

P C r: Q

A function on the surface H for fixed p (i.e. with fixed p1 ; : : : ; pr ) is defined to be a function f on H that is everywhere analytic up to poles. We say that f depends analytically on p1 ; : : : ; ps if in the neighbourhood of every point, f is a regularly analytic function of p1 ; : : : ; ps and t (for a permanent local parameter t ) except in the neighbourhoods of finitely many points of every single surface H. Of course the poles of f on H have to be excluded, perhaps even some more points. We already know that the system of principal parts of such a function f depends analytically on p1 ; : : : ; pr , but we can say even more: If t is a permanent local parameter and f is regular for 0 <  < jt j < R and if '.t / is a function that is regular for 0 <  < jt j < R and for fixed p is regular up to at most m poles for jt j < R, then the part of the system of principal parts of f .t /'.t / that is decreasing in jtj   depends analytically on p1 ; : : : ; ps . For a given ' this is a regularity statement on f . Let n.p1 ; : : : ; ps / be the degree of f , i.e. the number of a-places of f counted with multiplicities, that is independent of the choice of the constant a. The constant a may also be 1. (We do not consider the case of a constant function f here.) In general, n.p1 ; : : : ; ps / equals a constant n. Just for a few .p1 ; : : : ; ps / it may happen that n.p1 ; : : : ; ps / < n, but it is impossible that n.p1 ; : : : ; ps / > n. We consider f in the neighbourhood of a place .p1.0/ ; : : : ; ps.0/ ; t .0/ /. For fixed p D p.0/ , let f take the value a.0/ (that may be 1) at the place t .0/ exactly ˛ times. If .p1 ; : : : ; ps / lies sufficiently close to .p1.0/ ; : : : ; ps.0/ /, then f will take the fixed value a in an arbitrarily small neighbourhood of t D t .0/ for not exceptional .p1 ; : : : ; ps / either  times or

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 C ˛ times, depending on whether a ¤ a.0/ or a D a.0/ . Here  is a nonnegative integer that for fixed p.0/ depends only on t .0/ but not on a and we have X ; n  n.p1.0/ ; : : : ; ps.0/ / D where on the right-hand side we have to sum over all the ’s corresponding to the different t .0/ . In particular, there are only finitely many  different from zero. If all the  D 0, that means if n.p1 ; : : : ; ps / D n or if the degree of f does not decrease when specializing to p D p.0/ , then we say that f depends analytically of the first kind on p1 ; : : : ; ps . This is the case if and only if in a neighbourhood of every place either f or f1 is a regular analytic function of p1 ; : : : ; ps ; t . We don’t know whether for a given analytic family of Riemann surfaces there exist any functions at all depending analytically or even analytically of the first kind on the parameters. We define: An analytic family of Riemann surfaces H.p/ is called globally analytic if for a neighbourhood of every p a function on the surface H.p/ exists that depends analytically of the first kind on the parameters p1 ; : : : ; pr . Just for such families I can make global statements, i.e. statements referring to the behaviour of functions on the entire surface H. The dependence on p for the time being will be studied only locally, in the neighbourhood of a p.0/ . The same is true analogously for analytic families of topologically determined Riemann surfaces. We can now state the main result. For every g there exists a globally analytic family of topologically determined Riemann surfaces of genus g: HŒc where c runs through a  -dimensional complex analytic manifold C, with the following properties: • For every topologically determined surface H of genus g there exists one and only one conformally equivalent HŒc. • If H.p/ is any globally analytic family of topologically determined surfaces of genus g described by the parameters p1 ; : : : ; pr and the permanent local parameter t , there exists a map from the family H.p/ to the family HŒc with parameters c1 ; : : : ; c and permanent local parameter T , such that c1 ; : : : ; c become analytic functions of p1 ; : : : ; pr and T becomes an analytic function of p1 ; : : : ; pr ; t ¤ 0 and such that the topologically determined surface H.p/ is mapped with @T @t conformally onto HŒc. The family HŒc is essentially uniquely determined by these properties. The (complex) dimension  of C equals 8 ˆ0 if g D 0, < D 1 if g D 1, ˆ : 3.g  1/ if g > 1. It is easy to see that the family HŒc is uniquely determined by the stated properties: Let H0 Œc0  be a second family with the same properties, described by the parameters

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c10 ; : : : ; c0 0 and the permanent local parameter T 0 . Then the map c $ c0 , where HŒc and H0 Œc0  are conformally equivalent, is one-to-one and can be extended to an analytic map .c10 ; : : : ; c0 0 ; T 0 / ! .c1 ; : : : ; c ; T / as well as to a map .c1 ; : : : ; c ; T / ! .c10 ; : : : ; c0 0 ; T 0 /; that is not necessarily the inverse of the first map. But when omitting T and T 0 , the two maps are inverses of each other: c10 ; : : : ; c0 0 and c1 ; : : : ; c are related by an analytic 0 @T ¤ 0 and @T transformation with non vanishing functional determinant.6 Since @T 0 ¤ @T 0, both extended maps are analytic with non-vanishing functional determinant. So the two families can be mapped onto each other analytically. With this theorem the moduli problem is solved. Namely, we obtain a oneto-one correspondence between the space R of all classes of conformally analytic topologically determined surfaces of genus g and the  -dimensional complex analytic manifold C, by associating to every class h in R the element c in C such that HŒc corresponds to h. Hereby, we also provide R in a unique way with the structure of a  -dimensional complex analytic manifold. A conformal invariant of a topologically determined Riemann surface H, i.e. a function on R, is called analytic, if after transfer to C it depends analytically on c1 ; : : : ; c . We now also have a notion of neighbourhood in R. We will now state a second result that is needed for the proof of the first theorem. It consists of the determination of a function by generalized systems of principal parts. Let H.p/ be a globally analytic family of Riemann surfaces. For n separate parts of the surfaces H.p/ where p stays in the neighbourhood of p.0/ , we take permanent local parameters t and consider the circles jt j < R. Let 0 <  < R. Let ' .t / be a function of p1 ; : : : ; ps ; t that is analytic in  < jt j < R. Moreover, let ' for fixed p in jt j < R have no singularities except poles whose multiplicities sum up to at most m . Let Q .t / D tn C ˇn1; tn1 C    C ˇ0; : where the ˇi; depend analytically on the p and all the zeros of Q .t / lie in jt j < . Let P .t / D ˛0; C    C ˛n1; tn1 ; where the coefficients are yet indeterminate. We are looking for functions f on H.p/ P and is otherwise such that f ' in jt j < R has the same system of principal parts as Q  regular. First we deal with the homogeneous problem, where all ˛i; D 0. So here we ask for functions f such that f ' in jt j < R is regular and that f itself is otherwise regular. This is a question similar to the one underlying the theorem of RiemannRoch. Let h be the maximal number of linearly independent functions f with the 6 The

theorem of the invariance of the dimension shows that  D  0 .

Chapter 18. Variable Riemann surfaces

795

listed properties, so that any f is a linear combination with complex coefficients of h linearly independent f1 ; : : : ; fh . The number h still depends on p: h D h.p/. It turns out that the following holds: In general, h.p/ is constant. Just for special p it may grow but never decrease. If h.p/ does not grow when specializing to p D p.0/ , then one can choose the basic functions f1 ; : : : ; fh in a neighbourhood of p.0/ in such a way that they depend analytically on p1 ; : : : ; pr . We should add here that often one can say immediately that h is independent of p (e.g. because of the Riemann–Roch theorem). Now we come to the inhomogeneous problem and choose an enumeration for the ˛i; : ˛1 ; : : : ; ˛k . Moreover, we restrict to the case where h does not grow when specializing to p D p.0/ . Then the following holds: For given ˛1 ; : : : ; ˛k there exists a function f if and only if for an appropriate enumeration, ˛lC1 ; : : : ; ˛k are certain linear combinations of the free ˛1 ; : : : ; ˛l (0  l  k). The coefficients of these linear combinations depend analytically on p1 ; : : : ; ps . In this case one obtains f in the form f D ˛1 F1 C    C ˛l Fl C ˇ1 f1 C    C ˇh fh ; where the F are functions on H.p/ depending analytically on p1 ; : : : ; ps , whereas ˇ1 ; : : : ; ˇh are constants that can be chosen freely. All this I can only prove for globally analytic families of Riemann surfaces. I do not know whether every analytic family of Riemann surfaces is globally analytic or not. The assumption that there exists a function z of the surface H.p/ depending analytically of the first kind on p1 ; : : : ; pr , gives a hint how to start the proof of our claims: We put H.p/ over the z-plane as an n-sheeted covering Riemann surface Z.p/ where n is the degree of z that indeed does not depend on p. As is well-known, this is done as follows: z is a function of the place t on H.p/. Conversely, we now consider the point t of H.p/ (described by the local parameter t ) as a function of z. Since z takes every value n times, t becomes unique on an n-fold covering surface Z.p/ of the z-plane, and Z.p/ is mapped conformally onto H.p/. In doing so, we have to take care of the ), corresponding to exceptions in multiple poles and other crossing points (zeros of dz dt winding points of Z.p/. It is most suitable to describe Z.p/ by its winding points and its global structure. If z depends analytically of the first kind on p1 ; : : : ; pr , then how do the winding points of the surface Z depend on p1 ; : : : ; pr ? In this context it has to be taken into consideration that when specializing to p D p.0/ for example two two-sheeted winding points of Z could melt into one three-sheeted etc. Here we have reached the complex of questions that I started with at the beginning of my studies of variable Riemann surfaces. We now have to introduce the notion of a winding piece and the coordinates of a winding piece.

796

Oswald Teichmüller

Let C be a closed Jordan curve dividing the z-plane into an inner domain I and an outer domain A. If one cuts a closed Riemann surface Z over the z-plane over C, then the part of Z lying above I decomposes into finitely many pieces. The multi-sheeted but simply connected among those pieces are the winding pieces. So at first one needs to consider pieces W of the Riemann surface over I that are connected and contain only finitely many winding points but no boundary points over I and that cover every point n times (counting winding points with multiplicities): these are the n-sheeted coverings W of I . Such a covering is simply connected if and only if the sum of the multiplicities of its ramification points equals n  1 (a -sheeted winding point is counted as a .  1/-fold ramification point). In this case, the boundary of W consists of a single curve lying above C in n sheets. In contrast to this, for a multiply connected W the sum of the multiplicities of its ramification points is greater than n  1. However, a simply connected n-sheeted covering W of I is not yet a winding piece. If such a W may have dropped out when cutting a closed surface Z over C, one can put it back into the hole in n different ways, and in general one will hereby obtain n different surfaces Z (for n > 2). This is because one cannot distinguish on the n-sheeted boundary curve of W over C which boundary point of W corresponds to a certain boundary point of the rest of Z over the same place in the z-plane. It is not until we provide the simply connected n-sheeted coverings W of I with such a distinction, that we turn them into winding pieces W . Let z0 be any point of I . We consider the part lying above C C A of the Riemann p surface of the function n z  z0 : we call its inner part An and its boundary Cn . One p can distinguish the n points of Cn lying above the same z using that n z  z0 takes n different values on them. The boundary of a simply connected n-sheeted covering W of I has the same structure as Cn . We will identify the boundary of W in a certain way with Cn and thereby turn W into a winding piece W . Namely, if W has dropped out of a surface Z as above we just need to identify also the boundary of the hole in Z resulting from cutting out W in a certain way with Cn : now there can be no doubt about how to put W back into the hole. The piece W is not yet uniquely determined (except for n D 2) by the elementary functions of the z-values of its n  1 ramification points (counting multiplicities). We will describe W uniquely by n  1 extra complex numbers: those numbers will be the “winding piece coordinates”. By identifying the boundary of W in a certain way with the boundary Cn of An , we can view W as part of a closed surface F D W C Cn C An of genus 0 over the z-plane. Using the theory of uniformization, F can be mapped conformally onto the t -sphere. We choose this map such that 1 is mapped to 1 and that at 1 we have a development of the form p const const n z  z0 D t C C 2 C  t t

797

Chapter 18. Variable Riemann surfaces

p (note that n z  z0 is uniquely determined on An ). Then z is an everywhere (except at 1) regular function of t , hence a polynomial, and we have z D t n C bn2 t n2 C    C b1 t C b0 : So the winding piece W determines n  1 complex numbers b0 ; : : : ; bn2 . Conversely, these numbers determine W uniquely. We call b0 ; : : : ; bn2 the coordinates of the winding piece W . A winding piece depends analytically on parameters if its coordinates b are analytic functions of these parameters. t is called the permanent uniformizer 7 of W . If we had not moved from W to W then there would be n different boundary maps between W and An , and t would only be determined up to multiplication with an n-th root of unity. Accordingly, except in the case n D 2, the winding piece coordinates b0 ; : : : ; bn2 would allow a cyclic group of n transformations b !   b

. D 0; : : : ; n  2;  mod n;  D e

2 i n

/:

We don’t want to go into the details of the theory of winding piece coordinates here. In order to apply the winding pieces to variable Riemann surfaces Z.p/ over the zplane, we introduce the notion of a normed surface. That is a closed Riemann surface over the z-plane having no winding point above z D 1 and where among its sheets above 1 one is distinguished by definition as the “first” sheet. The point z D 1 on this “first” sheet we denote by 11 . A surface Z having been normed in this way, shall O If Z in addition is topologically determined, then we write Z. O be denoted by Z. O one can cut out winding pieces, e.g. by simply drawing a small circle Out of such a Z O as being constructed from l pairwise around each winding point. We want to think of Z disjoint (including the boundary) winding pieces W1 ; : : : ; Wl and an unramified offcut R with the distinguished point 11 . If n is the number of sheets of W , then l X

.n  1/ D V D 2n C 2g  2;

D1

where V is the total ramification number and g the genus of Z. The piece W has n 1 winding piece coordinates b;0 ; : : : ; b;n 2 , so altogether there are V coordinates O depend analytically on parameters p1 ; : : : ; pr by fixing R and 11 b; . We now let Z and varying W1 ; : : : ; Wl such that all b; depend analytically on the p . Of course the altered W have to be put back into the holes in R “correctly”. On every W we take the permanent uniformizer t as permanent local parameter and for the rest in all simple places z or z1 . One can prove that these definitions are independent of the O D R C W1 C    C Wl , and that Z O is an analytic family choice of the decomposition Z of Riemann surfaces whose parameter manifold is of complex dimension V . 7 Translator’s note: in the German original permanente Uniformisierende in contrast to Ortsuniformisierende, here translated as local parameter

798

Oswald Teichmüller

These definitions are justified by the following theorem: Let

N M X X

A z  w  D 0

D1 D1

be an equation for the algebraic function w of z whose coefficients A are analytic functions of p1 ; : : : ; pr (where .p1 ; : : : ; pr / lies in a simply connected domain P). We assume that the coefficient of w N vanishes identically (i.e. A1N ; : : : ; AMN do not have a common zero in P). Moreover, the equation shall be irreducible for all .p1 ; : : : ; pr /. Then automatically its discriminant D.z/ never vanishes identically. The Riemann surface Z of the algebraic function w.z/ shall have the same genus g for all .p1 ; : : : ; pr / in P. Above z D 1 there shall be no ramification point. Then O such that for all k the principal part of wz k one can turn Z into a normed surface Z O depends analytically on the p in at 11 depends analytically on the p , and then Z the sense of our definition. The main point of this long theorem is that with the coefficients of the equation also the coordinates of the winding pieces depend analytically on the parameters if we exclude those values of the parameters where the genus drops with specialization of parameters or where the surface decomposes. What does that mean? – We choose a winding piece W . It is easy to show that under our topological assumptions even after changing the p , the piece W at first stays a winding piece. On W we had z D t n C bn2 t n2 C    C b0 : The piece W was a part of the surface given by the algebraic function t .z/ defined by this equation. The ramification points of W correspond to the zeros of dz taking into dt account their multiplicities; from this one can see immediately that the elementary symmetric functions s1 ; : : : ; sn1 of the z-values a1 ; : : : ; an1 of the ramification points are polynomials in b0 ; : : : ; bn2 . It can be shown that b0 ; : : : ; bn2 are integral algebraic functions of s1 ; : : : ; sn1 . Now a1 ; : : : ; an1 are some of the zeros of the discriminant D.z/ of the above equation, and therefore one can expect that with the coefficients of D.z/ also s1 ; : : : ; sn1 depend analytically on the p . In fact, even the b , that are ramified as functions of s (except for n D 2), depend regularly on the p . To my knowledge up to now this phenomenon has not yet been pointed out. If for example by analytic specialization of parameters two two-sheeted winding points z D a1 , z D a2 of the surface melt into one three-sheeted winding point, then not only a1 C a2 and a1  a2 but also .a1  a2 /2=3 depends analytically on the parameters. Moreover, it turns out that the following holds: Let z be a function of the surface H.p/ depending analytically of first kind on the parameters. Let z have only simple separate poles and suppose we distinguish one of these poles as the “first” one in a O continuous manner. If we put H.p/ above the z-plane we obtain a normed surface Z.p/.

Chapter 18. Variable Riemann surfaces

799

This surface depends analytically on p1 ; : : : ; pr , and its permanent local parameters are analytically related to those of H.p/. O For any Z.p/, depending analytically on p (e.g. the one considered above), there P also exists an inducing equation A z  w  D 0 as above, whose coefficients depend analytically on p (at least in a neighbourhood of every p). The proof of this theorem contains the only difficulty; all the other claims on winding pieces so far follow more or less directly from the definitions. This theorem provides us for given z with a w depending analytically on the parameters and generating together with z the field of all functions on the surface H.p/. Using this idea one can already prove the above assertion on the analytic determination of a function by generalized systems of principal parts. Now I want to briefly sketch how I arrive at the family HŒc which is the main result. Let n > 2g  2: Then every Riemann surface of genus g is conformally equivalent to an n-sheeted O and Z O depends on normed surface Z, V D 2n C 2g  2 O by a neighbouring function complex coordinates b; . I now replace the function z of Z O over the -plane one  of degree n; it depends on 2n  g C 1 constants. If one puts Z O O obtains a neighbouring surface Z of Z. But Z in addition allows a continuous group of conformal maps onto itself whose element depends on 8 ˆ 1 O that can be constructed from Z O complex constants. Therefore the collection of all Z via the transformation z !  does not depend on 2n  g C 1, but only on 2n  g C 1  O decomposes in this way into families with constants. The neighbourhood of every Z O . The individual family .2n  g C 1  / parameters of conformally equivalent Z depends on 8 ˆ if g D 0, 1 parameters c1 ; : : : ; c . Of course one ought to give more details. O Now I introduce topological This holds in the neighbourhood of a certain Z. O and thereby also for the neighbouring surfaces. This means I determination for Z

800

Oswald Teichmüller

O When this is done the following can be proved: For every Z O O to Z. move from Z there is a neighbourhood with the property that any two topologically determined O 1 and Z O 2 , that can be mapped conformally onto each other as normed surfaces Z topologically determined surfaces, arise from each other as above by a small change O2 D Z O 1 . Again this theorem ought to be formulated in more details. Its proof of z: Z contains all the difficulties. Now we map bijectively the classes of conformally equivalent topologically deterO to certain points of the mined normed surfaces from the neighbourhood of a given Z space R of all classes of conformally equivalent topologically determined surfaces of genus g, and hence introduce a coordinate system in a subset of R. It turns out that R thus becomes a  -dimensional complex analytic manifold C. Now, to every c in C we associate analytically a corresponding topologically deO we introduce O as HŒc. In every winding piece W of Z termined normed surface Z O either z or 1 as the corresponding permanent uniformizer t , in all simple points of Z z permanent local parameter. One has to prove that now all requirements are fulfilled. The space R consists of at most countably many connected parts. I believe that R is in fact simply connected. At last I want to mention briefly what happens when abolishing the topological determination.8 Let H be a topologically determined Riemann surface of genus g. Let G be the group of all topological maps of the underlying surface H onto itself and let A be the normalizer of all elements of G that can be deformed into the identity. The factor group F D G=A is the mapping class group of H. If now H0 is a second topologically determined surface of genus g and if G 0 ; A0 ; F0 for H0 have the analogous meaning, then there is a map H from H onto H0 : H H D H0 . We associate to every element G of G the element G 0 D HGH 1 of G 0 . This is an isomorphism G ' G 0 , sending A to A0 and hence also inducing an isomorphism F ' F0 of the factor groups. But since H and H0 are topologically determined the map H is determined up to deformation (i.e. up to a right factor from A or up to a left factor from A0 ). Therefore not the isomorphism G ' G 0 but the isomorphism F ' F0 is uniquely determined. The mapping class groups of all topologically determined surfaces of genus g are isomorphic in a uniquely determined way. Thereupon we identify all of them and simply speak of the mapping class group F. 8 Compare

Nr. 141–144 of the article mentioned in footnote 2.

Chapter 18. Variable Riemann surfaces

801

Without the topological determination we could only identify F and F 0 up to an inner automorphism. Let K be the group of all conformal maps from H onto itself. One shows: The intersection K \ A, i.e. the group of all those conformal maps from H onto itself that can be deformed into the identity, is a continuous group with parts where 8 ˆ 1. In particular, K\AD1

for g > 1.

For all elements F of F D G=A let F H D GH;

if F $ G mod A.

So for G 2 G the topologically determined domain, GH that differs from H only by the topological determination, depends only on the residue class F of G modulo A and is therefore denoted by F H. If K is a conformal map from H onto H0 then K.GH/ D KGK 1 H0 : Hence, since the maps G from H onto itself and KGK 1 from H0 onto itself belong to the same residue class F of F, K.F H/ D F H0 : That means, if H and H0 are conformally equivalent then also F H and F H0 are conformally equivalent. So if h in R is the class of topologically determined surfaces that are conformally equivalent to H, then we may denote by F h the class of topologically determined surfaces that are conformally equivalent to F H without having to be afraid of a contradiction. In this way we have associated to every F in F a one-to-one map h ! F h from R onto itself. However, it may occur that F h D h for all h and yet F ¤ 1. For a fixed h, the class of H, we look for the subgroup of all F in F with F h D h. Then for one (and hence every) G in G whose residue class modulo A equals F , GH and H must be conformally equivalent, i.e. there must be a K in K with GH D KH. But that means G  K.mod A/ according to the definition of equality of topologically determined domains. One can see from this that the subgroup in question is the factor group modulo A of the group KA of all maps from H onto itself that are deformable into conformal maps; it equals KA=A ' K=K \ A; and is hence always finite. It is shown that the representation of F by maps h ! F h from R onto itself is properly discontinuous. Namely, for every h in R there is a sufficiently small

802

Oswald Teichmüller

neighbourhood U in R such that F h1 D h2 for h1 and h2 in U implies F h D h. It is clear that the maps h ! F h are analytic (R now is an analytic manifold). The space R of all classes h of conformally equivalent Riemann surfaces of genus g arises from R by identifying equivalent points via F (h1 and h2 are identified if and only if there is an F in F with F h1 D h2 ). R is an analytic manifold. In all points of R that are only invariant under those elements of F fixing all points of R, one can directly transfer a coordinate system from R to R and obtains also in R an analytic coordinate system. This is because the group is properly discontinuous. Here the “general” point of H is already taken care of because the exceptional points lie on certain analytic manifolds in R, as will result from the following. Let h be a given point of R. Let Fh D KA=A ' K=K \ A be the finite group of all F in F with F h D h. Without going back to the Riemann surfaces one can show easily that it is possible to choose an analytic coordinate system c1 ; : : : ; c of R in the neighbourhood of h such that h is the point c1 D    D c D 0 and that the coordinates c1 : : : : ; c transform linearly under the application of elements of Fh . That means we have a representation of the group Fh by linear transformations. Using the group K given on the surface H, I can also determine precisely which representation occurs here. But in order to explain that I would have to introduce quadratic and reciprocal differentials first. Now R arises from R in a neighbourhood of h by identifying those elements .c1 ; : : : ; c / that are equivalent via this linear group. Unless we have represented all elements of Fh by the identity, c1 ; : : : ; c is not a unique coordinate system in R. In some cases we succeed in introducing a coordinate system 1 ; : : : ;  in R that at least in general is analytically related to c1 ; : : : ; c ; in other cases this is impossible. As an example we take a hyperelliptic surface (g > 1) in general position. Fh consists of two elements: the identity and the interchanging of sheets. For suitable coordinates c1 ; : : : ; c ( D 3g  3), c1 ; : : : ; c2g1 stay unchanged, whereas the remaining .g  2/ coordinates c2g ; : : : ; c3g3 change sign with the “interchanging of sheets”. In the case g D 2 one can keep c1 , c2 , c3 as coordinates for R. In the case g D 3 the c1 ; : : : ; c5 stay unchanged and just c6 changes sign; here one can set

1 D c1 ; : : : ; 5 D c5 ; 6 D c62 and has a useful coordinate system for R. However, already in the case g D 4 the coordinates c1 ; : : : ; c7 stay unchanged and c8 and c9 change sign simultaneously; here one cannot define a coordinate system 1 ; : : : ; 9 in a sensible way. The space R here has a substantial singularity. This is an a posteriori justification for having dealt with R rather than R right from the beginning. Now for a given h in R, we look at least for a system of possibly too many analytic functions 1 ; : : : ;  0 of c1 ; : : : ; c that are invariant under the above linear group, that

Chapter 18. Variable Riemann surfaces

803

means functions of R, with the property that every analytic function that is invariant under the linear group can be written as a power series in 1 ; : : : ;  0 . The word “analytic” here of course refers only to the place c1 D    D c D 0. According to a method introduced by Witt one can estimate the minimal number  0 of such functions

i from below. In the case of a hyperelliptic surface in general position one obtains 0 

g.g C 1/ : 2

For example, one can take the functions c1 ; : : : ; c2g1 I ci ck

.2g  i  k  3g  3/

. This is apparently related to the fact that the Riewhose total number equals g.gC1/ 2 mannian period matrix of integrals of first kind also provides g.gC1/ constants. In fact, 2 the period matrix is invariant under a map F in F corresponding to an interchanging of sheets on the hyperelliptic surface.

Chapter 19

A commentary on Teichmüller’s paper Veränderliche Riemannsche Flächen (Variable Riemann surfaces) Deutsche Math. 7 (1944), 344–359

Annette A’Campo-Neuen, Norbert A’Campo, Lizhen Ji, and Athanase Papadopoulos

The article Veränderliche Riemannsche Flächen is the last one that Teichmüller wrote on the problem of moduli. At most places the paper contains ideas and no technical details. The author presents a completely new approach to Teichmüller space, compared to the approach he took in his first seminal paper Extremale quasikonforme Abbildungen und quadratische Differentiale [25] (and its sequel [26] in which he completed some of the results stated in [25]). In the paper [25], Teichmüller led the foundations of what we call today Teichmüller theory (but without the complex structure), defining its metric and introducing in that theory the techniques of quasiconformal mappings and of quadratic differentials as essential tools. In the present paper, the approach is more abstract, through complex analytic geometry. Teichmüller space, equipped with its complex-analytic structure, is characterized in the paper that we consider here by a certain universal property. A fibre bundle is constructed, an object which today bears the name Teichmüller universal curve. It seems that Teichmüller considered that the methods of his 1939 paper [25] could not lead to the definition of a complex structure for Teichmüller space.1 The present paper is very different in spirit from the preceding ones, and it did not attract much attention, compared to his previous papers on the subject, which were thoroughly 1 One can say a posteriori that Teichmüller was wrong in that, since we know now that the complex structure of Teichmüller space can be defined using quasiconformal mappings. Let us quote in this respect Ahlfors, who was the first to derive the complex structure of Teichmüller space from the quasiconformal theory. He writes in [4]: “Teichmüller states explicitly, in 1944, that his metrization is of no use for the construction of an analytic structure. The present author disagrees and will show that the metrization and the corresponding parametrization are at least very convenient tools for setting up the desired structure.” We add by the way that we also know that conversely, the Teichmüller metric can be recovered from the complex structure, since Royden showed that the Teichmüller metric coincides with the Kobayashi metric. Today, the complex structure of Teichmüller space is usually presented using the variational theory of the Beltrami equation, which is part of the quasiconformal theory. In this theory, Teichmüller space is realized as a quotient space of the unit ball in a Banach space of measurable Beltrami differentials over a fixed Riemann surface. The origin of the idea is in Teichmüller’s 1939 paper [25] although, as we said, Teichmüller did not recognize this fact. For a concise survey on this point of view on the complex structure of Teichmüller space, we refer the reader to the paper [14] by Earle.

806 Anette A’Campo-Neuen, Norbert A’Campo, Lizhen Ji, and Athanase Papadopoulos analyzed and commented by Ahlfors, Bers and the schools they founded. There are several reasons for which this paper did not attract much attention and is very rarely cited in the literature. The first reason is probably that the paper is very sketchy and the arguments are (still) very difficult to understand. Another reason is that Teichmüller’s papers were examined by analysts and not by geometers, and the approach in this paper is algebro-geometric. A third reason is that the paper appeared in Deutsche Mathematik, an ephemeral journal founded by Bieberbach to which very few libraries outside Germany subscribed. The journal contained, in the first two issues, articles presenting the Nazi viewpoint on the influence of race on mathematics. About the difficulty of reading Teichmüller’s papers, we can quote Ahlfors from his 1954 paper On quasiconformal mappings [3], referring to the results of Teichmüller’s 1939 paper [25]: “For the sake of completeness we have not hesitated to reproduce some of Teichmüller’s reasonings almost without change. One good reason for this is that the Teichmüller papers are not easily available. Another reason is that it requires considerable effort to extricate Teichmüller’s complete and incontestable proofs from the maze of conjectures in which they are hidden.” The new approach to the moduli problem that Teichmüller presents in the paper that is the subject of this commentary is so different from the previous ones that it seems that Teichmüller himself was not sure that the space he obtains through the present techniques is the same as the space he introduced in his previous papers through the quasiconformal map approach. That the paper was not read carefully by other mathematicians is testified by the fact that the paper is very rarely cited in the mathematical literature, and that there are results and methods in this paper that were rediscovered later on without always referring to Teichmüller. Among these, we mention: (1) The existence and uniqueness of the universal Teichmüller curve, rediscovered later on by Ahlfors and by Bers. At the same time, this paper introduced the first fibre bundle over Teichmüller space.2 (2) The proof of the fact that the automorphism group of the universal Teichmüller curve is the extended mapping class group.3 (3) The idea of a fine moduli space.4 (4) The idea that Teichmüller space represents a functor.5 2 In the three papers [15], [11] and [17] which concern the universal Teichmüller curve there is no mention of the work of Teichmüller on the subject, and it is likely that the authors were not aware of it. (This has been confirmed to us by Earle, in his case). 3A proof of this fact is contained in a paper by Andrei Duma (1975) who apparently was not aware of the fact that Teichmüller had already proved this result, see [10]. Duma’s proof does not use Royden’s theorem on the complex structure automorphism group of Teichmüller space. Earle and Kra had already given a proof of that theorem, without being aware that it was known to Teichmüller, using Royden’s theorem. We thank Bill Harvey and Cliff Earle for bringing this out to our attention. See the MR review of Duma’s paper by Earle (MR 53 #11099) and the Zentralblatt review by Abikoff (Zbl 0355.32021). 4A fine moduli space is a fiber space where the isomorphism type of the fibre determines the point below it. 5 The notion of a representable functor and the fact that Teichmüller space was one of the first (and may be the first) interesting example of an analytic space representing a functor was later on developed by Grothendieck, see [16], see also [2].

Chapter 19. Commentary

807

(5) The idea of using the period map to define a complex structure on Teichmüller space. While commenting more on these facts, we shall give a quick review of the content of the paper. Teichmüller starts by recalling that it was known before him, but only by heuristic arguments, that the number of complex parameters for the set of equivalence classes of Riemann surfaces of genus g is 8 ˆ if g D 0; 1: He declares that several authors came up with these numbers using different methods, but that in reality these authors were not capable of saying precisely what they were counting. He considers that the fact that the various counts lead to the same value is a sort of a miracle, since the methods that were used are not rigorous. Teichmüller then says that these numbers, in order to be meaningful, should represent a dimension, and that in order to discuss the dimension of a set (in the present case, of moduli space), one has to turn this set into some “space with a notion of neighborhood”.6 It appears here that Teichmüller was the first to formulate in such precise terms the moduli problem for Riemann surfaces. Teichmüller then emphasizes that one should not primarily ask for an explicit representation of points in the moduli space via numbers in a coordinate system (an approach which seems to have been suggested by Riemann’s work), but that one should rather study the inner structure of that space. He then goes on saying that not only one would like to have on the set of moduli R the structure of a topological space or of an algebraic variety, but one would also like to have the structure of an analytic manifold, that is to say, of a nonsingular complex space. Teichmüller says that this is not possible because R “contains certain singular manifolds”. In modern language, this corresponds to the fact that moduli space is an orbifold and not a manifold. He therefore constructs a covering R of R that has no singularities. The space R is the space that was called later on Teichmüller space.7 Teichmüller then gives a short overview of his results and methods, and he notes that he will not be able to publish details “in the near future”.8 He announces that his solution to the problem of moduli is based on three newly introduced notions: (1) The “topological determination” of Riemann surfaces: This is the notion that we call today a “marking” of a Riemann surface. Here, a Riemann surface is 6 Teichmüller writes that “as long as R is not a space with a notion of neighborhood, it does not have a dimension in the sense of analysis or set theory”. In this context, the expression “set theory” means topology. 7 It was probably André Weil who first proposed the name “Teichmüller space”; see Weil’s comments in his Collected Papers ([28] Vol. II, p. 546), where Weil writes: “[...] this led me to the decision of writing up my observations on the moduli of curves and on what I called ‘Teichmüller space’ [...]”; see also the historical comments in [18]. 8 Teichmüller died the same year, at the age of 30.

808 Anette A’Campo-Neuen, Norbert A’Campo, Lizhen Ji, and Athanase Papadopoulos equipped with a fixed homotopy class of homeomorphisms from a fixed Riemann surface. We note that Teichmüller had already introduced markings in his 1939 paper [25]. (2) The notion of an analytic family of Riemann surfaces: This notion plays a central role in this paper as well as in the later developments of Teichmüller theory. It has been reintroduced later on by several authors working on moduli spaces, and below, we shall mention in particular Grothendieck. (3) The notion of “winding piece coordinates”: This is an operation of modifying the complex structure in the regular neighborhood of a simple closed curve in a Riemann surface. The complex automorphism group of the annulus is S 1 and the winding piece deformation can be considered in some sense as an ancestor of a complex analogue of the Fenchel–Nielsen deformation. After this introduction, Teichmüller defines the moduli space R and the action of the mapping class group on that space. He formulates the problem of moduli as, a priori, the “problem of asking for the properties of the space R”. He says that however, it turns out that it is better to study, rather than the space R, its covering R. A formulation of the problem of moduli is again given in a more precise form. Teichmüller then introduces the notion of an analytic n-dimensional manifold, defined by coordinate charts with holomorphic coordinate changes. It seems that this is one of the first appearances of such a definition in the mathematical literature. We can quote here Remmert ([23] p. 225): It seems difficult to locate the first paper where complex manifolds explicitly occur. In 1944 they appear in Teichmüller’s work on “Veränderliche Riemannsche Flächen” (Collected Papers, p. 714); here we find for the first time the German expression “komplex analytische Mannigfaltigkeit”. The English “complex manifold” occurs in Chern’s work ([9] p. 103); he recalls the definition (by an atlas) just in passing. And in 1947 we find “variété analytique complexe” in the title of Weil’s paper [27]. Overnight complex manifolds blossomed everywhere.

In any case, it is an interesting fact that the first example of a complex manifold of higher dimension (other than the example of a domain of C n , n  2) that appeared in the literature is precisely a space of (equivalence classes) of marked complex manifolds of dimension one.9 Teichmüller then introduces the notion of an analytic family of Riemann surfaces. In modern language, this is a fiber bundle M over an analytic base space B, the fibers being Riemann surfaces. The fiber bundle is locally trivial from the differentiable point of view (but not from the analytic point of view, since in a trivializing product neighborhood, two fibers are generally not isomorphic as Riemann surfaces). A particularly interesting analytic family of Riemann surfaces is the one where the base space B is Teichmüller space and where the fibre above each point is a marked Riemann surface representing the point itself. In this case the fiber bundle is called, in modern language, the universal Teichmüller curve, or the Teichmüller curve. 9 Of course, Teichmüller space was not yet known to be a complex domain. An embedding of that space in a C N was discovered later on.

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The Teichmüller curve has been re-introduced later on in the mathematical literature, in general with no reference to Teichmüller’s paper. The fibre bundle approach to Teichmüller space was expanded and made precise by several authors, see e.g. Ahlfors [5], Bers [7], [16] and Earle and Eells [12]. In Ahlfors’ paper 1961 [5] and in Bers’ 1961 paper [7] this fiber bundle is used to define the complex structure on Teichmüller space.10 The Teichmüller curve turned out to be an extremely important object in Hodge theory and it had a clear impact on Kodaira’s work on the deformation of complex manifolds, since it provided an important instance of the principle he discovered linking infinitesimal deformations of a compact complex manifold to the sheaf of holomorphic vector fields which in this case is the space of quadratic differentials. We refer the reader to Chapter 5, in particular §5.2 (p. 226 of the new edition) of Kodaira’s book [19]. Teichmüller’s aim was to show that the Teichmüller curve is a complex manifold of dimension 3g  2. In the general setting where the base B is an analytic manifold with fibers being Riemann surfaces, Teichmüller introduces a notion he calls permanent uniformizing local parameter,11 which is a local analytic coordinate system that gives an analytic parameter on each of the fiber Riemann surfaces. Thus, Teichmüller gets a coordinate t that works locally for a family of Riemann surfaces that are above points in B, which, together with the local r-dimensional parameters of B, produces a system of .r C 1/-dimensional parameters of M as an .r C 1/-complex manifold. He states that such permanent uniformizing parameters exist. The surfaces that are the fibers of the bundle over the space B are a priori not marked. Teichmüller shows that from a marking on one fiber one can obtain a marking on nearby fibers. He then says that by well-known principles, from the space B one can construct a “relatively unramified” covering B of B and where the surfaces above points of B are marked surfaces. Teichmüller then states an existence and uniqueness theorem for a globally analytic family of marked (Teichmüller says “topologically determined”) surfaces HŒc, where 10Ahlfors, in his paper, says about his method (p. 171) that “this approach is essentially due to Bers”, and Bers writes (p. 356) that “the existence of a ‘natural’ complex structure in Tg has been asserted by Teichmüller; the first proof was given by Ahlfors after Rauch showed how to introduce complex-analytic co-ordinates in the neighborhood of any point which is not a hyperelliptic surface. Other proofs are due to Kodaira–Spencer and to Weil”. Bers writes, at the beginning of the section on the analytic structure: “The results of this and the following section confirm and extend some of Teichmüller’s assertions in the paper Veränderliche Riemannsche Flächen. They also show that the complex-analytic structure defined above is natural and coincides with that of RauchAhlfors.” The tangent bundle sequence of the bundle of Riemann surfaces is at the basis of the Kodaira–Spencer theory of infinitesimal deformations of Riemann surfaces [20], which also provides a description of the complex structure of Teichmüller space. Grothendieck’s work on Teichmüller space is based on the consideration of analytic fiber bundles over surfaces, whose fibers are Riemann surfaces of genus g, cf. [16], Exposé 7 and ff. Finally, let us note that the fiber bundle approach was also used to define analytic vector bundles over Teichmüller space, the most natural one being obtained by taking above each point of the base space the tangent bundle of the surface in the fiber. The study of characteristic classes of fiber bundles over moduli space led to several important developments. We mention in this context the Mumford conjecture (solved by Madsen) and the Witten conjecture (solved by Kontsevich). 11 This term, an English translation of the German “permanente Ortsuniformisierende” is also used by Bers in [7].

810 Anette A’Campo-Neuen, Norbert A’Campo, Lizhen Ji, and Athanase Papadopoulos c runs over a  -dimensional complex analytic manifold C such that for any marked Riemann surface H of genus g there is one and only one c such that the Riemann surface H is conformally equivalent to an HŒc and such that the family HŒc satisfies the following universal property: If HŒp is any globally analytic family of Riemann surfaces with base B, there is a holomorphic map f W B ! C such that the family HŒp is the pull-back by f of the family HŒc. Teichmüller states that such a family HŒc exists and that it is essentially unique. From the context, and stated in modern terms, essential uniqueness means that the family is unique up to the action of the mapping class group. The complex analytic manifold C, which is the base space of the family HŒc, is the object that we call today Teichmüller space. This existence and uniqueness result was rediscovered by Grothendieck, who gave a complete proof of it in an algebro-geometric language that is different from Teichmüller’s. Grothendieck gave a series of talks on this subject at Cartan’s seminar (1960–1961), and mimeographed notes of these talks were circulated. Grothendieck’s statement is more general than that of Teichmüller; it is expressed in terms of a universal property, concerning (using Grothendieck’s wording) a “rigidifying functor” P relative to a discrete group  , which can be taken in particular as the “Teichmüller rigidifying functor”, and where  is the mapping class group. Grothendieck’s statement is the following (in this statement, T is Teichmüller space): Theorem 3.1. —There exists an analytic space T and a P -algebraic curve V above T [denoted by V =T ] which are universal in the following sense: For every P -algebraic curve X above an analytic space S, there exists a unique morphism g from S to T such that X (together with its P -structure) is isomorphic to the pull-back of V =T by g.12

Grothendieck deduces the following corollary (in which  is, as before, the mapping class group): 12 ([16] p. 7-08) [Théorème 3.1.— Il existe un espace analytique T , et une P -courbe algébrique V au-dessus de T , qui soient universels au sens suivant : Pour toute P -courbe algébrique X au-dessus d’un espace analytique S , il existe un morphisme et un seul g de S dans T , tel que X soit isomorphe (avec sa P -structure) à l’image inverse par g de V =T ]. It seems that Grothendieck had heard of Teichmüller’s papers, but like many others, he did not read them. Teichmüller’s work on the problem of moduli had nevertheless an enormous influence on Grothendieck, who declares, in the introduction to this paper (Exposé 7, whose title is: “An axiomatic description of Teichmüller space and its variants” [Description axiomatique de l’espace de Teichmüller et de ses variantes]: “In doing this, the necessity of rewriting the foundations of analytic geometry will become manifest”. [Chemin faisant, la nécessité deviendra manifeste de revoir les fondements de la Géométrie analytique]. It seems that the analysts working on Teichmüller theory had heard about Grothendieck’s work, but did not understand it. We can quote here Abikoff, in a report he published in 1989 in the Bulletin of the AMS [1], on an book by Nag: “First, algebraic geometers took us, the noble but isolated practitioners of this iconoclastic discipline, under their mighty wings. We learned the joys of providing lemmas solving partial differential and integral equations and various other nuts and bolts results. These served to render provable such theorems as: The ?%]$! is representable.” Let us also quote Ahlfors, from his 1964 survey on quasiconformal mappings [6] (p. 152), talking about Teichmüller’s 1944 paper: “In a final effort Teichmüller produced a solution of the structure problem, by an entirely different method, but it was so cumbersome that it is doubtful whether anybody else has checked all the details. [...] It is only fair to mention, at this point, that the algebraists have also solved the problem of moduli, in some sense even more completely than the analysts. Because of the different language, it is at present difficult to compare the algebraic and analytic methods, but it would seem that both have their own advantages.”

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Proposition 3.3. — Let X, X 0 be two P -curves above S, defined respectively by morphisms f , f 0 from S in T . Assume that S is connected and nonempty. Then  the set of S-isomorphisms X  ! X 0 for the underlying curves (without the P structures), is in canonical one-to-one correspondence with the set of u 2  satisfying f 0 D uN B f .13

This is a rigidity statement concerning the mapping class group. The mapping class group is canonically identified with the group of isomorphism classes of P -curves. This statement about the existence and uniqueness of the Teichmüller curve up to the mapping class group action can be considered as the first among a series of results that were obtained later on the rigidity of mapping class group actions, the next (and probably the most famous) one being Royden’s result stating that the automorphism group of the complex structure of Teichmüller space is the extended mapping class group, cf. [24]. The existence and uniqueness of the universal family was later on constructed independently by Ahlfors and Bers, see the historical remarks in [13]. Teichmüller considers that this theorem solves the moduli problem,14 and on this occasion he formulates more precisely this problem: The space R of all classes of analytic marked surfaces of genus g is made into a complex analytic manifold by identifying it with the base space C of the universal analytic family. This endows Teichmüller space at the same time with a complex analytic structure and with a topology.15 Teichmüller’s proof of this theorem uses tools from algebraic geometry, and in particular the algebro-geometric language of divisors, principal parts and places. He states a second theorem which is needed in the proof of the first one, a result concerning the “determination of a function by generalized systems of principal parts”. In this context, Riemann surfaces are studied through function fields. A Riemann surface is viewed as a field with degree of transcendence 1 over C. The places in the field give back the surface. We finally note that Grothendieck gave a formulation of this result of Teichmüller in the setting of categories and functors. More precisely, the statement is that the functor whose objects are families of marked curves over a complex manifold and whose morphisms are bundle maps is representable, and that Teichmüller space is the analytic space which represents this functor, cf. [16], Exposé 7, see also [2]. 13 ([16] p. 7-10) [Proposition 3.3.— Soient X; X 0 deux P -courbes au-dessus de S , définies respectivement par des morphismes f; f 0 de S dans T . Supposons S connexe non vide. Alors l’ensemble des S -isomorphismes  X  ! X 0 pour les courbes sous-jacentes (sans P -structures), est en correspondance biunivoque canonique avec l’ensemble des u 2  tels que f 0 D uN B f ]. 14 It is interesting to note that Grothendieck, after stating his main theorem (Theorem 3.1, stated above), makes a remark similar to that of Teichmüller (p. 7-10 of [16]): “We shall see in the next sections that the analytic space T , equipped with the automorphism group  , can be considered as a satisfying solution to the ‘moduli problem’ for curves of genus g” [Nous verrons dans les paragraphes suivants que l’espace analytique T , muni du groupe d’automorphismes  , peut être considéré comme une solution satisfaisante du “problème des modules” pour les courbes de genre g]. 15 Let us note that it was indeed considered, although this was not clearly stated, that the problem of moduli consists of defining a complex structure on moduli space. We can quote here Ahlfors, from his 1960 paper [4], in which he writes the following: “The classical problem [of moduli] calls for a complex analytic structure rather than a metric [...] The problem is not a clear cut one, and several formulations seem equally reasonable”.

812 Anette A’Campo-Neuen, Norbert A’Campo, Lizhen Ji, and Athanase Papadopoulos Teichmüller then introduces the notion of winding piece and the one of winding piece coordinate of a winding piece. This is a method of varying the complex structure of a surface which is given, in the tradition of Riemann, as a branched covering of the plane associated to an algebraic function. The analytic functions are determined by a “generalized system of principal parts”. The analytic structure of Teichmüller space is defined by varying the coefficients of these functions. Teichmüller states that his space R “consists of at most countably many connected parts”, and he “thinks that R in fact is simply connected”. Thus, he was not sure that the space R he defines in this paper is the same as the Teichmüller space he defined in his 1939 paper [25] using the quasiconformal theory. It is interesting to note here that Grothendieck solved this issue. Indeed, after introducing the definition of Teichmüller space using the universal property, as we recalled above, Grothendieck writes in ([16], 7-08): It is also easy to check, using if needed a paper by Bers [8], that the space we introduce axiomatically here (if this space exists, and we shall prove this fact) is isomorphic to the Teichmüller space of the analysts. It follows that Teichmüller space is homeomorphic to a ball, and therefore contractible, in particular connected and simply connected. A fortiori, the Jacobi spaces of all levels are connected, as is the moduli space M introduced in Section 5 as a quotient space of Teichmüller space. It seems that for the time being there is no algebro-geometric proof even of the fact that moduli space is connected (which we can interpret in algebraic geometry by saying that two curves of the same genus g are part of a family of algebraic curves parametrized by a connected algebraic variety).16

Then Teichmüller goes on studying the moduli space, that is, the space obtained by forgetting the marking. The description is very brief. (He says: “At last I want to mention briefly what happens when releasing the topological determination”.) He recalls the definition of the mapping class group together with its action on the space of equivalence classes of marked surfaces, he shows that this action is properly discontinuous, and he defines the moduli space to be the quotient of Teichmüller space by this action. He considers what he calls the exceptional points of R; they lie on a certain analytic submanifold of R. These points are stabilized by nontrivial finite groups of the mapping class groups, and they are obstructions for the moduli space to be a complex manifold. He shows that some of these points are substantial singularities, that is, moduli space is not a manifold at these points. In the last part of the paper, Teichmüller introduces the idea that the period matrices of differentials of the first kind can be used to study the complex structure of moduli space, and he states that hyperelliptic points may cause problems. This program on 16 [Il est d’ailleurs facile, de vérifier, utilisant au besoin un exposé de Bers [8], que l’espace introduit axiomatiquement ici (s’il existe, ce que nous prouverons) est isomorphe à l’espace de Teichmüller des analystes. Il en résulte que l’espace de Teichmüller est homéomorphe à une boule, et par suite contractile, en particulier connexe et simplement connexe. A fortiori, les espaces de Jacobi de tout échelon sont connexes, ainsi que l’espace des modules M introduit au paragraphe 5 comme un espace quotient de l’espace de Teichmüller. Il semble qu’il n’existe pas à l’heure actuelle de démonstration, par voie algébrico-géométrique, même du fait que l’espace des modules est connexe, (ce qui s’interprète en géométrie algébrique en disant que deux courbes algébriques de même genre g font partie d’une famille de courbes algébriques paramétrée par une variété algébrique connexe)].

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defining the complex structure via the period map was carried on later by several authors including Rauch, who defined a complex structure away from hyperelliptic Riemann surfaces and showed that moduli space has non-uniformizable singularities at the surfaces which admit non-trivial conformal automorphisms, see [21] and [22]. Ahlfors [5] also used the period map to define the complex structure of Teichmüller space, hence completing the program of Rauch. Acknowledgements. The authors are grateful to Bill Abikoff, Cliff Earle and Bill Harvey for valuable comments during the preparation of this commentary.

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W. Abikoff, Review of The complex analytic theory of Teichmüller space by S. Nag. Bull. Amer. Math. (N.S.) 21 (1989), no. 1, 162–168. 810

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N. A’Campo, L. Ji, and A. Papadopoulos, On Grothendieck’s construction of Teichmüller space. In In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume V, EMS Publishing House, to appear. 806, 811

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L. V. Ahlfors, On quasiconformal mappings. J. Analyse Math. 3 (1954), 1–58; correction, ibid., 207–208. 806

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L. V. Ahlfors, The complex analytic structure of the space of closed Riemann surfaces. Princeton Math. Ser. 24, 45–66 (1960); Collected papers, Vol. II, Contemp. Mathematicians, Birkhäuser, Boston, MA, 1982, 123–145. 805, 811

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L. V. Ahlfors, Some remarks on Teichmüller’s space of Riemann surfaces. Ann. of Math. (2) 74 (1961), 171–191; Collected papers, Vol. II, Contemp. Mathematicians, Birkhäuser, Boston, MA, 1982, 156–176. 809, 813

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L. V. Ahlfors, Quasiconformal mappings and their applications. In Lectures on Modern Mathematics, Vol. II, Wiley, NewYork 1963, 151–164; Collected papers, Vol. II, Contemp. Mathematicians, Birkhäuser, Boston, MA, 1982, 301–314. 810

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L. Bers, Spaces of Riemann surfaces. In Proc. Internat. Congress Math. 1958, Cambridge University Press, New York 1960, 349–361. 809

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L. Bers, Uniformization and moduli. In Contributions to function theory (Internat. Colloq. function theory, Bombay, 1960), Tata Institute of Fundamental Research, Bombay 1960, 41–49. 812

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S. S. Chern, Characteristic classes of Hermitian manifolds. Ann. of Math. (2) 47 (1946), 85–121. 808

[10] A. Duma, Die Automorphismengruppe der universellen Familie kompakter Riemannscher Flächen vom Geschlecht g  3. Manuscripta Math. 17 (1975), no. 4, 309–315. 806 [11] C. J. Earle, On holomorphic families of pointed Riemann surfaces. Bull. Amer. Math. Soc. 79 (1973), 163–166. 806 [12] C. J. Earle and J. Eells, A fibre bundle description of Teichmüller theory. J. Differential Geom. 3 (1969), 19–43. 809

814 Anette A’Campo-Neuen, Norbert A’Campo, Lizhen Ji, and Athanase Papadopoulos [13] C. J. Earle and A. Marden, On holomorphic families of Riemann surfaces. In Conformal dynamics and hyperbolic geometry, Contemp. Math. 573, Amer. Math. Soc., Providence, RI, 2012, 67–97. 811 [14] C. J. Earle, Teichmüller spaces as complex manifolds. In Teichmüller theory and moduli problem, Lecture Notes Series 10, Ramanujan Mathematical Society, Mysore, 2010, 5–33. 805 [15] M. Engber, Teichmüller spaces and representability of functors. Trans. Amer. Math. Soc. 201 (1975), 213–226. 806 [16] A. Grothendieck, Techniques de construction en géométrie algébrique. Séminaire Cartan, Paris 1960–61, Exposés 1–20. 806, 809, 810, 811, 812 [17] J. H. Hubbard, Sur les sections analytiques de la courbe universelle de Teichmüller. Mem. Amer. Math. Soc. 166 (1976), 137 p. 806 [18] L. Ji and A. Papadopoulos, Historical development of Teichmüller theory. Arch. Hist. Exact Sci. 67 (2013), no. 2, 119–147. 807 [19] K. Kodaira, Complex manifolds and deformation of complex structures. Reprint of the 1986 edition, transl. from the Japanese by Kazuo Akao, Classics Math., Springer, Berlin 2005. 809 [20] K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. I, II. Ann. of Math. (2) 67 (1958), 328–401, 403–466. 809 [21] H. E. Rauch, A transcendental view of the space of algebraic Riemann surfaces. Bull. Amer. Math. Soc. 71 (1965), 1–39; errata ibid. 74 (1968), 767. 813 [22] H. E. Rauch, Weierstrass points, branch points, and moduli of Riemann surfaces. Commun. Pure Appl. Math. 12 (1959), 543–560; addendum ibid. 165 (1960), 13. 813 [23] R. Remmert, From Riemann surfaces to complex spaces. In Matériaux pour l’histoire des mathématiques au XXe siècle (Proceedings of the colloquium to the memory of Jean Dieudonné, Nice, France, 1996), Sémin. Congr. 3, Soc. Math. France, Paris 1998, 203–241. 808 [24] H. L. Royden, Automorphisms and isometries of Teichmüller space. In Advances in the theory Riemann surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Stud. 66, Princeton University Press, Princeton, NJ, 1971, 369–383. 811 [25] O. Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale. Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. 1939 (1940), no. 22, 1–197; Gesammelte Abhandlungen –Collected papers, Springer, Berlin 1982, 337–531. 805, 806, 808, 812 [26] O. Teichmüller, Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen. Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. 1943 (1943), no. 4, 42 p.; Gesammelte Abhandlungen – Collected papers, Springer, Berlin 1982, 635–676. 805 [27] A. Weil, Sur la théorie des formes différentielles attachées à une variété analytique complexe. Comment. Math. Helv. 20 (1947), 110–116. 808 [28] A. Weil, Œuvres scientifiques – Collected papers. Three volumes. Springer, Berlin 1979, 2nd printing, 2009. 807

List of Contributors

Norbert A’Campo, Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland; email: [email protected] Annette A’Campo-Neuen, Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland; email: [email protected] Oliver Baues, Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstraße 3–5, 37073 Göttingen, Germany; email: [email protected] Xavier Buff, Université de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, 31062 Toulouse, France; email: [email protected] Marc Burger, Department Mathematik, ETH Zentrum, Rämistrasse 101, 8092 Zürich, Switzerland; email: [email protected] Guizhen Cui, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China; email: [email protected] Ren Guo, Department of Mathematics, Oregon State University, Corvallis, OR 973314605, U.S.A.; email: [email protected] Alessandra Iozzi, Department Mathematik, ETH Zentrum, Rämistrasse 101, 8092 Zürich, Switzerland; email: [email protected] Lizhen Ji, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A.; email: [email protected] Ralph Kaufmann, Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907, U.S.A.; email: [email protected] Zhong Li, School of Mathematical Sciences, Peking University, Beijing 100875, P. R. China; email: [email protected] Lixin Liu, Department of Mathematics, Sun Yat-sen (Zhongshan) University, 510275, Guangzhou, P. R. China, email: [email protected] Gaven J. Martin, Institute for Advanced Study, Massey University, Auckland, New Zealand; email: [email protected] Katsuhiko Matsuzaki, Department of Mathematics, School of Education, Waseda University, Nishi-Waseda 1-6-1, Shinjuku, Tokyo 169-8050, Japan; email: [email protected] Hideki Miyachi, Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka, 560-0043, Japan; email: [email protected]

816

List of Contributors

Ken’ichi Ohshika, Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, 560-0043, Osaka, Japan; email: [email protected] Athanase Papadopoulos, Institut de Recherche Mathématique Avancée, CNRS et Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France; email: [email protected] Hugo Parlier, Département de mathématiques, Université de Fribourg, Chemin du Musée 23, 1700 Fribourg, Switzerland; email: [email protected] Hiroshige Shiga, Department of Mathematics, Tokyo Institute of Technology, O-okayama, Meguro-ku Tokyo, 152-8551, Japan; email: [email protected] Weixu Su, Department of Mathematics, Fudan University, 200433, Shanghai, P. R. China, email: [email protected] Lei Tan, Faculté des Sciences, LAREMA, Université d’Angers, 2 Boulevard Lavoisier, 49045 Angers cedex 01, France; email: [email protected] Cormac Walsh, INRIA Saclay & Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau, France; e-mail: [email protected] Anna Wienhard, Mathematisches Institut, Ruprecht-Karls Universität Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany; email: [email protected] Sumio Yamada, Department of Mathematics, Gakushuin University, 1-5-1 Mejiro, Toshima, Tokyo 171-8588, Japan; email: [email protected]

Index AB-removable, 449 adherence height, 252 adherence number, 252 admissible density, 626 A-harmonic functions, 655 Alexandrov angle, 95 Alexandrov tangent cone, 95 algebra BV, 317 Gerstenhaber, 317 pre-Lie, 317 almost-geodesic, 332 analytic definition of extremal length, 208 analytically finite, 684 Anosov representation, 599 Anosov structure, 599 arc essential, 260 family, 261 exhaustive, 263 multi-arc, 270 quasi-filling, 264 untwisted, 275 graph, 264 operad, 270 arc family, 261 Arc (operad), 270 Arens–Eells space, 408 arithmetic group S-, 164 Ascoli–Arzelà Theorem, 331 asymptotic cone, 163 asymptotic dimension mapping class group, 174 asymptotic Teichmüller distance, 701 asymptotic Teichmüller modular group, 701

asymptotic Teichmüller space, 701 asymptotically conformal, 701 conformal harmonic map, 82 conformal mapping class, 702 elliptic modular transformation, 702 equivalent, 701 trivial, 701 trivial mapping class group, 701 Atiyah–Singer, 648 attracting, 748 attracting basin, 723 augmented Teichmüller space, 88 b-group, 247, 441 bands, 262 Beltrami differential, 418 extremal, 419 equation, 49 system, 641 Benzécri theorem, 467 Bergman cocycle, 588 Bers boundary, 247 reduced, 251 compactification, 247 constant, 128 embedding, 247, 441 slice, 247 biset, 746 blowing up an arc, 745 Bolza curve, 122 boundary Bers, 247 Gardiner–Masur, 217, 386 horofunction, 223, 330 reduced Bers, 251 Thurston, 238, 328

818 boundary at infinity, 685 boundary group, 441 boundary map, 596 bounded cohomology, 561 bounded Euler class, 561 bounded Euler class of an action, 561 bounded Euler number, 563, 564 bounded geometry condition, 686 bounded Kähler class, 565, 577 bounded Toledo invariant, 576 bounded Toledo number, 567 bounded type, 689 building, 147, 153 algebraic group, 144 automorphism group, 153 Euclidean, 142 hyperbolic, 142 Tits, 137 Busemann point, 332, 337, 417 BV algebra, 278, 279 up to homotopy, 278 equation, 285, 286 operator, 284 Cact, 280 cacti, 280 spineless, 280 with stops, 296 Cacti, 280 Calabi–Markus phenomenon, 470 captures, 745 Carathéodory theorem, 99 Cartan–Hadamard theorem, 96 Casimir element, 303, 317 CAT(0) space, 88 Cauchy–Riemann system, 640 cellular chains for Arc, 272 central extension, 550 character variety, 480, 524 Chekhov–Fock algebra, 765 Chern conjecture, 468

Index

classification of mapping class subgroups, 92 closed geodesic simple, 115 closure equivalence, 709 cochain Hochschild, 302, 318 cohomology Hochschild, 319 collar lemma, 732 combinatorially equivalent, 728 compact C 1 -topology, 473 compact-open topology, 692 compactification Q, 395 Bers, 247 Gardiner–Masur, 217, 361, 386 geodesic, 139, 165 Hausdorff, 394 horofunction, 223, 361, 405 Lipschitz, 395 locally symmetric spaces, 155 moduli space, 181 of Hausdorff space, 394 Teichmüller, 244 Thurston, 238, 328, 333, 361 complete geodesic lamination, 336 conformal mapping class, 696 conformal structure, 486 measurable, 650 conical limit point, 670 conjecture Chern, 468 Deligne, 304 cyclic version, 306 ending lamination, 249 Lichnerowicz, 664 Martio, 644 Novikov, 175 integral, 178 convergence group, 668 convergence property, 668 convex core, 762

Index

convex map, 597 convex real projective structure, 607 coordinate change isomorphism, 765 coordinates Fenchel–Nielsen, 115 correlation function, 307, 308 Coxeter group, 96 Coxeter matrix, 96 cross ratio, 605, 633 cubical complex, 96 cup i-product, 300 current geodesic, 230 Liouville, 231 curve complex, 89, 137, 138, 144, 150, 168, 348 automorphisms, 168, 348 boundary of Teichmüller space, 181 compactification, 181 presentation of mapping class group, 186 subcomplex, 152 curve family, 626 modulus, 626 cusp, 442, 762 cyclic operad, 315 De Giorgi–Nash–Moser iteration, 79 de Leeuw map, 409 defect, 572 deformation of lattice, 483 deformation space, 726 branched covering of the sphere, 726 holonomy map, 462, 524 non-closed point of, 486, 519 of .X; G/-structures, 478 of complete .X; G/-structures, 482 of complete flat affine structures, 483, 488 of flat affine structures, 462, 483, 524

819

of lattice, 484 orientation components of, 479 stratification of, 484, 519, 523 deformation theorem, 475 degree-one monotone, 559 Dehn–Nielsen–Baer Theorem, 545 Deligne conjecture, 303, 304 A1 cyclic version, 306 A1 version, 305 cyclic version, 306 N @-energy, 81 detour cost, 337 detour metric, 341 development image, 462 of flat affine structure, 506 development map, 462 topological rigidity of, 476 development maps convergence of, 473 deformation of, 474 for .X; G/-structures, 469 for flat affine two-torus, 505, 509 space of, 473, 482 development pair, 469, 474 diagonal representation, 610 diagonal exchange, 761, 772 diagram automorphism, 100 diffeomorphism group action on flat affine structures, 487 contractible, 487 manifold, 478, 486 proper action of, 486 surface, 487 two-torus, 487 Diophantine equation, 447 discontinuous, 667 discontinuously, 688 weakly, 688 discrete quasiconformal group, 667 distortion function, 633 distortion tensor, 641 divergence, 748 divergence-free tensor, 54

820 divergent type, 689 domain of discontinuity, 476 Donaldson–Sullivan, 647 dual norm, 721 duality property arithmetic group, 159 earthquake, 240 elementary, 668, 685 elliptic modular transformation, 696 embedding Bers, 247 Gardiner–Masur, 386 end, 692 end compactification, 692 ending lamination, 248 ending lamination conjecture, 170, 249 energy functional, 603 ergodic component, 342 escaping sequence of mapping classes, 703 essential arc, 260 essentially trivial, 691 étale representation, 472, 485 affine, 492, 516 Euclidean building, 163 compactification, 165 Euler characteristic, 467 Euler class, 551, 552, 610 bounded, 561 Euler element, 307, 320 Euler number, 549, 552 bounded, 563, 564 extended mapping class group, 328 extremal Beltrami differential, 419 extremal length, 207, 358, 384 analytic definition, 208 generalized, 226 geometric definition, 208 of simple closed curves, 208 extremal metric, 207 Fat

Cact, 296

Index

Cacti, 296 Fatou component periodic, 723 wandering, 723 Fatou set, 661, 718 Fenchel–Nielsen coordinates, 115 finite rank, 97 finite topological type, 558, 685 flat G-bundles, 549 flat affine manifold, 461 complete, 463, 483 homogeneous, 491 flat affine surface, 467 flat affine torus, 467, 492 complete, 483 homogeneous, 490–492, 504, 516, 522 non-homogeneous, 511, 520 translation invariant, 516 flat structure, 452 flat surface, 452 foliation, 265 horizontal, 210 partial measured, 265 vertical, 210 formal invariants, 748 framed little discs operad, 279 Fricke space, 540, 544 Frobenius algebra, 302 comultiplication, 320 definition, 319 semi-simple, 313 Fuchsian representation, 610 Funk metric, 103 variational formulation of, 104 funnel, 762 Fat

Gardiner–Masur boundary, 217, 386 compactification, 217, 361, 386 embedding, 386 Gauß–Bonnet theorem, 467 .G; D/-structure, 543

Index

generalized extremal length, 226 geodesic, 331 geodesic compactification, 139, 165 geodesic completion, 95 geodesic current, 230, 333 geodesic lamination, 237 complete, 336 geometric definition of extremal length, 208 geometric limit, 249 geometric structure, 606 geometrically equivalent, 454 Gerstenhaber algebra, 278, 279, 317 bracket, 284 gluing foliation, 266 twisted, 276 weighted arc family, 266 grand orbit, 721 graph arc, 264 intersection, 304 loop, 265 ribbon, 264 Gromov norm, 572, 573 group b-, 441 boundary, 441 mapping class, 441 quasiconformal, 666 quasifuchsian, 441 symmetric, 269 totally degenerate, 442 Veech, 452 Grötzsch ring, 631 .G; X /-structure, see also .X; G/ structure, 598 Hamilton sequence, 420 harmonic map asymptotically conformal, 82 harmonic maps, 70

821

harmonic maps from Kähler manifolds, 82 harmonic polynomial, 743 Hausdorff compactification, 394 Hempel distance, 178 Heegaard splitting, 178 Herman ring, 723 Hermitian symmetric space, 552 Hermitian triple product, 589 Hermitian type, 576 Higgs bundle, 609 higher integrability, 641 higher Teichmüller space, 541, 575 Hilbert manifold, 62 Hilbert metric, 103 Hilbert transform, 62 Hitchin component, 593 representation, 593 Hochschild cochain, 280, 302, 318 cohomology, 319 holomorphic family of Riemann surfaces, 439, 442 holomorphic index, 748 holomorphic motion, 449 holomorphic quadratic differential, 55, 419 holonomy convergence of, 474 discontinuous, 476 non-discrete, 495 holonomy homomorphism, 462, 469, 476, 543 holonomy map, 462, 517 for flat affine structures, 481, 524 is a local homeomorphism, 480, 517, 524 is not a local homeomorphism, 481, 532 homogeneous quasimorphism, 562, 579 homomorphism positive, 587

822 tight, 586 Hopf torus, 486, 519 horizontal foliation, 210 horocyclic foliation, 336 horofunction, 330 horofunction boundary, 330 horofunction compactification, 361, 405 hybrid representation, 610 hyperbolic metric, 730 hyperbolic rational map, 718 hyperbolic shadow, 770 hyperbolic structure, 542 hyperboloid model in extremal length geometry, 228 in geodesic currents, 231 hyperelliptic surface, 122 ideal triangulation, 595, 760, 777 decorated, 771 inequality Milnor–Wood, 555 Milnor–Wood type, 576 infinite topological type, 685 intermediate Teichmüller space, 712 intersection number in extremal length geometry, 227 invariant component, 441 irrationally indifferent, 748 irreducible, 444 Jenkins–Strebel differential, 209 joint continuity, 332 Julia set, 661, 718 Kashaev algebra, 771 generalized, 777 Kashaev coordinates, 774 Kerckhoff formula, 215 k-fold loop space, 279 Kähler class, 554 bounded, 565, 577 Kähler form, 577

Index

lamination ending, 248 geodesic, 237 measured, 237 projective, 237 Lattès examples, 746 lattice deformation of, 483 rigidity, 484 smoothly rigid, 484 Lefschetz fibration, 94 length function, 115 length spectrum simple, 115 Levi-Civita connection for the L2 -pairing, 51 Lichnerowicz conjecture, 664 operator, 53 problem, 665 Lie group Hermitian type, 576 split, 592 limit point, 687 limit set, 668, 687 linear distortion, 625 Liouville current, 231 Liouville theorem, 639 Lipschitz algebra, 376 compactification, 395 constant, 381 function, 381 metric, 328 Lipschitz metric reversed, 329 little discs operad, 279 locally homogeneous space, 469 loop space, 280, 281 lower boundedness condition, 686 lower semicontinuous, 625 L2 -decomposition for TG M, 55

Index

L2 -decomposition theorem for the universal Teichmüller space, 58 L2 -pairing, 50 manifold .X; G/-, 469, 470 diffeomorphism group of, 478, 486 flat affine, 461 map convex, 597 positive, 597 mapping class asymptotically conformal, 702 conformal, 696 mapping class group, 48, 356, 441, 479, 542, 544, 601 asymptotic dimension, 174 asymptotically trivial, 701 cohomological dimension, 182 duality property, 182 pure, 260, 693 quasi-isometric rigidity, 173 quasiconformal, 685 stable, 691 mapping class subgroup closed, 692 discrete, 692 stationary, 693 mark rotation, 772 Martio conjecture, 644 Maslov cocycle, 589 mating of polynomials, 744 maximal measured lamination, 334 maximal representation, 582 measured foliation, 359, 384 space, 205 measured lamination, 237, 333 maximal, 334 uniquely ergodic, 334 meromorphic quadratic differential, 721

823

metric Funk, 103 Hilbert, 103 Thurston asymmetric, 106 variational formulation of Teichmüller, 105 Weil–Petersson, 178 Weil–Petersson Funk, 104 metric completion of Teichmüller space, 88 Milnor–Wood inequality, 555 Milnor–Wood type inequality, 576 minimal domains, 342 modified Teichmüller distance, 229 modular group, 328, 722 modular operad, 274, 316 modular torus, 118 modular transformation asymptotically elliptic, 702 elliptic, 696 modularly equivalent, 246 moduli space, 48, 289, 291, 441 enlarged, 712 geometric, 709 of .X; G/-structures, 478 of flat affine structures, 462 simplicial volume, 185 stable, 709 topological, 708 moduli space of a rational map, 722 moduli space of stable points, 708 modulus of annulus, 208, 384 modulus of an annulus, 730 monodromy, 439, 443 monodromy representation, 770 monoidal functor, 271 Montel theorem, 653 Mordell conjecture, 447 Mostow rigidity, 153 generalization, 155 multi-arc family, 270 multicurve, 729

824 multiplier of a cycle, 747 Mumford–Mahler compactness theorem, 80 Möbius transformation, 639 Nielsen realization theorem, 696 non-peripheral curve, 728 non-symmetric metric space, 330 norm-closed, 383 normal Lipschitz algebra, 383 normal family, 635 Novikov conjecture, 175 integral, 178 NPC space, 88 number Euler, 549 obstruction class, 549, 553 open/closed, 313 operad, 314, 318 Arc, 270 K-modular, 276 k-cubes, 279 cacti, 280 cellular for Arc, 272 cyclic, 315 framed little discs, 279 little discs, 279 modular, 274, 316 spineless cacti, 280 operation [i , 301 orbifold signature, 727 orbit, 718 order-complete, 383 ordinary set, 668 orientation cocycle, 566 pants decomposition, 115, 686 parabolic, 748 parabolic basin, 723 parabolic multiplicity, 748 parallel arcs, 260 Patterson Theorem, 329

Index

Penner coordinates, 774 periodic Fatou component, 723 peripheral curve, 728 Picard theorem, 652 Poincaré inequality, 655 Poincaré fundamental polygon theorem, 495 polynomial core, 770 positive map, 597 positive representation, 594 positivity, 595, 611 postcritically finite, 718, 725 preferred coordinate, 748 projective lamination, 237 projective measured foliation, 384 space, 205 projective structure, 607 PROP, 280, 291, 316 quasi, 293 proper group action, 486 proper metric space, 330 pseudo-periodic, 446 pure mapping class group, 260, 693 Q-compactification, 395 qr-elliptic, 665 quadratic differential holomorphic, 419 quantization of Teichmüller space, 760 quasi-Fuchsian group, 170 quasi-isometric rigidity lattices, 173 mapping class group, 174 quasi-Möbius, 635 quasi-transverse, 342 quasiconformal lim inf definition, 625 analytic definition, 637 elliptic, 668 geometric definition, 627 group, 666 linear dilatation, 624 loxodromic, 668

Index

parabolic, 668 quasiconformal extension, 648 quasiconformal group discrete, 667 quasiconformal mapping class group, 685 quasiconformal structure, 644 quasifuchsian group, 441 quasiisometric embedding, 602 quasimorphism, 556, 572 homogeneous, 562, 579 translation, 578 quasiregular, 638 quasisymmetry, 635 R-tree, 242 résidu itératif, 748 rational representative, 728 recurrent point, 687 reduced Bers boundary, 251 reduced Teichmüller space, 711 reducible, 444 refined Jordan decomposition, 579 region of discontinuity, 688 region of stability, 688 relative Teichmüller space, 711 removable singularity, 658 repelling, 748 representation étale, 472, 485 Anosov, 599 diagonal, 610 Fuchsian, 610 Hitchin, 593 hybrid, 610 maximal, 582 positive, 594 twisted diagonal, 610 representation variety, 544 Reshetnyak theorem, 638 reverse Hölder inequality, 642 reversed Lipschitz metric, 329 rigidity

lattice, 484 of development maps, 476 rotation number, 562, 580 Royden theorem, 329 self-adjoint, 382 semiconjugate, 560 separate points uniformly, 389 Shafarevich conjecture, 447 shear coordinates, 762 exponential, 763 Shilov boundary, 588 Siegel disk, 723 simple closed geodesic, 115 simple length spectrum, 115 simplicial volume, 160 moduli space, 185 slice diffeomorphism group, 486 slice theorem, 485 Slodkowski theorem, 449 space form, 470 split real Lie group, 592 stabilization, 298, 313 stable commutator length, 572, 574 stable mapping class group, 691 stable moduli space, 709 stable point of reductive group action, 485 stably, 688 weakly, 688 stationary, 693 Stiefel–Whitney class, 610 stops, 296 stratification, 90 Strebel point, 417 stretch line, 336 string topology, 280, 291, 309 t GTree.n/, 298 stump, 337 sub-additive property of extremal length, 213 Sullivan uniformisation theorem, 644

825

826 Sullivan quasi-PROP, 291 superattracting, 748 superattracting basin, 723 surface diffeomorphism group of, 487 flat affine, 467, 491 hyperbolic, 480 windowed, 260 symmetric group, 269 symmetrisation, 632 symmetrised metric, 330 symplectic group, 598 systole, 121 systole function, 121 tangent cone Teichmüller space, 183 tangent space, 735 Teichmüller compactification, 244 Teichmüller Coxeter complex, 95 Teichmüller curve, 453 Teichmüller disk, 452 Teichmüller distance, 204, 385, 440, 685 asymptotic, 701 modified, 229 Teichmüller geodesic, 214 Teichmüller geodesic ray, 214 Teichmüller metric, 358, 420, 489, 721 Teichmüller modular group, 48, 686 asymptotic, 701 Teichmüller ring, 631 Teichmüller space, 47, 204, 358, 384, 418, 440, 479, 480, 489, 685, 762 asymptotic, 701 augmented, 88 classifying space, 179, 182 decorated, 774 enhanced, 762 higher, 541 intermediate, 712

Index

isometry group, 169 partial compactification, 179 quantization, 760 quantum, 767 reduced, 711 relative, 711 tangent cone at infinity, 183 thick part, 182 universal, 48, 418 Teichmüller space of a rational map, 722 Teichmüller space of torus, 73 Thompson group, 707 Thurston metric, 106 Thurston algorithm, 730 Thurston boundary, 238, 328 Thurston classification theorem of mapping classes, 92 Thurston compactification, 238, 328, 333, 361 Thurston contraction principle, 727 Thurston deformation theorem, 475 Thurston metric, 328 Thurston obstruction, 728, 729 Thurston pullback map, 724 tight homomorphism, 586 Tits building, 137 axioms, 141 finite, 137 geometric realization, 140 types, 137 Toledo invariant, 576 bounded, 576 Toledo number, 553 bounded, 567 topological polynomial, 744 topological rigidity, 651 topology compact C 1 -, 473 torus diffeomorphism group of, 487 flat affine, 492 totally degenerate group, 442 totally transverse, 336

Index

trace-free tensor, 54 train track, 239 transfer map, 565 transition matrix, 729 translation quasimorphism, 578 transverse tensor, 54 triangulation ideal, 595 triple space, 669, 670 trivial near infinity, 691 truncation of Lipschitz function, 382 tube type domain, 585 twisted diagonal representation, 610 twisting, 275 .2; 2; 2; 2/-map, 727 uniformisation theorem Sullivan, 644 uniquely ergodic, 220, 360 measured lamination, 334 universal Teichmüller space, 48, 418 unmeasured lamination space, 251 upper boundedness condition, 686 variational formulation of Funk metric, 104 variational formulation of Teichmüller metric, 105 Veech group, 452 Veech surface, 453 vertical foliation, 210 wandering Fatou component, 723 weakly discontinuously, 688

827

weakly maximal, 584 weakly stably, 688 weight function, 263 Weil–Petersson geodesic completion, 95 geodesic equation, 64 Weil–Petersson completion of Teichmüller space, 88 Weil–Petersson complex structure, 61 Weil–Petersson convex body, 102 Weil–Petersson convexity of energy, 75 Weil–Petersson Funk metric, 104 Weil–Petersson metric, 178 Weil–Petersson Poisson structure, 779 Weil–Petersson potential, 84 well-displacing, 602 Whitehead equivalence, 336 wilted leaves, 277 windowed surface, 260 .X; G/-space form, 470 .X; G/-structure, see also .G; X /-structure complete, 470, 482 flat affine, 482, 483, 490 homogeneous, 489 hyperbolic, 482 marked, 488 on surfaces, 471 space of, 481 Yang–Mills, 647 Zalcman lemma, 654