Handbook of Teichmuller theory, Vol.5 978-3-03719-160-6, 3037191600

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IRMA Lectures in Mathematics and Theoretical Physics 26

This volume is the fifth in a series dedicated to Teichmüller theory in a broad sense, including the study of various deformation spaces and of mapping class group actions. It is divided into four parts:

The topics covered include identities for the hyperbolic geodesic length spectrum, Thurston’s metric, the cohomology of moduli space and mapping class groups, the Johnson homomorphisms, Higgs bundles, dynamics on character varieties, and many others. Besides surveying important parts of the theory, several chapters describe conjectures and open problems. The last part consists of two fundamental papers by Teichmüller, translated into English and accompanied by mathematical commentaries. The chapters, like those of the other volumes in this collection, are written by experts who have a broad view on the subject. They have an expository character (which fits with the original purpose of this handbook), but some of them also inlcude original and new material. The handbook is addressed to researchers and to graduate students.

ISBN 978-3-03719-160-6

www.ems-ph.org

Papadopoulos V | IRMA 26 | FONT: Rotis Sans | Farben: Pantone 287, Pantone 116 | 170 x 240 mm | RB: 34 mm

Volume V

Part A: The metric and the analytic theory. Part B: The group theory. Part C: Representation theory and generalized structures. Part D: Sources.

Handbook of Teichmüller Theory

Volume V Athanase Papadopoulos, Editor

Athanase Papadopoulos, Editor

Handbook of Teichmüller Theory

Handbook of Teichmüller Theory Volume V Athanase Papadopoulos Editor

IRMA Lectures in Mathematics and Theoretical Physics 26 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: 6 Athanase Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature 7 Numerical Methods for Hyperbolic and Kinetic Problems, Stéphane Cordier, Thierry Goudon, Michaël Gutnic and Eric Sonnendrücker (Eds.) 8 AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, Oliver Biquard (Ed.) 9 Differential Equations and Quantum Groups, D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah and R. Schäfke (Eds.) 10 Physics and Number Theory, Louise Nyssen (Ed.) 11 Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.) 12 Quantum Groups, Benjamin Enriquez (Ed.) 13 Handbook of Teichmüller Theory, Volume II, Athanase Papadopoulos (Ed.) 14 Michel Weber, Dynamical Systems and Processes 15 Renormalization and Galois Theory, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis (Eds.) 16 Handbook of Pseudo-Riemannian Geometry and Supersymmetry, Vicente Cortés (Ed.) 17 Handbook of Teichmüller Theory, Volume III, Athanase Papadopoulos (Ed.) 18 Strasbourg Master Class on Geometry, Athanase Papadopoulos (Ed.) 19 Handbook of Teichmüller Theory, Volume IV, Athanase Papadopoulos (Ed.) 20 Singularities in Geometry and Topology. Strasbourg 2009, Vincent Blanlœil and Toru Ohmoto (Eds.) 21 Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series, Kurusch Ebrahimi-Fard and Frédéric Fauvet (Eds.) 22 Handbook of Hilbert Geometry, Athanase Papadopoulos and Marc Troyanov (Eds.) 23 Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics, Lizhen Ji and Athanase Papadopoulos (Eds.) 24 Free Loop Spaces in Geometry and Topology, Janko Latschev and Alexandru Oancea (Eds.) 25 Takashi Shioya, Metric Measure Geometry. Gromov‘s Theory of Convergence and Concentration of Metrics and Measures Volumes 1–5 are available from De Gruyter (www.degruyter.de)

Handbook of Teichmüller Theory Volume V 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 978-3-03719-160-6 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.

© 2016 European Mathematical Society

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Foreword Teichmüller theory is a vast subject which encompasses ideas and techniques from complex analysis, hyperbolic geometry, topology, partial differential equations, algebraic geometry, Kähler geometry, geometric group theory, representation theory, dynamical systems, number theory and from other fields, with applications in mathematical physics (string theory, conformal and topological field theories, two-dimensional gravity, etc.) and, more recently, in biology. Besides the “classical” Teichmüller theory, there is a “quantum Teichmüller theory”, a “discrete Teichmüller theory”, and a “higher Teichmüller theory.” This Handbook project arouse from the desire of collecting in a systematic way a set of surveys covering all these theories and making them easily accessible to researchers and students. The result is due to the effort of many people, and above all the hundred authors who contributed to the various volumes which already exist in print. Let them all be thanked here. This collective work reflects the fact that Teichmüller theorists form a community. At the same time, skimming through all these pages gives a profound feeling of unity in mathematics. The present volume is divided into the following four parts:  Part A. The metric and the analytic theory, 5  Part B. The group theory, 4  Part C. Representation theory and generalized structures, 3  Part D. Sources, 2 The number after the name of each part indicates that it is a sequel to a part carrying the same name (with a different numbering) in a previous volume of the Handbook. Athanase Papadopoulos Strasbourg and Tokyo, November 2015

Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction to Teichmüller theory, old and new, V by Athanase Papadopoulos : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

1

Part A. The metric and the analytic theory Chapter 1. Identities on hyperbolic manifolds by Martin Bridgeman and Ser Peow Tan : : : : : : : : : : : : : : : : : : : : : : : : :

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Chapter 2. Problems on the Thurston metric by Weixu Su : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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Part B. The group theory Chapter 3. Meyer functions and the signature of fibered 4-manifolds by Yusuke Kuno : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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Chapter 4. The Goldman–Turaev Lie bialgebra and the Johnson homomorphisms by Nariya Kawazumi and Yusuke Kuno : : : : : : : : : : : : : : : : : : : : : : : : : :

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Chapter 5. A survey of the Johnson homomorphisms of the automorphism groups of free groups and related topics by Takao Satoh : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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Part C. Representation theory and generalized structures Chapter 6. Geometry and dynamics on character varieties by Inkang Kim : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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Chapter 7. Compactifications and reduction theory of geometrically finite locally symmetric spaces by Lizhen Ji : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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Chapter 8. Representations of fundamental groups of 2-manifolds by Lisa Jeffrey : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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Contents

Part D. Sources Chapter 9. Extremal quasiconformal mappings and quadratic differentials by Oswald Teichmüller : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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Chapter 10. A commentary on Teichmüller’s paper Extremale quasikonforme Abbildungen und quadratische Differentiale by Vincent Alberge, Athanase Papadopoulos, and Weixu Su : : : : : : : : : : : :

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Chapter 11. Determination of extremal quasiconformal mappings of closed oriented Riemann surfaces by Oswald Teichmüller : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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Chapter 12. A commentary on Teichmüller’s paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen by Annette A’Campo-Neuen, Norbert A’Campo, Vincent Alberge, and Athanase Papadopoulos : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 569 List of contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents 1 2

3

4

5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part A, The metric and the analytic theory, 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Identities on hyperbolic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Problems on Thurston’s metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part B. The group theory, 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Meyer cocycle and Meyer functions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Torelli–Johnson–Morita theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Johnson homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part C. Representation theory and generalized structures, 3 . . . . . . . . . . . . . . . . . . . 4.1 Dynamics of representations in Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Compactifications and coarse fundamental domains for locally symmetric spaces associated with Anosov subgroups of noncompact semisimple Lie groups . . . . 4.3 Moduli spaces of flat connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part D. Sources, 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 2 4 5 5 7 8 9 10 11 12 13

1 Introduction Teichmüller theory is a broad subject whose roots lie in the work of Bernhard Riemann who introduced in his doctoral dissertation (1851) an equivalence relation on the set of Riemann surfaces and stated that the number of (complex) “moduli” for the set of equivalence classes, in the case of a Riemann surface of genus g  2, is 3g  3. Several theories developed by major mathematicians during the two hundred years that followed this assertion were motivated to a large extent by the effort to give a precise meaning to this moduli count. We call this problem the “Riemann moduli problem.” It was given a huge impetus by Oswald Teichmüller who wrote several papers between 1937 and 1943, constructing a space (that was called later on “Teichmüller space”) equipped with various structures, including a Finsler metric and a complex-analytic structure of the right dimension, thus giving to Riemann’s count the status of a complex dimension and in this sense solving Riemann’s problem. At the same time, Teichmüller’s papers paved the way to several geometrical and analytical results on spaces of equivalence classes of Riemann surfaces and to

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the develoment of several theories by major mathematicians of the twentieth century, including Ahlfors, Bers, Weil, Grothendieck, Thurston, and there are many others. The present volume, like the preceding ones in this series, contains a collection of surveys on classical Teichmüller theory and on some of its modern developments. Certain topics that are considered were already treated from a different point of view in previous volumes. As a matter of fact, some of the chapters may be considered as sequels to chapters in the preceding volumes, surveying some particular subject in more detail or from a new perspective. For instance, Chapter 2, which consists of a collection of commented open problems on Thurston’s metric on Teichmüller space, is a natural sequel to Chapter 2 of Volume I which is a survey of that metric, written eight years ago, and which already contains some open problems. Some progress on that theory has been made during these years, and one of the main advances is the characterization of the horofunction boundary and the isometry group of this metric. These results are contained in Chapter 7 of Volume IV of this Handbook. Some of the material in Chapter 3 on the Meyer cocycle and Meyer functions was already mentioned in Chapter 6 of Volume II, whose subject is the cohomology of the mapping class group. Likewise, Chapter 5 which is a survey of the Johnson homomorphisms for automorphisms of free groups and more general groups, is a sequel to a section in Chapter 7 of Volume I where the Johnson homomorphisms for mapping class groups are studied. Chapter 6 on the geometry and dynamics on character varieties is related to Chapter 13 of Volume IV on higher Teichmüller theory. There are several other connections between chapters in this volume and other ones in preceding volumes. We present in some detail the topics treated this volume. They are grouped in four parts.

2 Part A, The metric and the analytic theory, 5 Part A of this volume includes two chapters. They concern identities on hyperbolic surfaces and the Thurston metric.

2.1 Identities on hyperbolic surfaces Chapter 1, written by Martin Bridgeman and Ser Peow Tan, concerns some remarkable identities on the length spectrum of hyperbolic surfaces (in particular lengths of simple closed curves and of properly embedded arcs) and generalizations to higherdimensional manifolds. The first such identities were discovered at the beginning of the 1990s by Greg McShane and Ara Basmajian. The theory was further developed by several authors who transformed, extended, generalized these identities and gave them new proofs.

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McShane’s identity, in its original form (1991), says that for any complete finitevolume hyperbolic structure on a once-punctured torus, we have X 1 1 D l. / 2 1Ce  where the sum is over all simple closed geodesics  , l. / being its length. Basmajian’s identities (1993) hold more generally for hyperbolic manifolds of finite volume with non-empty geodesic boundary, in any dimension  2. They concern the ortholength spectrum, that is, the set of lengths of properly immersed geodesic arcs which make right angles with the boundary. Such arcs are called orthogeodesics. Basmajian proved that a totally geodesic surface in a hyperbolic manifold M can be decomposed into embedded discs which are in one-to-one correspondence with the orthogeodesics of the manifold with boundary obtained by cutting M along that surface. He deduced an identity relating the volume of that surface and the ortholength spectrum of M . In the case where the manifold is a surface S with non-empty boundary @S , Basmajian’s identity becomes:   X l Length.@S / D 2 log coth 2 l2LS

where the sum is over the ortholength spectrum LS of S . More recently, in 2009, Bridgeman showed that the unit tangent bundle T1 .S / of a hyperbolic surface S with nonempty boundary @S can be decomposed into certain “drum-like” pieces which are in correspondence with the orthogeodesics of S . He obtained the following identity for the ortholength spectrum:   X 2l 4R sech Vol.T1 .S // D 2Area.S / D ; 2 l2LS

where T1 .S / is equipped with its canoncial volume form, where the sum is again over the ortholength spectrum LS , and where R is Rogers’ dilogarithm function. Soon after Bridgeman’s result, Bridgeman–Kahn and Calegari gave identities that are valid in any dimension and generalize Bridgeman’s identity. These identities have several applications. The most spectacular application of McShane’s identity is probably the one discovered by Mirzakhani who extended this identity to bordered surfaces and used it to show that the Weil–Petersson volumes of moduli spaces of such surfaces are polynomial functions of the lengths of the boundary components. The constant terms that appear in the polynomials she discovered are the Weil–Petersson volumes of the moduli spaces of complete hyperbolic surfaces of genus g with n punctures and with no boundary components. Furthermore, Mirzakhani gave recursive formulae for the other coefficients of the polynomials, which turned out to be the intersection numbers of the so-called tautological classes of the Deligne–Mumford compactification of moduli space. As a consequence, she obtained a completely new proof of the Witten conjecture (1991, proven by Kontsevich in 1992). We recall that this conjecture proposes a recursive formula for the

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tautological classes, stating that a certain generating function for their intersection numbers satisfies a series of KdV differential equations. In another direction, Luo and Tan, motivated by the works of McShane, Mirzakhani and Bridgeman, obtained an identity that generalizes the McShane identity, which is valid for surfaces with or without boundary and which involves the dilogarithms of the lengths of the simple closed geodesics in all 3-holed spheres and 1-holed tori embedded in the surface. In Chapter 1 of this volume, the authors survey the above identities, making connections between them and proposing a unified approach for the proofs. This involves the definition, for the hyperbolic manifold M whose length (respectively ortholength, etc.) spectrum is investigated, of a certain set X associated to M , equipped with a finite measure , and the study of a measure-theoretic decomposition of X into countably many disjoint subspaces Xi of finite non-zero measure and a subspace Z which has zero measure. Typically, X may be the boundary of M , or the set of geodesics in M embedded in the unit tangent bundle and equipped with the associated geodesic flow, or a set of random geodesics, and there are other possibilities. The general identity that the authors obtain has the form of a summation formula X .X / D .Xi / i

which the authors call a tautological identity. The subject of identities involving lengths of geodesics is very active now, and the authors in Chapter 1 mention works of Bowditch, Akiyoshi–Sakuma–Miyachi, Labourie–McShane, Luo–Tan, Hu, Hu–Tan–Zhang, Kim–Tan, Tan–Wong–Zhang and others, indicating at some places connections between the various results obtained.

2.2 Problems on Thurston’s metric Chapter 2, by Weixu Su, consists of a set of commented problems on Thurston’s asymmetric metric on Teichmüller space. This metric was already surveyed in Chapter 2 of Volume I of this Handbook, written by Théret and the author of this introduction. That chapter contained a commented list of 13 problems. Since then, some of these problems were solved, new results were obtained, and it seemed natural to us to include a new list of problems, especially since this metric has been recently the subject of intense investigation. The chapter contains 46 problems, divided into 5 sections: (1) Infinitesimal properties; (2) Geodesics; (3) Generalizations; (4) Infinitely-generated Fuchsian groups of the first kind; (5) Other general questions.

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The first section concerns the Finsler infinitesimal norm and its influence on the largescale properties, like quasi-isometries and the behavior of stretch lines and more general geodesics. The question of the relation of this infinitesimal norm with the complex structure is also addressed. Section (2) contains questions concerning the totality of geodesics joining two points, geodesic flows, the number of closed geodesics in moduli space of length  R and the (equi)-distribution of these geodesics, the existence of dense geodesics in moduli space, and the asymptoticity question (when do geodesic rays stay at a bounded distance from each other?). Questions on the group of quasi-isometries are again addressed. Section (3) contains questions on symmetrizations of Thurston’s metric (the so-called length-spectrum metric and others) and other metric generalizing this metric, like the arc metric for surfaces with boundary. Questions concerning the global and local properties, the boundaries and the isometries of these metrics are addressed. The questions in Section (4) concern generalizations to length spectra of surfaces of infinite type, namely, whether there exist quasiconformal deformations for which the lengths of all simple closed curves do not increase. In the case of a positive response, this would mean that the same formula for the Thurston metric, if it is used in the case of a surface of infinite type, does not define a metric. The same question is addressed for extremal length instead of hyperbolic length. Related questions concern the critical exponents of infinitelygenerated Fuchsian groups of the first kind. The last section contains some questions related to generalizations of the Thurston metric, including convex combinations of Thurston’s metric and it reverse, and metrics on spaces of n-tori. This problem list contained in Chapter 2 is an updated version of a list that was compiled after a workshop at the American Institute for Mathematics (Palo Alto) whose title was “Lipschitz metric on Teichmüller space.” The problems were contributed by several people. The study of Thurston’s metric became recently an active field of research, and several individuals and groups of researchers are presently working on problems related to that metric. However, it is fair to say that the most interesting results concerning this metric are still, by far, those contained in the first draft of Thurston, written in 1985.

3 Part B. The group theory, 4 Part B of this volume contains three chapters. They concern mapping class groups, and in particular their cohomology.

3.1 The Meyer cocycle and Meyer functions The cohomology of the mapping class group g of an orientable closed surfaces of genus g with rational coefficients is isomorphic to the cohomology of the Riemann moduli space of that surface. Thus, studying properties of the cohomology of the mapping class group is also a way of studying properties of Riemann’s moduli space.

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In 1982, Harer showed that the second cohomology group H 2 .g ; Z/ of g is isomorphic to Z, for all g  3. A nonzero element g in that group was already highlighted by Werner Meyer in 1973 and is now called the Meyer signature cocycle, or more simply the Meyer cocycle. We start by recalling a few facts about it. The Meyer cocycle is an integer-valued 2-cocycle that appeared in Meyer’s work as a characteristic class of surface bundles over surfaces. The word “signature” refers here to the signature of surface bundles as 4-manifolds. We recall that the signature of a compact oriented 4-manifold is defined as the signature of the intersection form on the second cohomology group. It is interesting for Teichmüller theorists to know that Meyer’s construction of the signature cocycle in the setting of fibered 4-manifolds uses the decomposition of the base surface into pairs of pants. The Meyer cocycle can also be obtained by pulling back by the homology representation g ! Sp.2g; Z/ a group 2-cocycle on the Siegel modular group Sp.2g; Z/ which represents the signature class in H 2 .Sp.2g; Z/; Z/. The intersection form of a 4-manifold is a powerful invariant. By a theorem attributed to Whitehead and Milnor, two simply-connected 4-manifolds are homotopy equivalent if and only if they have isomorphic intersection forms. This intersection form was used by Freedman in his classification of simply-connected 4-manifolds. Finally, we recall that the Hirzebruch signature theorem reduces the computation of the signature to that of the first Pontriaguine class. Meyer studied the signature of the total space of any Sg -bundle over an oriented closed surface of any genus g  1. He showed that for g  2 the signature is zero. For g  3, he showed that this signature is a multiple of 4. He also showed that for any g  4 and for any integer n which is a multiple of 4, there exists an Sg bundle whose signature is n. Turaev, in 1987, introduced a cocycle which turned out to be identical to Meyer’s signature cocycle, and he made relations with the Maslov index in symplectic geometry. It is also known that the Meyer signature cocycle is proportional to the first Mumford–Morita–Miller class and to the Weil–Petersson 2-form (works of Atiyah and of Wolpert). The Meyer functions are related to the Meyer cocycle. These functions play an important role in the study of the cohomology of the mapping class group g . They are defined in each genus g  1, as secondary invariants. Meyer showed that for g D 1 (where 1 ' SL.2; R/) and for g D 2, there exists a unique Q-valued cochain g W g ! Q whose coboundaries are respectively the Meyer signature cocycles 1 and 2 . He also gave an explicit form for 1 . Atiyah gave several geometric interpretations of the Meyer functions. He established relations of the value of the function 1 at a hyperbolic element of SL.2; Z/ with several invariants, including the Hirzebruch signature defect, the Shimizu L-function, and the Atiyah–Patodi–Singer  invariant of signature operators of Riemannian 3-manifolds. Morita and Morifuji established relations with the Casson invariant of homology 3-spheres. In Chapter 3 of this volume, Yusuke Kuno describes these developments as well as analogues for higher genera and higher dimensions. He also mentions the socalled local signature theory. This is based on the fact that given a closed oriented 4-manifold M fibered over a surface S , its signature is, under some conditions, localized at finitely many singular fibers of the fibration f W M ! S . The Meyer functions turn out to be useful for approaching this problem. Kuno also mentions works of Y.

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Matsumoto and E. Horikawa on this subject. Matsumoto gave the first examples of computation of local signatures of fibered 4-manifolds, in genus 1 and then in genus 2. His work uses the Meyer functions 1 and 2 . Horikawa considered local signature in the setting of algebraic geometry. The higher-dimensional analogue of the Meyer function 2 was also studied by Iida, who called this function the Meyer function for smooth theta divisors. The author of Chapter 3 also reviews his recent extension of Matsumoto’s work in the setting of projective varieties. Let us note that the Meyer cocycle was already considered in Chapter 6 of Volume II of this Handbook, by Nariya Kawazumi, and that an important class of fibered 4-manifolds, the Lefschetz fibrations, are surveyed in Chapter 7 of Volume III, by Mustafa Korkmaz and András Stipicz.

3.2 The Torelli–Johnson–Morita theory Chapter 4, by Nariya Kawazumi and Yusuke Kuno, is a survey on the so-called Torelli–Johnson–Morita theory, from a new point of view which uses heavily the Goldman Lie algebra. This theory is motivated by the study of the Torelli group, based on the Johnson homomorphism and its generalization by Morita who introduced the notion of k-th order Johnson homomorphism, the k-th component of an injective graded Lie algebra homomorphism. We recall that the first Johnson homomorphism was defined by Dennis Johnson, in his systematic study of the Torelli group which he started in the late 1970s. He introduced a homomorphism, denoted by  , which is useful in identifying the abelianization of that group. This homomorphism is now called the first Johnson homomorphism. Motivated by Johnson’s work, Morita, in 1993, generalized the Johnson homomorphism to a sequence of homomorphisms defined on a series of subgroups of the mapping class group into certain abelian subgroups. Morita’s homomorphisms are called higher degree Johnson homomorphisms. It is also natural to define Johnson homomorphisms in the setting of automorphisms of free groups. Morita revived early work on this subject done by Andreadakis, in 1965. This topic is surveyed in Chapter 5 of the present volume. The authors in Chapter 4 of the present volume consider an oriented compact surface S D Sg;1 of genus g  0 with one boundary component, with g;1 its mapping class group relative to the boundary (all homeomorphisms and isotopies fix the boundary pointwise), and Ig;1 its Torelli group, i.e. the subgroup of g;1 of elements that act trivially on the homology. The Johnson filtration is a central filtration of Ig;1 whose definition uses the action of g;1 on the fundamental group of S . There is a graded Lie algebra homomorphism whose k-th component is called the k-th Johnson homomorphism whose target is the infinite-dimensional Lie algebra of symplectic derivations of Lie type in the sense of Kontsevich. In fact, this graded Lie algebra homomorphism was defined by Morita by the end of the 1990s, and independently by Kontsevich. Its image is called the Johnson image. To get a characterization of this image is an important question in the study of the Torelli group Ig;1 .

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In 1986, Goldman introduced a Lie bracket on the free Z-module of homotopy classes of oriented loops on an oriented surface. The resulting Lie algebra is called the Goldman Lie algebra. The main object of Chapter 4 is to show how this Lie algebra appears in the Torelli–Johnson–Morita theory. This is based on the observation that the Goldman Lie algebra acts on the fundamental group of the surface by derivations. To make the relation with the action of the mapping class group by automorphisms, the authors construct completions of the Goldman Lie algebra and of the fundamental group. They introduce a morphism they call the “geometric Johnson homomorphism,”, whose graded quotients give Morita’s k-th Johnson homomorphisms. The geometric Johnson homomorphism turns out to be an intrinsic and choice-independent version of a map defined by Massuyeau in 2012 and called the “total Johnson map.” Turaev, in 1991, discovered a Lie cobracket on a certain quotient of the Goldman Lie algebra and he showed that this quotient algebra has the structure of a Lie bialgebra. This Lie bialgebra is called the Goldman–Turaev bialgebra, and it is also a central object of study in this survey. The Turaev cobracket induces a map from the target of the geometric Johnson homomorphism. The authors show that this gives a constraint on the Johnson image. They also mention relations with recent works of Turaev–Massuyeau and Church on related subjects. The chapter contains an overview of several aspects of the theory including the known results about the Johnson image, and on the Goldman–Turaev Lie bialgebra. The authors also discuss the extension theory of the Johnson homomorphism to the mapping class group. They also survey in detail the theory of (generalized) Dehn twists from this point of view, introduced by Kuno and extensively studied by Kawazumi and Kuno, showing that this generalized Dehn twist action on the completion of the fundamental groupoid of the surface has a canonical logarithm and specifying it as an element of the completion of the Goldman Lie algebra.

3.3 The Johnson homomorphisms In Chapter 5, Takao Satoh surveys the theory of the Johnson homomorphisms in the setting of automorphisms of free groups. These are in some sense generalizations of the Johnson homomorphisms for mapping class groups. It is a very well known fact now that there are close relations and analogies between mapping class groups and automorphisms of free groups. We recall in this respect that the mapping class group of a compact oriented surface Sg;1 of genus g with one boundary component is embedded in the outer automorphism group of its fundamental group, which is a finitely generated free group. The theory developed in this chapter may be considered as an instance of the analogies between the theory of mapping class groups and that of the (outer) automorphisms of free groups. As a matter of fact, the Johnson homomorphisms can be defined for a general class of groups that contains both groups, and they are useful in the study of the graded quotients of a certain descending filtration of the automorphism group of such

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a group. Let us also note that the Johnson homomorphisms for mapping class groups, that are considered in Chapter 4 of the present volume, are also surveyed in Chapter 7 of Volume I of this Handbook (by Morita) and in Chapter 6 of Volume II of this Handbook (by Kawazumi). In Chapter 5, the author surveys the Johnson homomorphisms of the automorphisms of free groups and of free metabelian groups. He also gives a general definition of the Johnson automorphisms for automorphisms of an arbitrary group G, and several other groups are involved in this theory. The Johnson filtration of the free group Fn of rank n appears as a certain descending central series of normal subgroups of Aut.Fn / whose first term is a group denoted by IAn and called the IAautomorphism group. This is the kernel of the natural homomorphism from Aut.Fn / into the automorphism group of its abelianization. The letters I and A stand for “Identity” and “Automorphism.” The study of some of these groups has a long history. For instance, Jakob Nielsen, in 1918, showed that the group IA2 , coincides with the inner automorphism group of the free group F2 . It is known that such a result does not hold for n  3. Wilhelm Magnus, in the 1930s, obtained a finite generating set for IAn for arbitrary n. Several basic questions on this group remain open. For instance, for n  4, it is unknown whether the group IAn is finitely presentable. In some sense, the Johnson filtration of Aut.Fn / and the associated Johnson homomorphisms n are tools for studying the structure of the group IAn . The author also reports on the applications of these results to the study of braid groups and mapping class groups of surfaces with boundary. A recent result presented is a description of the abelianization of the level-d congruence IA-automorphism group of a free group using the first Johnson map over Z=d Z defined by the Magnus expansion by Kawazumi (works of Putman and Satoh).

4 Part C. Representation theory and generalized structures, 3 Part C of this volume contains three chapters. They concern moduli spaces of generalized geometric structures on surfaces (the term “generalized” refers to the structures that are more general than the classical conformal and hyperbolic structures that are originally parametrized by Teichmüller space) and the theory of representations of their fundamental groups. The first chapter deals with representations of fundamental groups into Lie groups and the corresponding representation varieties, and moduli spaces of generalized geometric structures on surfaces. The representation varieties are modui spaces of such generalized structures, but they can also be considered as moduli spaces for groups, that is, they decribe groups varying in families. The stress in this chapter is on the dynamical aspects of this theory. The second chapter concerns compactifications and coarse fundamental domains for locally symmetric spaces associated with Anosov subgroups of noncompact semisimple Lie groups. Anosov groups play an important role in higher Teichmüller theory. The third chapter

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concerns the theory of Higgs bundles, which is a major component of higher Teichmüller theory and which establishes strong relations between Teichmüller spaces and physics.

4.1 Dynamics of representations in Lie groups It is a classical fact that the Teichmüller space of a closed surface S can be identified with a connected component of the character variety of representations of the fundamental group of S in PSL.2; R/, that is, the quotient of the representation space, that is, the space of homomorphisms 1 .S / ! PSL.2; R/, by the group PSL.2; R/ acting by conjugation. The group PSL.2; R/ plays a special role in this theory because it is the isometry group of the hyperbolic plane. This point of view leads to the consideration of the elements of Teichmüller space as equivalence classes of discrete group actions, and it is particularly useful in providing this space with a real algebraic structure. It is natural to study representations of more general groups  into more general Lie groups G. This is not generalization per se, but it turns out that it leads to several new interesting questions. It also brings new insight into Teichmüller space. In the general setting, the character variety of  in G, denoted by .; G/, is the quotient of the space of representations of  in G by the group G acting by conjugation. Some of these character varieties, for  D 1 .S /, but for G 6D PSL.2; R/, still contain a copy of Teichmüller space. One important such general setting is the case where G is the group PSL.3; R/, and its gives rise to projective structures on surfaces, which were studied by Goldman and collaborators, and after that by several other authors. Several important works were conducted in the last two decades to understand such generalized character varieties. We recall in this respect that higher Teichmüller theory was already surveyed in the previous volumes of this Handbook (cf. Chapter 13 by Burger, Iozzi and Wienhard in Volume IV, and there are several other chapters which concern real projective structures). In the general case, the representation variety has the structure of an affine variety and its study is made in the setting of algebraic group theory, using the formalism of geometric invariant theory. In particular, some care has to be taken to define the quotient space, which is called a categorical quotient. In general, the character variety .; G/ has the structure of an affine scheme and as such it contains several kinds of points, such as stable points, semistable points, isolated points, etc. The outer automorphism group of G acts by conjugation on the character variety generalizing the action of the mapping class group of the surface on the character variety, in the case of representations of 1 .S / into PSL.2; R/. In this setting, several dynamical questions naturally appear. The dynamical aspects of the actions of automorphism and outer automorphism groups of the source group on the character variety are manifested in particular in the nature of the closure of orbits of points. Chapter 6 of this volume, by Inkang Kim, concerns the geometry and dynamics of these character varieties. The author surveys the basic theory and also provides

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some background for the general theory of algebraic group actions. He also presents the relation with Higgs bundles, with a focus on rigidity and flexibility questions for free group and surface group representations and for the inclusion of lattices in semisimple Lie groups and of complex hyperbolic lattices into higher-dimensional complex hyperbolic groups. Several deformation spaces are mentioned. The notions of primitive stable and separable stable representation are discussed. The author also highlights the use of the Toledo invariant for representations into Hermitian semisimple Lie groups and in more general Lie groups, where the Killing form and the root space decomposition may be used instead of the Kähler form that is classically used in the definition of this invariant. The relation with bounded cohomology and with the notion of volumes of representations is mentioned. The familiar examples of representations of surface groups and of 3-manifold groups in SL.2; C/ in relation with Kleinian groups and the theories developed by Ahlfors–Bers and Thurston on this subject are presented in some detail.

4.2 Compactifications and coarse fundamental domains for locally symmetric spaces associated with Anosov subgroups of noncompact semisimple Lie groups In Chapter 7, Lizhen Ji surveys the geometry and the compactification theory of quotients of symmetric spaces of higher rank by the actions of Anosov subgroups of noncompact semisimple Lie groups. These quotients are called Anosov locally symmetric spaces. The name Anosov stems from a hyperbolicity property of the dynamics of these actions. The author proposes three conjectures regarding the compactification and the coarse fundamental domains for such spaces. The first two conjectures concern maximal Satake compactifications of Anosov locally symmetric spaces. These are real analytic manifolds with corners. The third conjecture concerns the characterization of Anosov subgroups by coarse fundamental domains. This is motivated by the reduction theory of arithmetic subgroups of semisimple Lie groups. The author describes results, motivations, and evidence for these three conjectures. In particular, in formulating the first conjecture, he introduces a special kind of coarse fundamental domain which is adapted to group structures, namely, unions of compact subsets and anti-Siegel sets, which are analogues, in some sense, of the Siegel sets which are essential in the reduction theory of arithmetic subgroups of semisimple Lie groups.1 Anosov subgroups of semisimple Lie groups of higher rank play an important role in higher Teichmüller theory. They appear in particular in the work of Labourie on the Hitchin component of SL.n; R/, where they are called Anosov representations. Anosov subgroups have also been generalized to convex cocompact subgroups of mapping class groups. In fact, there are several well-known conceptual similarities between Teichmüller spaces and symmetric spaces of rank one – this is discussed at various places in this Handbook – but less is known about actions of Anosov subgroups on symmetric spaces of higher rank. This is one of the outcomes of Chapter 7. 1 Added in proof: Some substantial work on these conjectures was done by Kapovich and Leeb; see the references in Chapter 7.

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4.3 Moduli spaces of flat connections Chapter 8, by Lisa Jeffery, is a survey of moduli spaces of flat connections over closed oriented surfaces. These are generalizations of Teichmüller spaces of surfaces: in a special case (the case where the gauge group of the connection is PSL.2; R/), a component of the moduli space is Teichmüller space. The theory of moduli spaces of flat connections is a variant of the so-called “higher Teichmüller theory” of representations of fundamental groups of surfaces in Lie groups, with a more differentialgeometric flavor, and with a strong relation with symplectic geometry. In fact, moduli spaces of flat connections carry a natural symplectic orbifold structure, and they were first constructed by Atiyah and Bott as symplectic quotients, using the process of symplectic reduction in the infinite-dimensional symplectic manifold of all connections. Furthermore, the Narasimhan–Seshadri theorem provides the moduli space of flat connections with a natural structure of complex manifold. In Chapter 8, the author starts by noting that depending on the description of the surface S in the topological, smooth or complex category, one associates to it moduli spaces of flat connections equipped with topological, smooth or complex structures respectively. She gives several examples that illustrate this general principle. The Jacobian J.S / of the surface is the simplest of such moduli spaces. It can be described topologically as the space of homomorphisms from the fundamental group 1 .S / into the Lie group U.1/. As a smooth object, it is the space of flat U.1/-connections modulo the gauge group. It is also a complex torus, classifying line bundles over S . The author then mentions generalizations of this picture where the group U.1/ is replaced by another compact nonabelian group, e.g. SU.n/, n  2. She then considers the theory of moduli spaces M.n; d / of (semi-stable) holomorphic vector bundles of rank n and degree d with fixed determinant line bundle. Such a space admits a description as a space of representations of the fundamental group of a surface with boundary obtained by removing an open disc D from the closed surface S . More precisely, assuming that n and d are coprime, this is the space of gauge equivalence classes of flat SU.n/-connections on S  D whose holonomy around the boundary is conjugate to e2id=n I. The space M.n; d / carries a Kähler structure. The author studies the moment map associated to the gauge group action on the space of flat connections and she explains why this moment map sends a connection to its curvature. She then mentions the relation with the work of Goldman on Hamiltonian flows on the spaces M.n; d / and describes recent works on Hamiltonian torus actions on open dense subsets of these spaces. Being generalizations of Teichmüller spaces is not the main reason for studying the theory of moduli spaces of flat connections. These moduli spaces have applications in the topology of 3-manifolds (Casson’s invariant for homology 3-spheres) and of 4-manifolds (Donaldson and Seiberg–Witten invariants), in symplectic geometry, in algebraic geometry (as moduli spaces of holomorphic vector bundles) and in geometric invariant theory (where they provide good examples of quotient constructions). There are also strong relations with mathematical physics. In fact, moduli spaces of flat connections over Riemann surfaces are, for appropriate Lie groups, phase spaces in Chern–Simons theory. They also appear in the theory of Yang–Mills equations on

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4-manifolds, in conformal and quantum field theories, and in several other theories in mathematical physics. There is also a quantization theory of moduli spaces of flat connections which is related to the quantization theory of Teichmüller space.

5 Part D. Sources, 2 Part D of this Handbook contains English translations, together with commentaries, of two papers by Teichmüller dealing with extremal quasiconformal mappings. The second one completes the first. Teichmüller’s papers are still very poorly known by Teichmüller theorists, despite the fact that a volume of his Collected works appeared in print a few years ago. One reason is that these papers are in German, and another one is the fact that they are written in an unconventional style, and hard to read.2 As a matter of fact, it took several years to Ahlfors and Bers to come to the conclusion that Teichmüller’s famous result, referred to as the Teichmüller existence and uniqueness theorem for extremal quasiconformal mappings, which is contained in the two papers that are translated here, is true and that his methods of proof are sound. The commentaries which accompany the translations we provide may be helpful for the reader to find his way in these papers, and they also make connections between them. We have started this enterprise in Volume IV, with the translation of the paper Veränderliche Riemannsche Flächen (Variable Riemann Surfaces). This translation, together with the commentary that accompanied it, received positive reactions from various mathematicians, and we are continuing now this editing project. The first of the two papers in this new series is Extremale quasikonforme Abbildungen und quadratische Differentiale (Extremal quasiconformal mappings and quadratic differentials), published in 1939. It is translated by Guillaume Théret, and the commentary is written by Vincent Alberge, Weixu Su and the author of this introduction. This is the most quoted article by Teichmüller, although it was read by very few people. It contains the foundations of what we call now the classical Teichmüller theory. Teichmüller works in the most general setting of surfaces of finite type: orientable or not, with boundary, or without boundary and with or without distinguished points in the interior or on the boundary. In this paper, we find for the first time the notions of marked surface, of Teichmüller space as a nonsingular covering of Riemann’s moduli space whereas the latter space is singular, the shift from the search for conformal invariants to that of invariants under quasiconformal mappings of bounded dilatation quotient and for precise estimates on this behavior in terms of this dilatation quotient, the introduction of the Teichmüller metric as the logarithm of the quasiconformal constant of an extremal map between two Riemann surfaces, the fact that this metric is Finsler and a study of its infinitesimal norm, the 2 The fact that Teichmüller was a nazi is another reason. Bers quotes in a 1960 paper a famous sentence by Plutarch (Perikles 2.2) which is often repeated: “It does not of necessity follow that, if the work delights you with its grace, the one who wrote it is worthy of your esteem.” We quote Gustave Flaubert: “L’homme n’est rien, l’œuvre est tout.” (Letter to George Sand, December 1875).

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definition of the Teichmüller maps and the relation with holomorphic (or meromorphic) quadratic differentials, the existence and uniquenes of extremal maps, the use of the trajectory structure of these differentials and the singular flat metric they define as an essential tool in the theory of moduli, the description of geodesics of the Teichmüller metric, the fact that Teichmüller space is homeomorphic to a cell whose dimension coincides with that given by Riemann’s count, the identification of the tangent space of Teichmüller space at any point with a space of equivalence classes of Beltrami differentials, the translation of problems concerning conformal structures into problems concerning Riemannian metrics and the use of these metrics to solve problems in the theory of conformal mappings, the introduction and the properties of Teichmüller discs, the comparison between length of closed geodesics and the quasiconformal dilatation of a map between hyperbolic surfaces, the question of whether there is a Hermitian metric on Teichmüller space, a study of convexity properties of the Teichmüller metric and in particular of totally geodesic subspaces for that metric, the quest for a solution of Nielsen’s realization problem as a question of finding a fixed point of the action of a finite subgroup of the mapping class group on Teichmüller space and the appeal to convexity properties for that solution, the introduction of the theory of non-reduced Teichmüller spaces, and many other fundamental ideas and results. Some of the results in this paper were rediscovered later on by other authors and published without any reference to Teichmüller. As a matter of fact, in several “comprehensive” books on Teichmüller theory, there is no reference to any of Teichmüller’s papers. It is true that some of the results in the paper are given with only sketches of proofs. Teichmüller himself writes on this fact, in the introduction of his paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen (Determination of Extremal quasiconformal mappings of closed oriented Riemann surfaces), which was published a few years later (1943) and which is also translated and commented on in the present volume: “In 1939, it was a risk to publish a lengthy article entirely built on conjectures. I had studied the topic thoroughly, was convinced of the truth of my conjectures and I did not want to keep back from the public the beautiful connections and perspectives that I had arrived at. Moreover, I wanted to encourage attempts for proofs. I acknowledge the reproaches, that have been made to me from various sides, even today, as justifiable but only in the sense that an unscrupulous imitation of my procedure would certainly lead to a barbarization of our mathematical literature. But I never had any doubts about the correctness of my article, and I am glad now to be able to actually prove its main part.” It took several years of work to Ahlfors, Bers and others to realize that all the results in this paper are sound and to provide complete proofs. The second paper that is translated and commented on in the present volume is Teichmüller’s Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen published in 1943, which we mentioned above. It is translated by Annette A’Campo, and the commentary is written by Annette A’Campo, Norbert A’Campo, Vincent Alberge and the author of this introduction. This paper is among the last ones that Teichmüller wrote. It may be considered as a supplement to the previous paper. It contains the proof of the so-called exis-

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tence theorem for quasiconformal mappings, for an arbitrary closed surface of genus g  2.3 This result was announced in the 1939 paper Extremale quasikonforme Abbildungen und quadratische Differentiale (Extremal quasiconformal mappings and quadratic differentials), which we just discussed, but without a complete proof. The proof that Teichmüller gives in the new paper uses in a fundamental way Brouwer’s theorem of invariance of domain, which is a precise version of the so-called “continuity method” which was introduced by Klein and Poincaré in their first attempts to prove uniformization.4 Teichmüller defines a mapping between a space of equivalence classes of marked Riemann surfaces (the Teichmüller space) and a space of equivalence classes of certain Fuchsian groups, which is called now the Fricke space. We recall that the latter has acquired in the recent literature a new status, as a subspace of the character variety of the representation space of the fundamental group of the surface in the Lie group PSL.2; R/. Teichmüller then defines a map between the Fricke space and a Euclidean space of dimension 6g 6. Using Brouwer’s theorem of invariance of domain, he shows that this map is a homeomorphism, and this implies the result he is seeking. This paper is important, for the result it contains, and for the relations that Teichmüller makes between several of his other papers. In the last section of the paper, he gives a clear formulation of the famous “problem of moduli” which arises from Riemann’s claim that the set of equivalence classes of Riemann surfaces of genus g  2 has 3g  3 moduli. We recall that giving a meaning to this statement was one of the most interesting research subjects during several decades after Riemann’s assertion, and it led to the development of several theories. Teichmüller writes: “In my opinion, the problem of moduli consists in turning the set R of all classes of conformally equivalent topologically determined closed oriented Riemann surfaces of genus g into an analytic manifold.” He then recalls that he outlined a solution to that problem in his paper Veränderliche Riemannsche Flächen published in 1944. An English translation together with a commentary of that paper appeared in Vol. IV of this Handbook. Teichmüller also announces, in the paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen, that there are close connections between the theory of complex conformal modules and the theory of extremal quasi-conformal mappings. It is also important to recall that in the paper Veränderliche Riemannsche Flächen, Teichmüller says that he thinks that the space R he builds (which is Teichmüller space equipped with a complex structure) “consists of at most countably many connected parts,” and he “thinks that R in fact is simply connected.” Thus, he is not sure that the space R he defines in this paper is the same as the Teichmüller space he defined in his 1939 paper Extremale quasikonforme Abbildungen und quadratische Differentiale using the quasiconformal theory. At the time Teichmüller wrote the paper under discussion, Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen, he already knew that the two spaces, defined using different approaches (the complex-analytic and the quasiconformal) are in fact the same. He 3 In the introduction of his paper, Teichmüller says that he restricts to the case of closed surfaces “in order to make the essential points clear,” and that “the more general statement shall be proved in a later article.” 4 Teichmüller calls this the “continuity argument.”

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ends this paper with the following: “ I cannot elaborate on these and similar questions and only express my opinion that all those aspects that have been treated separately in the discussion so far will appear as a great unified theory of variable Riemann surfaces in the near future.”

Part A

The metric and the analytic theory

Chapter 1

Identities on hyperbolic manifolds Martin Bridgeman1 and Ser Peow Tan2 Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Two orthospectra identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basmajian identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Bridgeman–Kahn identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The surface case of Theorems A and B . . . . . . . . . . . . . . . . . . . . . . . 2.4 Alternative derivations of the Bridgeman and Bridgeman–Kahn identity . 3 Two simple spectra identities for hyperbolic surfaces . . . . . . . . . . . . . . . . . . . 3.1 (Generalized) McShane identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Luo–Tan identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Luo–Tan identity for surfaces with boundary and non-orientable surfaces 4 Proofs of Theorems A and C – boundary flow . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Proof of the Basmajian identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Proof of the Generalized McShane identity . . . . . . . . . . . . . . . . . . . . . 4.3 Cusps and cone points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Index sets and the relation with Basmajian identity . . . . . . . . . . . . . . . 5 Proofs of Theorems B and D: interior flow . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Proof of the Bridgeman–Kahn identity . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Dilogarithm identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Finite identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Proof of the Luo–Tan identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Computing the measures of XP and XT . . . . . . . . . . . . . . . . . . . . . . 6 Moments of hitting function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Moments of the Bridgeman–Kahn identity . . . . . . . . . . . . . . . . . . . . . 6.2 Moments of the Basmajian identity . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The Bowditch proof of the McShane identity and generalizations . . . . . . . . . . . 8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 This 2 Tan

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work was partially supported by a grant from the Simons Foundation (#266344 to Martin Bridgeman). was partially supported by the National University of Singapore academic research grant R-146-000-

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1 Introduction In the last couple of decades, several authors have discovered various remarkable and elegant identities on hyperbolic manifolds, including Basmajian [3], McShane [32, 33], Bridgeman–Kahn [11, 13] and Luo–Tan [28]. Some of these identities have been generalized and extended, with different and independent proofs given in some cases. In addition to their intrinsic beauty and curiosity value, some of the identities have found important applications, in particular, the McShane identity as generalized by Mirzakhani to surfaces with boundary played a crucial role in Mirzakhani’s computation of the Weil–Petersson volumes of the moduli spaces of bordered Riemann surfaces, as well as some subsequent applications; the Bridgeman–Kahn identity gave lower bounds for the volumes of hyperbolic manifolds with totally geodesic boundary. Many of these identities were proven independently of each other, for example, although the Basmajian and McShane identities appeared at about the same time, the authors seemed unaware of each other’s work at that point; the Bridgeman–Kahn identity, although it appeared later was also proven somewhat independently of both of these works. The exception is the Luo–Tan identity which borrowed inspiration from the previous works. The aim of this article is to explore these identities, to analyse the various ingredients which make them work, and to provide a common framework to understand them. We hope that this will not only put the identities in a more natural setting and make them easier to understand, but also point the way towards a more unified theory with which to view these beautiful identities, and also point the direction towards possible applications for these identities. The unifying idea behind all of these identities is fairly simple. One considers a set X with a finite measure  associated with a hyperbolic manifold M , (for example X D @M , the boundary of M , or X D T1 .M /, the unit tangent bundle of M ) and look for interesting geometric/dynamical/measure-theoretic decompositions of X . Typically, by exploring some geometric or dynamical aspect of M , one can show that the set X decomposes into a countable union of disjoint subsets Xi of finite nonzero measure, and a set Z which is geometrically and dynamically complicated and interesting, but which has measure zero (this is what we mean by a measure-theoretic decomposition of X ). One deduces the fact that Z has measure zero from a deep but well-known result from hyperbolic geometry or dynamical systems. The identity is then just the tautological equation X .X / D .Xi / : i

The second part of the problem consists of analysing the sets Xi , in particular, computing their measures in terms of various geometric quantities like spectral data. This can be relatively simple, for example in the case of the Basmajian identity, somewhat more complicated, like for the McShane and Bridgeman–Kahn identities or considerably more involved, as in the Luo–Tan identities. A typical feature is that the sets Xi are indexed by either simple geometric objects on M like orthogeodesics, or simple subsurfaces of M , like embedded one-holed tori or thrice-punctured spheres. In

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particular, their measures depend only on the local geometry and data, and not on the global geometry of M . One can develop this viewpoint further, for example letting X be the set of geodesics on M with the Liouville measure and associating to X the length of the geodesic as a random variable and computing the moment generating function of this. In this way, for example, the Basmajian and the Bridgeman–Kahn identities can be viewed as different moments of the same generating function, see [14]. Alternatively, as in the case of Bowditch’s proof [8] of McShane’s original identity, one can adopt a different viewpoint, and prove it using a combination of algebraic and combinatorial techniques. This has been developed further in [9, 10, 46, 47, 49] etc., and provides an interesting direction for further exploration. Our main aim in this survey is to demystify these identities and to show that the basic ideas involved in deriving them are very simple. As such, the exposition will be somewhat leisurely, and where necessary, we will present slightly different proofs and perspectives than the original papers. We will refer the reader to the original papers for the more technical details of computing the measures .Xi /. The rest of the paper is organized as follows. In the next couple of sections we first state the four sets of identities, and then sketch the proofs for these identities from our perspective. Subsequently, we give a short discussion of the moment point of view adopted in [14] which allows one to view the Basmajian and Bridgeman– Kahn identities as different moments of the same variable and follow this with a short discussion of the Bowditch proof of the McShane identity and subsequent developments. We conclude the survey with some open questions and directions for further investigations.

1.1 Literature The literature on the subject is fairly large and growing. To aid the reader we will now give a brief synopsis by identity. McShane identity. The McShane identity first appeared in McShane’s 1991 thesis [32], “A remarkable identity for lengths of curves.” This was subsequently generalized (to higher genus surfaces) and published in [33]. In the papers [8, 9, 10], Bowditch gives a proof of the McShane identity using Markov triples, with extensions to punctured torus bundles and type-preserving quasi-fuchsian representations, see also [1, 2] by Akiyoshi, Miyachi and Sakuma for variations. The identity was extended to surfaces with cone singularities in Zhang’s 2004 thesis [52], see also [45]. A Weierstrass points version of the identity was derived by McShane in [34], and the identity was also generalized to closed surfaces of genus two in [35], using similar techniques. The last two generalizations can also be derived using the hyperelliptic involution on the punctured torus and on a genus two surface, and using the identity on the resulting cone surfaces, as explained by Tan, Wong and Zhang in [45]. Mirzakhani gave a proof of the general McShane identity for bordered surfaces in her 2004

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thesis [36] which was subsequently published in [37]. In [46], the identity was generalized to the SL.2; C/ case by Tan, Wong and Zhang and for non-orientable surfaces by Norbury in [39]. The identity was generalized to PSL.n; R/ for Hitchin representations by Labourie and McShane in [24]. A version for two-bridge links was given by Lee and Sakuma in [26]. Recent work of Hu, Tan and Zhang in [21, 22] have also given new variations and extensions of the identity to the context of Coxeter group actions on Cn . Basmajian identity. The Basmajian identity appears in the 1993 paper [3], “The orthogonal spectrum of a hyperbolic manifold.” A recent paper of Vlamis [50] analyses the statistics of the Basmajian identity and derives a formula for the moments of its associated hitting random variable. In a recent preprint [51], Vlamis and Yarmola generalized the identity to PSL.n; R/ for Hitchin representations. In the recent paper [41], Paulin and Parkkonen derive formulae for the asymptotic distribution of the orthogonal spectrum in a general negatively curved space. Bridgeman–Kahn identity. The Bridgeman–Kahn identity was first proven in the surface case by the first author in the 2011 paper [11] “Orthospectra and Dilogarithm Identities on Moduli Space.” An alternate proof was given by Calegari in [15]. The general case was proven by Bridgeman–Kahn in the paper [13]. The paper [16] of Calegari analyses the connections between the Bridgeman–Kahn identity and the Basmajian identity and gives an orthospectrum identity that has the same form as the Bridgeman–Kahn identity but arises out of a different decomposition. A recent paper of Masai and McShane [30] has shown that the identity obtained by Calegari is in fact the original Bridgeman–Kahn identity. In the paper [14] the authors consider the statistics of the Bridgeman–Kahn identity and derive a formula for the moments of its associated hitting random variable. We show that the Basmajian and Bridgeman– Kahn identities arise as the first two moments of this random variable. Luo–Tan identity. The Luo–Tan identity appears in the 2014 paper [28] “A dilogarithm identity on Moduli spaces of curves.” A version of the identity for small hyperbolic surfaces can be found in [20] and for surfaces with boundary and non-orientable surfaces in [29].

2 Two orthospectra identities We let M be a finite-volume hyperbolic manifold with non-empty totally geodesic boundary. In [3], Basmajian introduced the notion of orthogeodesics for hyperbolic manifolds. An orthogeodesic ˛ for M is an oriented proper geodesic arc in M which is perpendicular to @M at its endpoints (see Figure 1). Let OM be the collection of orthogeodesics for M and LM the set of lengths of orthogeodesics (with multiplicity). The set LM is the ortholength spectrum. Note that all multiplicities are even since we consider oriented orthogeodesics.

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Figure 1. Two orthogeodesics, one simple, one non-simple.

2.1 Basmajian identity In the 1993 paper [3], The orthogonal spectrum of a hyperbolic manifold, Basmajian derived the following orthospectrum identity: Theorem 2.1 (Theorem A: Basmajian’s Identity, [3]). Let M be a finite volume hyperbolic manifold of dimension n with non-empty totally geodesic boundary and Vk .r/ the volume of the ball of radius r in Hk . Then    X l : Vol.@M / D Vn1 log coth 2 l2LM

2.2 Bridgeman–Kahn identity In the 2011 paper [13], Hyperbolic volume of n-manifolds with geodesic boundary and orthospectra, Bridgeman and Kahn obtained the following identity for the volume of the unit tangent bundle T1 .M /, again in terms of the ortholength spectrum LM . Theorem 2.2 (Theorem B: Bridgeman–Kahn Identity, [13]). Let M be a compact hyperbolic manifold of dimension n with non-empty totally geodesic boundary, then X Vol.T1 .M // D Fn .l/ l2LM

where Fn W RC ! RC is an explicitly described smooth monotonically decreasing function depending only on the dimension n.

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We note that as Vol.T1 .M // D Vol.M /:Vol.Sn1 /, the above identity can also be thought of as an identity for the hyperbolic volume of the manifold M .

2.3 The surface case of Theorems A and B Theorems A and B are particularly interesting in the case of hyperbolic surfaces as they give identities for deformation spaces of Riemann surfaces with boundary. They are also related in this context to the McShane and Luo–Tan identities which we describe in the next section. The Bridgeman–Kahn identity in fact arose from a generalization of a previous paper of the first named author [11] which provided an explicit formula for the function F2 .l/ in Theorem B in terms of the Rogers dilogarithm. We have: Theorem 2.3 (Theorem A0 : Basmajian identity for surfaces). Let S be a hyperbolic surface with non-empty boundary @S . Then   X l Length.@S / D 2 log coth : 2 l2LS

Theorem 2.4 (Theorem B 0 : Bridgeman identity). Let S be a hyperbolic surface with non-empty boundary @S . Then   X l Vol.T1 .S // D 2Area.S / D 4R sech2 2 l2LS

where R is the Rogers dilogarithm function. The function R was introduced by Rogers in his 1907 paper [43]. This function arises in hyperbolic volume calculations; the imaginary part of R.z/ is the volume of an ideal tetrahedron with vertices having cross-ratio z.

2.4 Alternative derivations of the Bridgeman and Bridgeman–Kahn identity In the paper [15], Calegari gave an alternate derivation of the Bridgeman identity. Also in [16], Calegari derived an orthospectrum identity for all dimensions  2 which arose from a different decomposition of the unit tangent bundle. He showed in the surface case that it coincides with the Bridgeman identity. In a recent paper [30] by Masai and McShane, it was shown that for higher dimensions, it also coincides with the Bridgeman–Kahn identity.

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3 Two simple spectra identities for hyperbolic surfaces Let S be a finite area hyperbolic surface. We will consider various cases including when S has cusps, totally geodesic boundary, cone singularities (with cone angles  ), and finally, when S is a closed surface. We saw in the previous section that when S has non-empty totally geodesic boundary, we can define the collection of orthogeodesics OS , which provided an index set for the Basmajian and Bridgeman identities, which are then expressed in terms of the ortholength spectrum LS . This set can be extended in a natural way for surfaces which also have cusps or cone singularities. For the purposes of the next two classes of identities however, it is more useful to consider the smaller collection SOS of simple orthogeodesics, that is, orthogeodesics which do not have self-intersection, and also the collection S GS of simple closed geodesics on S . We will see that SOS , together with collections of certain subsets of S GS consisting of one, two or three disjoint geodesics satisfying some topological criteria, will be useful as index sets for the identities.

3.1 (Generalized) McShane identity Let Sg;n denote a hyperbolic surface of genus g with n cusps. In his 1991 thesis, A remarkable identity for lengths of curves, McShane proved an identity for the lengths of simple closed geodesics on any once-punctured hyperbolic torus S1;1 which he generalized later in [33] to more general cusped hyperbolic surfaces Sg;n , n  1. Theorem 3.1 (Theorem C: McShane Identity, [32, 33]). (1) If S1;1 is a hyperbolic torus with one cusp, then X 

1 1 D 2 1 C el. /

where the sum is over all simple closed geodesics  in S1;1 . (2) If Sg;n is a hyperbolic surface of genus g with n cusps, where n  1, then X 1 ;2

1 C exp



1 l.1 /Cl.2 / 2

D

1 2

where the sum is over all unordered pairs of simple closed geodesics f1 ; 2 g which bound together with a fixed cusp an embedded pair of pants in Sg;n . Here we adopt the convention that 1 or 2 may be one of the other cusps on Sg;n , considered as a (degenerate) geodesic of length 0. The case of the punctured torus can be regarded as a special case of the surface Sg;n where in the sum,  D 1 D 2 since any simple closed geodesic  on S1;1 cuts it into a pair of pants.

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In her 2005 thesis Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces, see [36, 37], Mirzakhani gave a general version of the McShane identity for hyperbolic surfaces with geodesic boundaries and cusps, which was an important tool for her computation of the Weil–Petersson volumes of the moduli spaces. Independently, Ying Zhang in his 2004 thesis Hyperbolic cone surfaces, generalized Markoff Maps, Schottky groups and McShane’s identity, see [52, 45], also gave a generalization of the McShane identity for surfaces, with (non-empty) boundary consisting of cusps, totally geodesic boundaries, or cone singularities with cone angle  , with slightly different forms for the functions involved in the identity. We first state Mirzakhani’s generalization and explain how to interpret the identity for cone surfaces later. Theorem 3.2 (Theorem C 0 : Generalized McShane–Mirzakhani Identity, [36, 37]). Let S be a finite-area hyperbolic surface with geodesic boundary components ˇ1 ; : : : ; ˇn , of length L1 ; : : : ; Ln . Then X

D.L1 ; l.1 /; l.2 // C

n X X

R.L1 ; Li ; l. // D L1

i D2 

f1 ;2 g

where the first sum is over all unordered pairs of (interior) simple closed geodesics bounding a pair of pants with ˇ1 , and the second sum is over simple closed geodesics bounding a pair of pants with ˇ1 ; ˇi and ! yCz x e2 Ce 2 ; D.x; y; z/ D 2 log yCz x e 2 C e 2  !   cosh y2 C cosh xCz 2     : R.x; y; z/ D x  log cosh y2 C cosh xz 2 The case where some of the other boundaries are cusps but ˇ1 is a geodesic boundary can be deduced from the above, by considering cusps to be boundaries of length 0, where again we adopt the convention that a cusp may be regarded as a geodesic of length 0 in the summands above. The case where ˇ1 is also a cusp, that is, L1 D 0, is more interesting. In this case, the original McShane identities can be deduced from the above by taking the limit as L1 ! 0, or taking the formal derivative of the above identity with respect to L1 and evaluating at L1 D 0. More interestingly, a cone singularity of cone angle may be regarded as a boundary component with purely imaginary complex length i , and the above identity is also valid if some of the boundary components are cone singularities of cone angles   as shown in [52, 45]. The restriction to cone angles   is necessary in the argument. This guarantees a convexity property and the existence of geodesic representatives for essential simple closed curves on the surface. For example, if S is a surface of genus g > 1 with one cone singularity  of cone angle , then we have X D.i ; l.1/; l.2 // D i

f1 ;2 g

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27

where f1 ; 2 g are unordered pairs of simple closed geodesics bounding a pair of pants with . We note that each of the summands of the identity in Theorem C 0 is the measure of some subset Xi of ˇ1 : the summands in the first sum of Theorem C 0 correspond to sets which are indexed by (a subset of) the simple orthogeodesics from ˇ1 to itself, the summands in the second sum correspond to sets which are indexed by simple orthogeodesics from ˇ1 to ˇi as we will see in the proof later.

3.2 Luo–Tan identity The Basmajian, Bridgeman and McShane identities for surfaces were in general only valid for surfaces with boundary; in the first two cases, for surfaces with geodesic boundary, in the third case, to surfaces with at least some cusp or cone singularity. They do not extend to general closed surfaces without boundary. However, for the genus 2 hyperbolic surface S2 , by considering S2 =hyp where hyp is the hyper-elliptic involution on S2 , one may lift the identity on the cone surface S2 =hyp to obtain identities on the closed genus two surface S2 , see [35, 45]. This method however does not generalize to higher genus. In their 2014 paper, A dilogarithm identity on moduli spaces of curves, F. Luo and the second author derived the following identity for closed hyperbolic surfaces. Theorem 3.3 (Theorem D: Luo–Tan identity, [28]). Let S be a closed hyperbolic surface. There exist functions f and g involving the dilogarithm of the lengths of the simple geodesic loops in a 3-holed sphere or 1-holed torus, such that X X Vol.T1 .S // D f .P / C g.T / P

T

where the first sum is over all properly embedded 3-holed spheres P  S with geodesic boundary, and where the second sum is over all properly embedded 1-holed tori T  S with geodesic boundary. The functions f and g are defined on the moduli spaces of simple hyperbolic surfaces (3-holed spheres and 1-holed tori) with geodesic boundary and given in terms of R, the Rogers dilogarithm function as follows: Suppose P is a hyperbolic 3-holed sphere with geodesic boundaries of lengths l1 ; l2 ; l3 . Let mi be the length of the shortest path from the li C1 -th boundary to the li C2 -th boundary (l4 D l1 , l5 D l2 ). Then     X 1  xi 1  yj 2R  2R  R.yj / f .P / WD 4 1  xi y j 1  xi y j i ¤j   .1  xi /2 yj (3.1) R .1  yj /2 xi where xi D eli and yi D tanh2 .mi =2/.

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Suppose T is a hyperbolic 1-holed torus with geodesic boundary. For any nonboundary parallel simple closed geodesic A of length a in T , let mA be the distance between @T and A. Then     X 1  xA 1  yA 2 2R  2R  2R.yA / g.T / WD 4 C 8 1  xA y A 1  xA y A A   .1  xA /2 yA (3.2) R .1  yA /2 xA where xA D ea and yA D tanh2 .mA =2/ and the sum is over all non-boundary parallel simple closed geodesics A in T .

3.3 Luo–Tan identity for surfaces with boundary and non-orientable surfaces The Luo–Tan identity also holds for surfaces S with geodesic boundary and cusps. However, in this case, the functions f and g need to be modified when P or T share some boundary component with S , similar to the McShane–Mirzakhani identity for surfaces with more than one boundary component, see [29]. In this case, the identity is trivial for the one-holed torus – however, one can obtain a meaningful identity involving the lengths of the simple closed geodesics in T by a topological covering argument, using the identity for a four-holed sphere, see [19, 20]. For example, for a once-punctured hyperbolic torus T , we obtain ! !!# " X 2 1  el. / 1 C el. / 2 R D R.sech .l. /=2// C 2 R 2 2 2  (3.3) where the sum extends over all simple closed geodesics  in T . For non-orientable surfaces, one also obtains an analogous identity. In this case, the summands include terms coming from embedded simple non-orientable surfaces, namely, one-holed Klein bottles and one-holed Möbius bands, see [29].

4 Proofs of Theorems A and C – boundary flow As remarked in the introduction, the proofs of all the results will be based on a decomposition of certain sets X associated to M . In particular for Theorems A and C, X will be subsets of T1 .M / associated to the boundary @M and the proofs arise from consideration of the boundary flow on X .

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29

4.1 Proof of the Basmajian identity Let T1 .M / be the unit tangent bundle of M and W T1 .M / ! M be the projection map. For the Basmajian identity, we let X be the set of unit tangent vectors v whose basepoints are on @M and which are perpendicular to @M and point into the interior of M , that is, X D fv 2 T1 .M /W .v/ 2 @M; v ? @M; v points into M g : Clearly, X can be identified with @M under , and we define the measure  on X to be the pullback of the Lebesgue measure on @M under . In particular, .X / D Vol.@M /. We consider the unit speed geodesic gv starting at p D .v/ 2 @M obtained by exponentiating v. Thus gv is the geodesic arc obtained by flowing from .v/ until you hit the boundary. To derive the Basmajian identity, we let Z D fv 2 X W Length.gv / D 1g : It follows from the fact that the limit set of M is of measure zero, that Z has zero volume. For each of the remaining vectors, gv is a geodesic arc of finite length with endpoints on @M . We define an equivalence relation on X n Z by defining v  w if gv ; gw are homotopic rel. boundary in M . Then each oriented orthogeodesic ˛ defines an equivalence class E˛ given by E˛ D fv 2 X n Z j gv is homotopic rel boundary to ˛g : By lifting to the universal cover and using a tightening argument, we see that for every v 2 X nZ, gv is homotopic rel. boundary to an orthogeodesic ˛, so fE˛ g˛2OM covers X n Z. Furthermore if ˛ ¤ ˇ then E˛ \ Eˇ D ; as no two F orthogeodesics are homotopic rel boundary. Thus we have the partition X D Z t ˛ E˛ and the associated identity X Vol.E˛ / : Vol.X / D Vol.@M / D ˛2OM

To calculate Vol.E˛ /, we lift to the universal cover so that ˛ lifts to a geodesic arc ˛Q orthogonal to two boundary hyperplanes P; Q. As ˛ is oriented, we assume ˛Q is oriented from P to Q. Thus any gv homotopic rel boundary to ˛ has a unique lift gQ v which is a geodesic arc perpendicular to P going from P to Q. Hence the base point of gQ v is in the orthogonal projection of Q onto P . Thus the set .E˛ / lifts to a disk of radius r.˛/ given by orthogonal projection of Q onto P . Let r.˛/; l.˛/ be the two finite sides of a hyperbolic quadrilateral with one ideal vertex and finite angles =2 (see Figure 2). By elementary hyperbolic geometry (see [5], for example), sinh.r.˛//: sinh.l.˛// D 1, giving r.˛/ D log.coth.l.˛/=2// and we obtain X X Vol.@M / D Vol.E˛ / D Vn1 .log.coth.l.˛/=2/// : ˛2OM

˛2OM

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Figure 2. Orthogonal Projection onto a plane.

Remark. The identity generalizes to hyperbolic manifolds with cusps, as long as the boundary contains some non-empty geodesic component. This is necessary to deduce that .Z/ D 0. Othogeodesics ending in a cusp have infinite length and do not contribute to the summands of the identity.

4.2 Proof of the Generalized McShane identity We consider a hyperbolic surface S with a finite number of geodesic boundary components, cusps and cone singularities (with cone angles  ). For simplicity we first consider the case where @S has only geodesic components ˇ1 ; : : : ; ˇn with lengths L1 ; : : : ; Ln . The basic idea of the proof is the same for the more general case. We will explain later how to modify the proof if some of the ˇi s are cusps or cone singularities of cone angle  . We derive the identity based at ˇ1 , as follows: X D fv 2 T1 .S /W .v/ 2 ˇ1 ; v ? ˇ1 ; v points into S g : Clearly,  induces a bijection from X to ˇ1 , so .X / D Length.ˇ1 / D L1 . Again, as in the proof of the Basmajian identity, we are going to consider the unit speed geodesic obtained by exponentiating v 2 X , However, this time, we are going to stop when the geodesic hits itself, or the boundary @S . More precisely, let Gv W Œ0; T ! S be the geodesic arc obtained by exponentiating v 2 X such that Gv is injective on

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31

Œ0; T ) and either Gv .T / D Gv .s/ for some s 2 Œ0; T /, or Gv .T / 2 @S . If Gv Œ0; t is defined and injective for all t > 0, then T D 1, that is, Gv is a simple geodesic arc of infinite length. A good analogy for the difference between gv in the Basmajian proof and Gv in the McShane proof is that we should consider gv as a laser beam starting from @M which is allowed to intersect itself any number of times, until it hits the boundary, whereas Gv should be thought of as a wall, which terminates when it hits itself, or the boundary. Now let Z  X be the set of vectors for which Gv has infinite length, that is, T D 1. Again, by the same argument as before, .Z/ D 0 since the limit set of S has measure zero, and the endpoints of the lifts of Gv must land on the limit set if v 2 Z. However, if ˇ is a cusp, then we need a stronger result, namely, the Birman–Series result [6] that the set of simple geodesics on the surface has Hausdorff dimension 1, which implies that .Z/ D 0. Similarly, if ˇ is a cone point, we require a generalization of the Birman–Series result, see [45]. We note that Z has a rather complicated Cantor set structure, and McShane analysed this set carefully in [33]. However, for the purposes of proving the identity, the structure of Z is irrelevant, and one only really needs to know that .Z/ D 0. We now look at Gv for v 2 X n Z. In this case, Gv is either a finite geodesic arc ending in a loop (a lasso), or a simple geodesic arc from ˇ1 to @S .  If Gv is a lasso, or a simple arc ending in ˇ1 , then a regular neighborhood N of ˇ1 [ Gv in S is topologically a pair of pants, where one of the boundary components is ˇ1 . The other two boundary components can then be tightened to geodesics 1 ; 2 which are disjoint and which bound together with ˇ1 an embedded pair of pants in S which contains Gv (note that in the case where S is a one-holed torus, then 1 D 2 WD  , where  is a simple closed geodesic on S disjoint from Gv , otherwise, 1 and 2 are distinct and disjoint).  If Gv is a simple arc from ˇ1 to ˇi where i ¤ 1, then a regular neighborhood of ˇ1 [ Gv [ ˇi is again a pair of pants, where ˇ1 and ˇi are two of the boundary components. The third boundary can again be tightened to a simple closed geodesic  , and again Gv is contained in the resulting pair of pants. One can prove the assertions in the previous paragraph by a cut and paste argument as follows: Cut S along Gv to obtain a (not necessarily connected) convex hyperbolic surface Scut with either two piecewise geodesic boundaries (if Gv is a lasso or simple arc from ˇ1 to itself), or one piecewise geodesic boundary (if Gv is a simple arc from ˇ1 to ˇi , i ¤ 1), and other geodesic boundaries. Note that if S is a one-holed torus then Scut is a cylinder whose core is a geodesic  disjoint from @Scut . Otherwise, in the first case, let 1 and 2 be the two disjoint geodesics which bound the convex core of Scut which again are disjoint from @Scut . Regluing along Gv , we see that 1 and 2 bound together with ˇ1 a pair of pants in S (basically the complement of the convex core of Scut ) which contains Gv , as asserted. The same argument applies to the second case to obtain a pair of pants bounded by ˇ1 ; ˇi and a geodesic  . To recap, for every v 2 X n Z, Gv is a geodesic arc contained in a unique pair of pants embedded in S bounded by ˇ1 and a pair of geodesics 1 , 2 (where one of

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1 ; 2 may be a different boundary component ˇi of S ). Let P be the set of all such pairs of pants embedded in S (equivalently, all unordered pairs of geodesics f1 ; 2 g in S which bound a pair of pants with ˇ1 ), and for each P 2 P, we define XP D fv 2 X n ZW Gv  P g ; then X is the disjoint union of Z and the XP s, hence X L1 D .XP / : P 2P

To derive the formulae for .XP / we consider a general pair of pants P with boundary a; b; c of lengths x; y; z. We consider perpendicular geodesics ˛p from points p on a. We let p1 ; p2 be the two points on a such that ˛pi is a simple geodesic spiraling towards b (one for each direction) and let q1 ; q2 be the two points on a such that ˛qi is a simple geodesic spiraling towards c. Also assume that p1 ; q1 ; q2 ; p2 is the cyclic ordering of the points on a, which divide a into the intervals Œp1 ; q1 , Œq1 ; q2 , Œq2 ; p2 and Œp2 ; p1 with disjoint interiors (see Figure 3). Each interval contains a unique point m such that ˛m is a simple orthogeodesic from a to a; c; a, and b respectively. We see that for p 2 .p2 ; p1 /, ˛p is a simple geodesic from a to b. Similarly, for p 2 .q1 ; q2 /, ˛p is a simple geodesic from a to c. For p 2 .p1 ; q1 / [ .q2 ; p2 /, ˛p is either a simple geodesic from a to itself or it has selfintersections. Furthermore, LŒp2 ; p1 is precisely the orthogonal projection of b to a. Similarly LŒq1 ; q2 is the orthogonal projection of c to a, and LŒp1 ; q1 D LŒq2 ; p2 by symmetry. Now let P 2 P where a D ˇ1 . We have:  If b D 1 and c D 2 are interior curves of S , then Gv  XP if and only if .v/ 2 .p1 ; q1 / [ .q2 ; p2 /. Then .XP / D D.x; y; z/ D 2LŒp1 ; q1 . By elementary hyperbolic geometry we have ! yCz x e2 Ce 2 ; D.x; y; z/ D 2 log yCz x e 2 C e 2  If say b D  is an interior geodesic and c D ˇi is a boundary geodesic of S , then Gv 2 XP if and only if .v/ 2 .p1 ; q1 / [ .q1 ; q2 / [ .q2 ; p2 /. We have LengthŒp2 ; p1 D log.coth.Z=2// where Z is the length of the perpendicular arc from a to b. Applying the hyperbolic cosine rule we get  !   cosh y2 C cosh xCz 2     : R.x; y; z/ D x  log cosh y2 C cosh xz 2 The generalized McShane identity now follows by substitution.

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Figure 3. Pants P with spiraling geodesics.

4.3 Cusps and cone points If ˇ1 is a cusp, we can take a horocycle C of length 1 about the cusp and remove the neighbourbood of the cusp bounded by C . We now take X to be the set of vectors in T1 .S / such that .v/ 2 C and v is perpendicular to C and pointing into S . Essentially the same analysis works to give a decomposition of X , with suitable modifications of the functions D and R, which now only depend on two variables. In the tightening argument, we need the fact that the horocycle chosen is sufficiently small so that it is disjoint from all simple closed geodesics, choosing length one as we did works. If all other boundaries are cusps, then we recover the original McShane identity since in this case, D.y; z/ D R.y; z/ D

1 1Ce

yCz 2

:

If ˇ1 is a cone point of cone angle 1 , we decompose the set of tangent vectors based at the cone point again in essentially the same way. We note that in order for the surface obtained after cutting to be convex, the restriction that the cone angle is   is necessary. We need this to perform the tightening argument. Similarly, we require all other cone angles to be   if we want every essential simple closed curve to be represented by a geodesic (or the double cover of a geodesic segment between two cone points of angle ).

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Here it is useful to regard a cone point as an axis through the point perpendicular to the hyperbolic plane, and use the complex measure of length between two skew axes in H3 . With this, the measure of the angle is purely imaginary. Similarly, other components ˇi which are cone angles should be regarded as axes perpendicular to the plane, and we recover exactly the same identity as that obtained by Mirzakhani, with the same functions, with the convention that cone points have purely imaginary lengths, see [45] for details.

4.4 Index sets and the relation with Basmajian identity The set X in the McShane identity is decomposed into the disjoint union of Z, a set of measure 0 and a countable union of disjoint open intervals X˛ , which from the previous observation is indexed by ˛ 2 SOS .ˇ1 /, the set of simple orthogonal geodesics on S with base point on ˇ1 . Each such simple orthogeodesic gives rise to an interval in ˇ1 , all of which are disjoint. The first sum consists of summands corresponding to the (two) intervals from the simple orthogeodesics from ˇ1 to itself contained in the pants P where 1 and 2 are interior geodesics, whereas a summand of the second sum consists of three intervals, the extra interval coming from the simple orthogeodesic contained in P from ˇ1 to ˇi . If ˛ 2 SOS .ˇ1 / goes from ˇ1 to itself, then .X˛ / D LŒp1 ; q1 D D.x; y; z/=2 (note that this length depends on the geometry of the pants P and not just the length of ˛) and if ˛ isa simple orthogonal geodesic from ˇ1 to another component ˇi , then  l.˛/ .X˛ / D 2 log coth 2 , the projection of ˇi to ˇ1 along ˛. The index set for the Basmajian identity is much larger, and strictly contains the index set for the McShane identity. In this sense, the Basmajian identity for surfaces, as restricted to ˇ1 , is a refinement of the McShane identity: the terms corresponding to simple geodesics from ˇ1 to a different component ˇi are the same for both identities. However, in the McShane identity, each set X˛ where ˛ is a simple geodesic from ˇ1 to itself contains infinitely many terms from the Basmajian identities, as infinitely many non-simple orthogeodesics have the non-intersecting beginning part (the geodesic segment Gv defined earlier) contained in the same pants P . We note also that the index set for the McShane identity can be regarded as the set of all embedded pairs of pants in S which contain ˇ1 as a boundary. These in turn split into two subsets, pairs of pants P for which @P \@S D ˇ1 or @P \@S D ˇ1 [ˇi for some i ¤ 1. The first type gives the first sum, the second type the second sum in Theorem B 0 . This point of view is useful as it generalizes to the Luo–Tan identity.

5 Proofs of Theorems B and D: interior flow For the Bridgeman–Kahn and Luo–Tan identities, we consider a hyperbolic manifold M and its unit tangent bundle T1 .M /. We let X D T1 .M / with  the volume on T1 .X /. We then consider for each v 2 T1 .M / the geodesic obtained by flowing

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35

in both directions. We will show that the two identities described are obtained by considering the dynamical properties of this geodesic.

5.1 Proof of the Bridgeman–Kahn identity Let M be a hyperbolic manifold with totally geodesic boundary. For each v 2 T1 .M / we let gv be the maximal geodesic arc tangent to v. To derive the Bridgeman–Kahn identity, we let Z be the set of v such that gv is not a proper geodesic arc (i.e. the flow does not hit the boundary in at least one direction). Once again, as the limit set of M is of measure zero, the set Z satisfies .Z/ D 0. For v 62 Z, gv is a proper geodesic arc, and as in the Basmajian identity, we define an equivalence relation by v  w if gv ; gw are homotopic rel boundary. Once again, each orthogeodesic ˛ defines an equivalence class E˛ and as before, they form a partition. Thus we have the associated identity X Vol.E˛ / : Vol.T1 .M // D ˛2OM

To calculate Vol.E˛ /, we lift ˛ to the universal cover such that it is perpendicular to two boundary planes P; Q. By definition, any v 2 E˛ has gv homotopic rel boundary to ˛. Thus gv has a unique lift to a geodesic gQ v which has endpoints on P; Q. Hence E˛ also has a unique lift to EQ ˛ where EQ ˛ D fv 2 T1 .Hn / j 9 a  0; b  0 such that gQ v .a/ 2 P; gQ v .b/ 2 Qg : The volume of this set only depends on l.˛/ D d.P; Q/ and the dimension. Therefore we have Vol.E˛ / D Fn .l.˛// for some function Fn which gives the Bridgeman–Kahn identity. To derive a formula for Fn , we let be the volume measure on the unit tangent bundle to the upper half space model for Hn , invariant under Isom.Hn /. We let G.Hn / be the space of oriented n1 geodesics in Hn and identify G.Hn / D .Sn1 1  S1 n Diagonal/ by assigning to g the pair of endpoints .x; y/. We have a natural fiber bundle pW T1 .Hn / ! G.Hn / by letting p.v/ be the oriented geodesic tangent to v. We obtain a parametrization of T1 .Hn / by choosing a basepoint on each geodesic. We let b 2 Hn and for each geodesic g, let bg be the nearest point of g to b. Then to each v 2 T1 .Hn / we n1 assign the triple .x; y; t/ 2 Sn1 1  S1  R where .x; y/ D p.v/ and t is the signed hyperbolic distance along the geodesic p.v/ from bp.v/ to v. In terms of this parametrization, 2n1 d Vx d Vy dt d D jx  yj2n2 where d Vx D dx1 dx2 : : : ; dxn and jx  yj is the Euclidean distance between x; y (see [38]). We choose planes P; Q such that d.P; Q/ D l to be given by the planes

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intersecting the boundary in the circles of radius 1; el about the origin. We define El D fv 2 T1 .Hn / j 9 a  0; b  0 such that ˛v .a/ 2 P; ˛v .b/ 2 Qg : If g D .x; y/ is a geodesic intersecting both planes P; Q, we let L.x; y; l/ D d.P \ g; Q \ g/, the length between intersection points. Alternatively we have L.x; y; l/ D Length.p1 .g/ \ El /. Then integrating over t we have Z  n1 Z Z 2 d Vx d Vy d D dt Fn .l/ D jx  yj2n2 El g2p.El / p 1 .g/\El giving

Z

Z

2n1 L.x; y; l/d Vx d Vy : (5.1) jx  yj2n2 jxjel This integral formula can be simplified to obtain a closed form in even dimensions and can be reduced to an integral over the unit interval of a closed form in odd dimensions. In particular, when n D 2, it takes on the explicit form given in the Bridgeman identity which we describe in the next subsection. Fn .l/ D

5.2 Dilogarithm identities We first describe the Rogers dilogarithm. We define the kth polylogarithm function Lik to be the analytic function with Taylor series Lik .z/ D

1 X zn nk nD1

for jzj < 1 :

Then we have

z Li1 .z/ D  log.1  z/ : 1z Also they satisfy the recursive formula Li0 .z/ D

Lik1 .z/ : z The function Li2 is the dilogarithm function and the Rogers dilogarithm function is a normalization of it given by Li0k .z/ D

1 log jzj log.1  z/ : 2 In the paper [11], the first author considered the surface case of the Bridgeman–Kahn identity. We let S be a finite-area surface with totally geodesic boundary. Then boundary components of S are either closed geodesics or bi-infinite geodesics (such as the case when S is an ideal polygon). A boundary cusp of S is a cusp on the boundary of S bounded by two bi-infinite geodesics contained in the boundary of S . Let NS be the number of boundary cusps. Then we have the following generalized version of Theorem B: R.z/ D Li2 .z/ C

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37

Theorem 5.1 (Bridgeman, [11]). Let S be a finite-area hyperbolic surface with totally geodesic boundary. Then   X 2 2 2l Vol.T1 .S // D 2Area.S / D 4:R sech C NS : 2 3 l2LS

where R is the Rogers dilogarithm function. In order to prove this, we once again take X D T1 .S / and  the volume measure. We again define Z to be the set of v such gv is not a proper geodesic arc. Then if v 62 Z, gv is a proper geodesic arc and we define v  w if gv ; gw are homotopic rel boundary. There are two cases, either gv is homotopic rel boundary to an orthogeodesic ˛ or gv is homotopic to a neighborhood of a boundary cusp c. Thus we have equivalence classes E˛ for each orthogeodesic and Ec for each boundary cusp. Therefore X X Vol.T1 .S // D Vol.E˛ / C Vol.Eci / : ˛

ci

We have by definition Vol.E˛ / D F2 .l.˛//. Also, the sets Eci are all isometric. Therefore X Vol.T1 .S // D F2 .l/ C NS Vol.Ec / : l2LS

As the above identity holds for S an ideal triangle T , we have Vol.T1 .T // D 2 2 D 3Vol.Ec / : Therefore Vol.T1 .S // D

X

F2 .l/ C

l2LS

2 2 NS : 3

The proof of the identity then follows by showing that F2 .l/ D 4:R.sech2 .l=2// where R is the Rogers dilogarithm (in [11] the index was over all unoriented orthogeodesics so the constant there was 8). In [11] this is done by directly computing the integral in the Formula 5.1 for F2 . We now describe an alternative approach given by Calegari in [15] that avoids this computation.

5.3 Finite identities The surface identity is only a finite identity when S is an ideal n-gon. In this case we have a finite orthospectrum l1 ; : : : ; lk . We then have 2 2 .n  2/ D

X i

F2 .li / C

2 2 n : 3

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Martin Bridgeman, Ser Peow Tan

We rewrite this as

X

F2 .li / D

i

4 2 .n  3/ : 3

We let R be the function defined by F2 .l/ D 8:R.sech2 .l=2//. Then R satisfies   X  .n  3/ 2 2 li R sech D : 2 6 i

If the ideal polygon S has cyclically ordered vertices xi , then the i th side can be identified with the geodesic with endpoints xi ; xi C1 . Then an orthogeodesic is the perpendicular between geodesic xi ; xi C1 and geodesic xj ; xj C1 where ji  j j  2 which we label ˛ij of length lij . We define the cross-ratio of four points by Œa; bI c; d D

.a  b/.d  c/ : .a  c/.d  b/

Then a simple calculation shows that Œxi ; xi C1 I xj ; xj C1 D Œ1; 1; elij ; elij D

.1 C 1/.elij C elij / D sech2 .lij =2/ : .1 C elij /.1 C elij /

Therefore we can rewrite the identity in the finite case as X ji j j2

  .n  3/ 2 : R Œxi ; xi C1 I xj ; xj C1 D 6

Ideal quadrilaterals and Euler’s identity. For n D 4, S is an ideal quadrilateral with two ortholengths l1 ; l2 and vertices x1 ; x2 ; x3 ; x4 . Thus as Œx1 ; x2 I x3 ; x4 D 1  Œx3 ; x4 I x1 ; x2 , for any 0 < x < 1 we have R.x/ C R.1  x/ D

2 : 6

This identity was proved for the dilogarithm function by Euler (see Lewin’s book [27] for details). Also by symmetry R.1=2/ D  2 =12 and similarly R.0/ D 0; R.1/ D  2 =6. Ideal pentagons and Abel’s identity. If S is a ideal pentagon then there are 5 orthogeodesics. We send three of the vertices to 0; 1; 1 and the other two to u; v with 0 < u < v < 1. Then the cross ratios in terms of u; v are u;

1v;

vu ; v

vu ; 1u

u.1  v/ v.1  u/

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Putting into the equation, we obtain the following equation.   v  u v  u 2 u.1  v/ R .u/ C R .1  v/ C R CR CR D : v 1u v.1  u/ 3 Letting x D u=v; y D v, we get 

y.1  x/ R .xy/ C R .1  y/ C R .1  x/ C R 1  xy





x.1  y/ CR 1  xy

 D

2 : 3

Now by applying Euler’s identity for x; y, we obtain Abel’s pentagon identity     x.1  y/ y.1  x/ CR : R .x/ C R .y/ D R .xy/ C R 1  xy 1  xy To show R D R we use the observation of Calegari in [15], that by a result of Dupont (see [17]), the Rogers dilogarithm is characterized by the Euler and Abel identities, therefore R D R.

5.4 Proof of the Luo–Tan identity We let S be a closed hyperbolic surface. We note that as @S D ;, the Basmajian and Bridgeman identities do not make sense as there are no orthogeodesics. Similarly, it is not clear how to extend the McShane identity to this case as there is no starting point, i.e., no boundary component or horocycle to decompose. Furthermore, since the generalization of the McShane identity to cone surfaces has a restriction that all cone angles are  , we cannot deform a cone singularity to a smooth point to obtain an identity. However, one can combine two key ideas from the Bridgeman–Kahn and the McShane proofs to obtain an identity, which is what Luo and the second author did. The key idea from the Bridgeman–Kahn identity is to start from any point and any direction, that is, to define X D T1 .S / with  the volume measure. The key idea we use from the proof of the McShane identity is that instead of flowing in both directions indefinitely, as in the Bridgeman identity, we flow until we get intersection points. In this way, for a generic vector v 2 T1 .S /, we construct a geodesic graph Gv with Euler characteristic 1 and use this to obtain a decomposition of X . From this decomposition we calculate the measures of the components to obtain the Luo–Tan identity. More precisely, given v 2 T1 .S /, consider the unit speed geodesic rays vC .t/ and  v .t/ .t  0/ determined by exponentiating ˙v. If the vector v is generic, then both rays will self-intersect transversely by the ergodicity of the geodesic flow, otherwise, we have v 2 Z  X where .Z/ D 0. Each v 2 X n Z will determine a canonical graph Gv as follows (see Figure 4). Consider the path At D v .Œ0; t / [ vC .Œ0; t / for t > 0 obtained by letting the geodesic rays gv and gvC grow at equal speed from time 0 to t. Let tC > 0 be the

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Figure 4. Graph Gv .

Figure 5. Subsurface †v , with Gv  †v .

smallest positive number so that AtC is not a simple arc. Without loss of generality, we may assume that vC .tC / ¤ v .tC / by ignoring a set of measure zero (i.e. putting it into Z). Say vC .tC / 2 v .Œ0; tC / [ vC .Œ0; tC //. Next, let t > tC be the next smallest time so that v .t / 2 v .Œ0; t // [ vC Œ0; tC /. Definition 5.2. The union v .Œ0; t / [ vC .Œ0; tC / is the graph Gv associated to v. From the definition, Gv has Euler characteristic is 1. We call an embedded pair of pants (three-holed sphere) or one-holed torus with geodesic boundary in S a simple geometric embedded subsurface. The following result allows us to decompose X n Z into subsets indexed by the set of simple geometric embedded subsurfaces. Proposition 5.3 (Proposition 3.1 of [28]). The graph Gv is contained in a unique simple geometric embedded subsurface †v  S . Proof. Cutting the surface S open along Gv , we obtain a surface whose metric completion SO is a compact hyperbolic surface with convex boundary. The boundary of SO

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41

consists of simple piecewise geodesic loops (corresponding to Gv ), and each boundary has at least one corner. If O is a simple piecewise geodesic loop in @SO , it is freely homotopic to a simple closed geodesic  in SO which is a component of the O Furthermore O and  are disjoint by boundary of core.SO /, the convex core of S. convexity. Therefore, O and  bound a convex annulus exterior to core.SO / and Gv is disjoint from core.SO /. The simple geometric subsurface †v containing Gv is the union of these convex annuli bounded by O and  (see Figure 5). The Euler characteristic of †v is 1 by construction. Furthermore, the surface †v is unique. Indeed, if †0 ¤ †  S is a simple geometric subsurface so that Gv  †0 , then †0 has a boundary component, say ˇ, which intersects one of the boundaries  of † transversely (we use here in an essential way the fact that the surfaces † and †0 are simple). Therefore, ˇ must intersect the other boundary O of the convex annulus described earlier, otherwise we have a hyperbolic bigon, a contradiction. Hence it intersects Gv which contradicts Gv  †0 .  The above discussion gives a decomposition of T1 .S /nZ indexed by these simple subsurfaces, namely, for any simple subsurface †  S , define X† WD fv 2 X W Gv  †g : P P Then .X / D .T1 .S // D P .XP / C T .XT / where the first sum is over all simple geometric embedded pairs of pants and the second sum is over all simple geometric embedded one-holed tori. It remains to calculate for a simple hyperbolic surface † D P or T the volume of the set of all unit tangent vectors v 2 T1 .†/ so that Gv is strictly contained in †, that is, Gv is a spine for †. This will give us the functions f and g in Theorem D.

5.5 Computing the measures of XP and XT By definition, we have f .P / D .XP / and g.T / D .XT /. It is complicated to compute the measures directly, and it turns out that it is easier to compute .XP / and .XT / by calculating the measure of the complementary set in T1 .†/ instead. The idea is that the vectors v 2 T1 .P / in the complementary set can be divided into a small number of disjoint types which can be described quite easily geometrically, hence the complementary set decomposes into a finite number of disjoint subsets whose measures can be computed in a similar way to the computation for the Bridgeman identity. Suppose P is a hyperbolic three-holed sphere, and i 2 f1; 2; 3g taken mod 3. Denote the boundary geodesics of P by Li , the simple orthogeodesics from Li C1 to Li C2 by Mi , and the simple orthogeodesic from Li to itself by Ni ; see Figure 6(a). Denote the lengths of the boundary geodesics and the orthogeodesics by the corresponding lower case letters, that is li , mi and ni respectively. For v 2 T1 .P /, we define v to be the maximal geodesic arc in P tangent to v. We modify the definition of Gv appropriately, to take into account the fact that

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Martin Bridgeman, Ser Peow Tan

Figure 6. 3-holed spheres (a) and 1-holed tori (b).

@P ¤ ;, that is,

GO v D v .Œ0; t / [ vC .Œ0; tC /

as before but the times tC and t may also occur when the geodesic hits @P in which case we stop generating the geodesic in that direction. From the definitions, it is clear that GO v  v , and v 2 XP if and only if GO v \@P D ;. In particular, if v 2 T1 .P / n XP , and v 62 Z, then GOv is either simple, with both endpoints on @P , or has one endpoint on Li and the other end is a loop freely homotopic to Lj , j ¤ i . We call this a lasso around Lj based at Li (note that the loop cannot be homotopic to Li ). The following gives a decomposition of T1 .P /nXP into finitely many types:  Define H.Mi / D fv 2 T1 .P /jv  M1 rel. boundaryg. If v 2 H.Mi /, then v is simple, GO v D v , and v 62 XP . The measure of these sets, computed by Bridgeman  in [11], depends only on mi and is given by .H.M1 // D  8:R sech2 m2i .  Define H.Ni / D fv 2 T1 .P /jv  N1 rel. boundaryg. If v 2 H.Ni /, then v intersects Mi exactly once, the point of intersection divides v into two components vC and v . This case is more complicated as v may have arbitrarily many self-intersections. However, vC and v are both simple. This can be seen by cutting P along Mi to obtain a convex cylinder bounded on one side by Li and the other by a piecewise geodesic boundary. Then both vC and v are geodesic arcs from one boundary of the cylinder to the other, so must be simple. In particular, in the construction of GO v , we see that in this case GO v is either a simple geodesic arc from Li to itself, or it must be a lasso based at Li . Liso v 62 XP . Again, by Bridgeman, the That is, for all such v, GO v intersects  measure is .H.Ni // D 8:R sech2 n2i . Note that these sets are disjoint from those in the first case.

1 Identities on hyperbolic manifolds

43

Figure 7. Universal Cover of P in H2 .

 The remaining case is when GO v is a lasso, but does not come from Case 2 above, we call these true lassos. For fi; j; kg D f1; 2; 3g distinct, let W .Li ; Mj / D fv 2 T1 .P /jGO v is a true lasso around Li based at Lk g : Suppose that GO v is a lasso based at the point q on L1 with (positive) loop around L2 , but such that v is not homotopic relative to the boundary to N1 . It is convenient to understand the set of v generating such GO v in the universal cover, PQ  H2 (we use the upper half space model). We work with the following setting: Consider a fundamental domain for P in H2 consisting of two adjacent right angled hexagons H and H 0 such that H is bounded by LQ 1 , LQ 2 and LQ 3 and Q 0 and LQ 3 , see Figure 7. Given x; y 2 H2 [ @H2 , H 0 is bounded by LQ 2 , L 1 x ¤ y, let GŒx; y denote the geodesic in H2 from x to y. Normalize so that LQ 1 D GŒc; d , LQ 2 D GŒ1; 0 , and LQ 3 D GŒe; f , where e; f; c; d 2 R satisfy 0 < e < f < c < d . Further normalize so that LQ 01 D GŒ1; a where 1 < a < e. By elementary calculations, we have c D el2 ; d D el2 coth2 .m3 =2/. We can choose a lift of GO v so that the endpoint q lies on LQ 1 D GŒc; d . Let GŒy; x , y; x 2 R2 be the complete geodesic in H2 containing this lift. In particular, by construction, y 2 .c; d /. We claim that there is a unique lift so that 0 < x < 1. The cases x D 0 are x D 1 are important limiting cases. When x D 0, GO v is a simple geodesic of infinite length which spirals around L2 in the positive direction. When x < 0, GO v is homotopic to M3 , so is a simple finite geodesic arc from L1 to L2 . When x is small and positive, GO v is

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Martin Bridgeman, Ser Peow Tan

a lasso with positive loop around L2 . When x D 1, v spirals around L1 with an infinite number of self-intersections, and GO v is still a lasso from L1 with positive loop around L2 . As x ranges between 1 and a, v is homotopic to N1 and GO v ranges from being a lasso with positive loop around L2 to being simple and then a lasso with negative loop around L3 . In particular, if GO v is a lasso with endpoint at L1 and positive loop around L2 , but v is not homotopic to N1 , we have a unique lift so that 0 < x < 1 and c < y < d . However, not all tangent vectors on T1 .GŒy; x / generate GO v . By the construction of GO v , the midpoint of the loop, p, of GO v is a critical point, all vectors which project to the longer part of GO v will generate GO v , those which project to the shorter part (consisting of half of the loop) will generate a different GO v , by construction. Also, the loop of GO v bounds with L2 a convex cylinder in P so p is the point on the loop which is closest to L2 . Hence, p is the point on GŒy; x such that the ray 0p is tangent to GŒy; x . To calculate the volume, we need to integrate the length of the geodesic from q to p over the set of all geodesics in H2 with endpoints x; y where 0 < x < 1 and c < y < d , with respect to the Liouville measure on the space of geodesics. Since both forward and backward vectors on this set generate GO v , we multiply the result by 2, that is, we want twice the volume of the set where

D fv 2 T1 .H2 /jv 2 T1 .GŒq; p / ; 0 < x < 1 ; c < y < d g :

(5.2)

The computation of Vol. / is elementary but somewhat messy, the final integral which needs to be computed is given by the following: Proposition 5.4 (Proposition 4.1 of [28]). The volume of is given by ˇ 1 ˇ 0 Z 1 Z d ln ˇˇ y.xc/.xd / ˇˇ x.yc/.yd / @ dy A dx : (5.3) .y  x/2 0 c Note that the above integral only depends on c and d which are given by c D el2 ; d D el2 coth2 .m3 =2/. More generally, define the Lasso function La.l; m/ to be the above integral where c D el ; d D el coth2 .m=2/. We have: Proposition 5.5 (Proposition 4.6 of [28]). The lasso function La.l; m/ is given by      1y 1x CR La.l; m/ D 2 R.y/  R 1  xy 1  xy where x D el and y D tanh2 .m=2/, R.z/ is the Rogers dilogarithm. As remarked earlier, we need to take twice the volume of to obtain the volume of the set of vectors generating true lassos from L1 with a positive

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loop around L2 . The volume for those with a negative loop around L2 is the same by symmetry, hence we have: Vol.W .Li ; Mj // D 4La.li ; mj / : Now .XP / can be computed by subtracting away from .T1 .P // the measures of the sets H.Mi /, H.Ni /, i D 1; 2; 3 and W .Li ; Mj /, i ¤ j . To obtain the expression in Theorem D, we use some of the pentagon relations satisfied by the Rogers dilogarithm function. For details, see [28]. The computation of .XT / for an embedded one-holed torus is similar, with an extra observation. Again, v 62 XT if GO v \ @T ¤ ; (@T has only one component). In this case GO v is either a simple geodesic arc from @T to itself, or is a lasso based at @T with a loop homotopic to an essential, non boundary parallel simple closed geodesic. In either case, GO v is disjoint from a unique simple closed geodesic A  T , see Figure 6b. Cutting along A produces a pair of pants PA , and from the previous calculations, we can calculate the set of all v 2 T1 .PA / such that GO v intersects @T but not the other two boundary components of PA . Summing up the measures over all possible simple closed geodesics A, we obtain the measure of the complement of XT in T1 .T /. Again, by manipulating the expressions using the identities for R, we obtain the expression in Theorem D for g.T /, see [28] for details. 

6 Moments of hitting function We consider all four identities and their associated measure space .X; /. Associated to this, we have a hitting function LW X ! RC where L.x/ is the length of the geodesic arc associated to x. The function L is measurable and we can consider it as a random variable with respect to the measure . For k 2 Z, the k t h moment of L with respect to the measure  is then Z k Mk .X / D .L /.x / D Lk .x/d : X

In particular, M0 .X / D .X /. Also the measurable decomposition X D Z [ gives us a formula XZ Mk .X / D Lk .v/d : i

S i

Xi

Xi

In each identity, it is easy to again show that each integral in the summation on the right only depends on the spectrum associated with the identity. Therefore one can find smooth functions Fk such that X Fk .l/ Mk .X / D l2S

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Martin Bridgeman, Ser Peow Tan

where S denotes the spectrum of the given identity. In particular, this formula is the original identity in the case k D 0. Also the average length of the geodesic associated with an element of X , called the average hitting time A.X /, is then given by A.X / D

M1 .X / M1 .X / D : .X / M0 .X /

6.1 Moments of the Bridgeman–Kahn identity In a recent paper, we consider the moments of the Bridgeman–Kahn identity and show that both the Bridgeman–Kahn and Basmajian identities arise as identities for its moments. The Bridgeman–Kahn identity obviously arises as the identity for the k D 0 moment. We show that the Basmajian identity appears as the identity for the k D 1 moment, giving a link between the two identities. We also derive an integral formula for the moments and an explicit formula for A.X / in the surface case. Theorem 6.1 (Bridgeman–Tan, [14]). There exist smooth functions Fn;k W RC ! RC and constants Cn > 0 such that if X is a compact hyperbolic n-manifold with totally geodesic boundary @X ¤ ;, then (1) The moment Mk .X / satisfies X Fn;k .l/ : Mk .X / D l2LX

(2) M0 .X / D Vol.T1 .X // and the identity is the Bridgeman–Kahn identity. (3) M1 .X / D Cn :Vol.@X / and the identity for M1 .X / is the Basmajian identity. (4) The average hitting time A.X / satisfies A.X / D

X X 1 Fn;1 .l/ D Gn .l/ : Vol.T1 .X // l2LX

l2LX

In the surface case we obtain an explicit formula for the function G2 and hence A.X / in terms of polylogarithms. Furthermore, besides compact surfaces obtained as quotients of Fuchsian groups, the identity holds more generally for finite-area surfaces with boundary cusps. Theorem 6.2 (Bridgeman–Tan, [14]). Let S be a finite-area hyperbolic surface with non-empty totally geodesic boundary. Then 1 0  X  1 l @ F sech2 C 6.3/CS A A.S / D 2Area.S / 2 l2LS

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where 4 2 F .a/ D  12.3/  log.1  a/ C 6 log2 .1  a/ log.a/  4 log.1  a/ log2 .a/ 3  2  a Li2 .a/ C 24Li3 .a/ C 12Li3 .1  a/ ;  8 log 1a for Lik .x/ the k t h -polylogarithm function, and  the Riemann -function.

6.2 Moments of the Basmajian identity In the recent paper [50], Vlamis considers the moments for the Basmajian identity. Theorem 6.3 (Vlamis, [50]). Let X be a compact hyperbolic manifold with totally geodesic boundary and mk .X / be the moments of the boundary hitting function with respect to Lebesgue measure on @M . Then X mk .X / D fn;k .l/ l2LX

where Z

log coth.l=2/

fn;k .l/ D n2 0





coth l C cosh r log coth l  cosh r

k sinhn2 rdr

and n2 is the volume of the unit .n  2/-dimensional sphere. Vlamis derives an explicit formula in odd-dimensions and further derives a formula for the first moment in the case of a surface S . Theorem 6.4 (Vlamis, [50]). Let S be a compact hyperbolic surface with non-empty totally geodesic boundary. Then      X  1 2 l 2 l 2 Li2  tanh  Li2 tanh C  =4 : A.S / D 2Area.S / 2 2 l2LS

7 The Bowditch proof of the McShane identity and generalizations Bowditch gave an algebraic-combinatorial proof of the original McShane identity for the punctured torus in [8], and extending the method, proved variations for punctured torus bundles in [9] and representations of the punctured torus group (including the

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quasi-fuchsian representations) satisfying some conditions he calls the Q-conditions in [10]. One advantage of the proof is that it avoids the use of the Birman–Series result on the Hausdorff dimension of the set of points on simple geodesics on a hyperbolic surface. Akiyoshi, Miyachi and Sakuma refined the identity for punctured torus groups in [1] and found variations for quasi-fuchsian punctured surface groups in [2]. Let T be a once-punctured hyperbolic torus, and  WD .T / D hX; Y i the fundamental group of T , a free group on two generators, and W  ! PSL.2; R/ the holonomy representation. Define an equivalence relation  on , by X  Y if X is conjugate to Y or Y 1 . The classes correspond to free homotopy classes of closed curves on T . Let  =  be the set of classes corresponding to essential, simple, non-peripheral simple closed curves on T . Classes in have representatives which form part of a generating pair for . We call them primitive classes. For X 2 , x WD tr.X / is well defined up to sign, and is related to the length l of the unique geodesic representing the class by cosh2 .l=2/ D x 2 =4 : McShane’s original identity for the once-punctured torus then has the form ! r X X 4 h.x/ D 1 1 2 D1 x X 2

(7.1)

X 2

q where we let x D tr.X /, and h.x/ D 1  1  x42 . This is the form which was proven by Bowditch, and generalized to type-preserving representations (i.e., tr.X YX 1Y 1 / D 2) of  into SL.2; C/ satisfying the following conditions which he calls the Q- conditions, and which we call here the BQ-conditions: Definition 7.1. A represention  from  into SL.2; C/ satisfies the Bowditch Qconditions (BQ-conditions) if (1) tr.X / 62 Œ2; 2 for all X 2 ; (2) jtr.X /j  2 for only finitely many (possibly none) X 2 . The basic idea of the proof was to represent the values taken by elements of in an infinite trivalent tree. This arises from the fact that can be identified with Q [ 1 by considering the slopes of the curves in T , and the action of the mapping class group of T on this set is essentially captured by the Farey tessellation, whose dual is an embedded infinite trivalent tree in H. In this way, is identified with the set of complementary regions of the dual tree, and the values tr.X / for X 2 satisfy vertex and edge relations which come from the Fricke trace identities. Using this function and the vertex and edge relations, Bowditch was able to cleverly assign a function on the directed edges of the tree which satisfied some simple conditions and applied a Stoke’s theorem type argument to prove the identity. Subsequently, Tan, Wong and Zhang extended the Bowditch method to prove versions of the identity for general representations of the free group  into PSL.2; C/

1 Identities on hyperbolic manifolds

49

satisfying the Q-conditions and closed hyperbolic three-manifolds obtained from hyperbolic Dehn surgery in [47, 46]. Specifically, let  2 Hom.; SL.2; C// be a representation satisfying the BQ-conditions and let  D tr.X YX 1 Y 1 /. We call  a  -representation. Define the function h WD h as follows: For  2 C, set  D cosh1 .=2/. We define p h D h W Cnf˙  C 2g ! C by 1

h.x/ D 2 tanh



sinh  cosh  C el.x/

 (7.2)

D log

e C el.x/ e C el.x/

(7.3)

D log

1 C .e  1/ h.x/ ; 1 C .e  1/ h.x/

(7.4)

where (7.2), (7.3) and (7.4) are equivalent (see [47] for details). We have: Theorem 7.2 (Tan–Wong–Zhang, Theorem 2.2, [47]). Let W  ! SL.2; C/ be a  representation (where  ¤ 2) satisfying the BQ-conditions. Set  D cosh1 .=2/. Then X h.x/ D  mod 2 i ; (7.5) X 2

where the sum converges absolutely, and x WD tr.X /. When  arises as the holonomy of a hyperbolic structure on a one-holed torus with geodesic boundary or with a cone point, then the above is equivalent to the Mirzakhani [37] and the Tan–Wong–Zhang [45] variations of the McShane identity. Recently, revisiting the original Bowditch proof, Hu, Tan and Zhang proved new variations of the identity for representations of  into PSL.2; C/ satisfying the BQconditions [21]: Theorem 7.3 (Hu–Tan–Zhang, [21]). Let W  ! SL.2; C/ be a  -representation (where  ¤ 2) satisfying the BQ-conditions. Let  D  C 2. Then ! r X X 4 3x 2  2 g.x/ D 1 2 D1; 1 (7.6) 3.x 2  / x X 2

X 2

where the sum converges absolutely, and x WD tr.X /. Note that the type-preserving case occurs when  D 2, so  D 0, and the above reduces to the original McShane identity in this case.

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The authors also extended this in [22] to identities for orbits of points in Cn under the action of the Coxeter group Gn generated by n-involutions which preserve the varieties defined by the Hurwitz equation. At this point, it is not clear what is the underlying geometric interpretation of these identities.

8 Concluding remarks We have shown that the identities obtained by Basmajian, McShane, Bridgeman– Kahn and Luo–Tan are obtained by considering decompositions of certain sets X with finite measure  associated to the manifold M obtained by considering some kind of geodesic flow, either from the boundary, or from the interior. Typically, there is a subset of measure zero which is complicated but which does not contribute to the identity, and the subsets of non-zero measure in the decomposition are indexed by some simple geometric objects on the manifold. The measures of each subset typically depend only on the local geometry and data, and not on the global geometry of M , which may be easy or fairly complicated to compute. One can apply this general philosophy to try to obtain other interesting identities. There are also many other interesting directions for further research and exploration in this area. We list a number of them: (1) Find good applications for the identities, for example use them to say something about the moduli space of hyperbolic surfaces. The McShane–Mirzakhani identity was an important ingredient in the work of Mirzakhani in the study of the Weil–Petersson geometry of the moduli space of bordered Riemann surfaces. It would be interesting to find similar applications for the Basmajian, Bridgeman and Luo–Tan identities. (2) Generalize the McShane/Luo–Tan identities to hyperbolic manifolds of higher dimension or to translation surfaces. (3) The Basmajian and Bridgeman–Kahn identities do not generalize easily to complete finite volume hyperbolic surfaces with cusps (or cone points) as in this case the limit set does not have measure zero. It would be interesting to find other interesting decompositions of horocycles which would generalize these identities. Some progress on this has been made recently by Basmajian and Parlier [4]. (4) It would be interesting to extend the Bowditch method to general surfaces of genus g with n cusps. Some progress has been made recently by Labourie and the second author [25]. (5) Give a geometric interpretation of the variation of the McShane identity (Theorem 7.3) obtained by Hu–Tan–Zhang using the extension of the Bowditch method, and also the identity for the n-variable case. (6) Extend the identities to other interesting representation spaces, see for example the work of Labourie and McShane in [24], and Kim, Kim and Tan in [23].

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(7) Extend the moment generating point of view, and derive formulae for the moments of the McShane and Luo–Tan identities. (8) Obtain identities for general closed hyperbolic manifolds. (9) Analyze the asymptotics of the functions f and g in the Luo–Tan identity and use it to study the asymptotics of embedded simple surfaces in a hyperbolic surface. Acknowledgements. We are grateful to Dick Canary, Francois Labourie, Feng Luo, Greg McShane, Hugo Parlier, Caroline Series, and Ying Zhang for helpful discussions on this material.

References [1] H. Akiyoshi, H. Miyachi and M. Sakuma, A refinement of McShane’s identity for quasifuchsian punctured torus groups, In the Tradition of Ahlfors and Bers, III. The Ahlfors–Bers Colloquium, Oct. 2001, Univ. of Connecticut at Storrs, W. Abikoff, A. Haas (Eds.), Contemporary Mathematics, 355, 2004, 21–40. [2] H. Akiyoshi, H. Miyachi and M. Sakuma, Variations of McShane’s identity for punctured surface groups, London Mathematical Society Lecture Notes, Y. Minsky, M. Sakuma & C. Series (Eds.), Cambridge University Press, 2006, 151–185. [3] A. Basmajian, The orthogonal spectrum of a hyperbolic manifold, Amer. J. Math. 115 (1993), 1139–1159. [4] A. Basmajian and H. Parlier, private communication. [5] A. Beardon, The Geometry of Discrete Groups, Springer-Verlag, 1995. [6] J. Birman and C. Series, Geodesics with bounded intersection number on surfaces are sparsely distributed. Topology 24 (1985), 217–225. [7] F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), 139–162. [8] B. Bowditch, A proof of McShane’s identity via Markov triples, Bull. London Math. Soc. 28 (1996), 73–78. [9] B. Bowditch, A variation of McShane’s identity for once-punctured torus bundles, Topology 36 (1997), 325–334. [10] B. Bowditch, Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. 77 (1998), 697–736. [11] M. Bridgeman, Orthospectra of geodesic laminations and dilogarithm identities on moduli space, Geom. Topol. 15 (2011), 707–733. [12] M. Bridgeman and D. Dumas, Distribution of intersection lengths of a random geodesic with a geodesic lamination, Ergodic Theory Dyn. Syst 27 (2007), 1055–1072. [13] M. Bridgeman and J. Kahn, Hyperbolic volume of n-manifolds with geodesic boundary and orthospectra, Geom. Funct. Anal. 20 (2010), 1210–1231.

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[14] M. Bridgeman and S. P. Tan, Moments of the boundary hitting function for the geodesic flow on a hyperbolic manifold, Geom. Topol. 18 (2014), 491–520. [15] D. Calegari, Bridgeman’s orthospectrum identity, Topology Proceedings 38 (2011), 173–179. [16] D. Calegari, Chimneys, leopard spots, and the identities of Basmajian and Bridgeman, Algebr. Geom. Topol. 10 (2010), 1857–1863. [17] J. Dupont, The dilogarithm as a characteristic class for flat bundles. J. Pure Appl. Algebra 44 (1987), 137–164. [18] W. Goldman, G. Margulis, and Y. Minsky, Complete flat Lorentz 3-manifolds and laminations on hyperbolic surfaces, in preparation. [19] H. Hu, Identities on hyperbolic surfaces, group actions, the Markoff and Hurwitz equations, PhD thesis, National University of Singapore, (2013). [20] H. Hu and S. P. Tan, New identities for small hyperbolic surfaces, Bull. London Math. Soc. 46 (2014), 1021–1031. [21] H. Hu, S. P. Tan, and Y. Zhang, A new identity for SL.2; C/- characters of the once punctured torus group, Math. Research Letters 22 (2015), 485–499. [22] H. Hu, S. P. Tan, and Y. Zhang, Polynomial automorphisms of Cn preserving the Markoff– Hurwitz polynomial, preprint, arXiv:1501.06955 (2015). [23] I. Kim, J. Kim, and S. P. Tan, McShane’s Identity in Rank One Symmetric Spaces Math. Proceedings of Cambridge Philo. Soc. 157 (2014) 113–137. [24] F. Labourie and G. McShane, Cross ratios and identities for higher Teichmüller-Thurston theory, Duke Math. J. 149 (2009), no. 2, 279–345. [25] F. Labourie and S. P. Tan, Topological coding of simple loops and harmonic measures, in preparation. [26] D. Lee and M. Sakuma, A variation of McShane’s identity for 2-bridge links, Geom. Topol. 17 (2013), 206–2101. [27] L. Lewin, Structural Properties of Polylogarithms, Mathematical Surveys and Monographs, AMS, Providence, RI, 1991. [28] F. Luo and S. P. Tan, A dilogarithm identity on moduli spaces of curves,J. Diff. Geom. 97 (2014), 255–274. [29] F. Luo and S. P. Tan, A dilogarithm identity on moduli spaces of compact surfaces with boundary and non-orientable surfaces, in preparation. [30] H. Masai and G. McShane, Equidecomposability, volume formulae and orthospectra, Algebr. Geom. Topol. 13 (2013), 3135–3152. [31] B. Maskit, Kleinian Groups, Graduate Texts in Mathematics, Springer-Verlag, 1987. [32] G. McShane, A remarkable identity for lengths of curves, PhD Thesis, Warwick, 1991. [33] G. McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math. 132 (1998), 607–632. [34] G. McShane, Weierstrass points and simple geodesics, Bull. London Math. Soc. 36 (2004), 181–187. [35] G. McShane, Simple geodesics on surfaces of genus 2, Ann. Acad. Sci. Fenn. Math. 31 (2006), 31–38.

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[36] M. Mirzakhani, Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces, PhD Thesis, Harvard, 2004. [37] M. Mirzakhani, Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007), 179–222. [38] P. Nicholls, The Ergodic Theory of Discrete Groups, volume 143 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1989. [39] P. Norbury, Lengths of geodesics on non-orientable hyperbolic surfaces, Geometriae Dedicata 134 (2008), 153–176. [40] A. Papadopoulos and G. Théret, Shortening all the simple closed geodesics on surfaces with boundary, Proc. Amer. Math. Soc. 138 (2010), 1775–1784. [41] J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature, preprint (2013). [42] H. Parlier, Lengths of geodesics on Riemann surfaces with boundary, Ann. Acad. Sci. Fenn. Math. 30 (2005), 227–236. P xn [43] L. J. Rogers, On function sum theorems connected with the series 1 1 n2 Proc. London Math. Soc. 4 (1907), 169–189. [44] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. IHES 50 (1979), 171–202. [45] S. P. Tan, Y. L. Wong, and Y. Zhang, Generalizations of McShane’s identity to hyperbolic cone-surfaces, J. Diff. Geom. 72 (2006), 73–112. [46] S. P. Tan, Y. L. Wong, and Y. Zhang, Necessary and sufficient conditions for McShane’s identity and variations, Geometriae Dedicata 119 (2006), 119–217. [47] S. P. Tan, Y. L. Wong, and Y. Zhang, Generalized Markoff maps and McShane’s identity, Adv. Math. 217 (2008), 761–813. [48] S. P. Tan, Y. L. Wong, and Y. Zhang, McShane’s identity for classical Schottky groups, Pac. J. Math. 237 (2008), 183–200. [49] S. P. Tan, Y. L. Wong, and Y. Zhang, End invariants for SL.2; C/ characters of the one-holed torus, Amer. J. Math. 130 (2008), 385–412. [50] N. Vlamis, Moments of a length function on the boundary of a hyperbolic manifold, Alg. and Geo. Top 15(4) (2015), 53–75. [51] N. Vlamis and A. Yarmola, Basmajian’s identity in higher Teichmüller–Thurston theory, preprint, arXiv:math/1504.00649. [52] Y. Zhang, Hyperbolic cone-surfaces, generalized Markoff maps, Schottky groups and McShane’s identity, PhD Thesis, National University of Singapore, 2004.

Chapter 2

Problems on the Thurston metric Weixu Su1 Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . 1.2 References . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . 2 Infinitesimal properties of the Thurston metric . . . . 3 Geodesics of the Thurston metric . . . . . . . . . . . . . 4 Generalizations of the Thurston metric . . . . . . . . . . 5 Infinitely-generated Fuchsian groups of the first kind 6 Other general questions . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction 1.1 Background Let S be an oriented surface of genus g with n punctures with negative Euler characteristic. The set of equivalence classes of marked conformal (or hyperbolic) structures on S is parametrized by the Teichmüller space of S . Teichmüller space has a canonical complex analytic structure, on which the mapping class group acts as the bi-holomorphic automorphism group. The quotient space of Teichmüller space under the mapping class group is the moduli space of Riemann surfaces. The study of Teichmüller space is a composite of diverse main streams of modern mathematics, such as quasiconformal mappings, hyperbolic geometry in two and three dimensions, geometric group theory, algebraic geometry and ergodic theory. It has also a lot of applications in complex dynamics, low-dimensional topology, projective geometry, representations of Lie groups, theoretical physics, etc. The two classical metrics on Teichmüller space are the Teichmüller metric and the Weil–Petersson metric. The Teichmüller metric measures the least maximal quasiconformal dilation between two conformal structures, while the Weil–Petersson metric is more related to the hyperbolic geometry of the surface. Thurston [61] defined 1 Work partially supported by Shanghai Center for Mathematical Sciences (CMSS), NSFC No: 11201078 and French ANR grant FINSLER.

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an asymmetric metric in analogy with the Teichmüller metric, as a solution to the extremal problem of finding the best Lipschitz map in the homotopy class of a homeomorphism between two hyperbolic surfaces. The Thurston metric2 on Teichmüller space is Finsler. Geodesics for this metric connecting two points in Teichmüller space are not unique. Any two points in Teichmüller space can be connected by a finite concatenation of a special kind of geodesics called stretch lines. Along a stretch line, extremal Lipschitz maps are represented by stretch maps between hyperbolic surfaces. The construction of stretch maps between hyperbolic surfaces provides a global parametrization of Teichmüller space called cataclysm or shear coordinates. These coordinates (and their complexification) generalize Thurston’s earthquake and they have many applications in the study of hyperbolic 3-manifolds [11], the Weil–Petersson symplectic form [12, 67], decorated Teichmüller thoery [46], quantization of Teichmüller space [21], higher Teichmüller theory [17], etc. Since the circulation of Thurston’s unpublished manuscript in 1986, research on the Thurston metric is still developing. Moreover, the original ideas of Thurston have influenced the study of outer spaces [15], geometrically finite Kleinian groups [23], Anti-de Sitter geometry [18] and iteration of rational maps on the Riemann sphere [60].

1.2 References The survey by Papadopoulos and Théret [42] provides background for the study of the Thurston metric, with a list of thirteen problems. A few more open questions, related to the work of Guéritaud and Kassel on a generalization of the Thurston metric to higher dimensional hyperbolic manifolds and the study of AdS geometry, can be found in [23, 6]. For research on outer space, on which an analogue of the Thurston metric is defined, we recommend Vogtmann [62], Bestvina [8] and [22]. For the marked length spectrum rigidity conjecture of Burns and Katok, see [13]. The conjecture is proved for negatively curved surfaces, see [64] for a recent exposition. For related results on strictly convex real projective manifolds and on symmetric spaces, see [27, 28].

1.3 Notation Let S be a hyperbolic surface of genus g with n punctures. We denote by T .S / the Teichmüller space of S . For simplicity, we shall consider a point in T .S / as a hyperbolic surface, without explicit reference to the marking or the equivalence relation. 2 In the literature, the asymmetric metric defined by Thurston is usually called Thurston’s asymmetric metric or Thurston’s Lipschitz metric. We call it Thurston metric because we think that this name is more appropriate. We also call a geodesic of the Thurston metric a Thurston geodesic.

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Let X be a point of Teichmüller space T .S /. The cotangent space to T .S / at X is canonically identified with the space of L1 integrable holomorphic quadratic differentials on X , denoted by Q.X /. We denote by M.X / the space of L1 measurable Beltrami differentials on X . A tangent vector  2 TX T .S / is represented by a Beltrami differential  2 M.X /. There is a natural paring between Q.X / and M.X / given by Z h; i D .z/.z/dzd zN : X

We have where

TX T .S / Š M.X /=Q.X /? Q.X /? D f 2 M.X / j h; i D 0; 8  2 Q.X /g :

We denote by ML the space of measured geodesic laminations on S . The geodesic length of any  2 ML on X 2 T .S / is denoted by ` .X /.

2 Infinitesimal properties of the Thurston metric The Thurston metric is Finsler, with an explicit asymmetric infinitesimal norm on each tangent space TX T .S /: F .X; / D sup d log ` ./ : 2ML

We call the above asymmetric norm the Thurston infinitesimal norm. The map d log `: (2.1)  7! d log ` embeds the space of projective measured laminations PML into TX T .S / as a convex sphere, dual to the unit sphere in TX T .S / endowed with the Thurston infinitesimal norm [61, 41]. Problem 2.1 (D. Dumas, K. Rafi). Does the Thurston infinitesimal norm on TX T .S /; X 2 T .S / determine X ? Remark 2.2. Problem 2.1 is motivated by Royden’s celebrated theorem saying essentially that the isometry group of Teichmüller space endowed with the Teichmüller metric is the extended mapping class group. Royden [48] proved his theorem (in the case of closed surfaces) by a detailed analysis of the fine structure of the space Q1 .X / of unit norm holomorphic quadratic differentials on X (the Finsler norm of the Teichmüller metric is at least of class C 1 on the tangent bundle). In particular, he used an embedding of X into Q1 .X /, the dual bicanonical map. Denote by S.X / the subset of Q1 .X / consisting of all quadratic differentials, which have a zero of order at least 3g  4. Then S.X / is an analytic variety which is the union of X (identified with

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the image of the bicanonical map) and a number of linear manifolds. The key point is that S.X /, as a subset of Q1 .X /, can be characterized by the degree of Hölder regularity of the norm on Q.X /. Problem 2.3 (K. Rafi). What properties of the hyperbolic surface are determined by the Thurston infinitesimal norm? Unlike the Teichmüller norm, the unit ball for the Thurston infinitesimal norm is not strictly convex. The unit sphere of the Thurston infinitesimal norm in TX T .S / is related to PML by the map d log ` defined by (2.1). Note that ML has a natural piecewise linear structure inherited from train tracks. The tangent cone structure of ML was explored in depth by Thurston [61]. For any  2 ML, the tangent space at  can be parametrized by a union of linear fragments and on each linear fragment the derivative of lengths of laminations (where we fix the hyperbolic structure and vary the laminations near ) can be computed. Problem 2.4 (F. Guértitaud). Does the unit sphere in TX T .S / endowed with the Thurston infinitesimal norm determine the global behaviour of stretch lines or geodesics? Remark 2.5. Thurston showed that the derivatives of lengths of laminations in ML is connected with the cataclysm coordinates for Teichmüller space. It was recently observed by Walsh [63] that the horofunction compactification of Teichmüller space with the Thurston metric is naturally identified with Thurston’s compactification. The study of the detour cost (a generalized metric defined on the horofunction boundary) led to the counterpart of Royden’s celebrated theorem: each isometry of Teichmüller space with the Thurston metric is induced by an element of the extended mapping class group. Thus a solution to Problem 2.1 may give a new proof of Walsh’s theorem. Moreover, we ask Problem 2.6. Is each local isometry of Teichmüller space with the Thurston metric induced by an element of the extended mapping class group? Remark 2.7. We don’t know whether a local isometry of the Thurston metric induces an isometry on the tangent bundle equipped with the infinitesimal norm. The equivalence of the two definitions of isometry in the Riemannian case is a result of Myers and Steenrod; and, in the Finsler case, proved by Deng and Hou [16] but under the hypothesis that the tangent bundle is C 1 . Problem 2.8 (D. Dumas). Is there any sense in which the Thurston metric is compatible with the complex structure of Teichmüller space? Remark 2.9. After all, all isometries of the Thurston metric turn out to be bi-holomorphic maps. We may ask the same question for the symplectic structure. We point out that all the above questions are related to the regularity of the Thurston infinitesimal norm.

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3 Geodesics of the Thurston metric To understand better the Thurston metric, we have to know its “curvature” and the behavior of it geodesics. Many investigations in this direction, inspired by properties of the Teichmüller metric, were done by Rafi and his collaborators. One major theme in the study of the Teichmüller metric is to know to what extent it resembles a metric of negative curvature. Like the Teichmüller metric, the Thurston metric has a product-like structure on thin parts of Teichmüller space [14]. One of the main tools in [30] is to capture the coarse geometry of the Thurston metric (or the Teichmüller metric) by (1) the short marking of a hyperbolic surface, which corresponds to a thin region of Teichmüller space. (2) relating the combinatorics of two distinct short markings, which corresponds to the intersection pattern of the thin regions and which is encoded in the curve graph, i.e., the one-skeleton of the complex of curves. Thurston showed that stretch lines are geodesics for the Thurston metric and that any two points in Teichmüller space can be joined by a geodesic path that is a finite concatenation of stretch segments (see Theorem 8.5 of [61] for a precise statement). However, there are other types of geodesics (some of them are described in [43]). Along the concatenation of stretch segments connecting two points in Teichmüller space, there is a unique maximally measured geodesic lamination all of whose leaves are stretched maximally [61]. Problem 3.1. Is there an algorithm to find the maximally stretch lamination? Generically, geodesics of the Thurston metric connecting two given points in Teichmüller space are not unique. Problem 3.2 (F. Guéritaud). Given X; Y 2 T .S /, describe the set Env.X; Y / D [fGg ; where G denotes a Thurston geodesic connecting X to Y . Remark 3.3. The shadow of a Thurston geodesic on the curve graph is a reparametrized quasi-geodesic. Since the curve graph is Gromov hyperbolic, the shadow of different Thurston geodesics connecting X to Y fellow travel each other. However, for any D > 0, there are Thurston geodesics G1 ; G2 with the same endpoints but G1 is not contained in the D-neighborhood of G2 . See [31] for details. Problem 3.4 (K. Rafi). Does Env.X; Y / depend continuously on X; Y ? Kassel and Rafi obtained a result to produce a preferred Thurston geodesic connecting two points. For any X; Y 2 T .S /, they prove that there is a Thurston geodesic GX;Y , parametrized linearly in Thurston’s shear (cataclysm) coordinates associated

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with a canonical lamination .X; Y / such that lengths of all simple closed curves along GX;Y are convex functions (up to reparametrization). Their argument depends on the fact that for any simple closed curve  on S , the length function of  is convex along any straight path in Thurston’s shear coordinates. Théret [59] independently proved a general result which states that the length function of a measured geodesic lamination is strictly convex in Thurston’s shearing coordinates over Teichmüller space. It was conjectured that the Thurston geodesic GX;Y constructed above does not back-track (this means that the projection of a geodesic to the complex of curves of any subsurface Y of S is an un-parametrized quasi-geodesic in the curve complex of Y ). The result is known to be true for the Teichmüller metric [47]. Rafi suggested to study this preferred path GX;Y instead of a generic Thurston geodesic, which may lead to an answer to the following question: Problem 3.5 (K. Rafi3 ). Identify curves that are short along the preferred path GX;Y . Remark 3.6. Papadopoulos and Théret [42] presented examples to study the behaviour of the lengths of some particular closed curves under the stretch and antistretch rays, in the case of the four-punctured sphere, punctured torus and closed surface of genus two. The picture they show permits us to understand the change in the geometry of the surface under a stretch map (in the case of closed surfaces of genus two, this is much more complicated). Problem 3.7 (K. Rafi). Assume that G .t/W RC ! T .S / is a stretch line directed by a maximal geodesic lamination . How does the geodesic G./ look like? Moreover, if the projection of G to the curve graph of a subsurface Y is large, is there an interval I of RC such that for G .t/; t 2 I , each boundary component ˇ of Y has bounded length and ˇ is close to a geodesic on Y ? The above question leads to Problem 3.8 (K. Rafi). Can we understand the behavior of Thurston geodesics inductively, going from surfaces to subsurfaces? Remark 3.9. The projection of stretch lines on subsurface is partially studied in [5], where the authors work on an asymmetric metric (an analogue of the Thurston metric) on Teichmüller spaces of surfaces with boundary. Another challenging problem is to to introduce a suitable notion of geodesic flow for the Thurston metric. There have also been a great deal of work done on the dynamical properties of the Teichmüller geodesic flow on moduli space or on the tangent bundle of moduli space, with applications to billiards and flat structures. However, the notion of geodesic flow does not exist naturally for the Thurston metric. 3 We

mention that Problem 9, Problem 12 and Problem 13 first appear in [45].

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Problem 3.10 (K. Rafi). Define the geodesic flow for the Thurston metric and study its properties, such as ergodicity or mixing. Let N.R/ be the number of closed Teichmüller geodesics of length at most R in the moduli space Mg . Note that N.R/ is also the number of conjugacy classes of pseudo-Anosov elements of the mapping class group of translation length at most R. Eskin and Mirzakhani [20] applied the properties of the Teichmüller geodesic flow to prove that e.6g6/R N.R/  ; as R ! 1 : .6g  6/R Problem 3.11 (K. Rafi). Find the asymptotic behavior for the number of Thurston geodesics of length at most R in the moduli space Mg . The action of the elements of the mapping class group on the Teichmüller space equipped with the Thurston metric is studied in [36]. The authors classified such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not. The above four types are related to Thurston’s classification of mapping classes. This study is parallel to the one made by Bers in the setting of the Teichmüller metric, and to the one made by Daskalopoulos and Wentworth in the setting of the Weil–Petersson metric. It was shown that, up to a finite order, a pseudo-Anosov map leaves a stretch line invariant. The projection on moduli space of the invariant stretch line provided by this result is a closed curve in moduli space, and it is homotopic to a Teichmüller closed geodesic of the same length. The question on determining the translation distance of a general pseudoAnosov mapping class remains open. Problem 3.12. If a mapping class is hyperbolic with respect to the Thurston metric, does it have an invariant Thurston geodesic? Remark 3.13. Unlike the action on the Teichmüller metric, there exist reducible mapping classes which leave a Thurston geodesic invariant [36]. By an argument of Bers, such a mapping class is hyperbolic. Problem 3.14. Is there a dense Thurston geodesic in moduli space? Remark 3.15. Masur [38] used closed Teichmüller geodesics to approximate a Teichmüller geodesic which is dense in moduli space. In general, we ask Problem 3.16 (K. Rafi). Is a stretch line typically recurrent/dense/equidistributed in moduli space?

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Remark 3.17. Thurston [61, Theorem 10.5] proved that the set of directions in the tangent sphere bundle of Teichmüller space which are tangent to a stretch path has Hausdorff dimension 0. On the other hand, we know more about the asymptotic behavior of Thurston geodesics near the Thurston boundary. A stretch line directed by a maximal geodesic lamination  converges to the projective class of the associated horocylic foliation, viewed as an element of Thurston’s boundary [40]. When the stump 0 of 4 is a union of closed curves or a uniquely ergodic measured lamination on a subsurface of S , the anti-stretch line converges to Œ0 as an element of Thurston’s boundary [57]. Walsh [63] proved that Thurston’s boundary is homeomorphic to the (Gromov) horofunctin boundary of Teichmüller space endowed with the Thurston metric. As a result, any Thurston geodesic ray converges to an element of Thurston’s boundary. The question of when two Teichmüller geodesic rays stay bounded distance apart has been answered completely: two Teichmüller geodesic rays stay bounded distance apart if and only if their corresponding vertical measured foliations are measure equivalent [29]. The same question for the Thurston metric remains open. Problem 3.18. Determine when two Thurston geodesic rays stay bounded distance apart. Remark 3.19. Théret [58] proved that two cylindrical stretch rays (i.e., stretch rays whose horocyclic laminations are weighted multi-curves) stay bounded distance apart if and only if they converge towards the same point in Thurston’s boundary. Such a behaviour is quite different from that of the Teichmüller geodesics. Problem 3.20 (K. Rafi). Given two points in Teichmüller space, do we have a comparison between the maximally stretched laminations (which define the concatenation of stretch segments) and the vertical (horizontal) foliations of the quadratic differential defining the Teichmüller geodesic connecting them? Remark 3.21. Minsky [39] studied the approximate behavior of high-energy harmonic maps when the domain surface varies, and he compared harmonic maps (which are stretched along Hopf foliations) with Teichmüller maps and stretch maps. A quasi-geodesic in a metric space .M; d / is a map rW I ! M such that js  tj=L  L  d.r.s/; r.t//  Ljs  tj C L 8 s; t 2 I where I  R is a closed interval and L is positive constant. A quasi-geodesic is stable if it is contained in a uniformly bounded neighborhood of a geodesic. Lenzhen, Rafi and Tao [30] proved that if G is a Thurston geodesic whose endpoints have bounded combinatorics (which implies that X and Y is cobounded), then any quasi-geodesic with the same endpoints as G fellow travels G. See [30] for more 4

0 is the unique maximal measured geodesic laminations that is supported on .

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precise definitions and descriptions. The above result is an extension of a theorem of Hamenstädt [24] for the Teichmüller metric. For the Teichmüller metric, the notions of bounded and cobounded combinatorics are equivalent. It is unknown whether for the Thurston metric coubounded implies bounded combinatorics. Eskin, Masur and Rafi have work in progress on understanding the quasi-isometry group of Teichmüller space equipped with the Teichmüller metric. The geometric rank of a metric space is the largest integer n such that there exists a quasi-isometric embedding from Rn into the space. In a recent paper [19], the above authors studied quasi-Lipschitz maps from large boxes in Rn into a metric space quasi-isometric to the Teichmüller space equipped with the Teichmüller metric or the Weil–Petersson metric, or the mapping class group quipped with the word metric. One of their results is that the geometric rank of the above three metrics are equal. Problem 3.22. What is the quasi-isometry group of Teichmüller space equipped with the Thurston metric?

4 Generalizations of the Thurston metric Problem 4.1 (A. Papadopoulos). What is a good way to symmetrize the Thurston metric? The length-spectrum metric on Teichmüller space is defined as

 ` .Y / ` .X / dls .X; Y / D max log sup ; log sup  ` .X /  ` .Y / It is a symmetrization of the Thurston metric. We have dls .X; Y / D maxfdTh .X; Y /; dTh .Y; X /g : This metric was first defined by Sorvali [53], more than ten years before the appearance of Thurston’s preprint. The topology of the length-spectrum metric was studied by [54, 32, 33], by comparing it with the Teichmüller metric. From an inequality of Sorvali [55] (also obtained by Wolpert [66]), we have dls  2dTeich on T .S /. Choi and Rafi [14] (and independently Liu–Sun-Wei [34]) proved that the two metrics are quasi-isometric on any thick part of T .S /, but that there are sequences .Xn /; .Yn/ in the thin part with dls .Xn ; Yn / ! 0 while dTeich .Xn ; Yn / ! 1. The divergence between dls and dTeich disappears in moduli space [37]. Problem 4.2 (A. Papadopoulos). Is the length-spectrum metric Finsler? If yes, give a formula for the infinitesimal norm of a vector on Teichmüller space with respect to this Finsler structure. Problem 4.3 (A. Papadopoulos). Is the isometry group of the length-spectrum metric the extended mapping class group?

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Problem 4.4 (K. Rafi). Is the Thurston metric a degeneration of some better vectorvalued symmetric distance on Teichmüller space? The reverse Thurston metric d  is defined by d  .X; Y / D dTh .Y; X / : Anti-stretch lines are geodesics for the reverse Thurston metric. Problem 4.5 (C. Walsh). What is the horofunction boundary of the reverse Thurston metric? Remark 4.6. Walsh conjectured that, fixing a base point b 2 T .S /, the Busemann points of the reverse Thurston metric are of the form .x/ D log max j

`j .x/ `j .p/

 log max j

`j .b/ `j .p/

where  is a complete geodesic lamination with stump given by X j .ergodic decomposition/ j

and p is some point along the stretch line directed by . Let S be a surface of finite type with nonempty boundary. The reduced Teichmüller space T .S / is the space of equivalence classes of marked bordered hyperbolic structures on S . The term reduced refers to the fact that in the definition of the equivalence relation between two hyperbolic metrics or conformal structures on the surface, one allows a free homotopy on the boundary components. There exist distinct hyperbolic structures X and Y on S such that for any simple closed curve  , l .Y /=l .X / < 1, which implies l .Y / 0:  2C.S/ l .X /

log sup

Here, C.S / is the set of homotopy classes of essential simple closed classes on S . Problem 4.7 (F. Guéritaud). Describe the cone of directions in the tangent space TX T .S / which shorten the lengths of all simple closed curves on S . The following metric, called the arc metric on T .S /, is a natural analogue of the Thurston metric for surfaces with boundary [35]: dA .X; Y / D log

sup S

 2C.S/

` .Y / : A.S/ ` .X /

Here, A.S / is the set of homotopy classes of essential properly embedded arcs in S . By doubling, there is a natural isometric embedding from .T .S /; dA / to .T .S d /; dTh /. We shall identify T .S / with its image in T .S d /.

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Problem 4.8. Is the metric space .T .S /; dA / a length space? Given X; Y 2 T .S /, is there a geodesic  of .T .S d /; dTh / connecting X and Y such that   T .S /? Problem 4.9. Is the metric dA Finsler? If yes, what is the Finsler norm? Problem 4.10. What is the relation between the metric dA and the extremal Lipschitz maps between hyperbolic structures? A related question is to construct families of lines between any two points on Teichmüller space, analogous to concatenations of stretch lines in the case without boundary. In [5], the authors proved that Thurston’s compactification of T .S / is homeomorphic to the horofunction compactification of .T .S /; dA /. This is a generalization of Walsh’s Theorem for surfaces without boundary [63], and it might be viewed as a step towards a positive answer to the following question: Problem 4.11. Is the isometry group of the metric dA on T .S / the extended mapping class group of S ? Problem 4.12. Let S be a hyperbolic surface with ideal boundary. Can we define the Thurston metric on the non-reduced Teichmüller space of S ? Problem 4.13 (D. Dumas). Can we define the Thurston metric on the universal Teichmüller space? Problem 4.13 is a special case of Problem 4.12.

5 Infinitely-generated Fuchsian groups of the first kind It is known that the Teichmüller metric can be defined on the Teichmüller space of any Fuchsian group (even on the universal Teichmüller space). Let 0 be an infinitely-generated Fuchsian group of the first kind, that is, the limit set ƒ.0 / of the action of 0 on the upper-half plane H is R. The Poincaré metric on H descends to the hyperbolic metric on X0 D H= 0 , a complete hyperbolic surface without boundary and of infinite area. We shall denote the reduced quasiconformal Teichmüller space of X0 by Tqc .X0 /. Problem 5.1. Can we deform an infinitely-generated Fuchsian group of the first kind (or, a closed hyperbolic surface of infinite area) by quasiconformal maps such that the lengths of all simple closed curves are not increased? Problem 5.1 is related to

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Problem 5.2. Can we define the Thurston metric on Tqc .X0 / by showing that K.X; Y / D log sup 

` .Y / > 0; if X ¤ Y ‹ ` .X /

(5.1)

Remark 5.3. For finitely-generated Fuchsian groups of the first kind, Thurston’s original proof of (5.1) is, like his many other proofs, based on first principles. He first assumed that  is an area-preserving diffeomorphism (in the homotopy class) between the hyperbolic surfaces X and Y . The map  lifts to a homoeomorphism Q of the universal cover. Using a version of Mostow’s rigidity theorem in dimension two, Thurston showed that Q is D-unbounded, that is, for any p 2 H, the function Q Q D.q/ D d..p/; .q//  d.p; q/ has no upper bound. The D-unbounded property of Q and the Poincaré recurrence lemma imply that there is a long closed curve  for Y with sufficiently small curvature, and whose preimage under  is homotopic to a shorter geodesic for X . Since  is homotopic to a geodesic for Y which has about the same length, we have ` .Y / >1: ` .X / If  is not simple, then a cut-and-paste argument of Thurston, named Shrinking at the waist, allow us to construct a simple closed curve ˛ with value `˛ .Y /=`˛ .X / greater than ` .Y /=` .X /. Another way to prove (5.1), for finitly-generated Fuchsian groups of the first kind, would be to use the earthquake theorem and the first variational formula of geodesic length given by Kerckhoff [26]. We ask the same question for extremal length: Problem 5.4. Do we have log sup 

Ext .Y / > 0; if X ¤ Y ‹ Ext .X /

Kapovich suggested that the following question may be related to Problem 5.1. Problem 5.5 (M. Kapovich). Can we construct examples of 0 (infinitely-generated and of the first kind) where the critical exponent of some elements in Tqc .0 / are distinct? The answer to Problem 5.5 may depend on the ergodic properties of 0 . Recall that, given a Fuchsian group , the critical exponent of , is defined as ı./ D lim sup R!1

log nR R

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where nR is the cardinality of the intersection of a fixed orbit of  in the hyperbolic plane with balls of hyperbolic radius R with fixed center. The critical exponent of  thus measures the exponential growth rate of the orbit of . There is an equivalent definition of ı./ given by the Poincaré series, that is, 8 9 < = X ı./ D inf s  0 j esd.0; 0/ < 1 : : ;  2

The notion of critical exponent applies to Kleinian groups and some lattices in Lie groups. Relations between ı./ and other geometric notions may be found in the introduction of Sullivan’s paper [56]. Bishop and Jones [10] proved that the critical exponent of a non-elementary Kleinian group is equal to the Hausdorff dimension of the canonical limit set ƒc ./ (the points in ƒc ./ correspond to recurrent geodesics, i.e., geodesics which return to some compact set infinitely often). Bishop [9] studied a question related to Problem 5.5. We say that a Fuchsian group  is ı-stable if ı. 0 / D ı./ for every quasi-Fuchsian deformation  0 of . It is well known that every finitely generated Fuchsian group has this property. Bishop proved that there is a Fuchsian group  (infinite-generated and of the first kind) with ı./ D 1 which is not ı-stable. Futhermore he gave sufficient condition under which a Fuchsian group is ı-stable. One idea to understand surfaces of infinite type is to approximate them by subsurfaces of finite type. Let 0 be an infinitely-generated Fuchsian group of the first kind. Denote by n the fundamental group carried by the n-ball in H= 0. Problem 5.6 (F. Guéritaud). Does ı.n / ! ı.0 / as n ! 1? If the answer to the above question is yes, then we ask Problem 5.7 (F. Guéritaud). Can we have a quasiconformal deformation of 0 which increases (or decreases) all ı.n / uniformly independently of n? In the case where the hyperbolic surface X0 D H=0 admits a geodesic pants decomposition, we know that ˇ ˇ ˇ ` .Y / ˇˇ ˇ dls .X; Y / D sup ˇlog ` .X / ˇ  defines a metric on Tqc .X0 /, which in general is not complete [52]. One way to understand the completion is to introduce the notion of length-spectrum Teichmüller space Tls .X0 / WD fX j dls .X0 ; X / < 1g : The length-spectrum Teichmüller space endowed with the length-spectrum metric is a complete metric space, containing Tqc .X0 / as a subset [2]. The inclusion of Tqc .X0 / into Tls .X0 / and their relation to earthquakes was investigated in [4]

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and [50, 51], with some additional assumptions on the boundedness of the pants decomposition of X0 . The following vague question of Hubbard [25] was a motivation for the above study. Problem 5.8 (J. Hubbard). For a Riemann surface of infinite type and some fixed maximal multicurve, there are Fenchel–Nielsen coordinates for the Teichmüller space. By equicontinuity of quasiconformal mappings, the length coordinates lie in `1 , but in which function space should these generalized twist parameters be understood? Remark 5.9. Some answers to the above question are given in the papers [1, 3]. For a surface of infinite type X0 , Saric [49] used geodesic currents to define a Thurston-type boundary of Tqc .X0 /, to which the action of the quasiconformal mapping class group extends continuously, and such that the boundary is homeomorphic to the set of bounded measured geodesic laminations on X0 . Problem 5.10. Define a Thurston-type metric on Teichmüller space and relate the horofunction boundary of such a metric to the Thurston-type boundary defined by Saric.

6 Other general questions Problem 6.1 (A. Papadopoulos). Study families of metrics on Teichmüller space that include the Thurston metric. One such family is tdT h .X; Y / C .1  t/dT h .Y; X /; t 2 Œ0; 1 : Problem 6.2 (A. Papadopoulos). Study the locus in T .S /  T .S / defined by f.X; Y / j dT h .X; Y / D dT h .Y; X /g : Problem 6.3. What is the Thurston metric on H2 , viewed as the Teichmüller space of punctured tori? Remark 6.4. Belkhirat, Papadopoulos and Troyanov [7] studied the Thurston metric on the Teichmüller space of flat tori. Problem 6.5 (D. Dumas). Does the Thurston metric have any relation with the following unsolved problem: Consider all conformal metrics  on a Riemann surface which satisfy inf ` . /  1 : 

Is there always one of such metrics with least area?

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Remark 6.6. See Wolf and Zwiebach [65] for a study of the above question and an application to string theory. Problem 6.7. Define and study the Thurston metric on the space of flat n-tori SLn .R/= SLn .Z/. Problem 6.8 (D. Dumas). How does the “Lipschitz constant function" .g; h/ 7! inf Lip./ Wg!h

behave on a class of metrics larger than the set of hyperbolic metrics, for example, the set of negatively curved Riemannian metrics or its closure? Problem 6.9 (S. Yamada). Study the relation between the Weil–Petersson Funk metric and the Thurston metric. Remark 6.10. The Weil–Petersson Funk metric is defined by d.Y; T / : d.X; T / P When X; Y are close to a noded surface , we have d.X; T / ` .X /1=2 d.Y; T / ` .Y /1=2 , thus F .X; Y / D log sup

F .X; Y /

1 ` .Y / log : 2 ` .X /

We refer to the paper [68] for the notation T .

Acknowledgements. We would like to thank the organizers and all the participants in the AIM workshop, especially the contributions of D. Dumas, F. Guéritaud, M. Kapovich, A. Papadopoulos and K. Rafi to this problem list.

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[49] D. Saric, Geodesic currents and Teichmüller space. Topology 44 (1) (2005), 99–130. [50] D. Saric, Fenchel–Nielsen coordinates on upper bounded pants decompositions. Math. Proc. Cambridge Philos. Soc. 158 (03) (2015), 385–397. [51] D. Saric, Earthquakes in the length-spectrum Teichmüller spaces. Proc. Amer. Math. Soc. 143 (2015), 1531–1543. [52] H. Shiga, On a distance defined by length spectrum on Teichmüller space. Ann. Acad. Sci. Fenn. Math. 28 (2003), 315–326. [53] T. Sorvali, The boundary mapping induced by an isomorphism of covering groups. Ann. Acad. Sci. Fenn. AI 526 (1972), 1–31. [54] T. Sorvali, On the dilatation of isomorphisms between covering groups. Ann. Acad. Sci. Fenn. AI 551 (1973), 1–15. [55] T. Sorvali, On isomorphisms between covering groups containing parabolic elements. Ann. Acad. Sci. Fenn. AI 572 (1974), 1–6. [56] D. Sullivan, Related aspects of positivity in Riemannian geometry. J. Diff. Geom. 25 (3) (1987), 327–351. [57] G. Théret, A propos de la métrique asymétrique de Thurston sur l’espace de Teichmüller d’une surface. Prépublication de l’Institut de Recherche Mathématique Avancée, 2005. [58] G. Théret, Divergence et parallélisme des rayons d’étirement cylindriques. Algebraic and geometric topology 10 (2010) 2451–2468. [59] G. Théret, Convexity of length functions and Thurston’s shear coordinates, arXiv:1408.5771. [60] D. Thurston, From rubber bands to rational maps. arXiv:1502.02561. [61] W. P. Thurston, Minimal stretch maps between hyperbolic surfaces. 1986 preprint. arxiv:math/9801039v1. [62] K. Vogtmann, Automorphisms of free groups and outer space. Geometriae Dedicata 94 (1) (2002), 1–31. [63] C. Walsh, The horoboundary and isometry groups of Thurston’s Lipschitz metric. In Handbook of Teichmüller theory (A. Papadopoulos, ed.) Vol. IV, EMS Publishing House, Zürich, 2014, 327–353. [64] A. Wilkinson, Lectures on marked length spectrum rigidity. IAS/Park City Mathematics Series 21 (2012). [65] M. Wolf and B. Zwiebach, The plumbing of minimal area surfaces. J. Geom. Phys. 15 (1) (1994), 23–56. [66] S. Wolpert, The length spectra as moduli for compact Riemann surfaces. Ann. Math. 109 (1979), 323–351. [67] S. Wolpert, Products of twists, geodesic-lengths and Thurston shears. Compositio Mathematica 151 (02) (2014), 313–350. [68] S. Yamada, Convex bodies in Euclidean and Weil–Petersson geometries. Proc. Amer. Math. Soc. 142 (2014), 603–616.

Part B

The group theory

Chapter 3

Meyer functions and the signature of fibered 4-manifolds Yusuke Kuno1 Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . The signature cocycle and Meyer’s theorem . . . . . 2.1 Prehistory . . . . . . . . . . . . . . . . . . . . . . . 2.2 The signature cocycle . . . . . . . . . . . . . . . 2.3 Evaluation of the signature class . . . . . . . . 2.4 Meyer’s theorems . . . . . . . . . . . . . . . . . . 2.5 Atiyah’s theorem . . . . . . . . . . . . . . . . . . 3 Local signatures . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Local signatures and Horikawa index . . . . . 3.2 Matsumoto’s formula . . . . . . . . . . . . . . . 4 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Hyperelliptic mapping class group . . . . . . . 4.2 Family of smooth theta divisors . . . . . . . . 4.3 The Meyer functions for projective varieties References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction In this chapter, we give a survey on secondary invariants called Meyer functions with emphasis on their application to the signature of fibered 4-manifolds. These secondary invariants are associated to the vanishing of the primary invariant called the first Mumford–Morita–Miller class e1 , the first in a series of characteristic classes of surface bundles [36, 38, 41]. There have been various known representatives of e1 coming from different geometric contexts, as group 2-cocycles on the mapping class group or differential 2-forms on the moduli space of curves (see [26], especially for the latter). The viewpoint we take here is the signature of surface bundles over surfaces, and we work with the signature cocycle g introduced by W. Meyer [35] (and by Turaev [46] independently), a Z-valued 2-cocycle of the mapping class group Mg of a closed oriented surface of genus g whose cohomology class is proportional to e1 . 1

Work partially supported by JSPS Research Fellowships for Young Scientists (224810).

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The computation of the cohomology of Mg is one of the big problems in the study of Mg . J. Harer’s result [18, 19] that the second homology group H2 .Mg I Z/ is isomorphic to Z if g  5 was a landmark theorem in this direction. However, the non-triviality of H2 .Mg I Z/ has been known since the late 1960’s by examples of surface bundles over surfaces with non-zero signature. It is in this context that Meyer introduced the signature cocycle. He already found that the cohomology class Œg 2 H 2 .Mg I Z/ is an element of infinite order if g  3. On the other hand, as was shown by Meyer, if g D 1 or 2 the cocycle g is the coboundary of a unique Q-valued 1-cochain g of Mg . The existence of such a 1cochain implies that over the rationals, e1 of a surface bundle with fiber a surface of genus 1 or 2 vanishes. The uniqueness of g follows from the fact H 1 .Mg I Q/ D 0. These 1-cochains are called the Meyer functions of genus 1 or 2. Meyer [35] extensively studied the case of genus 1 and gave an explicit formula for 1 which involves the Dedekind sums. In [6], Atiyah reproved Meyer’s formula by a quite different method and he also showed various number theoretic and differential geometric aspects of 1 . In §2, we recall basic results of Meyer and Atiyah with a sketch of proof for several assertions. In §3, we mention an application of Meyer functions to the localization of the signature of fibered 4-manifolds. This topic has been studied also from an algebro-geometric point of view, which we shall mention in §3.1. Recently, various higher genera or higher-dimensional analogues of 1 have been considered and a part of Atiyah’s result has been generalized to these settings. In §4, we present three examples of these generalizations. Some conventions about surface bundles follow. Throughout this chapter, g is an integer  1. Let †g be a closed oriented C 1 -surface of genus g. By a †g -bundle we mean a smooth fiber bundle W E ! B over a C 1 -manifold B with fiber †g such that the fibers are coherently oriented: the tangent bundle along the fibers T  WD fv 2 TEI  .v/ D 0g is oriented. The transition functions of such bundles take values in DiffC .†g /, the group of orientation preserving diffeomorphisms of †g endowed with the C 1 -topology. The mapping class group Mg WD 0 .DiffC .†g // is the group of connected components of DiffC .†g /. In other words, Mg is the quotient group DiffC .†g /=Diff0 .†g /, where Diff0 .†g / is the group of diffeomorphisms isotopic to the identity. To a †g -bundle W E ! B over a path-connected space B, is associated (the conjugacy class of) a homomorphism W 1 .B/ ! Mg called the monodromy. This correspondence is defined as the composition f†g -bundles over Bg=isom D ŒB; BDiffC .†g / ! Hom.1 .B/; Mg /=conj :

(1.1)

Namely, if f W B ! BDiffC .†g / is a classifying map of W E ! B, then D f , the induced map from 1 .B/ to 1 .BDiffC .†g // D 0 .DiffC .†g // D Mg . To be more careful about the base points and to give a more direct description, choose a base point b0 2 B and fix an orientation-preserving diffeomorphism 'W †g !  1 .b0 /. Let `W Œ0; 1 ! B be a based loop. Since Œ0; 1 is contractible, the pullback ` .E/ ! Œ0; 1 of W E ! B is a trivial †g -bundle. Hence there exists a

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trivialization ˆW †g  Œ0; 1 ! ` .E/ such that ˆ.x; 0/ D '.x/. In this setting, W 1 .B; b0 / ! Mg is given by .Œ` / D Œˆ.x; 1/1 ı ' . Here our convention is: 1) for any two mapping classes f1 and f2 , their multiplication f1 ı f2 means that f2 is applied first, 2) for any two homotopy classes of based loops `1 and `2 , their product `1 `2 means that `1 is traversed first. By the result of Earle–Eells [14], if g  2 the space Diff0 .†g / is contractible, so the classifying space BDiffC .†g / is a K.Mg ; 1/-space. Hence the map (1.1) is a bijection. If g D 1, then †1 D T 2 , the 2-torus. The embedding T 2 ,! Diff0 .T 2 / as parallel translations is a homotopy equivalence, and M1 is isomorphic to SL.2; Z/. Thus we have a fibration BDiffC .T 2 / ! BSL.2; Z/ D K.SL.2; Z/; 1/ with fiber BT 2 D CP 1  CP 1 . In particular, by elementary obstruction theory, it follows that if the base space B has the homotopy type of a 1-dimensional CW complex, then the isomorphism class of T 2 -bundles over B is also classified by monodromies: (1.1) is bijective.

2 The signature cocycle and Meyer’s theorem In this section we begin by explaining how the signature of 4-manifolds is related to the second cohomology of the mapping class group. Then we review the signature cocycle, its variants, and the original version of Meyer functions, i.e., the Meyer functions of genus 1 and 2. We also mention Atiyah’s theorem which shows various aspects of the Meyer function of genus 1.

2.1 Prehistory In the study of the topology of fiber bundles, a basic question is how the topological invariants of the total space, the base space and the fiber are related. In the 1950’s Chern, Hirzebruch and Serre studied the signature of the total space of a fiber bundle, by an application of the Serre spectral sequence. Recall that the signature of a compact oriented manifold M of dimension 4n (possibly with boundary), denoted by Sign.M /, is the signature of the intersection form H2n .M I R/  H2n .M I R/ ! R, which is a symmetric bilinear form. If the dimension of M is not a multiple of 4, we understand that the signature of M is zero. Theorem 2.1 (Chern–Hirzebruch–Serre [12]). Let E and B be closed oriented manifolds and E ! B a fiber bundle with fiber a closed oriented manifold F . We arrange that the orientation of F is compatible with those of E and B. If 1 .B/ acts trivially on the homology H .F I R/, then the signature of E is the product of the signatures of B and F : Sign.E/ D Sign.B/Sign.F /. The assumption that 1 .B/ acts trivially on the homology of the fiber is crucial, and the conclusion of the theorem does not hold in general. Indeed, Atiyah [5] and

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Kodaira [27] independently constructed an algebraic surface with non-zero signature, which is the total space of a complex analytic family of compact Riemann surfaces over a compact Riemann surface. Their method uses branched covering of algebraic surfaces, and it can be used to produce examples where the genus of the fiber can take arbitrary integer values  4. One important consequence of the examples by Atiyah and Kodaira is that there are non-trivial characteristic classes of surface bundles. In fact, since the signature of a manifold which is the boundary of some manifold is zero, the map SignW 2 .BDiffC .†g // ! Z;

Œf 7! Sign.f  /

is well-defined. Here 2 .X / is the second oriented bordism group of a space X (hence its element is represented by some continuous map f from a closed oriented surface to X ) and  is a universal †g -bundle over the classifying space BDiffC .†g /. Since 2 .X / is naturally isomorphic to H2 .X I Z/, the map Sign becomes an element in Hom.H2 .BDiffC .†g //; Z/, and the examples by Atiyah and Kodaira show that the map Sign is non-trivial. Hence H 2 .BDiffC .†g /I Z/ Š H 2 .Mg I Z/ is nontrivial and contains an element of infinite order, provided g  4. As we recall in the following, Meyer showed that this non-triviality holds when g  3.

2.2 The signature cocycle W. Meyer [34] [35] studied the signature of surface bundles over surfaces and introduced the signature cocycle. The basic idea of Meyer is to decompose the base space into simple pieces: pairs of pants. Let †0;n be a compact surface obtained from the 2-sphere by removing n open disks with embedded disjoint closures. Specifying an orientation of †0;n and a base point  2 Int.†0;n /, we take n based loops `1 ; : : : ; `n 2 1 .†0;n ; / such that each `i is freely homotopic to one of the boundaries with the counter-clockwise orientation, and the relation `1 `n D 1 2 1 .†0;n ; / holds. The group 1 .†0;n ; / is free of rank n  1, generated by any n  1 of `1 ; : : : ; `n . The surface P D †0;3 is called a pair of pants. Given f1 ; : : : ; fn1 2 Mg , consider a †g -bundle W E.f1 ; : : : ; fn1 / ! †0;n with  1 ./ D †g whose monodromy W 1 .†0;n ; / ! Mg sends `i to fi (i D 1; : : : ; n  1). Since †0;n is homotopy equivalent to a 1-dimensional CW complex, such a bundle exists and is unique up to isomorphism (see §1). The total space E.f1 ; : : : ; fn1 / is a compact oriented 4-manifold with boundary. Definition 2.2. The signature cocycle g W Mg  Mg ! Z is defined by g .f1 ; f2 / WD Sign.E.f1 ; f2 //;

f1 ; f2 2 Mg :

The map g is actually a normalized 2-cocycle of Mg .

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Lemma 2.3. For f1 ; f2 ; f3 2 Mg , we have (1) g .f1 f2 ; f3 / C g .f1 ; f2 / D g .f1 ; f2 f3 / C g .f2 ; f3 /; (2) g .f1 ; 1/ D g .1; f1 / D g .f1 ; f11 / D 0; (3) g .f11 ; f21 / D g .f1 ; f2 /; (4) g .f1 ; f2 / D g .f2 ; f1 /; (5) g .f3 f1 f31 ; f3 f2 f31 / D g .f1 ; f2 /. Sketch of proof. Recall the Novikov additivity of signature. Let M1 and M2 be compact oriented manifolds of the same dimension, Y1 and Y2 closed and open submanifolds of @M1 and @M2 , respectively, and 'W Y1 ! Y2 an orientation-reversing homeomorphism. Then the signature of the glued manifold M1 [' M2 is the sum of the signatures of M1 and M2 . We only give the proof of (1), the cocycle condition for g . Consider a †g -bundle W E.f1 ; f2 ; f3 / ! †0;4 and let C1 ; C2  †0;4 be essential simple closed curves intersecting each other in two points, such that C1 cuts †0;4 into two pairs of pants and the boundary of one of the two contains the free homotopy class of `1 and `2 . According to the decomposition of the base space, the total space E.f1 ; f2 ; f3 / can be written as a connected sum of E.f1 f2 ; f3 / and E.f1 ; f2 /. By the Novikov additivity of signature, we obtain Sign.E.f1 ; f2 ; f3 // D g .f1 f2 ; f3 / C g .f1 ; f2 /. On the other hand, cutting along C2 and arguing similarly, we obtain Sign.E.f1 ; f2 ; f3 // D g .f1 ; f2 f3 / C g .f2 ; f3 /.  The signature cocycle has a purely algebraic description. We denote by In the n  n identity matrix. The integral symplectic group Sp.2g; Z/, also called the Siegel modular group, is defined by Sp.2g; Z/ WD fA 2 GL.2g; Z/I t AJA D J g ;   0 Ig where J D and Ig is the g  g identity matrix. Fix a symplectic basis Ig 0 of H1 .†g I Z/, i.e., elements A1 ; : : : ; Ag ; B1 ; : : : ; Bg 2 H1 .†g I Z/ whose algebraic intersection numbers satisfy .Ai Bj / D ıij ;

.Ai Aj / D .Bi Bj / D 0 :

In terms of a symplectic basis, the (left) action of Mg on H1 .†g I Z/ is expressed by matrices and we get a (surjective) group homomorphism W Mg ! Sp.2g; Z/ : Given A; B 2 Sp.2g; Z/, consider an R-linear space VA;B WD f.x; y/ 2 R2g ˚ R2g I .A1  I2g /x C .B  I2g /y D 0g

(2.1)

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and a bilinear form h ; iA;B W VA;B  VA;B ! R defined by h.x; y/; .x 0 ; y 0 /iA;B WD t .x C y/J.I2g  B/y 0 : It turns out that h ; iA;B is symmetric, hence its signature Sign.VA;B ; sp h ; iA;B / is defined. We denote by g the map Sp.2g; Z/  Sp.2g; Z/ ! Z; .A; B/ 7! sp Sign.VA;B ; h ; iA;B /. Note that g is naturally defined on the Lie group Sp.2g; R/. sp

Theorem 2.4 (Meyer [34]). The signature cocycle on Mg is the pull-back of g on Sp.2g; Z/, i.e., for any f1 ; f2 2 Mg , we have g .f1 ; f2 / WD Sign.V .f1 /; .f2 / ; h ; i .f1 /; .f2 / / : Sketch of proof. The proof proceeds following the proof of Theorem 2.1. Consider the Serre cohomology spectral sequence of E.f1 ; f2 / ! P . The E2 page is E2p;q D H p .P; @P I Hq .†g I R//, where Hq .†g I R/ denotes the local system on P whose stalk at b 2 P is the cohomology of  1 .b/. On the other hand each page Er is a Poincaré ring in the sense of [12], in particular its signature Sign.Er / is defined. The proof is done through three steps: (1) to show that Sign.Er / D Sign.ErC1 /, (2) to show that Sign.E1 / D Sign.E.f1 ; f2 //, and (3) to show that Sign.E2 / D Sign.V .f1 /; .f2 / ; h ; i .f1 /; .f2 / /. To prove the last step, by taking a simplicial decomposition of P , Meyer [34] observed that E21;1 D H 1 .P; @P I H1 .†g I R// is isomorphic to V .f1 /; .f2 / , and the cup product on the former corresponds to h ; i .f1 /; .f2 / .  The signature cocycle was independently introduced by Turaev [46]. Turaev gave sp sp another algebraic description for g and directly proved that g is a normalized 2cocycle. He also discussed a relation with the Maslov index. For the coincidence of sp the definitions of g by Meyer and Turaev, see Endo–Nagami [16] Appendix. Remark 2.5. Let M be a closed oriented manifold of dimension 4n  2 and W E ! B an oriented M -bundle with B path-connected. By mimicking Definition 2.2, i.e., by constructing an M -bundle over P and taking the signature of the total space, we obtain a normalized 2-cocycle cM W 1 .B/  1 .B/ ! Z. In another direction, Atiyah [6] introduced the signature cocycle on the Lie group U.p; q/, the unitary group of the Hermitian form with signature .p; q/. The restriction to Sp.2g; R/  sp U.g; g/ is g .

2.3 Evaluation of the signature class The cocycle g 2 Z 2 .Mg I Z/ determines a cohomology class Œg 2 H 2 .Mg I Z/, which we call here the signature class. We give a combinatorial method to compute the order of Œg . Following Meyer [35], we consider the following slightly general situation: let G be a group and kW G  G ! Z a normalized 2-cocycle satisfying

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z.x; x 1 / D 0 for any x 2 G. Suppose a presentation of G is given. Namely G fits into an exact sequence $ 1!R!F !G!1 where F is the free group generated by a set fei gi 2I . Any x 2 F can be written as x D x1 x2 xm , where xj 2 fei g [ fei1 g. Define cW F ! Z by c.x/ WD

m X

z.$ .x1 xj 1 /; $ .xj // :

j D1

It follows that c is well-defined and ıc D $  z, i.e., c.xy/ D c.x/ C c.y/ C z.$ .x/; $ .y// for x; y 2 F . Moreover, c is a class function: c.yxy 1/ D c.x/ for x; y 2 F . The 1-cochain c is involved in a commutative diagram

Š

ev.Œz /

/Z pp8 p p pp pppc p p p

H2 .GI Z/ O

R \ ŒF; F =ŒR; F

where the vertical isomorphism is due to Hopf’s formula (see [11]) and the upper right arrow is the evaluation map ev.Œz /W H2 .GI Z/ ! Z by Œz . For i 2 I , let ei W F ! Z be the map counting the total exponents of ei in elements of F . Proposition 2.6 (Meyer [35]). For m 2 Z n f0g, the order of Œz 2 P H 2 .GI Z/ divides  m if and only if there exists fmi gi 2I  Z such that mcjR D i 2I mi ei jR . In 2 particular, if R is the normal closure of a set P frj gj 2J  F , then Œz D 0 2 H .GI Q/  if and only if the liner equation c.rj / D i 2I mi ei .rj /; j 2 J , has a solution fmi gi 2I  Q. The proof is straightforward, but we briefly mention the ‘if’ part. Take fmi gi 2I satisfying the condition. Consider the .1=m/Z-valued 1-cochain c1 WD c  .1=m/ P  m e i i 2I i of F . Then it turns out that c1 descends to a 1-cochain c 1 W G D F=R ! .1=m/Z. In fact, for x 2 F and r 2 R, we have c1 .xr/ D c.x/ C c.r/ C $  z.x; r/ 

1 X mi .ei .x/ C ei .r// m i 2I

1 X mi ei .x/ D c1 .x/ D c.x/  m i 2I

(we use $ .r/ D 1). Since $ is surjective, it follows that ıc 1 D z. In a special situation, this criterion becomes simpler. Let Art.G/ be a (small) Artin group associated to a connected graph G without loops. This means that Art.G/ is generated by the vertex set fai gi 2I of G, subject to the defining relations ai aj ai D

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aj ai aj if ai and aj are adjacent, and ai aj D aj ai if not. Further let frj gj 2J be a set of words in fai gi . We shall consider the case where G is the group obtained by adding relations rj D 1, j 2 J to Art.G/. Suppose there exists fmi gi 2I  Q satisfying the condition of Proposition 2.6, and let ak and a` be adjacent vertices of G. Now we have rk;` WD ak a` ak a`1 ak1 a`1 2 R, and c.rk;` / D c.ak / C c..a`ak /a`1.a` ak /1 / C z.$ .ak /; $ .ak /1 / D c.ak / C c.a`1/ D 0 : 1 Here we use the condition P z.x; x / D 0 and the fact that c is a class function. On the other hand, we have i 2I mi ei .rk;` / D mk  m` . Therefore we obtain mk D m` . Since G is connected, we conclude that mk D m` for any k; ` 2 I . In summary, we have the following.

Proposition 2.7. Suppose G is the quotient of an Artin group as above, and let z 2 Z 2 .GI Z/ be a normalized 2-cocycle with z.x; x 1 / D 0 for any x 2 G. (1) For n 2 N, nŒz D 0 2 H 2 .GI Z/ if and only if there exists m 2 Z such that n c.rj / D m ˛.rj / for all j 2 J . (2) In the situation of (1), the 1-cochain W G ! .1=n/Z defined by .$ .x// D c.x/ C .m=n/˛.x/, x 2 F is well-defined. Moreover, ı D z. Here ˛W F ! Z is a homomorphism given by ˛.ai / D 1 for i 2 I . For example, the mapping class group admits a presentation as the quotient of an Artin group where the relation ai aj ai D aj ai aj corresponds to the braid relation among two Dehn twists. Thus we can apply this proposition.

2.4 Meyer’s theorems Using the combinatorial criterion in the previous section, Meyer determined the order of the cohomology class Œg 2 H 2 .Mg I Z/. Theorem 2.8 (Meyer [35], Satz 2). The order of Œ1 is 3, the order of Œ2 is 5, and the order of Œg is infinite if g  3. To settle the case g D 1 and 2, Meyer used a classical presentation of M1 Š SL.2; Z/ and a presentation of M2 by Birman–Hilden [8]. For g  3, no finite presentation of Mg was known at that time. Still, using some of the known relations and showing that Œg is divisible by 4, Meyer proved that the image of ev.Œg / is 4Z. We remark that by the Hirzebruch signature formula, we have e1 D 3Œg 2 H 2 .Mg I Z/. Remark 2.9. Today several finite presentations of Mg for g  3 are known. Using one of them, say the one due to Wajnryb [47], one can directly show that the image of ev.Œg / is 4Z.

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The following is an immediate consequence of Theorem 2.8. Theorem 2.10 (Meyer [35], Satz 3). (1) If g  2, the signature of the total space of any †g -bundle over a closed oriented surface is zero. (2) If g  3, the signature of the total space of a †g -bundle over a closed oriented surface is a multiple of 4. Conversely, for any g  3 and n 2 4Z, there exists a †g -bundle E ! B over a closed oriented surface with Sign.E/ D n. As a consequence of Theorem 2.8, there exist 1-cochains 1 W M1 ! .1=3/Z and 2 W M2 ! .1=5/Z such that ı1 D 1 and ı2 D 2 . Here for a 1-cochain W G ! A with coefficient in an abelian group A, its coboundary ı is a map from G  G to A given by ı.x; y/ D .x/  .xy/ C .y/ (for terminologies of cohomology of groups, see for example, [11]). Thus the condition ıg D g (g D 1 or 2) is equivalent to g .x; y/ D g .x/  g .xy/ C g .y/;

x; y 2 Mg :

(2.2)

Moreover, since H 1 .M1 I Q/ D H 1 .M2 I Q/ D 0, such 1-cochains are unique and characterized by (2.2). The 1-cochain 1 (resp. 2 ) is called the Meyer function of genus 1 (resp. of genus 2). The following lemma can be directly proved by Lemma 2.3 and (2.2). Lemma 2.11. The Meyer functions 1 and 2 satisfy the following properties: for x; y 2 Mg (g D 1 or 2), (1) g .1/ D 0; (2) g .x 1 / D g .x/; (3) g .yxy 1/ D g .x/. Consider a surface bundle over a compact oriented surface. Then the values of g around a boundary circle (which is well-defined by Lemma 2.11 (3)) is interpreted as signature defects. Proposition 2.12. Suppose g D 1 or 2 and let W E ! B be a †g -bundle over a compact oriented surface B with boundary components @Bi , i 2 I . Then X g .xi / ; Sign.E/ D i 2I

where xi 2 Mg is the monodromy along the boundary component @Bi with the counter-clockwise orientation. Sketch of proof. Take a pants decomposition of B. By the Novikov additivity of signature, Sign.E/ is the sum of the signatures of the components, which is expressed in terms of g . Using (2.2), we obtain the formula. 

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Meyer extensively studied the function 1 and gave for it an explicit formula. Note that the mapping class group M1 is isomorphic to SL.2; Z/ D Sp.2; Z/ by the homomorphism (2.1). To state his result, let us prepare some notation. The Rademacher function [43] is a map ‰W SL.2; Z/ ! Q defined by 8 aCd   ˆ <  12sign.c/s.a; c/  3sign.c.a C d // if c ¤ 0 ; a b ‰ D b c c d ˆ : if c D 0 : d Here sign.x/ 2 f0; ˙1g is the sign of x if x ¤ 0, 0 if x D 0, and s.a; c/ is the Dedekind sum X  ak   k  s.a; c/ WD c c kmod jcj

(

where

if x 2 R n Z ; if x 2 Z   a b (Œx denotes the integer part of x). Also, for ˛ D 2 SL.2; Z/, set c d  .˛/ D 1 .˛; 1/, which by a direct computation turns out to be the signature of   2c a  d the symmetric matrix . ad 2b ..x// D

x  Œx  0

1 2

 Theorem 2.13 (Meyer [35], Satz 4). For any ˛ D

a b c d

 2 SL.2; Z/, we have

1 1 1 .˛/ D  ‰.˛/ C  .˛/ .1 C sign.a C d // : 3 2 In particular, if a C d ¤ 0; 1; 2, then 1 .˛/ D .1=3/‰.˛/. Meyer’s proof is based on a certain cocycle identity of ‰, behind which is the transformation law under SL.2; Z/ of the logarithm of the Dedekind -function . / D ei =12

1 Y

.1  e2i n /;

 2 fz 2 CI Im.z/ > 0g :

nD1

Atiyah [6] gave another proof of Theorem 2.13 of a more topological nature.

2.5 Atiyah’s theorem Atiyah [6] showed that the value of 1 on hyperbolic elements coincides with various invariants. Recall that ˛ 2 SL.2; Z/ is called hyperbolic if jTr.˛/j > 2.

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Theorem 2.14 (Atiyah [6]). For a hyperbolic element ˛ 2 SL.2; Z/, the following quantities coincide. (1) 1 .˛/; (2) Hirzebruch’s signature defect ı.˛/; (3) the transformation law of the logarithm of the Dedekind -function under ˛; (4) the logarithmic monodromy of Quillen’s determinant line bundle of the mapping torus of ˛; (5) the value L˛ .0/ of the Shimizu L-function; (6) the Atiyah–Patodi–Singer invariant .˛/ of the mapping torus of ˛; (7) the adiabatic limit 0 .˛/. We give a brief explanation of the notions appearing in the statement of Theorem 2.14. In the below we denote by M˛ the mapping torus of ˛. Namely, M˛ WD Œ0; 1  T 2 =.0; x/  .1; ˛x/. The natural projection M˛ ! S 1 D Œ0; 1 =0  1 is a T 2 -bundle. First of all, Hirzebruch’s signature defect ı.˛/ [20] is defined by the formula ı.˛/ D .1=3/p1.Z; @Z/  Sign.Z/. Here Z is any compact oriented 4-manifold with boundary diffeomorphic to M˛ , and p1 .Z; @Z/ is the relative Pontrjagin class with respect to a certain natural connection on @Z D M˛ . The invariant (3) is closely related to the Rademacher function ‰, and it essentially describes the relationship between log .˛ / and log . /. Let E ! B be a smooth F -bundle where F is compact, orientable, spin, and even-dimensional. Assume that a metric on the relative tangent bundle T .E=B/, a metric on B, and a connection on TE are given. Then we have a family of Diractype operators on F parametrized by B. In this situation, extending the construction of Quillen [42], Bismut–Freed [10] defined the Quillen determinant line bundle as a complex line bundle on B endowed with a natural unitary connection. In our situation we have a T 2 -bundle M˛ ! S 1 . Then the assertion that .˛/ coincides with the invariant (4) means that the monodromy of the Quillen determinant line bundle of M˛ ! S 1 is given by exp. i .˛//.   a b The invariant (5) is defined as follows. For ˛ D , consider the quadratic c d form N.x; y/ defined by N.x; y/ D sign.a C d /.cx 2 C .d  a/xy  by 2 /. The Shimizu L-function was introduced by Hirzebruch [20] and is defined by L˛ .s/ D

X signN.u/ u

jN.u/js

;

where the sum is taken over non-zero ˛-orbits of the integral points in R2 . It is known that L˛ .s/ converges in the region Re.s/ > 0, and has a meromorphic continuation in s and is holomorphic at s D 0. Finally, we mention the invariants (6)(7). The Atiyah–Patodi–Singer invariant [7], also called the -invariant, is a spectral invariant of a closed oriented odd-dimensional

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Riemannian manifold .M; g/ and is denoted by .M; g/ or .M /. Further, let E and B be closed oriented C 1 -manifolds and W E ! B an oriented M -bundle where the dimension of E is divisible by 4. Once a metric g E=B on the relative tangent bundle T .E=B/, a metric g B on B, and a connection r on TE are given, the metric on E is given by g E WD g E=B ˚   g E according to the decomposition TE D T .E=B/ ˚   TB induced from r. Then the one-parameter family of metrics on E is defined by g"E WD g E=B ˚ "1   g B , " 2 R>0 . By Bismut–Cheeger [9], it is shown that the limit lim"!0 .E; g"E / exists. The limit is called the adiabatic limit of the -invariants and is denoted by 0 .E/. In our situation, a suitable metric is chosen for M˛ and we denote .˛/ D .M˛ / and 0 .˛/ D 0 .M˛ /. In fact, Atiyah also showed the following result, giving an analytic expression of the value of 1 on an arbitrary element of SL.2; Z/. Theorem 2.15 (Atiyah [6]). For ˛ 2 SL.2; Z/, we have 1 .˛/ D 0 .˛/. A generalization of this result to 2 will be dealt with in §4.2.

3 Local signatures Consider a closed oriented 4-manifold M admitting a fibration f W M ! B onto a closed oriented surface B. Under some conditions, the signature of M happens to localize to finitely many singular fibers of f . This phenomenon is called the localization of the signature, and has been studied from several points of view. In this section we review some of these treatments, and recall an approach using Meyer functions.

3.1 Local signatures and Horikawa index Let E and B be compact oriented C 1 -manifolds of dimension 4 and 2 respectively, f W E ! B a proper surjective C 1 -map having the structure of a †g -bundle in the complement of finitely many points fbi gi 2I  Int.B/. We call such a triple .E; f; B/ a fibered 4-manifold (of genus g). For b 2 B, we denote by Fb the fiber germ of f around b. If b 2 B n fbi gi 2I , Fb is called a general fiber. If b D bi for some i 2 I , Fb is called a singular fiber. Typical examples of fibered 4-manifolds are elliptic surfaces and Lefschetz fibrations. For Lefschetz fibrations, we refer to [24]. When we work in the holomorphic category, then E is a complex surface, B is a Riemann surface, and f is a holomorphic map. In this case if we say, for example, that f W E ! B is a hyperelliptic fibration, then the general fibers are hyperelliptic Riemann surfaces. Here note that a compact Riemann surface R is called hyperelliptic if there exists a holomorphic map R ! CP 1 of degree two. If the genus of R is g, this is equivalent to that R is the compactification of an algebraic curve y 2 D F .x/ where F .x/ is a polynomial of

3 Meyer functions and the signature of fibered 4-manifolds

87

degree 2g C 1 or 2g C 2 with distinct root. Then the projection .x; y/ 7! x defines a holomorphic map R ! CP 1 of degree two. Among the topological invariants of such an E, the topological Euler number .E/ is easy to compute. For simplicity we assume that E and B are closed, and we denote by g.B/ the genus of B. Let i  BS be a small closed disk with center bi and we set Ei D f 1 .i / and E0 D f 1 .B n i Int.i //. Since the topological Euler number is multiplicative in fiber bundles, we have .E0 / D .22g/.22g.B/jI j/. Moreover, since f is proper we have .Ei / D .f 1 .bi //. Thus X .E/ D .2  2g/.2  2g.B// C ".Fb / ; b2B

where the number ".Fb / WD .f 1 .bi //  .2  2g/ is called the topological Euler contribution. In short, we can compute .E/ by the contributions ".Fb /. On the other hand, the signature of E is not so easy to compute and in general one cannot compute it from the data of singular fiber germs. Nevertheless, under some conditions on the general fibers, it happens that we can assign a rational number  .Fb / to each fiber Fb satisfying the following two conditions: (1) if Fb is a general fiber, then  .Fb / D 0. P (2) if E is closed, then Sign.E/ D b2B  .Fb /. The assignment  is called a local signature, and when such a phenomenon happens, we say that the signature of E is localized. The first example of a local signature is the one for fibered 4-manifolds of genus 1 due to Y. Matsumoto [32]. He called such an assignment a fractional signature. Later he also gave a local signature for Lefschetz fibrations of genus 2 [33]. In both the examples, he used the Meyer functions 1 and 2 to construct a local signature. See the next subsection for details. In the algebro-geometric setting, local signatures are closely related to an invariant of fiber germs which originates in the work of Horikawa [21] [22]. Horikawa studied a global family of curves of genus 2 f W E ! B. He defined an invariant H.Fb /  0 to each fiber germ, and showed the equality X 2 KE D 2 .OE /  6 C 6g.B/ C H.Fb / : (3.1) b2B 2 is the self intersection number of the canonical Here g.B/ is the genus of B, KE bundle of E, and .OE / is the Euler characteristic number of the structure sheaf of E. In the geography of complex surfaces of general type, one often studies complex 2 surfaces with the pair of specified numerical invariants .KE ; .OE //. We have the 2 Hirzebruch signature formula Sign.E/ D .1=3/.KE  2 .E// and the Noether for2 2 C .E//, therefore to fix .KE ; .OE // is equivalent to mula .OE / D .1=12/.KE 2 fix .Sign.E/; .E//. The inequality KE  2 .OE /  6 is called the Noether inequality, a lower bound for the numerical invariants of complex surfaces of general

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type. Thus H.Fb / is regarded as a local contribution of each fiber germ to the dis2 tance from the geographical lower bound for .KE ; .OE //. The invariant H.Fb / is called the Horikawa index. There are several situations in which the Horikawa index exists. M. Reid [44] defined it for fiber germs of non-hyperelliptic fibrations of genus 3. This is generalized by Konno [29] to Clifford general fibrations of odd genus. Arakawa and Ashikaga [1] introduced the Horikawa index for hyperelliptic fibrations, which is regarded as a direct generalization of the work of Horikawa. Let f W E ! B be a hyperelliptic fibration of genus g with B closed. They introduced an invariant H.Fb /  0 for each fiber germ satisfying 2 D KE=B

X 4.g  1/ H.Fb / ; f C g

(3.2)

b2B

2 2 D KE  8.g  1/.g.B/  1/ and f D .OE /  .g  1/.g.B/  1/. where KE=B Moreover, they defined a local signature for hyperelliptic fibrations of genus g by

galg .Fb / WD

1 .gH.Fb /  .g C 1/".Fb // : 2g C 1

(3.3) alg

Here ".Fb / is the topological Euler contribution, as above. That g is a local signature follows from (3.2). More generally, if we find a Horikawa index in a class of fibrations (say non-hyperelliptic fibrations of genus 3), then a formula of type (3.3) gives a local signature for such fibrations. For more details about local signatures, we refer to the survey articles Ashikaga– Endo [2] and Ashikaga–Konno [3]. We also refer to recent works by Ashikaga– Yoshikawa [4] and Sato [45].

3.2 Matsumoto’s formula For a while, we assume g is 1 or 2. Let .E; f; B/ be a fibered 4-manifold of genus g. For each b 2 B, take a small closed disk neighborhood   B of b and consider the restriction of f to  n fbg. Let xb 2 Mg be the monodromy of this †g -bundle along the boundary @ with the counter-clockwise orientation, and set g .Fb / WD g .xb / C Sign.f 1 .// 2 Q :

(3.4)

Here g is the Meyer function of genus g. Note that although xb is only defined up to conjugacy, g .xb / is well defined by Lemma 2.11 (3). Proposition 3.1 (Y. Matusmoto [32] [33]). Let g D 1 or 2. The assignment g .Fb / is a local signature for fibered 4-manifolds of genus g.

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89

Proof. The property (1) is clear since xb is trivial if Fb is non-singular. To prove (2), for each i , let i be a small closed disk neighborhood of bi . By Proposition 2.12, we have X Sign.E/ D Sign.f 1 .B0 // C Sign.f 1 .i // D

X i 2I

g .xbi / C

X

i 2I

Sign.f 1 .i // D

i 2I

X

g .Fbi / :



i 2I

Matsumoto [32] [33] also gave some computations of his local signatures. Using the Meyer function on the hyperelliptic mapping class group and applying the formula (3.4), Endo [15] introduced a local signature for hyperelliptic fibrations (see §4.1). By Terasoma, it was shown that Endo’s local signature and Arakawa–Ashikaga’s local signature (3.3) coincide. See [15], Appendix. The formula (3.4) implies that the local signature is only determined by topological data. But as Konno [28] observed, there exists a topologically non-singular fiber germ of non-hyperelliptic fibrations of genus 3 which has a non-zero Horikawa index. In fact, in the central fiber f 1 .b/ of Konno’s example is a non-singular hyperelliptic curve of genus 3. From the viewpoint of local signature, this fiber germ should be thought as a singular fiber. A modification of the formula (3.4) for such situations will be explained in §4.3.

4 Variations In this section we review higher genera analogues and higher-dimensional analogues of Meyer’s 1 or 2 . First note that by Theorem 2.8, Meyer functions do not exist on Mg for g > 2. But the signature cocycle happens to be a coboundary when it is pulled back to some group, for example, a subgroup of Mg . The examples in §4.1 and §4.3 are of this kind. The example in §4.2 is in a situation of Remark 2.5, and can be regarded as a generalization of Theorem 2.15.

4.1 Hyperelliptic mapping class group Let  2 Mg be a hyperelliptic involution, i.e., (the class of) an involution of †g acting on H1 .†g I Z/ as id. It is known that a hyperelliptic involution is unique up to conjugacy. It has 2g C 2 fixed points, and the quotient space †g =hi is homeomorphic to the 2-sphere. If R is a hyperelliptic Riemann surface obtained as the compactification of an algebraic curve y 2 D F .x/, the map .x; y/ 7! .x; y/ defines a hyperelliptic involution of the underlying C 1 -surface of R. The hyperelliptic mapping class group Hg is the centralizer of : Hg WD ff 2 Mg I f  D f g :

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Let gH 2 Z 2 .Hg I Z/ be the restriction of g to the subgroup Hg  Mg . Using a finite presentation of Hg by Birman–Hilden [8] and Proposition 2.6, Endo [15] proved the following theorem. Theorem 4.1 (Endo [15]). The order of ŒgH 2 H 2 .Hg I Z/ is 2g C 1. Furthermore, there exists a unique function gH W Hg ! .1=2g C 1/Z such that ıgH D gH . The 1-cochain gH is called the Meyer function for the hyperelliptic mapping class group of genus g. Remark 4.2. The existence and uniqueness of gH also follow from the Q-acyclicity of Hg which is independently proved by Cohen [13] and Kawazumi [25]. Remark that Hg D Mg if g D 1 or 2. Thus the series gH , g  3 could be a higher genus analogue of Meyer’s 1 and 2 . The values of gH on Dehn twists are given as follows ([15, 37]). Let C be an -invariant simple closed curve on †g . We denote by tC the right handed Dehn twist along C , which is an element of Hg . If C is non-separating, then gH .tC / D .g C 1/=2g C 1; if C is separating and separates †g into surfaces of genus h and g  h, then gH .tC / D 4h.g  h/=2g C 1. A fibered 4-manifold .E; f; B/ is called hyperelliptic if the monodromy of the †g -bundle over B n fbi gi can take value in Hg by a suitable identification of a reference fiber with †g . For example, if f is a hyperelliptic fibration in the sense of the second paragraph of §3.1, then .E; f; B/ is hyperellptic. Replacing g with gH in (3.4), Endo [15] introduced a local signature for hyperelliptic fibered 4-manifold. Morifuji [37] studied geometrical aspects of gH . He showed that if f 2 Hg is of finite order, then gH .f / equals .f /, the -invariant (see §2.5) of the mapping torus †g  Œ0; 1 =.x; 0/  .f .x/; 1/. Further, he showed that gH .f / D d0 .f / if f belongs to the hyperelliptic Torelli group, where d0 is the so-called core of the Casson invariant introduced by Morita [39] [40].

4.2 Family of smooth theta divisors Iida [23] gave a higher-dimensional analogue of Meyer’s 2 , which he called the Meyer function for smooth theta divisors. Let Sg WD f 2 M.gI C/I t  D ; Im. / > 0g be the Siegel upper half space of degree g and f W Ag ! Sg the universal family of principally polarized Abelian varieties. The fiber of f at  2 Sg is the complex torus A D Cg =ƒg , where ƒg is the lattice spanned by the column vectors of the g  2g matrix .Ig  /. We set p e.t/ D exp.2 1t/. The Riemann theta function  X 1 t t n n C n z ; z 2 Cg ;

.z;  / WD e 2 g n2Z

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3 Meyer functions and the signature of fibered 4-manifolds

converges absolutely and uniformly on compact subsets of Cg  Sg . It enjoys the property .z C l;  / D .z;  / and .z C  m/ D e.m t z  .1=2/m t m/ .z;  / for l; m 2 Zg . Thus we can regard .z;  / as a holomorphic section of a certain holomorphic vector bundle on A , and its zero locus ‚  A is called the theta divisor. Set ‚ WD f.z;  /I  2 Sg ; z 2 A ; .z;  / D 0g and let pW ‚ ! Sg be the natural projection. This is the universal family of theta divisors whose fiber at  is ‚ . The group Sp.2g; Z/, which for simplicity we denote here by g , naturally acts on Sg . Iida introduced a g -action on ‚ so that p is g -equivariant. The Zariski closed set Ng WD f 2 Sg I Sing.‚ / ¤ ;g is called the Andreotti– Mayer locus. It is a divisor of Sg . The group g acts on the complement Sıg WD Sg n Ng properly discontinuously. Let Sg be the orbifold fundamental group of the quotient orbifold g nSıg . In other words, Sg is the fundamental group of the Borel construction .Sıg / g WD Eg  g Sıg , where Eg is the total space of the classifying space of g . The group Sg fits into an exact sequence 1 ! 1 .Sıg / ! Sg ! g ! 1 :

(4.1)

If g D 1, Ng D ; and 1 nSı1 is the moduli space of curves of genus 1, hence S1 D M1 . By the Torelli theorem, 2 nSı2 is the moduli space of curves of genus 2 and S2 D M2 . The projection p induces a fiber bundle over .Sıg / g . The fiber is diffeomorphic to a smooth theta divisor. By the construction given in Remark 2.5, we get the signature cocycle cg W Sg  Sg ! Z. If g is odd, cg 0 since the real dimension of a sp smooth theta divisor is 2g  2. When g D 2, c2 is the pull back of 2 by (4.1). But if g  3, this is not the case. Using adiabatic limits of -invariants and a certain automorphic form, Iida constructed a 1-cochain of Sg which cobounds cg . Suppose g is even. An element  2 Sg can be written as  D .˛;  /, where ˛W Œ0; 1 ! Sıg is a continuous map with ˛.0/ a specified basepoint of Sıg and  2 g such that ˛.1/ D  ˛.0/. Consider the mapping torus M WD Œ0; 1 ˛ ‚=.0; x/  .1; x/ and the projection W M ! S 1 D Œ0; 1 =0  1. He introduced a metric on the relative tangent bundle T .M =S 1 / and a connection on M . Then the adiabatic limit 0 .M / is defined (see §2.5). Set g

ˆg . / WD 0 .M / C

.1/ 2 2gC3 .2gC2  1/ B g C1 2 .g C 3/Š

Z S1

˛  d c log jjg . /jj :

Here g . / is a Siegel cusp form of weight .g C 3/gŠ=2 with zero divisor Ng and Bk is the k-th Bernoulli number. Theorem 4.3 (Iida [23]). The 1-cochain ˆg cobounds cg , i.e., cg .1 ; 2 / D ˆg .1 /  ˆg .1 2 / C ˆg .2 /;

 1 ;  2 2 Sg :

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It should be remarked that the uniqueness of ˆg does not hold. In fact, Iida proved that H 1 .Sg I Z/ D Z for g  4 ([23] Theorem 13). The 1-cochain cg actually takes values in Q ([23] Theorem 15). As a special case, Iida obtained an analytic expression of the Meyer function of genus 2. Corollary 4.4 (Iida [23]). For  D .˛;  / 2 S2 D M2 , we have Z 2 2 . / D 0 .M /  ˛  d c log jj 2 . /jj2 : 15 S 1 Here 2 . / is a Siegel modular form of weight 5 called the Igusa modular form.

4.3 The Meyer functions for projective varieties We mention an approach [30] [31] to extend Matsumoto’s formula (3.4) for generic non-hyperelliptic fibrations of small genera. Let X ¨ PN be a smooth projective variety of dimension n  2, embedded in a complex projective space of dimension N . The intersection of X and a generic plane in PN of codimension n  1 is non-singular of dimension 1. Set k WD N  n C 1 and let Gk .PN / be the Grassmann manifold of k-planes of PN . The set DX WD fW 2 Gk .PN /I W meets X not transversallyg is called the k-th associated subvariety of X [17]. Over the complement U X WD Gk .PN / n DX , there is a family of compact Riemann surfaces pX W C X ! U X whose fiber at W 2 U X is X \ W . Let g be the genus of the fibers and let X W 1 .U X / ! Mg be the monodromy of this family. Theorem 4.5 ([31]). There exists a unique Q-valued 1-cochain X W 1 .U X / ! Q  whose coboundary equals the pull-back X g . The 1-cochain X is called the Meyer function associated to X  PN . The fundamental group 1 .U X / is normally generated by a single element called a lasso, which is represented by a loop “going once around DX .” By X , a lasso is mapped to a Dehn twist. By a certain extension of the theory of Lefschetz pencils, the value of X on a lasso is given in terms of invariants of X . Under a mild condition on X , it follows that X is an unbounded function. As a consequence, we can show that the group 1 .U X / is non-amenable for such X . As an application, we can define a local signature for generic non-hyperelliptic fibrations of small genera. Let us illustrate this by an example. Let .E; f; B/ be a fibered 4-manifold of genus 3, such that the restriction of f to B n fbi gi 2I is a continuous family of Riemann surfaces with non-hyperelliptic fiber. We call such .E; f; B/ a non-hyperelliptic fibration of genus 3. Note that we assume a fiberwise complex structure on the general fibers, but do not assume a global complex structure.

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93

The idea is to construct a certain universal family and to lift the monodromy to the fundamental group of its base space. Hereafter let X be the image of the Veronese embedding v4 W P2 ! P14 of degree 4. A generic hyperplane section of P14 corresponds to a smooth plane curve of degree 4 in P2 , which is non-hyperelliptic of genus 3. The group G D P GL.3/ naturally acts on P14 preserving DX . This induces G-actions on CX and U X , making pX W C X ! U X a G-equivariant map. Therefore we have a continuous family of non-hyperelliptic Riemann surfaces of genus 3 over the Borel construction UGX WD EG G U X , which we denote by pu W CGX ! UGX . This family has a certain universal property: if pW E ! B is a continuous family of non-hyperelliptic Riemann surfaces of genus 3, then there exists a continuous map gW B ! UGX such that the fiber product C X g B and the original family are isotopic. Moreover, such a g is unique up to homotopy. The fundamental group 1 .UGX / fits into an exact sequence 1 .P GL.3// Š Z=3Z ! 1 .U X / ! 1 .UGX / ! 1 : From this and the existence of X on 1 .U X /, we can deduce that there exists a unique Q-valued 1-cochain 3NH W 1 .UGX / ! Q which cobounds the pull-back of 3 by the monodromy u W 1 .UGX / ! M3 . Now, let Fb be a fiber germ of non-hyperelliptic fibration of genus 3. Take a small closed disk  with center b, so that there is no singular fiber on  n fbg. By the universality of pu , there is a continuous map gFb W  n fbg ! UGX . Set xFb WD .gFb / .@/ 2 1 .UGX /, where we give @ the counter-clockwise orientation. Note that xFb is uniquely determined up to conjugacy. Set 3NH .Fb / WD 3NH .xFb / C Sign.f 1 .// : By applying the proof of Proposition 3.1, we have the following. Theorem 4.6 ([30]). The assignment 3NH is a local signature for non-hyperelliptic fibrations of genus 3. The formulation of 3NH gives a topological interpretation of Konno’s example in §3.2. While the monodromy around b is trivial, its lift xFb 2 1 .UGX / is nontrivial and contributes to 3NH . Similar constructions are possible for generic nonhyperelliptic fibrations of genus 4 and 5. For details, see [31]. Acknowledgments. The author would like to express his gratitude to Athanase Papadopoulos for careful reading of the first version of this chapter. Furthermore he would like to thank Tadashi Ashikaga for helpful comments on an earlier version.

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[42] D. Quillen, Determinants of Cauchy–Riemann operators over a Riemann surface. Funct. Anal. Appl. 19 (1985), 31–34. [43] H. Rademacher, Theorie der Dedekindschen Summen. Math. Zeitschrift 63 (1955/56), 445– 463. [44] M. Reid, Problems on pencils of small genus. Preprint 1990. [45] M. Sato, A local signature for fibrations with a finite group action. Tohoku Math. J. 65 (2013), 545–568. [46] V. G. Turaev, First symplectic Chern class and Maslov indices. J. Soviet Math. 37 (1987), 1115–1127. [47] B. Wajnryb, A simple presentation for the mapping class group of an oriented surface. Israel J. Math. 45 (1983), 157–174.

Chapter 4

The Goldman–Turaev Lie bialgebra and the Johnson homomorphisms Nariya Kawazumi1 and Yusuke Kuno2 Contents 1 2

3

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5

6

7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Torelli–Johnson–Morita theory . . . . . . . . . . . . 2.1 Lower central series and the higher Torelli groups . 2.2 The Johnson homomorphisms and their images . . . 2.3 Extensions of the Johnson homomorphisms . . . . . Dehn–Nielsen embedding . . . . . . . . . . . . . . . . . . . . . . 3.1 Groupoids and their completions . . . . . . . . . . . . . 3.2 Derivations and their exponentials . . . . . . . . . . . . 3.3 Fundamental groupoid . . . . . . . . . . . . . . . . . . . . 3.4 Dehn–Nielsen homomorphism . . . . . . . . . . . . . . 3.5 Cut and paste arguments . . . . . . . . . . . . . . . . . . Operations on curves on surfaces . . . . . . . . . . . . . . . . . 4.1 Goldman–Turaev Lie bialgebra . . . . . . . . . . . . . . 4.2 The action of free loops on based paths . . . . . . . . 4.3 Intersection of based paths . . . . . . . . . . . . . . . . . 4.4 Self intersections . . . . . . . . . . . . . . . . . . . . . . . 4.5 Completions of the operations . . . . . . . . . . . . . . Dehn twists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The logarithms of Dehn twists . . . . . . . . . . . . . . 5.2 Generalized Dehn twists . . . . . . . . . . . . . . . . . . 5.3 Criterion using the self intersection . . . . . . . . . . . Classical theory revisited . . . . . . . . . . . . . . . . . . . . . . . 6.1 Symplectic expansions . . . . . . . . . . . . . . . . . . . 6.2 The Lie algebra of symplectic derivations . . . . . . . 6.3 Algebraic interpretation of the Goldman bracket . . 6.4 The Turaev cobracket and the Morita trace . . . . . . Compact surfaces with non-empty boundary . . . . . . . . . . 7.1 The “associative” Lie algebra for a compact surface 7.2 A tensorial description of the Goldman Lie algebra 7.3 The geometric Johnson homomorphism . . . . . . . .

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Other topics and applications . . . . . . . . . . . . . . . . . . . . . 8.1 The action of Dehn twists on the nilpotent quotients . 8.2 Lie algebras based on chord diagrams . . . . . . . . . . 8.3 The center of the Goldman Lie algebra . . . . . . . . . . 8.4 The homological Goldman Lie algebra . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction The purpose of this chapter is to survey a new aspect of the topological study of Riemann surfaces via infinite-dimensional Lie algebras. More concretely, we discuss a geometric approach to the Torelli–Johnson–Morita theory using an infinitedimensional Lie algebra called the Goldman Lie algebra, which comes from a global structure of the surface. The Torelli–Johnson–Morita theory, initiated by Johnson [23, 24] and elaborated later by Morita [57], is a place where infinite-dimensional Lie algebras appear in the study of mapping class groups and Torelli groups. In this chapter, surfaces are always assumed to be differentiable. Let † D †g;1 be a compact oriented surface of genus g > 0 with one boundary component, and Mg;1 the mapping class group of † relative to the boundary. The Torelli group Ig;1 is a normal subgroup of Mg;1 consisting of mapping classes acting trivially on the homology of †. There are many motivations from various fields of mathematics for studying this group, but we are still very far from a full understanding of it. To study Ig;1 we consider a central filtration of Ig;1 called the Johnson filtration, which is defined by the action of mapping classes on the lower central series of the fundamental group of the surface. A central object of the theory is an injective, graded Lie algebra homomorphism W

1 M kD1

grk .Ig;1 / !

1 M

hZg;1 .k/ :

(1.1)

kD1

hZg;1 .k/

The k-th component k W gr .Ig;1 / ! is called the k-th Johnson homomorphism. Here the target is an infinite-dimensional Lie algebra called the Lie algebra of symplectic derivations of type “Lie” in the sense of Kontsevich [40]. It is purely algebraically defined and was introduced by Kontsevich [40] and Morita [53, 54] independently. The image of  is called the Johnson image. Getting a characterization of this image is a hard but very important problem in the study of Ig;1 . Morally, the problem asks to describe the best approximation of the Torelli group by a pronilpotent Lie algebra. An infinite-dimensional Lie algebra related to an oriented surface S also arises in the following way. Let .S O / be the set of homotopy classes of oriented loops on S . k

1 Partially supported by the Grant-in-Aid for Scientific Research (B) (No. 24340010) from the Japan Society for Promotion of Sciences. 2 Partially supported by the Grant-in-Aid for Research Activity Start-up (No. 24840038) from the Japan Society for Promotion of Sciences.

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Motivated by the study of the symplectic structure of the moduli space of flat bundles over a surface, Goldman [17] introduced a Lie bracket called the Goldman bracket on the free Z-module Z.S O / with basis the set .S O /. The definition of the Lie bracket involves the intersections of two loops and this Lie algebra is called the Goldman Lie algebra. Later Turaev [79] found a Lie cobracket called the Turaev cobracket on the quotient Lie algebra ZO 0 .S / D Z.S O /=Z1, where 1 is the class of a constant loop, and showed that ZO 0 .S / has a structure of a Lie bialgebra. The definition of the Lie cobracket involves the self-intersections of a loop. This Lie bialgebra is called the Goldman–Turaev Lie bialgebra and will be our central object of consideration. In this chapter we work over the rationals Q and we consider Q.S O / D Z.S O / ˝Z Q. Our primary goal is to show that the Goldman Lie algebra appears naturally in the Torelli–Johnson–Morita theory. This relation was first found in [31] and further developed in [33] [34]. Here we explain the main idea briefly. To start with, the mapping class group Mg;1 acts on the fundamental group of † by automorphisms, where we take a base point on the boundary of †. This is a point of view illustrated in the Dehn–Nielsen theorem. A basic observation is that the Goldman Lie algebra Q.†/ O also acts on the fundamental group, but this time the action is by derivations. As for the relationship between Lie algebras and Lie groups, derivations and automorphisms are related by the exponential map. Here we come to a technical but inevitable point; since we work with the exponential map, we have to consider completions of objects and care about convergence. In any case we can construct suitable completions of the Goldman Lie algebra and the fundamental group, and we have the exponential map from a subset of the completion of Q.†/ O to the automorphism group of the completion of the fundamental group. After that we introduce a Lie subalgebra LC .†/ of the completion of Q.†/, O and show that the automorphisms of the completion of the fundamental group of † induced by elements of Ig;1 are in the image of the derivations coming from LC .†/ by the exponential map. Taking the logarithm, we obtain an injective group homomorphism  W Ig;1 ! LC .†/ ;

(1.2)

where the group structure of the target is described by the Hausdorff series. We call (1.2) the geometric Johnson homomorphism, since taking its graded quotients we can recover (1.1). Actually,  is essentially the same as Massuyeau’s total Johnson map [47]. However, our construction is free from any choice and is more intrinsic. A practical advantage of our construction is that it can be applied to other compact surfaces with more than one boundary component. The Turaev cobracket induces a map ı from the target of the geometric Johnson homomorphism  . By the fact that any diffeomorphism of a surface preserves the self-intersections of loops on the surface, we show that ı ı  D 0. This gives a non-trivial geometric constraint on the Johnson image. Indeed, we show that all the Morita traces [57], which are obstructions to the surjectivity of (1.1), can be derived from ı. We hope that some of other known constraints on the Johnson image also can admit interpretations from our geometric context. This survey is organized as follows. In §2, we give an overview of the construction of the Johnson homomorphisms and known results about the Johnson image. We

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also discuss how to extend the Johnson homomorphisms to the Torelli group or to the whole mapping class group. The main body of this chapter is from §3 to §7. We always consider mapping class groups relative to the boundary; all the diffeomorphisms and the isotopies that we consider are required to fix the boundary pointwise. Therefore, when the surface S has more than one boundary component, it is natural to consider that the mapping class group acts on the fundamental groupoid of the surface with base points chosen from each boundary component, instead of the fundamental group. In §3, we provide some language to deal with such a situation. In §4, we give the definition of the Goldman–Turaev Lie bialgebra, and explain how it interacts with the homotopy set of based paths on the surface. In particular, we show that Q.S O / acts on the fundamental groupoid of S by derivations. We also discuss other operations on curves on surfaces. In §5, we investigate Dehn twists from our point of view in detail. We show that the action of a Dehn twist on the completion of the fundamental groupoid has a canonical logarithm, and specify it as an element of the completion of the Goldman Lie algebra. This was first observed in [31], and leads us to introducing a “generalized Dehn twist”, which is an automorphism of the completion of the fundamental groupoid associated to a loop on the surface which is not necessarily simple. Generalized Dehn twists were first introduced in [41] and are further studied in [33, 34]. Massuyeau and Turaev [49] also study them from a slightly different point of view. In §5.2 and §5.3 we present basic properties of generalized Dehn twists. In §6 and §7, we define the geometric Johnson homomorphism. We first treat the case S D † in §6, then the general case in §7. In the general case, we obtain an injective group homomorphism  W I L .S / ! LC .S / :

(1.3)

Here I L .S / is the “largest” Torelli group in the sense of Putman [68], and LC .S / is a Lie subalgebra of the completion of Q.S O /. Note that when S has more than one boundary component, there are natural choices for the Torelli group, see [68]. Recently Church [7] constructed the first Johnson homomorphism for all kinds of Putman’s Torelli groups. We do not know any relation between Church’s construction and ours. In §6 and §7, we also give an algebraic description of the Goldman bracket. When S D †, this description says that the completion of Q.S O / is isomorphic to (the degree completion of an enhancement of) the Lie algebra of symplectic derivations of type “associative” in the sense of Kontsevich [40]. Note that the tensorial description of the Goldman bracket is also obtained by Massuyeau and Turaev [49, 50] by a different approach. In the case S D †, we also mention a partial result about a tensorial description of the Turaev cobracket based on a result of [49]. In §8, we discuss other related topics. Finally, we note something less relevant to the main part of the text but still worth mentioning. As for an infinite-dimensional Lie algebra coming from local structures of surfaces, there is an observation due to Kontsevich [39] and Beilinson, Manin and Schechtman [3], who discovered that the Lie algebra of germs of meromorphic vector fields at the origin of C, i.e., a complex analytic version of the Lie algebra Vect.S 1 /, acts on the moduli space of compact Riemann surfaces with local coordinates in an infinitesimally transitive way. This enables us to regard the Lie algebra as the Lie

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algebra of the stable mapping class group. This idea has been well-understood for the last few decades. For example, from this fact, we can derive some topological information on the stable cohomology of the mapping class group of a surface. For details, see [1, 26, 27, 28]. Compared with this idea, our approach, which will be presented in this chapter, suggests a quite new interaction between the mapping class group and an infinite-dimensional Lie algebra coming from the global nature of surfaces.

2 Classical Torelli–Johnson–Morita theory We describe the Torelli–Johnson–Morita theory for a once-bordered surface and its recent developments. This theory, initiated by Johnson [23, 24] and elaborated later by Morita [57], studies a certain filtration of the mapping class group and a graded Lie algebra associated to it. The treatment here is brief and limited. In particular, we confine ourselves to compact surfaces with one boundary component. For more details and other aspects, we refer to the chapters of Habiro and Massuyeau [18], Morita [61], Sakasai [70] and Satoh [73] in this handbook.

2.1 Lower central series and the higher Torelli groups Let † D †g;1 be a compact connected oriented surface of genus g > 0 with one boundary component, and Mg;1 the mapping class group of † relative to the boundary, i.e., the group of diffeomorphisms of † fixing the boundary @† pointwise, modulo isotopies fixing @† pointwise. Taking a base point  on @†, we set  D 1 .†; /. The group Mg;1 acts naturally on . Let k D k ./, k  1, be the lower central series of , i.e., the series of normal subgroups ofT  successively defined by 1 D  and k D Œk1 ;  for k  2. The intersection 1 kD1 k is trivial since  is a free group. Since k is characteristic, i.e., k is preserved by any automorphism of , Mg;1 acts naturally on the k-th nilpotent quotient Nk D Nk ./ D = kC1 . For k  1, the k-th Torelli group is defined as Mg;1 .k/ D f' 2 Mg;1 j' acts trivially on Nk g : of normal subgroups of Mg;1 Then we obtain a decreasing filtration fMg;1 .k/g1 kD1 called the Johnson filtration. The first term Mg;1 .1/ is nothing but the Torelli group Ig;1 since N1 D =Œ;  is canonically isomorphic to the first homology group HZ D H1 .†I Z/. The second term Mg;1.2/ is known as the Johnson kernel Kg;1 , which is by definition the kernel of the first Johnson homomorphism 1 (see §2.2). Due to a deep result by Johnson [25], Kg;1 is equal to the group generated by Dehn twists along separating simple closed curves on †. It is known that the filtration fMg;1 .k/g1 is central, i.e., kD1 ŒMg;1 .k/; Mg;1.`/  Mg;1 .k C `/

for k; `  1

(2.1)

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(see [55] Corollary 3.3). Thus the commutator product induces a structure of a graded L k Lie algebra on the graded module 1 gr .I where grk .Ig;1 / D Mg;1 .k/= g;1 /, T kD1 1 M .k C 1/. On the other hand, the intersection g;1 kD1 Mg;1 .k/ is trivial since T1  D f1g. We can regard the quotient groups Mg;1=Mg;1.k/ and Ig;1 = kD1 k Mg;1 .k/ as approximations of the whole group Mg;1 and the Torelli group Ig;1 . From this point of view it is important to understand grk .Ig;1 / for a specific k or the L k whole graded Lie algebra 1 kD1 gr .Ig;1 /. The Johnson homomorphisms are a key tool to study them.

2.2 The Johnson homomorphisms and their images We briefly recall the definition of the Johnson homomorphisms. Let us fix k  1 and consider an Mg;1 -equivariant exact sequence 0 ! kC1 = kC2 ! NkC1 ! Nk ! 1. Since  is free, the quotient k = kC1 is canonically isomorphic to LZ .k/, the degree k-part of the free Lie algebra generated by N1 D HZ (see e.g., [46] [75]). Thus the exact sequence becomes a central extension 0 ! LZ .k C 1/ ! NkC1 ! Nk ! 1 :

(2.2)

Take ' 2 Mg;1 .k/. Since ' acts trivially on Nk , for any x 2  the image of '.x/x 1 in NkC1 is actually an element of LZ .k C 1/ in view of (2.2). Then we obtain a mapping  ! LZ .k C 1/, x 7! Œ'.x/x 1 . One can show that this mapping is a homomorphism, thus induces a homomorphism k .'/W HZ ! LZ .k C 1/. The mapping k W Mg;1.k/ ! Hom.HZ ; LZ .k C 1//, ' 7! k .'/ is in fact a homomorphism, and is called the k-th Johnson homomorphism. It was introduced by Johnson [23] [24]. Note that using the intersection form . /W HZ  HZ ! Z on the surface, we can identify HZ and its dual HZ D Hom.HZ ; Z/ by HZ ! HZ , X 7! .Y 7! .Y X //, where X; Y 2 HZ . This induces an isomorphism Hom.HZ ; LZ .k C 1// D HZ ˝ LZ .k C 1/ Š HZ ˝ LZ .k C 1/ ; through which we can also write k as k W Mg;1.k/ ! HZ ˝ LZ .k C 1/ :

(2.3)

One can easily see that the kernel of k is Mg;1.k C 1/, hence k induces an injective group homomorphism k W grk .Ig;1 / ,! HZ ˝ LZ .k C 1/

(2.4)

(using the same letter k ). In particular the graded quotient grk .Ig;1 / is isomorphic to Im.k /. Remark 2.1. Let Sp.HZ / be the group of Z-linear automorphisms of HZ preserving the intersection form. Fixing a symplectic basis of HZ , we have an isomorphism

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Sp.HZ / Š Sp.2gI Z/. The group Sp.HZ / acts on both the domain and the target of (2.4). First of all for each k  1 the group Mg;1 acts on Mg;1 .k/ by conjugation, hence on grk .Ig;1 /. From (2.1) we see that the subgroup Mg;1 .1/ D Ig;1 acts trivially on grk .Ig;1 /. Since we have an exact sequence 1 ! Ig;1 ! Mg;1 ! Sp.HZ / ! 1, the action of Sp.HZ / on the domain is induced. The action of Sp.HZ / on the target is naturally induced by the action of Sp.HZ / on HZ . Then one can see that the map (2.4) is Sp.HZ /-equivariant. This point of view is particularly important when we study k ˝Z Q, since we can apply representation theory of Sp.2gI Q/. Johnson [23] proved that 1 .Ig;1/ D ƒ3Z HZ ¨ HZ ˝ LZ .2/. Morita [57] found constitutes a graded that the target of k can be smaller and the collection fk g1 kD1 Lie algebra homomorphism. He introduced a submodule hZg;1 .k/  HZ ˝ LZ .k C 1/ defined by hZg;1 .k/ D Ker.Œ ; W HZ ˝ LZ .k C 1/ ! LZ .k C 2// : L1 When k D 1, we have hZg;1 .1/ D ƒ3Z HZ . Let LZ D kD1 LZ .k/ be the free Lie Z algebra generated by HZ . Any element of hg;1 .k/ can be considered as a symplectic derivation of LZ as follows. For u 2 hZg;1 .k/, we define a Z-linear map Du W HZ D LZ .1/ ! LZ .k C 1/ by Du .X / D C12 .X ˝ u/, where C12 W H ˝kC3 ! H ˝kC1 , X1 ˝X2 ˝X3 ˝ ˝XkC3 7! .X1 X2 /X3 ˝ ˝XkC3 is the contraction of the first and the second factor by the intersection form. Then we can extend Du uniquely to a derivation Du W LZ ! LZ (using the same letter), that is, a Z-linear map satisfying the Leibniz rule Du .Œv; w / D ŒDu .v/; w C Œv; Du .w/ for any v; w 2 LZ . The derivation Du is of degree k in the sense that Du .LZ .`//  LZ .k C `/ for any `  1, and is symplectic in the sense that Du .!/ D 0, where ! 2 LZ .2/ D ƒ2 HZ  HZ˝2 is the tensor called the symplectic form, corresponding to 1H 2 Hom.HZ ; HZ / D HZ ˝PHZ D HZ ˝ HZ . Note that if fAi ; Bi ggiD1  HZ is a symplectic basis, then ! D giD1 Ai ˝ Bi  Bi ˝ Ai , cf. §6.2. The correspondence u 7! Du is injective. On the other hand any symplectic derivation of LZ of degree k can be written as the form Du for some u 2 hZg;1 .k/. Thus we can identify hZg;1 .k/ with the Z-module L Z of symplectic derivations of LZ of degree k. Then the graded module 1 kD1 hg;1 .k/ is the Z-module of symplectic derivations of LZ and naturally has a structure of a graded Lie algebra. We will discuss more details of the Lie algebra of symplectic derivations in §6.2. Theorem 2.2 (Morita [57]). (1) The image of (2.3) is contained in hZg;1 .k/. (2) The maps fk g1 induce an injective homomorphism of graded Lie algebras kD1 W

1 M kD1

grk .Ig;1 / !

1 M kD1

hZg;1 .k/ :

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By the result of Morita, we can write k as k W Mg;1.k/ ! hZg;1 .k/ :

(2.5)

In this chapter, we understand the k-th Johnson homomorphism on the k-th Torelli group to be (2.5). L k As posed [61], the characterization of 1 kD1 gr .Ig;1 / as a Lie subalL1in Morita Z gebra of kD1 hg;1 .k/ is one of the big and basic problems in the Torelli–Johnson– Morita theory. Actually, k is not surjective in general. This was observed first by Morita [57]. One often considers the problem over Q to make use of the representation theory of Sp.2gI Q/, (see Remark 2.1), but still raised by L the question k Morita is very hard. In this chapter we call the subalgebra 1 gr .I / g;1 tensored kD1 by the rationals Q the Johnson image. In his monumental paper [20], Hain gave an explicit presentation of the Malcev completion of the Torelli group Ig;1 when g  6. In particular, from the presentation together with his other result, Proposition 7.1 in [19], the Johnson image is generated by the first degree component gr1 .Ig;1 / ˝ Q D ƒ3 HZ ˝ Q. Now what we want is aL complete system of definZ ing equations of the Johnson image in the Lie algebra 1 kD1 hg;1 .k/ ˝ Q. Such an equation is called a Johnson cokernel or an obstruction of the surjectivity of the Johnson homomorphism. First of all, Morita [57] found an obstruction for the surjectivity of k . Let k S HZ be the k-th symmetric power of HZ , C12 W HZ˝kC2 ! HZ˝k the contraction of the first and the second factor, and sW HZ˝k ! S k HZ the natural projection. Let Trk W hZg;1.k/ ! S k HZ be a Z-linear map defined by C12

s

Trk W hZg;1.k/  HZ ˝ LZ .k C 1/  HZ˝kC2 ! HZ˝k ! S k HZ : The map Trk is called the k-th Morita trace. Theorem 2.3 (Morita [57]). (1) If k  2, we have Trk ı k D 0W grk .Ig;1 / ! S k HZ . (2) If k is odd, then Trk is non-trivial. In fact, Trk ˝Z Q is surjective. (3) If k is even, then Trk D 0 on the whole hZg;1 .k/. In the original definition [57] the map Tr is defined as a map hZg;1 .k  1/ ! S k1 HZ , while we follow the grading in [59], p. 376. In §6.4, we will give a topological interpretation of the Morita traces by the Turaev cobracket [34]. In a natural way the absolute Galois group Gal.Q=Q/ of the rational number field Q acts on the arithmetic fundamental group of a pointed algebraic curve defined over the rationals Q, which is a group extension of the Galois group Gal.Q=Q/ by the (geometric) fundamental of the curve. This induces an image of the Galois group L group Z in the Lie algebra 1 kD1 hg;1 .k/ ˝ Q, which is called the Galois image. The origin of this construction is in Grothendieck, Ihara and Deligne. For its precise description, see [64] and references therein. The relation between the Johnson image and the

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105

Galois image has been studied by T. Oda, H. Nakamura, M. Matsumoto and others. For example, H. Nakamura [64] introduced some explicit Johnson cokernels coming from the Galois image. Such Johnson cokernels are called Galois obstructions. In his study of the IA-automorphism group of a free group, Satoh [71, 72] discovered a refinement of the Morita traces. Let Fn be a free group of rank n  2, and HZ its abelianization as in §2.1. We denote by HZ its dual HomZ .HZ ; Z/, and by LZ .k/ the degree k part of the free Lie algebra generated by HZ . The cyclic group of degree k acts on the tensor space HZ ˝k by cyclic permutation of the components. Following Satoh, we denote by Cn .k/ the coinvariants of the action, i.e., Cn .k/ WD HZ ˝k =hX1 ˝ X2 ˝ ˝ Xk  X2 ˝ X3 ˝ ˝ Xk ˝ X1 I Xi 2 HZ i : brk W HZ ˝ LZ .k C 1/ ! Cn .k/ is defined to be the composite of The Satoh trace T  0 W HZ ˝ the inclusion HZ ˝ LZ .k C 1/ ,! HZ ˝ HZ ˝.kC1/ , the contraction map C12 ˝.kC1/ ˝k ! HZ , f ˝ X2 ˝ X3 ˝ ˝ XkC2 7! f .X2 /X3 ˝ ˝ XkC2 , HZ .f 2 HZ ; Xi 2 HZ /, and the quotient map HZ ˝k ! Cn .k/. Satoh [71, 72] proved that the images of the lower central series of the IA-automorphism group under the brk up Johnson homomorphisms stably coincide with the kernels of the Satoh traces T to torsion. For details, see his chapter in this volume, [73]. The fundamental group 1 .†g;1 ; / is free of rank 2g, so that we can consider L Z 0 brk on the Lie algebra 1 the Satoh traces T kD1 hg;1 .k/. Then the contraction map C12 is exactly the same as the map C12 under Poincaré duality. From Satoh’s result [71] brk ı k D 0 on Mg;1.k/ for any k  2. together with Hain’s result [20], we have T brk is a refinement of the Morita trace Trk . Enomoto and Satoh [13] carried Hence T L Z brk ’s on 1 out some explicit computation of T kD1 hg;1 .k/˝Q, to prove that they have many non-trivial components of the Johnson cokernels other than the Morita traces. L Z brk to 1 Thus the restriction of T kD1 hg;1 .k/ is called the Enomoto–Satoh trace.

2.3 Extensions of the Johnson homomorphisms From Theorem 2.2 (2) by Morita, the totality of the Johnson homomorphisms (tensored by the rationals Q) W

1 M kD1

grk .Ig;1 / ˝ Q !

1 M

hZg;1 .k/ ˝ Q

kD1

is L an injective homomorphism of graded Lie algebras. Hence the Johnson image  1 k  gr .I / ˝ Q can be regarded as the “Lie algebra” of the Torelli group g;1 kD1 Ig;1 . But the map  is not defined on the Torelli group itself, but on the graded quotients. So it is desirable to find a lift of  , or equivalently, an extension of  to the Torelli group or to the whole mapping class group Mg;1 . As will be stated below, there are various ways to construct extensions of the Johnson homomorphisms.

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The diversity of the constructions comes from that of the realizations of the Malcev completion of the free group  D 1 .†g;1 ; /. The first result on this problem was given by Morita [56, 58] through an explicit construction of the automorphism group of the group Nk , a truncated Malcev completion. Here it should be remarked that the abelianization Mg;1 abel is trivial (g  3) or finite (g D 2). Hence there exists no non-trivial homomorphism from Mg;1 to any rational vector space if g  2. In [56] Morita gave an extension as a crossed Q Mg;1 ! gr1 .Ig;1 / ˝ Q D ƒ3 HZ ˝ Q of the first Johnson hohomomorphism kW momorphism 1 . More precisely, he proved that there is a unique cohomology class 2kQ 2 H 1 .Mg;1I ƒ3 HZ / whose restriction to Ig;1 is twice the first Johnson homomorphism 21 . Here ƒ3 HZ is a non-trivial Mg;1 -module in an obvious way. Let 0 W Mg;1 ! Sp.HZ / be the natural action of Mg;1 on the first homology group HZ . The crossed homomorphism kQ defines a group homomorphism   1 3 ƒ HZ Ì Sp.HZ / ; 1 W Mg;1 ! 2 which induces a homomorphism of the cohomology groups    Sp.HZ /  

1 1 3 1 3 ! H ƒ HZ I Q ƒ HZ Ì Sp.HZ /I Q ! H  .Mg;1I Q/: kQ  W H  2 2 Theorem 2.4 (Kawazumi–Morita [37]). The image Image.kQ  / equals the subalgebra of H  .Mg;1I Q/ generated by the Morita–Mumford classes ei D .1/i C1 i , i  1. We remark that the theorem holds also for the unstable range. So it is not covered by the Madsen–Weiss theorem [45]. The original proof of Theorem 2.4 is obtained by interpreting the extended first Johnson homomorphism kQ as the .0; 3/-twisted Morita– Mumford class m0;3 [28]. As for the second Johnson homomorphism 2 , Morita [58] constructed a group homomorphism     1 Z 1 Z Q Ì Sp.HZ / h .2/  h .1/ 2 W Mg;1 ! 24 g;1 2 g;1 Q means some central extension of extending the homomorphisms 1 and 2 , where  1 Z 1 Z h .1/ by h .2/. From the Madsen–Weiss theorem, all the rational cohomolg;1 g;1 2 24 ogy classes coming from 2 in the stable range are generated by the Morita–Mumford classes. From Hain’s theorem [20] stated above follows the existence of an extension of the k-th Johnson homomorphism to the whole Mg;1 for any k  1. On the other hand, Kawazumi [29] gave an explicit recipe for constructing extensions of the totality of the Johnson homomorphisms from a generalized Magnus expansion of a free group. For any k  1, Kitano [38] described the k-th Johnson homomorphism in terms of the standard Magnus expansion of the free group  D 1 .†g;1 ; / associated to a

4 The Goldman–Turaev Lie bialgebra and the Johnson homomorphisms

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symplectic generating system. Moreover Perron [67] constructed an extension of the k-th Johnson homomorphism k for any k  1 in terms of the standard Magnus expansion. In general, consider a free group Fn of rank n  2 with a generating system fx1 ; x2 ; : : : ; xn g. Let H be the abelianization of Fn tensored by the rationals b D T b.H / the completed tensor algebra generated Q, H WD Fn abel ˝Z Q, and T Q1 ˝p bm g1 of b D T b.H / WD , which has a decreasing filtration fT by H , T pD0 H mD1 Q1 ˝p b1 is a subgroup of bm WD H . The set 1 C T two-sided ideals defined by T pm b. The standard Magnus expansion of Fn the multiplicative group of the algebra T associated to fx1 ; x2 ; : : : ; xn g is the group homomorphism stdW Fn ! 1 C b T 1 defined by std.xi / WD 1 C Œxi , 1  i  n. Here Œ WD . mod ŒFn ; Fn / ˝ 1 2 H D Fn abel ˝Z Q is the homology class of  2 Fn . On the other hand, Bourbaki [5] b1 . See also [75]. So developed a basic theory of group homomorphisms Fn ! 1 C T we define the notion of a (generalized) Magnus expansion of the free group Fn by the minimum conditions for describing the Johnson homomorphisms. b is a (Q-valued) Magnus expansion of the Definition 2.5 ([29]). A map W Fn ! T b1 and satisfies the free group Fn if it is a group homomorphism of Fn into 1 C T b condition . / 1 C Œ .mod T 2 / for any  2 Fn . The standard Magnus expansion std is a Magnus expansion in this sense. Let QFn be the rational group ring of the group Fn , and QFn its completion

b

b

QFn WD lim QFn =.IFn /m :  m!1

b b WD Ker.QF b ! QF =.IF / QF

Here IFn is the augmentation ideal, or equivalently, the kernel of the augmentation P P map augW QFn ! Q,  2Fn a  7!  2Fn a . The algebra QFn has a natural de-

b

m /. creasing filtration fFm QFn g1 n n n n mD1 defined by Fm Fix an arbitrary Magnus expansion of the free group Fn . Then its Q-linear P P b, extension W QFn ! T  a  7!  a . /, induces an algebra isomorphism

b

b

Š

b

W QFn ! T

(2.6)

bm for any m  1. See, for example, [29] Theorem 1.3. such that .Fm QFn / D T For any automorphism ' 2 Aut.Fn / of the group Fn , we define an automorphism ' Š Š  b by T  .'/ WD ı ' ı 1 W b b, T .'/ of the algebra T T ! QFn ! QFn ! T  bm / D T bm for any m  1. Denote by Aut.T b/ the group which satisfies T .'/.T bm b bm / D T of all automorphisms U of the algebra T satisfying the condition U.T

b

b

b

for any m  1. Since the completion map QFn ! QFn is injective, the group homomorphism b/; ' 7! T  .'/ ; T  W Aut.Fn / ! Aut.T

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is injective. All the Johnson homomorphisms come from the homomorphism T  . So we call T  the total Johnson map of the automorphism group Aut.Fn / [29]. There are at least two ways to extract an extension of the k-th Johnson homomorphism k from the map T  . One way [29] was prepared for the group cohomology of Aut.Fn /, and the other [47] suitable for the Malcev completion of the group Fn . b/ be the First we explain the original Johnson map introduced in [29]. Let IA.T b2 D H . Then the restriction b/ on the space T b1 = T kernel of the natural action of Aut.T b to the subspace H  T induces a bijective correspondence b/ Š Hom.H; b IA.T T 2/ D

1 Y

Hom.H; H ˝.kC1/ / ;

kD1

by which we identify these sets. For any ' 2 Aut.Fn / the induced map j'j on b in an obvious way, so that we may regard the H D Fn abel ˝ Q acts on the algebra T Q  1 ˝.kC1/ b/ D 1 /. We composite T .'/ ı j'j as an element of IA.T kD1 Hom.H; H define the k-th Johnson map k W Aut.Fn / ! Hom.H; H ˝.kC1/ /, k  1, by T .'/ ı j'j 

1

D

.k .'//1 kD1

b/ D 2 IA.T

1 Y

Hom.H; H ˝.kC1/ / :

kD1

The maps k ’s are no longer group homomorphisms. Instead they satisfy an infinite sequence of coboundary equations. For example, we have  d 1 .'/ D 0 2 C 2 .Aut.Fn /I Hom.H; H ˝2 // ;  d 2 .'/ D .1 ˝ 1H C 1H ˝ 1 / [ 1 2 C 2 .Aut.Fn /I Hom.H; H ˝3 // : (2.7) Here C  .Aut.Fn /I M / is the normalized cochain complex of the group Aut.Fn / with values in an Aut.Fn /-module M , d the coboundary operator, and [ the Alexander– Whitney cup product. From the equation (2.7) we obtain a straightforward proof of Theorem 2.4. Let IAn be the kernel of the natural action of Aut.Fn / on the abelianization Fn abel , which is called the IA-automorphism group, and an analogue of the b/. Torelli group. Then we have an injective group homomorphism T  W IAn ! IA.T  In the case n D 2g and Fn D  D 1 .†g;1 ; /, the restriction k jMg;1 .k/ equals the (original) k-th Johnson homomorphism k . In other words, the graded quotient of the restriction T  jIg;1 equals the totality of the (original) Johnson homomorphisms gr.T  jIg;1 / D  W

1 M kD1

grk .Ig;1 / !

1 M kD1

hZg;1 .k/ 

1 M

Hom.H; H ˝.kC1/ / : (2.8)

kD1

For details, see [29]. Next we discuss Massuyeau’s total Johnson map [47] b// :   W IAn ! Hom.H; LC .T

4 The Goldman–Turaev Lie bialgebra and the Johnson homomorphisms

109

b

We need to recall some general facts on complete Hopf algebras to explain the definition of the target. The completed group ring QFn and the completed tensor alb D T b.H / are complete Hopf algebras, whose coproducts  are given by gebra T b˝ b b 1 C 1˝ bX 2 T bT b b QFn for  2 Fn , and by .X / D X ˝ . / D  ˝ 2 QFn ˝ for X 2 H , respectively. We denote by Gr.R/ the set of all group-like elements in a complete Hopf algebra R, or equivalently Gr.R/ WD fr 2 R n f0gI .r/ D b r 2 R˝ b Rg, which is a subgroup of the multiplicative group of the algebra R. r˝ The group Gr.QFn / is, by definition, the Malcev completion of the group Fn . Simib 1 C 1˝ b ug the set of all primitive larly we denote by L.R/ WD fu 2 RI .u/ D u˝ elements, which is a Lie subalgebra of the associative algebra R. The Lie algebra L b/ equals the degree completion of 1 LZ .k/ ˝ Q. As is known [69], the L.T kD1 P 1 k exponential expW L.R/ ! Gr.R/, exp.u/ WD kD0 .1=kŠ/u , and the logarithm P1 k1 k logW Gr.R/ ! L.R/, log.r/ WD kD1 ..1/ =k/.r  1/ , are the inverses of each other. b/ be the stabilizer of the coproduct  in the group IA.T b/, and LC .T b/ Let IA .T L1 the degree completion of the Lie algebra kD2 LZ .k/ ˝ Q. Then the Lie algebra b which stabilize the coproduct  and vanish consisting of continuous derivations of T b2 is naturally identified with Hom.H; LC .T b//, which b1 =T on the quotient H D T b// ! IA .T b/, is the target of the map   . The exponential expW Hom.H; LC .T P1 k C b b exp.D/ WD kD0 .1=kŠ/D , and the logarithm logW IA .T / ! Hom.H; L .T //, P k1 log.U / WD 1 =k/.U  1/k , are inverses of each other. Following MaskD1 ..1/ suyeau [47], we consider the notion of a group-like expansion of the group Fn .

b b

b

b of the free group Definition 2.6 (Massuyeau [47]). A Magnus expansion W Fn ! T b b . / for any Fn is group-like if .Fn /  Gr.T /, or equivalently . . // D . /˝  2 Fn .

b

Š

b (2.6) preserves Fix a group-like expansion . Then the isomorphism W QFn ! T the coproduct, so that the Malcev completion of Fn is isomorphic to the group of the b/. Massuyeau b through , and we have T  .IAn /  IA .T group-like elements of T introduced the composite 1 X .1/k1  .T .'/  1/k jH ; k kD1 (2.9) which we call Massuyeau’s total Johnson map. From (2.8) the graded quotient of   equals the totality of the (original) Johnson homomorphisms (for Aut.Fn /). In the case n D 2g and Fn D  D 1 .†g;1 ; /, it is desirable that   .Ig;1 /  lC g WD Q1 Z C b kD1 hg;1 .k/ ˝ Q ¤ Hom.H; L .T //. A symplectic expansion introduced by Massuyeau [47] makes it possible as will be stated in §6.1. Our purpose is to re-construct the map   in a geometric context with no use of Magnus expansions. We conclude this subsection by reviewing some other approaches to extending the Johnson homomorphisms or their enlargement to the whole mapping class group

b// ;   WD log ıT  W IAn ! Hom.H; LC .T

' 7!

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Nariya Kawazumi, Yusuke Kuno

or some wider objects. The fatgraph decompositions of the surface †g;1 define the Ptolemy groupoid of †g;1 , which includes the mapping class group Mg;1 . Morita and Penner [62] introduced an explicit 1-cocycle on the Ptolemy groupoid representQ Bene, Kawazumi and Penner [4] ing the extended first Johnson homomorphism k. discovered a canonical way to associate a group-like expansion to any bordered trivalent fatgraph with one tail. Unfortunately it is not symplectic. But the 1-cocycle in [62] is the first term of the difference of two group-like expansions associated to two fatgraphs adjacent by one Whitehead move. Contracting the coefficients ƒ3 H ! H by the intersection form on the homology group H D H1 .†g;1 I Q/, we have the Earle class k 2 H 1 .Mg;1I HZ /. Kuno, Penner and Turaev [43] introduced an explicit 1-cocycle on the Ptolemy groupoid representing the Earle class, which is simpler than the contraction of the Morita–Penner cocycle. On the other hand, in [57], Morita introduced a refinement of the k-th Johnson homomorphism Mg;1 .k/ ! H3 .Nk / for any k  1. Here H3 .Nk / is the third homology group of the nilpotent group Nk in §2.2. Massuyeau [48] discovered a canonical way to attach a 3-chain of the group  modulo the boundaries to each marked trivalent fatgraph, which is an extension of Morita’s refinement of the Johnson homomorphisms to the Ptolemy groupoid. It is unknown whether the cocycle representing the Johnson homomorphisms induced from Massuyeau’s and that in [4] coincide with each other or not. As was proved by Massuyeau [47] Theorem 4.4, Morita’s refinement is equivaL  lent to the sum 2k1 j Dk j . In [47], he gave an extension of all of Morita’s refinements to the monoid of homology cylinders, which includes the mapping class group Mg;1. See the chapter by Habiro and Massuyeau [18] in volume III of this handbook. M. Day [8, 9] realized truncations of the Malcev completion of the group  in the framework of general Lie theory of nilpotent groups [65] to present two geometric ways to extend Morita’s refinements to the whole mapping class group Mg;1 . It is also unknown whether one of them coincides with any of those we stated above.

3 Dehn–Nielsen embedding In study of the mapping class group of a surface, it is often useful to consider its action on curves on the surface. In the classical case, the surface is † D †g;1 as in §2 and the mapping class group Mg;1 acts on  D 1 .†; /. This action induces an injective group homomorphism

DNW Mg;1 ! Aut./ ;

(3.1)

whose image is characterized as the automorphisms of  preserving the boundary loop of †. This is the Dehn–Nielsen theorem. In this section we work with general oriented surfaces and consider an analogue of (3.1) for their mapping class groups. Instead of the fundamental group as in the case † D †g;1 , we consider the fundamental groupoid of the surface with suitably chosen base points and the action of the mapping class group on it.

4 The Goldman–Turaev Lie bialgebra and the Johnson homomorphisms

111

3.1 Groupoids and their completions We begin by some general discussions about groupoids. Let us recall a classical construction associated to a group G (see [69]). The group ring QG is a Q-vector space with basis the set G. Extending Q-bilinearly the product of G, it becomes a Q-algebra. Also it is a Hopf algebra with respect to the coproduct W QG ! QG ˝ QG, G 3 g 7! g ˝ g and the antipode W QG ! QG, G 3 g 7! g 1 . The augmentation ideal IG is the kernel of the Q-algebra homomorphism QG ! Q; G 3 g 7! 1. The powers .IG/n , n  0, are twosided ideals of QG. We denote by QG the projective limit lim QG=.IG/n . The n product, the coproduct, and the antipode of QG are induced by those of QG. It is called the completed group ring of G, and is naturally a complete Hopf algebra with respect to the filtration Fn QG D Ker.QG ! QG=.IG/n /, n  0. The set of groupb D fg 2 QGI .g/ D g ˝ b g; g ¤ 0g is a group with respect to the like elements G product of QG, and is called the Malcev completion of G. We have a canonical group b This map is not injective in general. Note that in §2.3 we homomorphism G ! G. have already seen the above construction for a free group. Let us consider an analogous construction for groupoids. Let G be a groupoid such that the set of objects is Ob.G/ and the set of morphisms from p0 2 Ob.G/ to p1 2 Ob.G/ is G.p0 ; p1 /. First we consider the “group ring” for G. Let QG be the following small category. The set of objects of QG is the same as that of G, i.e., Ob.G/. The set of morphisms from p0 2 Ob.G/ to p1 2 Ob.G/ is QG.p0 ; p1 /, the Q-vector space with basis the set G.p0 ; p1 /. By an obvious manner the product of morphisms in QG is induced from that in G. For any p0 ; p1 ; p2 2 Ob.G/ the product QG.p0 ; p1 /  QG.p1 ; p2 / ! QG.p0 ; p2 / is Q-bilinear. We define the coproduct, the antipode, and the augmentation of QG as the collections

b

b

b b

b

b

fp0 ;p1 W QG.p0 ; p1 / ! QG.p0 ; p1 / ˝ QG.p0 ; p1 /gp0 ;p1 2Ob.G/ ; fp0 ;p1 W QG.p0 ; p1 / ! QG.p1 ; p0 /gp0 ;p1 2Ob.G/ ; and faugp0 ;p1 W QG.p0 ; p1 / ! Qgp0 ;p1 2Ob.G/ ; of Q-linear maps respectively, where p0 ;p1 is defined by p0 ;p1 .`/ D ` ˝ ` for ` 2 G.p0 ; p1 /, p0 ;p1 is induced by taking the inverse of morphisms in G, and augp0 ;p1 is defined by augp0 ;p1 .`/ D 1 for ` 2 G.p0 ; p1 /. For ` 2 G.p0 ; p1 /, we denote ` WD p0 ;p1 .`/. If there is no fear of confusion, we simply write , , and aug instead of p0 ;p1 , p0 ;p1 , and augp0 ;p1 , respectively. Clearly  is a contravariant functor from QG to itself. Let QG ˝ QG be the following small category. The set of objects of QG ˝ QG is Ob.G/, the set of morphisms from p0 2 Ob.G/ to p1 2 Ob.G/ is QG.p0 ; p1 / ˝ QG.p0 ; p1 /, and the product of morphisms in QG ˝ QG is the tensor product of morphisms in QG. We call QG ˝ QG the tensor product. Then we can regard the coproduct  as a covariant functor from QG to QG ˝ QG. We next consider a concept corresponding to the augmentation ideal IG and its powers. Notice that for any p 2 Ob.G/ the set Gp D G.p; p/ is a group. Let p0 ; p1 2 Ob.G/ and n  0. If there is no morphism from p0 to p1 , i.e., G.p0 ; p1 / D

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;, we set Fn QG.p0 ; p1 / D 0. Otherwise, taking a morphism ` 2 G.p0 ; p1 / we set Fn QG.p0 ; p1 / D .I Gp0 /n `. Here I Gp0 is the augmentation ideal of the group Gp0 . We understand that Fn QG.p0 ; p1 / D QG.p0 ; p1 / for n < 0. Proposition 3.1. (1) The subspace Fn QG.p0 ; p1 / is independent of the choice of `, and fFn QG.p0 ; p1 /gn0 is a decreasing filtration of QG.p0 ; p1 /. The augmentation induces an isomorphism QG.p0 ; p1 /=F1 QG.p0 ; p1 / Š Q. (2) For any p0 ; p1 ; p2 2 Ob.G/ and n1 ; n2  0, we have Fn1 QG.p0 ; p1 / Fn2 QG.p1 ; p2 /  Fn1 Cn2 QG.p0 ; p2 / : (3) For any p0 ; p1 2 Ob.G/ and n  0, we have X Fn QG.p0 ; p1 /  Fn1 QG.p0 ; p1 / ˝ Fn2 QG.p0 ; p1 / ; n1 Cn2 Dn

Fn QG.p0 ; p1 /  Fn QG.p1 ; p0 / : Clearly Fn QG.p; p/ D .I Gp /n for any p 2 Ob.G/ and n  0, and F1 QG.p0 ; p1 / D Ker.aug/ for any p0 ; p1 2 Ob.G/. Now we construct a completion of QG. Let QG be the following small category. The set of objects of QG is Ob.G/. For p0 ; p1 2 Ob.G/ we set

b b bG.p ; p / WD lim QG.p ; p /=F QG.p ; p / ; Q 0

1

0

n

1

n

0

1

b

and define the set of morphisms from p0 to p1 to be QG.p0 ; p1 /. By Proposition 3.1 (2), the product of morphisms in QG is induced from that in QG. Also, by Proposition 3.1 (1)(3) the coproduct, the antipode, and the augmentation of QG are induced naturally. We shall use the same letters , , " for them. For example the coproduct of QG is the collection of maps  D p0 ;p1 , p0 ; p1 2 Ob.G/, where  is a map from QG.p0 ; p1 / to the completed tensor product

b

b b bG.p ; p /˝bQbG.p ; p / Q 0

1

0

1

D lim.QG.p0 ; p1 / ˝ QG.p0 ; p1 //=  n

b

X

Fn1 QG.p0 ; p1 / ˝ Fn2 QG.p0 ; p1 / :

n1Cn2 Dn

b b b b b

b QG in an obvious manner, we can regard Again, introducing a small category QG ˝ b QG.  as a covariant functor from QG to QG ˝ We call QG the completion of QG. We define a filtration of QG.p0 ; p1 / by

b

b

b

b

Fn QG.p0 ; p1 / WD Ker.QG.p0 ; p1 / ! QG.p0 ; p1 /=Fn QG.p0 ; p1 //;

for n  0 :

4 The Goldman–Turaev Lie bialgebra and the Johnson homomorphisms

b

b

b b

113

There is a canonical isomorphism QG.p0 ; p1 / Š lim QG.p0 ; p1 /=Fn QG.p0 ; p1 /. n This filtration enjoys a property similar to Proposition 3.1, and endows QG.p0 ; p1 / b u and with a topology. We shall say that u 2 QG.p0 ; p1 / is group-like if .u/ D u˝ u ¤ 0. Note that the set of group-like elements of QG is closed under the product of morphisms and the antipode, and any group-like element of QG is an isomorphism. Thus the set of group-like elements of QG constitutes a subcategory Gr.QG/ of QG and is in fact a groupoid. There is a natural homomorphism of groupoids from G to Gr.QG/. We call Gr.QG/ the Malcev completion of the groupoid G. We end this subsection by recording the following fact which will be used later. Let n  1 and p0 ; p1 ; : : : ; pn N 2 Ob.G/, and assume G.pi 1 ; pi / ¤ ; for 1  i  n. Then the multiplication niD1 F1 QG.pi 1 ; pi / ! Fn QG.p0 ; pn / is surjecN tive, and the sum of the multiplication and the inclusion niD1 F1 QG.pi 1 ; pi / ˚ FnC1 QG.p0 ; pn / ! Fn QG.p0 ; pn / is surjective.

b

b

b

b

b

b

b

b b

b

b

3.2 Derivations and their exponentials Let G be a groupoid. Recall that a derivation of an associative Q-algebra A is a Q-endomorphism DW A ! A satisfying the Leibniz rule D.ab/ D .Da/b C a.Db/ for any a; b 2 A. We generalize this notion to QG and QG. We define a derivation of QG to be a collection D D fDp0 ;p1 gp0 ;p1 2Ob.G/ of Q-endomorphisms Dp0 ;p1 W QG.p0 ; p1 / ! QG.p0 ; p1 / satisfying the Leibniz rule in the sense that

b

Dp0 ;p2 .uv/ D .Dp0 ;p1 u/v C u.Dp1 ;p2 v/ for any p0 ; p1 ; p2 2 Ob.G/, u 2 QG.p0 ; p1 / and v 2 QG.p1 ; p2 /. To simplify the notation we often write D instead of Dp0 ;p1 . The derivations of QG form a Lie algebra Der.QG/ with Lie bracket ŒD1 ; D2 D D1 D2  D2 D1 , D1 ; D2 2 Der.QG/. Similarly, we define a derivation of QG to be a collection of continuous Q-endomorphisms of QG.p0 ; p1 /, p0 ; p1 2 Ob.G/, satisfying the Leibniz rule in the same sense as before. We denote by Der.QG/ the set of derivations of QG. This is a Lie algebra in an obvious manner. For later use we introduce a filtration of Der.QG/. For n 2 Z, we define Fn Der.QG/ to be the set of D 2 Der.QG/ such that

b

b

b

b

b b bG.p ; p //  F QbG.p ; p / D.F Q 0

`

1

0

`Cn

1

b

b

for any p0 ; p1 2 Ob.G/ and `  0. We say that a derivation D 2 Der.QG/ stab 1 C 1˝ b D/W QG.p0 ; p1 / ! QG.p0 ; p1 / for bilizes the coproduct if D D .D ˝ any p0 ; p1 2 Ob.G/. The derivations of QG stabilizing the coproduct form a Lie subalgebra Der .QG/ of Der.QG/. We show that a derivation of QG naturally induces a derivation of QG. Let D 2 Der.QG/. We claim that for any p0 ; p1 2 Ob.G/ and n  0 we have

b

b

b

b

D.Fn QG.p0 ; p1 //  Fn1 QG.p0 ; p1 / :

b

b

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To prove this, we may assume that G.p0 ; p1 / ¤ ;. By the remark at the end of §3.1 there exist u1 ; : : : ; un1 2 F1 QG.p0 ; p0 / and un 2 F1 QG.p0 ; p1 / such that u D u1 un1 un . Then D.u1 un / D

n X

u1 ui 1 .Dui /ui C1 un 2 Fn1 QG.p0 ; p1 / ;

i D1

as desired. This shows that D D Dp0 ;p1 induces a continuous Q-endomorphism of QG.p0 ; p1 /, and there is a natural Lie algebra homomorphism Der.QG/ ! Der.QG/. We next discuss the exponential of derivations. Recall that if A is an associative D is a derivation of A, then the formal power series exp.D/ D P1Q-algebra and n .1=nŠ/D is a Q-algebra automorphism of A, provided it converges. To prove nD0 this, note that for any a; b 2 A and n  0, we have X nŠ D n1 .a/D n2 .b/ D n .ab/ D n Šn Š 1 2 n ;n 0 ;

b

b

1

2

n1 Cn2 Dn

b

by the Leibniz rule. Now let us consider the exponential of derivations of QG.

b

Lemma 3.2 ([33] §1.3). Suppose D 2 Der.QG/ satisfies the following conditions. (1) For any p0 ; p1 2 Ob.G/, n  0, we have D.FnQG.p0 ; p1 //  Fn QG.p0 ; p1 /.

b

b

b

b

(2) For any p0 ; p1 2 Ob.G/, we have D.QG.p0 ; p1 //  F1 QG.p0 ; p1 /.

b

(3) For any p0 ; p1 2 Ob.G/, there exists  > 0 such that D  .F1 QG.p0 ; p1 //  F2 QG.p0 ; p1 /. P n Then for any p0 ; p1 2 Ob.G/ the series exp.D/ D 1 nD0 .1=nŠ/D converges and is a Q-linear homeomorphism of QG.p0 ; p1 /. Moreover, if D 0 2 Der.QG/ satisfies the above conditions and exp.D/ D exp.D 0 /, then we have D D D 0 .

b

b

b b

b

Assume that D 2 Der.QG/ satisfies the assumption of Lemma 3.2. It is a formal consequence of the Leibniz rule that exp.D/.uv/ D .exp.D/u/.exp.D/v/ for any p0 ; p1 ; p2 2 Ob.G/, u 2 QG.p0 ; p1 /, and v 2 QG.p1 ; p2 /. Thus exp.D/ is an automorphism of the small category QG acting on the set of objects as the identity. Moreover, by the condition (2) of the assumption of Lemma 3.2 the automorphism exp.D/ preserves the augmentation in the sense that aug ı exp.D/ D augW QG.p0 ; p1 / ! Q for any p0 ; p1 2 Ob.G/.

b

b

b

3.3 Fundamental groupoid Let us consider the construction in §3.1 for surfaces. Let S be an oriented surface. For p0 ; p1 2 S , let …S.p0 ; p1 / D Œ.Œ0; 1 ; 0; 1/; .S; p0; p1 /

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be the homotopy set of paths from p0 to p1 . Throughout this chapter, we often ignore the distinction between a path and its homotopy class. Let E be a non-empty closed subset of S , which is the disjoint union of finitely many simple closed curves and finitely many points. We denote by …S jE the fundamental groupoid of S based at E. Namely, the set of objects of …S jE is E, and the set of morphisms from p0 2 E to p1 2 E is …S.p0 ; p1 /. The product of morphisms is induced by conjunction of paths. For p0 ; p1 2 E we use the notation Q…S.p0 ; p1 / and Q…S.p0 ; p1 / instead of Q…S jE .p0 ; p1 / and Q…S jE .p0 ; p1 /, respectively.

2

1

3.4 Dehn–Nielsen homomorphism Let S and E be as in §3.3. We define the mapping class group of the pair .S; E/, denoted by M.S; E/, as the group of diffeomorphisms of S fixing E [ @S pointwise, modulo isotopies fixing E [@S pointwise. If E  @S , we use the notation M.S; @S / or M.S / instead of M.S; E/. Unless otherwise stated we ignore the distinction between a diffeomorphism and its mapping class in M.S; E/. The mapping class group M.S; E/ acts naturally on the groupoid …S jE . Let Aut.…S jE / be the group of automorphisms of the groupoid …S jE acting on the set of objects as the identity. If ' is a diffeomorphism fixing E [ @S pointwise, then for any p0 ; p1 2 E and for any path ` from p0 to p1 the path '.`/ is from p0 to p1 . Moreover the homotopy class '.`/ 2 …S.p0 ; p1 / depends only on the isotopy class of ' and the homotopy class of `. In this way (the mapping class of) ' induces an automorphism of …S jE , giving a group homomorphism

DNW M.S; E/ ! Aut.…S jE / :

(3.2)

We call it the Dehn–Nielsen homomorphism. We are interested in the case where DN is injective. We say S is of finite type, if S is a compact oriented surface, or a surface obtained from a compact oriented surface by removing finitely many points in the interior. Theorem 3.3. Suppose S is of finite type and any component of S has non-empty boundary, E  @S , and suppose that any connected component of @S contains an element of E. Then the homomorphism DNW M.S; @S / ! Aut.…S jE / is injective. To prove Theorem 3.3, we argue as follows. Let ' 2 M.S; @S / and suppose that DN.'/ D 1. Take a system of proper arcs in S such that the surface obtained from S by cutting along these arcs is the union of disks and punctured disks. Since DN.'/ D 1, we may assume that ' is the identity on these arcs. Finally we deform ' outside of these arcs to the identity to conclude that ' D 1. For more details, see the proof of [33] §3.1. Let Aut.Q…S jE / be the group of automorphisms of the small category Q…S jE acting on the set of objects as the identity and on the set of morphisms Q-linearly. Furthermore, let Q…S jE be the completion of Q…S jE introduced in §3.1. We

2

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2

introduce the group Aut.Q…S jE / in the same manner as for Q…S jE except that we consider only the automorphisms acting on the set of morphisms continuously. Then we have natural group homomorphisms Aut.…S jE / ! Aut.Q…S jE / and Aut.Q…S jE / ! Aut.Q…S jE /. By post-composing them to DN, we get a group homomorphism DNW M.S; E/ ! Aut.Q…S jE /

2 b

2

which we call the completed Dehn–Nielsen homomorphism. For technical reasons and topological considerations, we introduce a subgroup of Aut.Q…S jE / in which the homomorphism DN takes its values.

2

b

2

Definition 3.4. Define the group A.S; E/ as the subgroup of Aut.Q…S jE / consisting of automorphisms U satisfying the following conditions. (1) If  2 …S.p0 ; p1 / is represented by a path included in E, then U. / D  .

1 1 1 1 bQ b U /W Q …S.p ; p / ! Q …S .p ; p /˝ …S .p ; p / (3) We have U D .U ˝ (2) We have aug ı U D augW Q…S.p0 ; p1 / ! Q for any p0 ; p1 2 E. 0

for any p0 ; p1 2 E.

1

0

1

0

1

2

By (3), any U 2 A.S; E/ preserves the group-like elements of Q…S jE . For any ' 2 M.S; E/, the element DN.'/ satisfies the three above conditions. Note that DN.'/ satisfies (1) since ' fixes E [ @S pointwise. Thus we can write

b

b

b

DNW M.S; E/ ! A.S; E/ :

(3.3)

If S and E satisfy the assumption of Theorem 3.3, the fundamental group of each component of S is a finitely generated free group. Then for any p 2 E the natural T n map 1 .S; p/ ! Q1 .S; p/ is injective, since 1 nD1 I 1 .S; p/ D 0 (see [46]). It follows that for any p0 ; p1 2 E, the natural map …S.p0 ; p1 / ! Q…S .p0 ; p1 / is also injective.

4

1

Corollary 3.5. If S and E satisfy the assumption of Theorem 3.3, the completed Dehn–Nielsen homomorphism (3.3) is injective.

3.5 Cut and paste arguments

b

Notice that the construction of QG and QG for a groupoid G is functorial. If S is a subsurface of an oriented surface S 0 , and E  S and E 0  S 0 are closed subsets as in §3.3 such that E  E 0 , then the inclusion map S ,! S 0 induces a groupoid homomorphism from …S jE to …S 0 jE 0 , called the inclusion homomorphism. In this

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subsection we study certain kinds of cut and paste arguments associated to the inclusion homomorphism. First we show the easier half of the van Kampen theorem for …S jE . Let S and E be as in §3.3, and let S1 and S2 be closed subsurfaces of S such that S1 [ S2 D S and S1 \ S2 is a disjoint union of finitely many simple closed curves on S . We further assume that for i D 1; 2, the set Ei WD Si \ E is a disjoint union of finitely many simple closed curves and finitely many points, and that any connected component of S1 \ S2 has an element of E. We denote Ci WD …Si jEi , i D 1; 2. We claim that …S jE is “generated by C1 and C2 .” To formulate this claim we prepare some notation. For p0 ; p1 2 E, we denote by E.p0 ; p1 / the set of finite sequences of points in E,  D .q0 ; q1 ; : : : ; qn / 2 E nC1 , n  0, such that (1) We have q0 D p0 and qn D p1 , (2) For 1  j  n, either fqj 1 ; qj g  S1 or fqj 1 ; qj g  S2 . Further let E.p0 ; p1 / be the set of pairs .; /,  D .q0 ; q1 ; : : : ; qn / 2 E.p0 ; p1 /,  D .1 ; : : : ; n / 2 f1; 2gn such that fqj 1 ; qj g  Sj for any 1  j  n. Nn For .; / 2 E.p0 ; p1 /, we set QC.; / WD j D1 QCj .qj 1 ; qj /. Then the multiplication map QC.; / ! Q…S.p0 ; p1 / is defined. Proposition 3.6 (the easier half of the van Kampen theorem, [33]). We keep the above notation. For any p0 ; p1 2 E the multiplication map O QC.; / ! Q…S.p0 ; p1 / .;/2E.p0 ;p1 /

is surjective. We next consider the forgetful homomorphisms. Let S and E be as in §3.3, and we assume that S is of finite type and has non-empty boundary. If C is a simple closed curve on Int.S / n E, we can consider the forgetful homomorphism M.S; E [ C / ! M.S; E/. Its kernel is generated by pushing maps along simple closed curves on Int.S / n .E [ C / parallel to C . This can be proved by a standard argument found in e.g., [15] §3.6. We shall give a corresponding result for A.S; E/. Let Ci  Int.S / n E, 1 Si  n, be disjoint simple closed curves not nullhomotopic in S . Set E1 WD niD1 Ci . The inclusion homomorphism …S jE ! …S jE [E1 naturally induces the forgetful homomorphism W A.S; E [ E1 / ! A.S; E/. We study the kernel of . For definiteness, we fix an orientation of each curve Ci , 1  i  n. For 1  i  n and p 2 Ci , we denote by i;p the loop Ci based at p. We can assume that i;p 2 1 .S; p/. Then for a rational number a 2 Q, we can define ai;p WD exp.a log i;p / 2 Q1 .S; p/ :

4

Proposition 3.7 ([33]). We keep the above notation. Let U 2 A.S; E [ E1 / and suppose .U / D 1 2 A.S; E/. Then there exist rational numbers ai D aiU 2 Q,

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1

1  i  n, such that for any p0 ; p1 2 E [ E1 and v 2 Q…S .p0 ; p1 /, we have 8 v; ˆ ˆ ˆ ˆ a0 ˆ ˆ ˆi0i;p v; < 0  a 1 Uv D i1 ˆ ; ˆ ˆv i1 ;p1 ˆ ˆ   ˆ ˆ :ai0 v ai1 1 ; i0 ;p0 i1 ;p1

if p0 ; p1 2 E, if p0 2 Ci0 , p1 2 E, if p0 2 E, p1 2 Ci1 , if p0 2 Ci0 , p1 2 Ci1 .

Morally, this proposition says that the kernel of  is generated by “rational pushing maps” along Ci .

4 Operations on curves on surfaces Let S be an oriented surface. In this section we consider several operations on (homotopy classes of) curves on S . Here a curve on S means a loop or a path on S . These operations are first defined for curves in general position, then shown to be homotopy invariant. A quite natural and important property of these operations is that they are equivariant with respect to the action of the mapping class group. In later sections we will see applications of this fact. A rather technical but worth mentioning fact is that these operations are compatible with filtrations on the Q-vector spaces based on curves coming from the augmentation ideal of Q1 .S / (see §3). This point will be explained in §4.5. We say that a curve on S is generic if it is an immersion and its self-intersections consist of transverse double points. Likewise, we say that finitely many curves on S are in general position if each of the curves is generic and their intersections consist of transverse double points. We often identify a generic curve, which is a map into S , with its image, which is a subset of S . For simplicity, we will work over the rationals Q. However, all the constructions in this section works well over the integers Z as well as over any commutative ring with unit.

4.1 Goldman–Turaev Lie bialgebra Let .S O / D ŒS 1 ; S be the homotopy set of oriented free loops on S . For p 2 S we O / the map obtained by forgetting the base denote by j jW 1 .S / D 1 .S; p/ ! .S point of a based loop. If S is connected, j j is surjective. Let Q.S O / be the Q-vector space with basis the set .S O /. The map j j extends Q-linearly to j jW Q1 .S / ! Q.S O /.

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(!1) birth-death of monogons !

(!2) birth-death of bigons !

(!3) jumping over a double point !

Figure 1. The three moves.

Let us recall the definition of the Goldman bracket. We use the intersection of two generic oriented loops on S . Let ˛ and ˇ be oriented loops on S in general position. For each p 2 ˛ \ ˇ, let ".pI ˛; ˇ/ 2 f˙1g be the local intersection number of ˛ and ˇ at p. Also let ˛p be the loop ˛ based at p and define ˇp similarly. Then the conjunction ˛p ˇp 2 1 .S; p/, and j˛p ˇp j 2 .S O / are defined. The Goldman bracket [17] of ˛ and ˇ is X Œ˛; ˇ WD ".pI ˛; ˇ/j˛p ˇp j 2 Q.S O /: (4.1) p2˛\ˇ

Theorem 4.1 (Goldman [17]). The Goldman bracket (4.1) induces a Lie bracket Œ ; W Q.S O / ˝ Q.S O / ! Q.S O /. We call Q.S O / the Goldman Lie algebra of S . Goldman introduced this Lie algebra along the study of the Poisson bracket of two trace functions on the moduli space of flat G-bundles Hom.1 .S /; G/=G, where G is a Lie group satisfying very general conditions. The proof of Theorem 4.1 goes as follows. (1) To prove that the Goldman bracket is well-defined, it suffices to check that Œ˛; ˇ is unchanged under the three local moves in Figure 1. For, every pair of free loops on S is homotopic to a generic pair of free loops, and if two generic pairs of free loops on S are homotopic to each other, then they are related by a sequence of such moves. For another proof using twisted homology, see [31] Proposition 3.4.3.

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(2) To prove that the Goldman bracket is a Lie bracket, one needs to check that it is skew-symmetric and satisfies the Jacobi identity. The skew-symmetry is clear from (4.1) since j˛p ˇp j D jˇp ˛p j and ".pI ˛; ˇ/ D ".pI ˇ; ˛/ for p 2 ˛\ˇ. To prove the Jacobi identity, take three free loops ˛, ˇ,  in general position. Then one can directly check that Œ˛; Œˇ;  C Œˇ; Œ; ˛ C Œ; Œ˛; ˇ D 0, using (4.1). In this section we will see statements similar to Theorem 4.1, e.g., Theorems`4.2, 4.3, 4.5, and 4.8. They can be proved by the above method. Note that ifL S D  S is O  / as the decomposition of S into connected components, then Q.S O / D  Q.S Lie algebras. Next let us recall the definition of the Turaev cobracket. We use the self-intersection of a generic oriented loop on S . We assume that S is connected, in view of a direct sum decomposition given in the last sentence of the preceding paragraph. Let 1 2 .S O / be the class of a constant loop. The Q-linear subspace Q1 is an ideal of Q.S O /. We denote by QO 0 .S / the quotient Lie algebra Q.S O /=Q1, and we let $ W Q.S O / ! QO 0 .S / be the projection. We write j j0 WD $ ı j jW Q1 .S / ! QO 0 .S /. Let ˛W S 1 ! S be a generic oriented loop. Set D D D˛ WD f.t1 ; t2 / 2 1 S  S 1 I t1 ¤ t2 ; ˛.t1 / D ˛.t2 /g. For .t1 ; t2 / 2 D, let ˛t1 t2 (resp. ˛t2 t1 ) be the restriction of ˛ to the interval Œt1 ; t2 (resp. Œt2 ; t1 )  S 1 (they are indeed loops since ˛.t1 / D ˛.t2 /). Also, let ".˛.t P 1 /; ˛.t P 2 // 2 f˙1g be the local intersection number of the velocity vectors ˛.t P i / 2 T˛.ti / S , i D 1; 2. The Turaev cobracket [79] of ˛ is ı.˛/ WD

X

".˛.t P 1 /; ˛.t P 2 //j˛t1t2 j0 ˝ j˛t2 t1 j0 2 QO 0 .S / ˝ QO 0 .S / :

(4.2)

.t1 ;t2 /2D

Theorem 4.2 (Turaev [79], the involutivity is due to Chas [6]). The Turaev cobracket (4.2) induces a Lie cobracket ıW QO 0 .S / ! QO 0 .S / ˝ QO 0 .S /. Moreover, the Qvector space QO 0 .S / is an involutive Lie bialgebra with respect to the Goldman bracket and the Turaev cobracket. To be more precise we have the following. (1) The space QO 0 .S / is a Lie algebra with respect to the Goldman bracket. (2) The space QO 0 .S / is a Lie coalgebra with respect to the Turaev cobracket. (3) We have ıŒu; v D  .u/.ıv/ .v/.ıu/ for any u; v 2 QO 0 .S /. Here  .u/.v˝ w/ D Œu; v ˝ w C v ˝ Œu; w for u; v; w 2 QO 0 .S /. (4) We have Œ ; ı ı D 0W QO 0 .S / ! QO 0 .S /. The condition (3) is called the compatibility, and (4) is called the involutivity. We call QO 0 .S / the Goldman–Turaev Lie bialgebra. Turaev introduced this Lie bialgebra along the study of a skein quantization of Poisson algebras of loops on surfaces, and he showed that some skein bialgebra of links in S  Œ0; 1 quantizes the Goldman– Turaev Lie bialgebra ([79] Theorem 10.1).

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4.2 The action of free loops on based paths We introduce an operation denoted by  , using the intersection of an oriented loop and a based path in general position. Let S and E be as in §3.3. Put S  D S n.En@S /. Note that S  D S if E  @S . We show that the Goldman Lie algebra Q.S O  / acts on Q…S jE by derivations. Take two points 0 ; 1 2 E which are not necessarily distinct. Let ˛ be an oriented loop on S  and ˇW Œ0; 1 ! S a path from 0 to 1 , and assume that they are in general position. For p 2 ˛ \ ˇ, let ".pI ˛; ˇ/ be the local intersection number as before. Also let ˇ0 p be the path from 0 to p traversing ˇ, and define ˇp1 similarly. Then the conjunction ˇ0 p ˛p ˇp1 2 …S.0 ; 1 / is defined. Set X ".pI ˛; ˇ/ˇ0 p ˛p ˇp1 2 Q…S.0 ; 1 / : (4.3)  .˛ ˝ ˇ/ WD p2˛\ˇ

Theorem 4.3 ([31]). The formula (4.3) induces a Q-linear map  W Q.S O / ˝ Q…S.0 ; 1 / ! Q…S.0 ; 1 /. Moreover, with respect to  and the Goldman bracket, the vector space Q…S.0 ; 1 / is a left Q.S O  /-module. Recall that 1 2 .S O  / denotes the class of a constant loop. We have  .1 ˝ v/ D 0 for any v 2 Q…S.0 ; 1 /. Thus  naturally induces a map QO 0 .S  / ˝ Q…S.0 ; 1 / ! Q…S.0 ; 1 /, which we denote by the same letter  . For u 2 Q.S O  / and m 2 Q…S.0 ; 1 / we often write  .u/m or um for short instead of  .u ˝ m/. That Q…S.0 ; 1 / is a left Q.S O  /-module means that Œu; v m D u.vm/  v.um/ for u; v 2 Q.S O  / and m 2 Q…S.0 ; 1 /. If we consider not only a single pair .0 ; 1 / but also all the ordered pairs of elements of E, we obtain a derivation of Q…S jE . First notice that the operation  satisfies the Leibniz rule in the following sense. For any 0 ; 1 ; 2 2 E and ˛ 2 Q.S O  /, ˇ1 2 Q…S.0 ; 1 /, ˇ2 2 Q…S.1 ; 2 /, we have  .˛/.ˇ1ˇ2 / D . .˛/ˇ1/ˇ2 C ˇ1 . .˛/ˇ2/ :

(4.4)

This shows that for any ˛ 2 Q.S O  / the collection  .˛/ D  .˛/0 ;1 , 0 ; 1 2 E, determines a derivation of Q…S jE in the sense of §3.2. Thus we get a Q-linear map  W Q.S O  / ! Der.Q…S jE / ;

(4.5)

and by the second sentence of Theorem 4.3, this is a Lie algebra homomorphism. As a special case, if E D fg is a singleton with  2 @S , the group ring Q1 .S; / is a Q.S O /-module and we have a Lie algebra homomorphism  W Q.S O / ! Der.Q1 .S; //. Note that for any u 2 Q.S O / and v 2 Q1 .S; / we have Œu; jvj D j .u ˝ v/j :

(4.6)

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4.3 Intersection of based paths Take points 1 ; 2 ; 3 ; 4 on the boundary of S . We define a Q-linear map W Q…S.1 ; 2 / ˝ Q…S.3 ; 4 / ! Q…S.1 ; 4 / ˝ Q…S.3 ; 2 / ; using the intersection of two based paths in general position. Then we show that this is closely related to an operation called the homotopy intersection form by Massuyeau and Turaev [49]. First we discuss the most generic case. Namely, we assume f1 ; 2 g \ f3 ; 4 g D ;. Let xW Œ0; 1 ! S be a path from 1 to 2 and yW Œ0; 1 ! S a path from 3 to 4 , and assume that they are in general position. Set X ".pI x; y/.x1 p yp4 / ˝ .y3 p xp2 / (4.7) .x; y/W D  p2x\y

2 Q…S.1 ; 4 / ˝ Q…S.3 ; 2 /: Here x1 p is the path from 1 to p traversing x, etc. One can show that (4.7) gives rise to a well-defined Q-linear map W Q…S.1 ; 2 / ˝ Q…S.3 ; 4 / ! Q…S.1 ; 4 / ˝ Q…S.3 ; 2 / : The operation  is introduced in [34]. It satisfies the following product formula. Lemma 4.4. (1) Let 1 ; 2 ; 02 ; 3 ; 4 be points on the boundary of S such that f1 ; 2 ; 02 g \ f3 ; 4 g D ;. Then for any u 2 Q…S.1 ; 2 /, v 2 Q…S.2 ; 02 / and w 2 Q…S.3 ; 4 /, we have .uv; w/ D .u; w/.1 ˝ v/ C .u ˝ 1/.v; w/ :

(4.8)

(2) Let 1 ; 2 ; 3 ; 4 ; 04 be points on the boundary of S such that f1 ; 2 g \ f3 ; 4 ; 04 g D ;. Then for any u 2 Q…S.1 ; 2 /, v 2 Q…S.3 ; 4 / and w 2 Q…S.4 ; 04 /, we have .u; vw/ D .u; v/.w ˝ 1/ C .1 ˝ v/.u; w/ :

(4.9)

Here, .u; w/.1 ˝ v/ is the image of .u; w/ ˝ v by the map Q…S.1 ; 4 / ˝ Q…S.3 ; 2 /˝Q…S.2 ; 02 / ! Q…S.1 ; 4 /˝Q…S.3 ; 02 /, a˝b˝c 7! a˝bc, etc. Next we consider the degenerate case. Let 1 ; 2 ; 3 ; 4 2 @S be points on the boundary of S and assume f1 ; 2 g \ f3 ; 4 g ¤ ;. To define  for this case, we move the points 1 ; 2 slightly along the negatively oriented boundary of S to achieve f1 ; 2 g \ f3 ; 4 g D ;, then apply the formula (4.7). For more precise explanation

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S @S

Figure 2. The base points  and .

we use an example, which is the most extreme. Namely, let us consider the case 1 D 2 D 3 D 4 . Take a base point  2 @S and pick an orientation-preserving embedding W Œ0; 1 ! @S such that .1/ D . Set .0/ D . See Figure 2. Then we have three isomorphisms 1 .S; / Š 1 .S; /, x 7! x, 1 .S; / Š …S.; /, x 7! x, and 1 .S; / Š …S.; /, x 7! x. Now we define W Q1 .S; / ˝ Q1 .S; / ! Q1 .S; / ˝ Q1 .S; /

(4.10)

so that the diagram 

Q1 .S; / ˝ Q1 .S; / ! Q1 .S; / ˝ Q1 .S; / ? ? ? ? Šy Šy 

Q1 .S; / ˝ Q1 .S; / ! Q…S.; / ˝ Q…S.; / commutes. Here the vertical maps are via the above isomorphisms, and the bottom horizontal arrow is the map already defined. To write down  in (4.10) explicitly, let ˛ be a loop based at , and ˇ a loop based at  and assume that they are in general position. By the isomorphism 1 .S; / Š 1 .S; / given by , we regard that ˛ represents an element of 1 .S; /. Then X .˛; ˇ/ WD  ".pI ˛; ˇ/.˛p ˇp / ˝ .ˇp ˛p / : (4.11) p2˛\ˇ

Note this  satisfies (4.8) (4.9) for any u; v; w 2 Q1 .S; /. By a similar way for any four points 1 ; 2 ; 3 ; 4 2 @S , which are not necessarily distinct, we can define the operation . Since we use only the most extreme case (4.10), we omit the details of the construction. Post-composing 1 ˝ augW Q1 .S; / ˝ Q1 .S; / ! Q1 .S; / ˝ Q Š Q1 .S; / to (4.10), we obtain a Q-linear map W Q1 .S; / ˝ Q1 .S; / ! Q1 .S; / : By (4.11), an explicit formula for  is given by X ".pI ˛; ˇ/˛p ˇp 2 Q1 .S; / ; .˛; ˇ/ WD p2˛\ˇ

(4.12)

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where the notation is the same as in the preceding paragraph. The map  is introduced by Massuyeau and Turaev [49], and is called the homotopy intersection form. It is actually a modification of the operation W Q1 .S; /  Q1 .S; / ! Q1 .S; / introduced by Papakyriakopoulos [66] and Turaev [78] independently. The relationship between  and  is given by .˛; ˇ/ D .˛; ˇ/ˇ 1 for ˛; ˇ 2 1 .S; /. By (4.8) (4.9), we have the following, which is essentially due to Papakyriakopoulos [66] and Turaev [78]. Proposition 4.5 ([49]). The homotopy intersection form satisfies the following identities: .˛1 ˛2 ; ˇ/ D .˛1 ; ˇ/aug.˛2 / C ˛1 .˛2 ; ˇ/; .˛; ˇ1 ˇ2 / D .˛; ˇ1 /ˇ2 C aug.ˇ1 /.˛; ˇ2/ ;

(4.13)

where ˛; ˛1 ; ˛2 ; ˇ; ˇ1 ; ˇ2 2 Q1 .S; /. Here augW Q1.S; / ! Q is the augmentation map. In [49], a bilinear pairing on the group ring satisfying (4.13) is called a Fox pairing. Given a Fox pairing one can consider its derived form. Actually the derived form of  turns out to be  . Let u; v 2 Q1 .S; /. The elementP v is uniquely written as v D P v 1 c x where c 2 Q. We use the notation u D ux. We define x x2 x x2 cx x  a Q-linear map  W Q1 .S; / ˝ Q1 .S; / ! Q1 .S; / by setting   .x ˝ y/ D y.x .x;y/ / for x; y 2 1 .S; / and extending Q-linearly to Q1 .S; / ˝ Q1 .S; /. In [49],   is called the derived form of . From (4.13), we have   .u; vw/ D   .u; v/w C v  .u; w/;   .uv; w/ D   .vu; w/

(4.14)

for u; v; w 2 Q1 .S; /. Lemma 4.6 (Massuyeau–Turaev [49]). The composition of j j ˝ 1W Q1 .S; / ˝ Q1 .S; / ! Q.S O / ˝ Q1 .S; / and  W Q.S O / ˝ Q1 .S; / ! Q1 .S; / coincides with the map   . We end this subsection by a remark that one can recover  from . Proposition 4.7. Let  be the map in (4.10). We have  D .1 ˝ m/.1 ˝ 1 ˝ m/P2431 .1 ˝ ..1 ˝ // ˝ 1/. ˝ / : Here, 1 is the identity map, , , and m are the coproduct, the antipode, and the product of the group ring Q1 .S; /, and P2431 W Q1 .S; /˝4 ! Q1 .S; /˝4 is the Q-linear map given by P2431 .x1 ˝ x2 ˝ x3 ˝ x4 / D x2 ˝ x4 ˝ x3 ˝ x1 .

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4.4 Self intersections Take two points 0 ; 1 on the boundary of S . We define a Q-linear map W Q…S.0 ; 1 / ! Q…S.0 ; 1 / ˝ QO 0 .S / ; using the self intersections of a generic path from 0 to 1 . Then we mention a certain product formula for  and a relationship with the Turaev cobracket. First we consider the general case 0 ¤ 1 . Let  W Œ0; 1 ! S be a generic path from 0 to 1 . We denote by   S the set of double points of  . For p 2 , we p p p p p p use the notation  1 .p/ D ft1 ; t2 g, so that t1 < t2 . Let "..t P 1 /; .t P 2 // be the local intersection number as in §4.1. We also define 0t p to be the restriction of  to the 1 interval Œ0; t1p , and define t p 1 and t p t p similarly. Set 2 1 2 X p p . / WD  "..t P 1 /; .t P 2 //.0t p t p 1 / ˝ jt p t p j0 2 Q…S.0 ; 1 / ˝ QO 0 .S / : 1

2

1 2

p2

(4.15) One can show that this gives rise to a well-defined Q-linear map W Q…S.0 ; 1 / ! Q…S.0 ; 1 /˝QO 0 .S /. Next we consider the case 0 D 1 . Let  2 @S , W Œ0; 1 ! @S , and  D .0/ be as in §4.3. Then we have an isomorphism W Q1.S; / D Q…S.; / Š Q…S.; /, u 7! u. We define W Q1 .S; / ! Q1 .S; / ˝ QO 0 .S / so that the diagram 

Q1 .S; / ! Q1 .S; / ˝ QO 0 .S / ? ? ? ? y ˝1y 

Q…S.; / ! Q…S.; / ˝ QO 0 .S / commutes. Theorem 4.8 ([34]). The Q-vector space Q…S.0 ; 1 / is an involutive right QO 0 .S /bimodule with respect to  and . To be more precise we have the following. (1) The space Q…S.0 ; 1 / is a left QO 0 .S /-module with respect to  (see Theorem 4.3). (2) The space Q…S.0 ; 1 / is a right QO 0 .S /-comodule with respect to . That is, the diagram Q…S.0 ; 1 / ? ? y



!

Q…S.0 ; 1 / ˝ QO 0 .S / ? ? 1˝ı y

.1˝.1T //.˝1/

Q…S.0 ; 1 / ˝ QO 0 .S / ! Q…S.0 ; 1 / ˝ QO 0 .S / ˝ QO 0 .S / commutes. Here ı is the Turaev cobracket and T W QO 0 .S / ˝ QO 0 .S / ! QO 0 .S / ˝ QO 0 .S /, u ˝ v 7! v ˝ u is the switch map.

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(3) The operations  and  satisfy a compatibility property in the sense that  .u/.m/  . .u/m/  . ˝ 1/.1 ˝ ı/.m ˝ u/ D 0 for u 2 QO 0 .S /, m 2 Q…S.0 ; 1 /. Here W Q…S.0 ; 1 / ˝ QO 0 .S / ! Q…S.0 ; 1 / is given by .m ˝ u/ D  .u ˝ m/, and  .u/.m/ D . ˝ 1/.u ˝ .m// C .1 ˝ ad.u//.m/. (4) The operations  and  satisfy the following involutivity condition:  D 0W Q…S.0 ; 1 / ! Q…S.0 ; 1 / : The operation  is introduced in [34], and inspired by Turaev’s self intersection  D T W 1 .S; / ! Z1 .S; / in [78] §1.4. Indeed, for any  2 1 .S; / we have T . / D .1 ˝ "/. /, where "W QO 0 .S / ! Q is the Q-linear map given by ".$ .˛// D 1 for ˛ 2 .S O / n f1g. We end this subsection by stating two results about . The first one is a certain product formula, and the second one is a relation with the Turaev cobracket. Lemma 4.9. For any 1 ; 2 ; 3 2 @S and u 2 Q…S.1 ; 2 /, v 2 Q…S.2 ; 3 /, we have .uv/ D .u/.v ˝ 1/ C .u ˝ 1/.v/ C .1 ˝ j j0 /.u; v/ : Here .u/.v ˝ 1/ is the image of .u/ ˝ v by the map Q1 .S; / ˝ QO 0 .S / ˝ Q1 .S; / ! Q1 .S; / ˝ QO 0 .S /, a ˝ b ˝ c 7! ac ˝ b, etc. As a corollary, for any n  2 and 0 ; : : : ; n 2 @S , ui 2 Q…S.i 1 ; i /, 1  i  n, we have .u1 un / D

n X

..u1 ui 1 / ˝ 1/.ui /..ui C1 un / ˝ 1/

i D1

C

X

..u1 ui 1 / ˝ 1/Ki;j ..uj C1 un / ˝ 1/ ;

i 0, define the thick part .nHn /" D fx 2 nHn j inj.x/  "g ; where inj.x/ is the injectivity radius of nHn at x. When " is sufficiently small, .nHn /" is a real analytic compact submanifold with boundary and is diffeomorphic BS

to nHn . In general, for an arithmetic locally symmetric space  0 nX of higher rank, there is also a truncated subspace . 0 nX /T [61] which is a real analytic compact submanifold BS

and is diffeomorphic to  0 nX , where T is a sufficiently large truncation parameter. For example, when  0 nX is a product 1 nH2 2 nH2 of two noncompact hyperbolic surfaces of finite areas,     . 0 nX /T D 1 nH2 "  2 nH2 " ; 1

2

and the corner structure of the truncated subspace . 0 nX /T is clearly visible. Let nX be an Anosov locally symmetric space. For every small " > 0, define its “non-fat” part,

 1 .nX / 1 D x 2 nX j x 2 inj.x/  : " " One question is whether for every sufficiently small ", .nX / 1 is a compact " subspace and is a deformation retract of nX . This is certainly true when the rank of X is equal to 1, i.e., when  is a convex cocompact subgroup acting on a symmetric space of rank 1. It seems to be true too in general, at least when  is Anosov with respect to a minimal parabolic subgroup of G. Remark 5.30. Close connections between anti-Siegel sets and the maximal Satake compactifications in Propositions 5.24, 5.16 and 5.17 give another indication, besides those to be discussed in the next section, that maximal Satake compactifications S

nX max of Anosov locally symmetric spaces nX are special among all possible compactifications, even among those which are manifolds with corners.

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6 Compactifications and spectral theory of locally symmetric spaces S

As we explained in §2.3, the maximal Satake compactification X max , rather than the geodesic compactification X [ X.1/, can be used to understand joint eigenfunctions of the algebra of invariant differential operators D.X / on X . In this section, we briefly explain how boundaries or corners of maximal Satake compactifications of X and nX are related to their spectral analysis, or the spectral resolution of their Laplace operator . When the rank of X is equal to 1, D.X / is generated by the Laplace operator  of the invariant Riemannian metric on X . In the category of Riemannian manifolds, the Laplace operator is essentially the only natural differential operator acting on functions. For higher rank symmetric spaces, considering all invariant differential operators gives a more refined decomposition of function spaces on X such as L2 .X / (see Remark 6.1). For simplicity, we concentrate on the Laplace operator . Let .M; g/ be a Riemannian manifold, and L2 .M / be the space of square-integrable measurable functions on M with respect to the Riemannian measure. Let  be the Laplace operator associated with the Riemannian metric g on M . Then  acts on C01 .M /, the space of smooth functions on M with compact support, and  is a symmetric differential operator: for f; g 2 C01 .M /, hf; gi D hf; gi : To study the spectral theory of , we need to extend it to a self-adjoint operator on the Hilbert space L2 .M /. It is a basic fact in geometric analysis that when .M; g/ is a complete Riemannian manifold, its Laplace operator  has a unique self-adjoint extension to L2 .M /. The extension is still denoted by . A nonzero function f 2 L2 .M / is called an eigenfunction of eigenvalue  if f belongs to the domain of the extended Laplace operator, and f D f :

(6.1)

By the regularity of elliptic equations, f 2 C 1 .M /. It is also common to call any function f satisfying Equation 6.1 an eigenfunction whether it is square-integrable or not. To distinguish between these two cases, f is called a square-integrable eigenfunction in the former case. It is also a basic fact from the elliptic theory of differential equations that when .M; g/ is a compact Riemannian manifold, there exists an orthonormal basis '1 ; '2 ; : : :, of L2 .M / consisting of eigenfunctions, 'j D j 'j , whose eigenvalues j form a discrete sequence going to C1. Note that every function f 2 L2 .M / can be written as 1 X f D aj 'j ; aj D hf; 'i : (6.2) j D1

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291

When f belongs to the domain of the extended Laplace operator , we have f D

1 X

j aj 'j :

j D1

Therefore, in terms of this basis f'j g, the Laplace operator  becomes a multiplication operator defined by the sequence of eigenvalues j . This gives a concrete spectral resolution of . If M is noncompact, considering square-integrable eigenfunctions alone is not enough to obtain a spectral resolution of  in general. Take the example of M D Rn with the standard Euclidean metric. For every y 2 Rn , we have an exponential function e2i xy . It satisfies the eigenequation e2i xy D 4 2 jyj2 e2i xy ; but e2i xy is not a square-integrable function in x. Note that the Laplace operator is normalized to be nonnegative here,   @ @ C:::C 2 : D @xn @x12 On the other hand, the Fourier transform allows us to write f .x/ 2 L2 .Rn / as an integral of such exponential functions: Z (6.3) f .x/ D fO.y/e2i xy d y ; Rn

where fO.y/ is the Fourier transform of f , Z O f .x/e2i xyd x : f .y/ D Rn

In terms of the Fourier decomposition of f in Equation 6.3, the Laplace operator  becomes a multiplication operator by the function 4 2 jyj2 : Z f .x/ D 4 2 jyj2 fO.y/e2i xy d y : Rn

Therefore, the Fourier decomposition gives a spectral resolution of . Though the functions e2i xy are not square-integrable eigenfunctions, they play a role similar to that of square-integrable eigenfunctions 'j in Equation 6.2 in decomposing L2 .Rn /. Note that the spectrum of the Laplace operator  of Rn is equal to Œ0; C1/, a continuous interval, unlike the case of a compact Riemannian manifold where the spectrum of  consists of a sequence of discrete points going to infinity. It can be made rigorous that the spectrum (or rather the spectrum measure) of  of Rn is absolutely continuous.

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For a general Riemannian manifold .M; g/, the spectrum of  is defined to be the subset of R: Spec.M / D f 2 R j .  /1 does not exist as a bounded operator on L2 .M /g : By the convention here,   0 and hence Spec.M /  Œ0; C1/. For this self-adjoint operator , there is a spectral measure, and Spec.M / is the support of the spectral measure and is decomposed according to the spectral measure into two parts: discrete and continuous, Spec.M / D Specdis .M / [ Speccon .M / : This decomposition corresponds to a decomposition of the function space L2 .M / D L2dis .M / C L2con .M / :

(6.4)

For Riemannian manifolds whose geometry at infinity is modeled on some special spaces, the continuous part of the spectral measure can often be absolutely continuous, and the above decomposition of function spaces in Equation 6.4 can be described more explicitly in the following way: There exists a sequence, which could be finite, of orthonormal square-integrable eigenfunctions '1 ; '2 ; : : :, 'j D j 'j , and a family of functions E.x; b/, where b belongs to a parameter space B and E.x; b/ is a non-square-integrable function and satisfies an eigenequation E.x; b/ D .b/E.x; b/ ; such that for every f 2 L2 .M /, we have the following decomposition Z X f .x/ D a j 'j C a.f; b/E.x; b/d.b/ ;

(6.5)

B

j 1

where aj and a.f; b/ are constants depending on f , and .b/ is a measure on the parameter space B. When f belongs to the domain of , Z X j aj 'j C .b/a.f; b/E.x; b/d.b/ ; f D j 1

B

i.e.,  is resolved spectrally by the eigenfunction 'j and the functions E.x; b/. The eigenfunctions 'j span L2d i s .M /, while integrals (or superpositions) of E.x; b/ produce functions in L2con .M /. In some sense, the above decomposition in Equation 6.5 is a combination of the spectral decompositions in Equations 6.2 and 6.3. Because of the above consideration, for a complete noncompact Riemannian manifold .M; g/ which admits a spectral decomposition like Equation 6.5, we roughly define its generalized eigenfunctions to be those functions f such that (1) f satisfies the eigenequation f D f , for some  2 Speccon .M /, (2) f is not square-integrable but appears as a multiple of some E.x; b/ in the spectral resolution of the Laplace operator  in Equation 6.5.

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7 Compactifications and reduction theory

One of the basic points of this section is that for symmetric spaces X and some locally symmetric spaces nX which admit natural compactifications, a spectral decomposition like Equation 6.5 holds, and the parameter space B for generalized eigenfunctions is often related to their geometric boundaries. The most basic example is Rn , which is a flat symmetric space. Its generalized eigenfunctions can be parametrized as follows: For every point  at the sphere at infinity Rn .1/, which can be identified with the unit sphere S n1 , and every t  0, define E.x; ; t/ D e2t x : Therefore, in the spectral decomposition of the Laplace operator of Rn according to Equation 6.3 or Equation 6.5, the parameter space B for generalized eigenfunctions is Rn .1/  Œ0; C1/. Such a clean and direct relation between the parameter space for generalized eigenfunctions and geometric boundaries also holds for symmetric spaces and some locally symmetric spaces discussed in this chapter. We will take a brief look at them.

6.1 Spectral decomposition of the upper half-plane and symmetric spaces Consider the upper half-plane H2 . The Poincaré metric is given by ds 2 D 2

2

dx 2 Cdy 2 , y2 s

@ @ and the Laplace operator is  D y 2 . @x 2 C @y 2 /. For any s 2 C, the function y satisfies the eigenequation y s D s.1  s/y s :

For y s to be a generalized eigenfunction, s.1  s/ must be a nonnegative number, which implies that either s 2 Œ0; 1 or s D 12 C i r, r 2 R. It turns out that     1 ; C1 ; Spec H2 D 4 1

and is absolutely continuous, and y 2 Ci r , r 2 R, are generalized eigenfunctions. Actually we have more generalized eigenfunctions. For any g 2 SL.2; R/, Im g.z/s defines a function on H2 which also satisfies f D s.1  s/f ; since  commutes with the action of g. Note that y s D Im .g.z//s if and only if g 2 P1 , the parabolic subgroup of upper triangular matrices. It turns out that the 1 functions Im g.z/ 2 Ci r , g 2 SL.2; R/=P1, r 2 R, are all the generalized eigenfunctions of H2 . Note that SL.2; R/=P1 can be identified with the boundary H2 .1/. To understand the above parametrization of generalized eigenfunctions better and to generalize it to other symmetric spaces of noncompact type, we recall that x; y-coordinates on

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H2 correspond to the horospherical decomposition of H2 with respect to the parabolic subgroup P1 . For every parabolic subgroup P of SL.2; R/, we have a horopsherical decomposition H2 Š NP  AP ; z 7! .nP .z/; aP .z// : Let aP be the Lie algebra of AP , and logW AP ! aP the inverse of the exponential map. There is a unique positive root ˛ which is a linear functional on aP and takes positive values on the positive chamber. Then we can define a function   1 e.z; P; r/ D exp C i r ˛.log aP .z// : 2 1

It turns out that when P D P1 , e.z; P; r/ D y 2 Ci r , and when P D gP1 g 1 , 1

e.z; P; r/ D Img.z/ 2 Ci r : For a general symmetric space X D G=K, for every minimal parabolic subgroup P of G, we can also consider the horospherical decomposition X Š NP  AP ;

x ! .nP .x/; aP .x// :

Note that in this case, the boundary symmetric space XP is reduced to a point. Let aP be the Lie algebra of AP , and logW AP ! aP the inverse of the exponential map. Let P be the half sum of all positive roots of the aP -action on the Lie algebra of NP . Then for any  in the real dual space aP of aP , we can define a function on X : e.x; P; / D exp.P C i /.log aP .x// :

(6.6)

Note that when G D SL.2; R/, P D 12 as expected. With the above definitions, it turns out that the following is true for the symmetric space X : (1) e.x; P; / satisfies the eigenequation   e.x; P; / D jj2 C jj2 e.x; P; / : (2) The spectrum is X is equal to Œjj2 ; C1/ and is absolutely continuous. (3) When P ranges over all minimal parabolic subgroups of G and  2 aP , e.x; P; / gives all generalized eigenfunctions of X . Therefore, the maximal Furstenberg boundary G=P (or rather G=P  R) is a parameter space for all the generalized eigenfunctions of X . This is related to the so-called Helgason– Fourier transform. See [34] for details. Remark 6.1. In the above discussion in this section, we only looked at the spectral decomposition of the Laplace operator . For a symmetric space X of higher rank, using all invariant differential operators can refine the decomposition.

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We briefly defined generalized eigenfunctions before. But even for symmetric spaces X , a rigorous definition of generalized eigenspaces is not easy. For any character ƒW D.X / ! C, we can consider the function space Eƒ .X / D ff 2 C 1 .X / j Lf D ƒ.L/f; L 2 D.X /g : This can be considered as a generalization of eigenspaces. If we require f 2 L2 .X /, then f is a square-integrable eigenfuncton, and we obtain a usual eigenspace Eƒ .X /\ L2 .X /. For the symmetric space X , there are no eigenfunctions, i.e., Eƒ .X / \ L2 .X / D 0 ; since the spectrum of X is absolutely continuous as pointed out before. On the other hand, if we impose no bound on f , there are many such functions from the formulation of the Helgason conjecture mentioned above: the Poisson transform of any nonzero hyperfunctions on the maximal Furstenberg boundary G=B of X gives many nonzero functions in Eƒ .X /. The positive solution of the Helgason conjecture in [45] shows that every function in Eƒ .X / is of this form. From the point of view of the spectral decomposition of , most such functions in Eƒ .X / are not needed. This raises a natural question: what are natural conditions we should impose on functions in Eƒ .X / in order to obtain the generalized eigenfunctions and their superpositions. This is a bit subtle. The functions which give generalized eigenfunctions have moderate growth in some sense. See [34] for details.

6.2 Spectral theory of locally symmetric spaces of finite volume If   G is a non-uniform arithmetic subgroup, then the spectral theory of nX is more complicated and interesting. In this case, Spec.nX / has both a nonempty discrete part and continuous part. The former corresponds to square-integrable Maass modular forms and are very subtle to determine, and the generalized eigenfunctions are given by Eisenstein series. For example, constant functions are squareintegrable eigenfunctions of eigenvalue 0. But it is often difficult to obtain other square-integrable eigenfunctions. We cannot get into detail here since it is a well-developed and sophisticated subject, but only mention briefly how Eisenstein series are related to the boundary of nX through the example of  D SL.2; Z/ and X D H2 . In this case, Speccon .SL.2; Z/nH2 / D Œ 14 ; C1/, and it is absolutely continuous. Though the Borel–Serre compactification SL.2; Z/nH2 is useful for applications in topology, it is too large for the spectral theory of the Laplace operator, since the Laplace operator is not of the regular singular type at the boundary points, and we need to blow down the unipotent part NP in the Borel–Serre compactification. Therefore, the right compactification of SL.2; Z/nH2 for the spectral theory and other analysis problems is obtained as follows: In the boundary H2 .1/, consider only the rational boundary points Q [ f1g. The space H2 [ Q [ f1g is given the Satake

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topology, where a basis of each boundary point is given by horoballs of that point. Then SL.2; Z/ acts on H2 [ Q [ f1g with a compact quotient. If   SL.2; Z/ is a subgroup of finite index, then  also acts on H2 [ Q [ f1g with a compact quotient nH2 [ Q [ f1g, which is obtained by adding one point to each end or cusp neighborhood of nH2 . For every rational boundary point, say 1, we have a family of generalized eigenfunctions e.z; P1 ; 12 C i r/, r 2 R. To obtain generalized eigenfunctions on the quotient SL.2; Z/nH2 , we need to average over an SL.2; Z/-orbit of such functions in order to achieve SL.2; Z/-invariance so that they define functions on the quotient. Such functions are the Eisenstein series E.z; P; s/. For example, when P D P1 , X   e z; P  1 ; s : (6.7) E.z; P1 ; s/ D  2SL.2;Z/=P1 \SL.2;Z/

In the above sum, we have used the fact that e.z; P1 ; s/ is invariant under P1 and hence under P1 \ SL.2; Z/, and therefore, we need to remove this redundancy in order to remove the obvious obstruction to convergence of the infinite sum. This gives the right idea to introduce Eisenstein series, but such an infinite sum does not converge in general. In fact, when s D r 2 R, the above series definitely does not converge. It is highly nontrivial to make the above definition of E.z; P; s/ rigorous and to derive a spectral decomposition of L2 .SL.2; Z/nH2 /. This problem was proposed and solved by Selberg in [66], and was the starting point of the celebrated Selberg trace formula. See [3] for a summary of many applications of the trace formula in automorphic representations. One important conclusion of our brief discussion here is that for every subgroup   SL.2; Z/ of finite index, a natural parameter space for generalized eigenfunctions of nH2 is the geometric boundary of the above compactification nH2 [ Q [ f1g. The spectral decomposition and generalized eigenfunctions of general arithmetic locally symmetric spaces nX were achieved and described by Langlands in [49], which was the basis of the celebrated Langlands program. The generalized eigenfunctions are also given by Eisenstein series, which are suitable averages of generalized eigenfunctions e.x; P; / (Equation 6.6) of X as in the special case of H2 above (Equation 6.7). But how to define them properly and to show that they give all the generalized eigenfunctions is extremely difficult. A geometric parametrization of the generalized eigenfunctions of nX in terms BS

of a suitable blow-down of the Borel–Serre compactification nX , the so-called reductive Borel–Serre compactification of nX , was given in [40]. Note that for nH2 , the reductive Borel–Serre compactification is the same as the Baily–Borel compactification. But the reductive Borel–Serre compactification strictly dominates the Baily–Borel compactification for general nX .

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6.3 Spectral theory and geometric scattering theory of convex cocompact quotients of rank one symmetric spaces If X is a rank one symmetric space and  is a non-lattice convex cocompact subgroup, then nX has nonempty continuous spectrum. The generalized eigenfunctions have been described by analogues of Eisenstein series, and their relation with the geometric boundary of nX is more direct than that for locally symmetric spaces of finite volume. We briefly take a look at the case X D H2 . By assumption,  is a convex cocompact subgroup acting on H2 . Let ./ be the domain of discontinuity. Then nH2 [ ./ is the compactification which is a real analytic manifold with boundary (Proposition 2.11). Every point  2 ./ corresponds to a parabolic subgroup P D P . By the above discussion, we have a family of generalized eigenfunctions e.x; P ; s/ associated with P . Consider the -orbit of  in ./ and define X   E.x; P ; s/ D e x; P  1 ; s :  2

If this series converges, then it only depends on the -orbit   and hence only on the image of  in the boundary n ./, which is denoted by Π. It turns out that E.x; Π; s/ can be defined rigorously, and that when Πruns over all points of n ./ and s D r, r 2 R, E.x; Π; r/ gives all the generalized eigenfunctions of n ./. See the book [15] and references there for details. For a general rank one symmetric space X and a convex cocompact subgroup  acting on X , similar results on the spectral decomposition of L2 .nX / and the parametrization of generalized eigenfunctions by the geometric boundary n ./ hold. See [20] and references there.

6.4 Spectral theory and geometric scattering theory of Anosov quotients of higher rank symmetric spaces Suppose that X is a symmetric space of noncompact type of rank at least 2, and  is an Anosov subgroup acting on X . It can be seen that the spectrum of the Laplace operator  of nX contains a nonempty continuous part. The spectral decomposition of L2 .nX / and the generalized eigenfunctions of nX are not completely understood yet. On the other hand, given the above summary of results on the spectral theory for X , arithmetic locally symmetric spaces nX , and convex cocompact groups acting on rank one symmetric spaces, it is reasonable to conjecture that a similar picture holds for Anosov locally symmetric spaces: generalized eigenfunctions of nX are suitable averages of generalized eigenfunctions of X , S

and the corners of the maximal Satake compactifications nX max in Conjecture 2.15 should serve as parameter spaces for the generalized eigenfunctions.

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The structure of real analytic manifolds with corners on the compactifications S

nX max in Conjecture 2.15 provides a natural class of manifolds with corners such that the Laplace operator of the interior Riemannian manifolds nX has regular sinS

gularities along the corners of the compactifications nX max . Therefore, Anosov locally symmetric spaces and their maximal Satake compactifications fit into the geometric scattering theory in the book [54]. In particular, the expected spectral decomposition and parametrization of the generalized eigenfunctions of nX in terms S

of the boundaries of nX max provide some support for Conjecture 7.1 in [54] on the geometric scattering theory of manifolds with corners. We also note that in order to apply methods from microlocal analysis as in [54] to understand the spectral theory S

of nX , it is crucial that the real analytic corner structure on nX max is compatible with the Laplace operator of nX .

6.5 Martin compactification and horofunction compactification In the previous subsection, we discussed the problem of parametrizing generalized eigenfunctions in terms of some geometric boundaries. A closely related problem is to study the cone of positive harmonic functions. As we shall see, this discussion will raise several natural questions about three other compactifications of Anosov locally symmetric spaces. First, we introduce some general concepts. See [31] for more details and references. Let .M; g/ be a noncompact, complete Riemannian manifold. For every  2 R, define C .M / D ff 2 C 1 .M / j f D f; f > 0g :

(6.8)

It is clear that each C .M / is a convex cone in C 1 .M / if it is nonempty. Let 0 .M / be the bottom of the spectrum of M : 0 .M / D inff j  2 Spec.M /g. Then it is known that C .M / ¤ ; if and only if   0 . For example, if M D Rn with the standard Euclidean metric, then 0 .Rn / D 0, since Spec.Rn / D Œ0; C1/. When  D 0, C0 .Rn /pconsists of positive constant functions. When  < 0, C .Rn / contains functions e jjx , where  belongs to the unit sphere S n1 of Rn , or the sphere at infinity Rn .1/. In this case, it can be shown that every extremal function of the cone C .Rn / is a multiple of such a function p jjx e , and the cone C .Rn / is spanned by these exponential functions in a suitable sense. If M D H2 , then 0 .H2 / D 14 , since Spec.H2 / D Œ 14 ; C1/. When  D 0, C .H2 / is the space of positive harmonic functions on H2 . The boundary point 1 2 H2 .1/ gives the positive harmonic function K1 .z/ D y. We note that this function takes value 0 at all boundary points of H2 .1/ except at 1, where it converges to C1. Similarly, for every boundary point  2 H2 .1/, there is also a positive harmonic function K .z/ which can be obtained from K1 .z/ and satisfies a similar boundary behavior: K .z/ takes value 0 at all points of H2 .1/ except at

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. The Poisson integral formula for H2 (or equivalently the unit disc) (see [35]) says that every harmonic function on H2 with a continuous boundary value on H2 .1/ is an integral of these functions K .z/. A generalization of the Poisson integral formula holds for every noncompact, complete Riemannian manifold .M; g/. Briefly, for every   0 .M /, there is a Martin compactification M [ @ M such that: (1) For each boundary point  2 @ M , there is a function K .x; / which belongs to C .M /. This function K .x; / is called the Martin kernel function associated with the boundary point . (2) Every function in C .M / is an integral of these functions K .x; / on the Martin boundary @ M with respect to some nonnegative measure. Every extremal function f .x/ of the cone C .M / is a multiple of some K .x; /, but not every K .x; / is an extremal function. When  D 0, besides clarifying the structure of the cone C .M /, the Martin compactification M [ @ M also allows one to understand asymptotic behavior of paths of Brownian motion on M . It can be shown that for Rn , when  < 0, Rn [ @ Rn Š Rn [ Rn .1/ : Similarly, for H2 , when   14 , H2 [ @ H2 Š H2 [ H2 .1/ : The following result was proved in [31] Theorem 6.2. For any symmetric space X of noncompact type, when  D 0 .X /, S

X [ @ X Š X max I S

and for every  < 0 .X /, X [ @ X is the least common refinement of X max and X [ X.1/, or equivalently, X [ @ X is isomorphic to the closure of X under the diagonal embedding S X ,! X max  X [ X.1/ : In particular, if the rank of X is equal to 1, then for every   0 .X /, S

X [ @ X Š X max Š X [ X.1/ : One natural problem is to identify the Martin compactification of Anosov locally symmetric spaces nX . We note that 0 .nX / > 0, and hence the Martin compactification for  D 0 will contribute to a better understanding of the Brownian motion on nX . We start with the rank 1 case. Proposition 6.3. Assume that  is a convex cocompact subgroup acting on a rank 1 symmetric space X . Then for every  < 0 .nX /, the Martin compactification nX [ @ nX is canonically homeomorphic to the compactification nX [ ./ in Proposition 2.11, where ./  X.1/ is the domain of discontinuity.

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Proof. This follows from the general results in [1, Theorem 8, p. 518]. Basically, the crucial point in identifying the Martin compactification with a given compactification in [1] and [2] is the boundary Harnack principle. Since the sectional curvature X is bounded between two negative constants, by [2, Theorem 5.1] and [1], the boundary Harnack principle holds for every point in the geodesic compactification X [ X.1/. Since every boundary point in the compactification nX [ ./ has small neighborhoods which are the same as neighborhoods of points in X [ X.1/, the boundary Harnack principle also holds for every boundary point of nX [ ./. Therefore the conditions in [1, Theorem 8, p. 518] are satisfied, and the Martin compactification nX [ @ nX is canonically homeomorphic to the compactification nX [ ./.  For a symmetric space X of higher rank, its sectional curvature is equal to 0 at maximal flats. Therefore, the boundary Harnack inequality does not necessarily hold. If we want to generalize the results of [31] in Theorem 6.2 on the Martin compactification of symmetric spaces to Anosov locally symmetric spaces, we need two S compactifications corresponding to X max and X [ X.1/. Maximal Satake compactifications of nX are provided by Conjecture 2.15. Then a natural question is the following one. Question 6.4. What is the correct analogue of the geodesic compactification X [ X.1/ for an Anosov locally symmetric space nX ? One guess is to use rays in nX to compactify nX . Briefly, a ray in nX is an isometric embedding  W Œa; C1/ ! nX , for some a 2 R. We can similarly define an equivalence relation on rays to obtain the set of equivalence classes of rays in nX , denoted by nX.1/. The results of [24] imply that if  is a convex cocompact subgroup acting on a symmetric space X of rank 1, then there is a topology on nX [ nX.1/ such that it is a compactification of nX and is homeomorphic to nX [

./. On the other hand, for the diagonal action of Z D h;  i on H2  H2 discussed in §3, it can be shown that the set of equivalence classes of rays does not have a natural topology so that it becomes a compact space, and hence it cannot be used as the ideal boundary to define a compactification of nX in this case. Briefly, the reason is that for a ray in H2  H2 to be projected to a ray in nH2  H2 , its image in each factor h inH2 must be a ray. One way to overcome this difficulty is to use a related compactification. For any complete Riemannian manifold .M; g/ (or more generally any noncompact proper metric space), there is a horofunction compactification [6]. Briefly, let C 0 .M / be the space of continuous functions on M , and C 0 .M /=const the quotient space by constant functions. The image of any continuous function f in C0 .M /==const is denoted by Œf . Let d.x; y/ be the distance function on M . Then we have an embedding i W M ,! C 0 .M /==const;

y 7! Œd.x; y/ :

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The closure i.M / is compact and called the horofunction compactification of M , hor denoted by M .14 It is known that for any symmetric space X of noncompact type, we have an idenhor tification X Š X [ X.1/ [6]. (In fact, the same result holds for every complete simply connected nonpositively curved Riemannian manifold). If X is of rank 1 and  is a convex cocompact subgroup acting on X , then results of [24] also imply that hor nX Š nX [ nX.1/. In view of the above results, another natural question is the following. Question 6.5. Let nX be an Anosov locally symmetric space of higher rank. How hor is the horofunction boundary @nX related to the set nX.1/ of equivalence classes of rays? The example of the above diagonal action of h;  i on H2  H2 suggests that hor nX.1/ is a proper subset of @nX in general. The above discussions also suggest that when  < 0 .nX /, one candidate for the Martin compactification nX [ @ nX of an Anosov locally symmetric space S

nX is the least common refinement of a maximal Satake compactification nX max hor

and the horofunction compactification nX . Another reason to make such a guess is that we should be able to define Martin kernel functions for boundary points on such a compactification of nX by suitably averaging the Martin kernel functions of X as in the case of Eisenstein series. If this guess is true, then one puzzling thing is that both nX [ @ nX and hor nX are uniquely defined, but nX usually admits several different maximal Satake compactifications as we observed before. Of course, this does not necessarily lead to a contradiction due to stability phenomena in topology. Therefore, there are many questions about the geometry and analysis of Anosov locally symmetric spaces, and interactions between them. A proper understanding of them will enrich these Anosov locally symmetric spaces. Added in Proofs: After this chapter was sent to the printer, Kapovich and Leeb expanded substantially their original paper cited as [41] in this chapter and posted a new version on arxiv on August 24, 2015. In this later version, they proved Conjectures 2.15, 5.25, 5.27 formulated in this chapter. Therefore, results of this chapter, in particular Proposition 5.17, imply that every Anosov subgroup of a semisimple Lie group G admits a coarse fundamental domain in the symmetric space X D G=K which consists of a finite union of anti-Siegel sets and a compact subset. 14 A less intrinsic but probably more computational way is to fix a basepoint x 2 M , and for each point Œf 0 in the quotient space C 0 .M /=const, chose a unique function f in the coset Œf such that f .x0 / D 0. Then hor

we can define the compactification M by using the space Cx00 D ff 2 C 0 .M / j f .x0 / D 0g, and the map y 2 M 7! d.x; y/  d.x0 ; y/ 2 Cx00 .

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Acknowledgments. I would like to thank O. Guichard, M. Kapovich, B. Leeb, A. Wienhard for helpful conversations about Anosov subgroups and compactifications of their associated locally symmetric spaces. I learned Proposition 3.2 from a conversation with M. Kapovich, and Remark 5.26 resulted from a conversation with A. Wienhard. Especially, I would like to thank O. Guichard for explaining the results in the upcoming paper [28], for reading this chapter carefully, and for suggesting several improvements. Finally, I would like to thank Athanase Papadopoulos for carefully reading this chapter.

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[53] G. Margulis, Discrete subgroups of semisimple Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17. Springer-Verlag, Berlin, 1991, x+388. [54] R. Melrose, Geometric scattering theory. Stanford Lectures. Cambridge University Press, Cambridge, 1995, xii+116. [55] D. W. Morris, Introduction to Arithmetic Groups, Deductive Press, 2015. [56] G. Mostow, Strong rigidity of locally symmetric spaces. Annals of Mathematics Studies, No. 78. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973, v+195. [57] Y. Namikawa, Toroidal compactification of Siegel spaces. Lecture Notes in Mathematics, 812. Springer, Berlin, 1980, viii+162. [58] T. Oda, Torus embeddings and applications. Based on joint work with Katsuya Miyake. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 57. Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin-New York, 1978, xi+175. [59] J.-F. Quint, Groupes convexes cocompacts en rang supérieur. Geometriae Dedicata 113 (2005), 1–19. [60] M.S. Raghunathan, Discrete subgroups of Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68. Springer-Verlag, New York-Heidelberg, 1972, ix+227. [61] L. Saper, Tilings and finite energy retractions of locally symmetric spaces. Comment. Math. Helv. 72 (2) (1997), 167–202. [62] L. Saper and M. Stern, L2 -cohomology of arithmetic varieties. Ann. Math. 132 (1) (1990), 1–69. [63] P. Sarnak, Notes on thin matrix groups. Thin groups and superstrong approximation, 343–362, Math. Sci. Res. Inst. Publ., 61, Cambridge Univ. Press, Cambridge, 2014. [64] I. Satake, On compactifications of the quotient spaces for arithmetically defined discontinuous groups. Ann. Math. 72 (3) (1960) 555–580. [65] I. Satake, On representations and compactifications of symmetric Riemannian spaces. Ann. Math. 71 (1) (1960) 77–110. [66] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. [67] C. L. Siegel, Zur Theorie der Modulfunktionen n-ten Grades. Comm. Pure Appl. Math. 8 (1955), 677–681. [68] C. L. Siegel, Zur Reduktionstheorie quadratischer Formen. Publications of the Mathematical Society of Japan, Vol. 5. The Mathematical Society of Japan, Tokyo, 1959, ix+69. [69] R. Zimmer, Ergodic theory and semisimple groups. Monographs in Mathematics, 81. Birkhäuser Verlag, Basel, 1984, x+209.

Chapter 8

Representations of fundamental groups of 2-manifolds Lisa Jeffrey1 Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Representations into a nonabelian Lie group . . . . . . . . . . . . . . 1.4 Moduli spaces of vector bundles of nonzero degree . . . . . . . . . 2 The moment map of the gauge group action on the space of connections on a 2-manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Spaces of flat connections on oriented 2-manifolds . . . . . . . . . . . . . . 3.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Hamiltonian flows on moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction Moduli spaces of flat connections on oriented 2-manifolds have played a central role in geometry over the past thirty years, since the seminal paper of Atiyah and Bott [1] on their topology. When the gauge group is PSL.2; R/ a component of this moduli space is identified with Teichmüller space. They are an easier analogue of the study of the solutions of the Yang–Mills equations on 4-manifolds. Atiyah and Bott constructed these spaces as symplectic quotients. The layout of this chapter is as follows. Section 1.2 treats the Jacobian (which is the simplest example of the class of objects treated throughout this chapter, corresponding to the group U.1/). Section 1.3 treats the case corresponding to compact nonabelian Lie groups (such as SU.n/). Section 1.4 treats the case where compact manifolds without boundary are replaced by compact manifolds with one boundary component (corresponding to replacing vector bundles of degree 0 with vector bundles of nonzero degree). Section 2 explains why the moment map for the gauge group action on the space of connections is the map that sends a connection to its curvature. Section 3 treats spaces of flat connections on oriented 2-manifolds; Section 3.1 treats 1

Partially supported by an NSERC grant.

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the general properties of these spaces, and Section 3.2 treats spaces of connections. Section 4 treats recent work on Hamiltonian torus actions on open dense subsets of these spaces.

1.1 Background material Let † be a compact two-dimensional orientable manifold. Unless otherwise specified, the dimension refers to the dimension as a real manifold. The space † can be described in different ways depending on how much structure we choose to specify. One may form a class of topological spaces homeomorphic to † by gluing together (in pairs) the edges of a polygon with 4g sides. These spaces are classified by their fundamental groups (in other words, by the genus g, for which the Euler characteristic of the space is 2  2g: a space with genus g is a g-holed torus): * + g Y g 1 1  D 1 .† / D a1 ; b1 ; : : : ; ag ; bg W a j bj a j bj D 1 : (1.1) j D1

The aj ; bj provide a basis of H1 .†/, chosen so that their intersection numbers are a j \ bj D 1 and all other intersections are zero. The objects described in this section may be endowed with structures of smooth orientable manifolds of dimension 2, and all smooth structures on a compact orientable 2-manifold of genus g are equivalent up to diffeomorphism. The objects in this section may be endowed with additional structure. If we choose a complex structure on our 2-manifold, the Hodge star operator gives rise to an almost complex structure on the space of connections, and this almost complex structure turns out to be integrable. Thus the space of gauge equivalence classes of flat connections inherits the structure of complex manifold.

1.2 The Jacobian One may associate certain spaces with † which admit different descriptions depending on the amount of structure we have given to our 2-manifold. A prototype is the Jacobian, which can be described in several different ways. If we view † as a topological space and retain only the structure of its fundamental group , we may define Jac.†/ D Hom.; U.1// D U.1/2g :

(1.2)

If we view † as a smooth orientable 2-manifold, the Jacobian has a gauge theory description Jac.†/ D flat U.1/ connections /gauge group : (1.3)

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The space A of U.1/ connections is the space 1 .†/ of 1-forms on †. In terms of local coordinates y1 ; y2 on a neighbourhood U in †, A is identified with two copies of C 1 .U /; (1.4) A D fA D A1 dy1 C A2 dy2 W A1 ; A2 2 C 1 .U /g : See Section 3.1 for a more general description of the space of connections and the gauge group, and an explanation of how the gauge group acts on the space of connections. We can impose the condition that a connection A be flat: this leads to the space Aflat defined in terms of local coordinates as follows.

 @A1 @A2  D0 : (1.5) Aflat D A D A1 dy1 C A2 dy2 W @y2 @y1 The gauge group is G D C 1 .†; U.1//; its Lie algebra is Lie.G/ D C 1 .†/. The space of connections is given a more detailed treatment in Section 3.2. If we endow † with a complex structure, Jac.†/ is identified with an algebraic variety which classifies holomorphic line bundles over †: this is how the Jacobian arises naturally in algebraic geometry.

1.3 Representations into a nonabelian Lie group When we replace U.1/ by a compact nonabelian group G (e.g. G D SU.2/, or more generally G D SU.n/), some complications arise. The natural generalization of Jac.†/ is M.†/ D Hom.; G/=G where G acts on Hom.; G/ by conjugation. M.†/ has a natural gauge theory description which generalizes the description of the Jacobian: M.†/ Š flat G connections on †=G

(1.6)

where G D C 1 .†; G/ is the gauge group. M.†/ also has a description in algebraic geometry: it is the moduli space of holomorphic G C bundles over †, with an appropriate stability condition from geometric invariant theory. The identification between the space of representations and that of holomorphic vector bundles was established by Narasimhan and Seshadri [15]. Example. For G D U.n/, M.†/ is identified with the moduli space of (semistable) holomorphic vector bundles of rank n and degree 0 over †. When G D SU.n/ we obtain the moduli space of (semistable) holomorphic vector bundles of rank n with trivial determinant line bundle over †.

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1.4 Moduli spaces of vector bundles of nonzero degree More generally, we can consider the moduli spaces M.n; d / of (semistable) holomorphic vector bundles of rank n and (possibly nonzero) degree d with fixed determinant line bundle L over †. Replace  by ˝ ˛ (1.7)  0 D 1 .† n D/ D x1 ; : : : ; x2g (the free group on 2g generators), where D is a small disc in †. The group  0 is the fundamental group of a compact orientable 2-manifold with one boundary component, which may be obtained by removing D from †. For G D SU.n/, choose c D e2id=n I which generates the center Z.G/ (in other words n and d are coprime). For example, when G D SU.2/ we choose c D I. 8 9 0 1 g < = Y  0  1 1 A =G (1.8) M.n; d / D  2 Hom  ; G W  @ x2j 1x2j x2j x D c 1 2j : ; j D1

where G acts by conjugation. M.n; d / has a description as the space of gauge equivalence classes of flat G connections on †  D, whose holonomy around the boundary of D is conjugate to e2id=n I. M.n; d / is the moduli space of (semistable) holomorphic vector bundles of rank n, degree d and fixed determinant over † (see [15]). Provided n and d are coprime, the space M.n; d / is a smooth manifold with a symplectic form (a closed nondegenerate 2-form ! 2 2 .M.n; d // – see [13] for more on symplectic geometry). The space M.n; d / is also a complex manifold, and the complex structure is compatible with the symplectic form. In other words, M.n; d / is a Kähler manifold. The space M.n; d / is smooth. Singularities arise when the stabilizer of an element under the conjugation action is not finite. The condition that ..a1 ; : : : ; ag ; b1 ; : : : ; bg // is a generator of the center of SU.n/ has the following consequence. Suppose the stabilizer of .a1 ; : : : ; ag ; b1 ; : : : ; bg / is not finite. Call it J . If all .a1 ; : : : ; ag ; b1 ; : : : ; bg / belong to a subgroup H of G (where H is the subgroup of elements commuting with J ) then .a1 ; : : : ; ag ; b1 ; : : : ; bg / 2 ŒH; H (the commutator subgroup of H ). This is not possible if c is not in H . For example if the stabilizer of all ai ; bi is T (the maximal torus of G) then all a1 ; : : : ; ag ; b1 ; : : : ; bg 2 T . (The only elements whose stabilizer is T are elements that are themselves members of T .) In this case .a1 ; : : : ; ag ; b1 ; : : : ; bg / must be the identity element, so it cannot be a generator of Z.G/. So the stabilizer of all points for which the product of commutators equals such a generator of the center of G must be finite. So that point in the moduli space is smooth (or at worst an orbifold singularity). (For G D SU.2/Qthe condition that aj ; bj are stabilized by a subgroup H – U.1/ or SU.2/ – means j Œaj ; bj D 1, whereas the nontrivial element of the center of SU.2/ is 1.)

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2 The moment map of the gauge group action on the space of connections on a 2-manifold A Lie group action preserving the symplectic form is called Hamiltonian if the 1form obtained by contracting the fundamental vector field with the symplectic form is not only closed but also exact. The condition that the symplectic form is preserved tells us that LX # ! D 0 for the vector field X # associated to the action of X 2 Lie.G/. By Cartan’s formula this means iX # ! is closed (because ! itself is closed). The group action is called Hamiltonian if in addition iX # ! is exact for all X in the Lie algebra, in other words iX # ! D dX for some function X . The function X is then called the moment map (it is the Hamiltonian vector field generating the group action). One also imposes the condition that the moment map is equivariant with respect to the group action. This implies that the group action preserves the zero level set 1 .0/ of the moment map. The symplectic quotient is the quotient of the zero level set by the group action. For more details see [2] or [3]. Atiyah and Bott [1] exhibited the spaces M.n; d / as symplectic quotients via an infinite dimensional construction; the space of all connections A on † is acted on by the gauge group G with moment map the curvature W A 7! FA : To see why the curvature is the moment map for the action of the gauge group, we proceed as follows. Let A be the space of connections on an orientable 2-manifold †. The curvature of a connection A is 1 FA D dA C ŒA; A : 2 Define a map W A ! 2 .†/ ˝ g by Then we have

.A/ D FA : .d/A .a/ D da C ŒA; a :

We would like to show that  is the moment map for the action of the gauge group on A. The Lie algebra of the gauge group is Lie.G/ D 0 .†/ ˝ g : The vector field associated to  2 Lie.G/ is . # /A D dA . We have Z # !. ; a/ D hdA ; ai : †

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This can be rewritten as

Z

Z hd; ai C †

hŒA;  ; ai :

(2.1)

†

By Stokes’ theorem, the first term in (2.1) becomes Z  h; dai : †

The second term in (2.1) becomes

Z



h; ŒA; a i : †

Z

Thus we have  By definition this is which identifies

h; dA ai :

 hdA .a/; i dA .a/ D da C ŒA; a

which integrates to 1 .A/ D dA C ŒA; A : 2 Thus the symplectic quotient (in other words 1 .0/=G) is the space of flat connections up to equivalence under the action of the gauge group. These spaces are interesting examples of quotient constructions in symplectic geometry and geometric invariant theory. It has been established [1, 11, 12, 14, 16] that the symplectic quotient of a Kähler manifold by a (compact) group G is equivalent to the geometric invariant theory quotient by the complexification G C . See for instance [14] Chapter 8.2, Theorem 8.3 for details, and [14] Chapter 8.9 for a discussion of the relation between the symplectic and geometric invariant theory approaches to moduli spaces of vector bundles.

3 Spaces of flat connections on oriented 2-manifolds 3.1 General properties The objects of interest are the flat connections on a (topologically trivial) principal G-bundle over †. A connection specifies a way to do parallel transport in a principal bundle over † with structure group G. If the bundle can be trivialized, it is equivalent to the product bundle †  G. There are many different ways to specify the trivialization. After one fixed choice of trivialization has been made, the choice of another

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trivialization is equivalent to the choice of an element of the gauge group G D C 1 .†; G/ ; the group of (smooth) maps from † to G. The Lie algebra of G is the space of smooth maps from † to Lie.G/. The gauge group acts on the space of connections in the following way: if G is a matrix group, a choice of a trivialization of the principal bundle identifies the space A of connections with 1 .†/ ˝ Lie.G/, and an element  2 G sends A 2 A to  1 A C  1 d . For any closed loop  in †, a connection determines a holonomy, which is the group element h such that the image of parallel transport around  starting at a point .0/ Q in the fiber above .0/ is obtained by multiplying .0/ Q by h. If the connection is flat (the curvature FA is zero), then the parallel transport is not changed by continuous deformations of the loop (as long as these deformations keep the beginning point of the loop fixed). It depends only on the class of the loop as an element of the fundamental group  (equivalence classes of loops under deformation). In fact the action of the gauge group takes the subspace of flat connections to itself. It is not hard to prove the following: Theorem 3.1. We have: fflat connections/gauge group g Š frepresentations of  into Gg=conjugation Proof. Let AF be the space of flat connections and R be the space of representations of  into G. We define a map ‰ from AF to R by sending a flat connection A to the representation W  ! G determined by the holonomy of A. We define a map ˆ from R to AF as follows. A representation  of  determines a principal G bundle over e be the universal † with a distinguished flat connection in the following way. Let † e e cover of †. The product bundle †  G over † is equipped with an action of  as follows: an element  2  acts by    W .x; h/ 7! x; . /1h : (3.1) e  G so when we take the quotient This action preserves the product connection on † by the action of  we get a principal bundle P over † equipped with a flat connection e  G. We define ‰./ D A./. A./ which comes from the product connection on † It can be shown that ˆ and ‰ descend to maps between the spaces AF =G and R=fconjugationg which are the inverses of each other.  The space described in Theorem 3.1 may be called the space M of flat connections modulo gauge transformations. According to Theorem 3.1, this is the same as the space of representations of the fundamental group of † modulo conjugation. Example 1. Representations into an abelian group. An oriented 2-manifold of genus g (g-holed torus) is formed by taking a polygon with 4g sides and gluing the sides together in pairs according to the prescription given by (3.2) below. The sides

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of the polygon become the generators of the group. If we take representations into an abelian group T , the representation space is T 2g , and the conjugation action is trivial. Now, however, the fundamental group is not commutative: from the information that the loop around the outside of the polygon can be shrunk to a point we learn only that the generators satisfy the relation a1 b1 .a1 /1 .b1 /1 : : : ag bg .ag /1 .bg /1 D 1 :

(3.2)

In order to specify a representation  of  into a compact Lie group G we must specify the elements Ai ; Bi in G to which  sends each loop ai , bi . In order that it should be a representation we insist that the relation is preserved: A1 B1 .A1 /1 .B1 /1 : : : Ag Bg .Ag /1 .Bg /1 D 1 :

(3.3)

We must also take the quotient by the action of G by conjugation on the space of representations: h 2 GW Ai 7! h1 Ai hI Bi 7! h1 Bi h : (3.4) Example 2. † D S 1  S 1 , G D U.n/ (a nonabelian group). In this case if we choose elements A and B in G to represent the two loops a and b in S 1  S 1 , we need to insist that AB D BA (because ab D ba in ). Every element of G is conjugate to a diagonal matrix with unit complex numbers ei along the diagonal (i.e. it can be diagonalized). If we have diagonalized A, and A is a generic element (in other words it has distinct eigenvalues), then the only elements B which commute with it are the diagonal matrices with unit complex number entries. Call the space of such matrices T . Since we are interested in representations up to conjugation by G, we may assume A 2 T . If A and B are both in T , we may ask what is left over of the conjugation action (in other words, what elements of G will conjugate T into itself). In general the elements of G that will do this act via a finite group isomorphic to the permutation group Sn on n letters, which acts by permuting the diagonal entries. This is the Weyl group W D N.T /=T . So we find that M D .T  T /=W :

(3.5)

For G D SU.2/ and T D U.1/, this space is sometimes referred to as the “pillowcase” because of the four singularities (corners) that arise from the four elements in Z.G/  Z.G/ whose isotropy group is G. Example 3. The general case. Let G D U.n/, and suppose † is an orientable 2manifold with genus g > 1. In the case when G is a nonabelian group (such as U.n/ when n > 1) the space is more complicated than when G is abelian. The action of the group on the space of representations by conjugation is now nontrivial. In this case the space M is usually not smooth. We may replace it by a smooth analogue obtained

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by cutting out a small disc D in † and requiring that the representation send the loop around the boundary of the disc not to 1 but to the product of the identity matrix and a root of unity e2id=n which generates the n-th roots of unity. This space (denoted M.n; d /) is in fact smooth (see below) and its dimension is .2g  2/dim.G/; it shares many properties with the more natural space M (the smooth locus of M has the same dimension as M.n; d /). We have described some of the properties of M.n; d / in Section 1.4.

3.2 Connections The space A of all connections is simply the vector space of 1-forms tensored with g. To determine the tangent space to the space of flat connections, we use the fact that the curvature is the quantity 1 FA D dA C ŒA; A : 2

(3.6)

Infinitesimally, if FA D 0 then the condition that FACa D 0 translates to da C ŒA; a D 0 : We write this as

(3.7)

dA a D 0 :

It turns out that one can generalize this definition (in a way that can easily be described) to give an operator dA which maps g-valued differential forms of degree p to g-valued differential forms of degree p C 1 and satisfies dA ı dA D 0 : At the infinitesimal level, the image of the g-valued 0-forms under dA is the tangent space to the orbits of the group of gauge transformations. Thus the tangent space to the space M is the space TA M D H 1 .†; dA / D

fa 2 1 .†/ ˝ g j dA a D 0g : fdA  j  2 0 .†/ ˝ gg

(3.8)

We can see from this that M has a symplectic form, a closed 2-form which induces a nondegenerate skew-symmetric pairing on each tangent space. At the level of the vector space A of all connections, this just comes from the wedge product on differential forms, combined with an Ad-invariant inner product h; i on g; choosing a basis fe˛ g for g we have Z X !.A; B/ D he˛ ; eˇ i A˛ ^ B ˇ : (3.9) ˛;ˇ

†

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P ˛ ˛ Here we have written A D ˛ A e˛ for 1-forms A (and similarly for B). One can see (using Stokes’ theorem) that ! descends to a skew-symmetric pairing on H 1 .†; dA / (because !.dA ; b/ D 0 for all b 2 1 .†; g/ and any  2 0 .†; g/.) In fact this pairing is nondegenerate. This specifies a symplectic structure on M. For more details on the construction of this symplectic structure, see [6, 10, 17].

4 Hamiltonian flows on moduli spaces In [8] Goldman studied a collection of Hamiltonian flows on M.n; d /. For a compact Lie group G these flows are associated to a collection of simple closed curves C in †. Given C we cut † along C , forming a surface †0 with two boundary components CC and C . If A is a flat connection on †, denote by A0 its restriction to †0 . Assume the stabilizer of A on C is T . Choose a maximal torus T in G. For t 2 T let gt be a gauge transformation which restricts to the identity on CC but to the constant element t on C . This gauge transformation does not correspond to a global gauge transformaion on †, and so if A is a flat connection on †, A0 ı gt restricts to the same value on CC as on C (because the restriction of g to CC and C are in the stabilizer of A on C ). So A0 ı gt glues back together to form a flat connection on † which we will denote by A ı t. However A ı t is not gauge equivalent to A because AjCC and AjC are not gauge equivalent. This defines an action of T on an open dense set in M.n; d /. Goldman proved in [8] that if two curves C1 and C2 are disjoint, then the corresponding actions of T commute. It should be noted that the open dense set in M.n; d / where the action of T is defined is the set where the holonomy around C is regular. Otherwise the maximal torus in which the holonomy around C lives is not unique and the above construction is not well defined. The number of homotopy equivalence classes of disjoint simple closed curves on † is 3g  3, so the above construction specifies an action of T 3g3 on an open dense set in M.n; d /. The only case where the dimension of this torus is half the dimension of M.n; d / is G D SU.2/. The author and J. Weitsman have studied these Hamiltonian torus actions in the case G D SU.2/, using them to derive formulas for the symplectic volume of M.n; d / and M.†/. See [9]. This construction also works for noncompact groups, so Goldman studies SL.2; C/ and PSL.2; C/ (see [7]). In the case of G D PSL.2; R/, he recovers results on Fenchel–Nielsen coordinates on Teichmüller space, which is the identity component of the space of conjugacy classe of representations of the fundamental group.

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References [1] M.F. Atiyah and R. Bott, The Yang–Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. Lond. A308 (1983), 523–615. [2] M. Audin, The topology of torus actions on symplectic manifolds. Springer-Verlag, 2004. [3] A. Cannas da Silva, Lectures on symplectic geometry. Springer-Verlag, 2001. [4] S.K. Donaldson, Gluing techniques in the cohomology of moduli spaces, in Topological methods in modern mathematics, Publish or Perish, 1994, 137–170. [5] J.-M. Drezet and M.S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semistables sur les courbes algébriques, Invent. Math. 97 (1989), 53–94. [6] W. Goldman, The symplectic nature of fundamental groups of surfaces. Adv. Math. 54 (1984), 200–225. [7] W. Goldman, The complex–symplectic geometry of SL(2)-characters over a Riemann surface. In volume dedicated to M.S. Raghunathan on his sixtieth birthday, 2003. Available from http://www.math.umd.edu/~wmg [8] W. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations. Invent. Math. 85 (1986), 263–302. [9] L. Jeffrey and J. Weitsman, Bohr–Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula. Commun. Math. Phys. 150 (1992), 593–630. [10] Y. Karshon, An algebraic proof for the symplectic structure of moduli space, Proc. AMS. 116 (1992), 591–605. [11] G. Kempf and L. Ness, The length of vectors in representation spaces. Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), 233–243, Springer-Verlag (Lecture Notes in Math. 732), 1979. [12] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry. Princeton University Press (Mathematical Notes vol. 31), 1984. [13] D. McDuff and D. Salamon, Introduction to symplectic topology, Oxford University Press, 1998. [14] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory (Third Enlarged Edition), Springer-Verlag, 1994. [15] M.S. Narasimhan and C.S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface. Ann. Math. 82 (1965), 540–567. [16] L. Ness, A stratification of the null cone via the moment map. Amer. J. Math. 106 (1984), 1281. [17] A. Weinstein, The symplectic structure on moduli space. The Floer memorial volume, 627– 635, Progr. Math., 133, Birkhäuser, 1995.

Part D

Sources

Chapter 9

Extremal quasiconformal mappings and quadratic differentials Oswald Teichmüller Extremale quasikonforme Abbildungen und quadratische Differentiale Abhandlungen der Preussischen Akademie der Wissenschaften Math-Naturw. 22 (1939). Translated from the German by Guillaume Théret

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of conformal invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-orientable regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The notion of principal region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of bordered orientable principal regions . . . . . . . . . . . . . . . . . . . . . The torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of non-orientable principal regions . . . . . . . . . . . . . . . . . . . . . . . . A wrong track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extremal quasiconformal mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Riemann–Roch theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological determination of principal regions . . . . . . . . . . . . . . . . . . . . . . . Definition of principal regions through metrics . . . . . . . . . . . . . . . . . . . . . . . Infinitesimal quasiconformal mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extremal infinitesimal quasiconformal mappings . . . . . . . . . . . . . . . . . . . . . . The equation wx C iwy D B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The linear metric space L of classes of infinitesimal quasiconformal mappings Doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regular quadratic differentials and inverse differentials for principal regions . . . The equation wx C iwy D B for arbitrary principal region . . . . . . . . . . . . . . . The dimension of the linear metric space L . . . . . . . . . . . . . . . . . . . . . . . . Passing to finite mappings. R as a Finsler space . . . . . . . . . . . . . . . . . . . . . Simple cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Oswald Teichmüller Proof of the extremality property . . . . . . . . . . . . . . Conformal mappings of a principal region onto itself Remarks on the torus . . . . . . . . . . . . . . . . . . . . . . Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . A metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extremal Problem of conformal mappings . . . . . . .

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1 Introduction 1. In the present study, the behaviour of conformal invariants under quasiconformal mappings shall be examined. This will lead to the problem of finding the mappings that deviate as little as possible from conformality under certain additional conditions. We shall give the solution of this problem, without however being able to give a rigorous proof. The solution relies on the notion of quadratic differentials (function times the square of a differential) from the theory of algebraic functions. As I have done it before, I shall here also examine quasiconformal mappings not exclusively for their own sake, but chiefly because of their connections with notions and questions that interest function theorists (see 164ff.). It is true that quasiconformal mappings have been for only very few years systematically applied to purely function theoretical questions, and so this method has managed to gain until now only a limited number of friends. It may nevertheless be mentioned that the contribution I have been able to make some time ago in this journal to the Bieberbach coefficients problem relies in fact on ideas that shall be developed here.1 Besides fundamental notions and methods from function theory, we shall also often use those from differential geometry; in addition, algebraic functions and topology will contribute. We shall use different methods and tools going from computation with infinitely small quantities to uniformization theory, from length and area estimates to the invariance of domain theorem, from integration of partial differential equations to Galois theory; we shall compute the number of conformal invariants with the Riemann–Roch theorem. Of course, too much at the same time cannot be assumed to be known here; much of what has been already known for a long time will be gathered and partly established once more. As I said before, the main results will not be proven in the strictest sense. I only hope to establish them in a way that any serious doubts are practically excluded, and to encourage finding the proofs. Thus when possible, the ideas that led to discovering the solution are conveyed in direct chain. As far as I can see, attempts of exact proofs find their right place only where this train of thought ends, according to the paradox “proving is reversing the train of thought.” So it is expected that a systematic theory progressing toward exact proofs would cause for the time being bigger difficulties in understanding than the present heuristic introduction. 1 O. Teichmüller, Ungleichungen zwischen den Koeffizienten schlichter Funktionen, these proceedings, 1938.

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Finally, let me quickly point out the topological and algebraic side results and problems contained in 123 and in 141ff.

2 Examples of conformal invariants 2. As the easiest example, we consider a planar doubly-connected domain whose boundary components are not collapsed to points. Any such ring domain can be mapped conformally2 onto a round ring 1 < jwj < R .1 < R < 1/, and any two such round rings with different R cannot be mapped upon each other conformally. Hence, two ring domains can be mapped upon each other conformally if and only if they give, after mapping them onto a concentric round ring, the same radius quotient R. In this sense, R is the characteristic conformal invariant of the ring domain. The only conformal self-mappings of 1 < jwj < R are w 0 D ei # w and w 0 D

Rei # : w

Thus, conformal self-mappings of a ring domain fall into two connected classes. Among them, only the principal class (that contains the identity) remains if a winding direction is distinguished on the ring domain and if one requires that it is not reversed. Instead, one can also insist that the mappings do not swap both boundary components of the ring domain. 3. A simply-connected domain has no appropriate conformal invariant, for any two planar domains bounded by a continuum can be mapped upon each other conformally. Besides, one is also allowed to arbitrarily specify the image of an arbitrary inner point. But if two different inner points of a simply-connected domain are distinguished, then there is a conformal invariant: the Green function. If the domain is mapped onto the unit disc so that one chosen point goes to 0 and the other to r .0 < r < 1/, then the Green function equals log 1r . The domain can be mapped onto another simplyconnected domain with non-degenerate boundary so that the two points go to two prescribed inner points of the other domain if and only if the Green functions agree. The Green function is therefore the characteristic conformal invariant of a simplyconnected domain with two distinguished inner points. 4. Two simply-connected domains with, say, piecewise analytic boundary curves, can be mapped upon each other conformally such that three boundary points of the first domain go to three given boundary points of the other domain in the same order. But four boundary points of a simply-connected domain possess a characteristic conformal invariant. We enumerate the four boundary points in such a way that they 2 By conformal mapping we always mean a mapping which is also one-to-one. Later, we shall also consider indirect conformal mappings.

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follow each other as we go around the domain in the positive direction and we map the domain onto the upper half-plane so that the four boundary points go to 0; 1; ; 1 (1 <  < 1); the cross-ratio  is then the characteristic conformal invariant. But  is only well-defined if the boundary points have been enumerated. If the sequence of numbers 1; 2; 3; 4 of the boundary points is changed to 3; 4; 1; 2, then  remains  unchanged; however, passing to 2; 3; 4; 1 or to 4; 1; 2; 3 turns  into 1 . We can therefore either introduce, instead of , the invariant C

 2  D 

D 1 1 1

which is independent of how the distinguished boundary points are numbered, or regard boundary point numbering as essential and keep on with . The second option shall be more suitable for us. Matching ’s is therefore the necessary and sufficient condition for two piecewise analytically bordered simply-connected domains being mapped upon each other so that each of the four given boundary points of the first domain goes to one prescribed boundary point of the other domain (of course, the prescribed image boundary points must be ordered properly). Similarly, for simply-connected, piecewise analytically bordered regions with five distinguished boundary points, a characteristic pair of conformal invariants is given by two cross-ratios 1 , 2 . 5. For conformal invariants of the closed surface of genus 1 – the torus – a special theory has been developed. The simply-connected relatively unramified covering of the torus can be mapped conformally onto the punctured u-plane, thus yielding a pair of primitive periods .!1 ; !2 / such that two points u; u0 of the u-plane correspond to the same point on the torus if and only if u0 D u C m!1 C n!2 (m; n integers). We 1 can take = !1 !2 > 0 or = ! > 0. However, !1 ; !2 are not uniquely defined by the !2 surface, but only up to the substitutions !10 D .a!1 C b!2 /, !20 D .c!1 C d!2 /, where a; b; c; d are integers with determinant ad  bc D 1 and  a complex factor different from zero. The period quotient ! D !2 =!1 with = ! > 0 is therefore only . Now the “absolute invariant” defined up to the modular substitutions ! 0 D cCd! aCb! J.!/ is a regular function on the upper half-plane = ! > 0 for which J.! 0 / D J.!/ ; if and only if ! and ! 0 are related through some modular substitution ! 0 D cCd! aCb! J. !!12 / is therefore the characteristic conformal invariant of the torus. Nevertheless, our forthcoming considerations will be easier if we use ! instead of J as a conformal invariant. Let us draw in the u-plane the segments from 0 to !1 and from 0 to !2 ; to these segments correspond straight closed curves C1 , C2 on the torus which we think of as determined only up to continuous deformation. Now if a pair of curves C01 , C02 is likewise given on another torus, then the first torus can be mapped upon the other conformally so that the image of C1 can be deformed to C01 and the image of C2 into C02 if and only if ! D ! 0 ; here, ! is, after mapping the covering surface of the first torus onto the u-plane, the quotient of the growth of u around C2 by that around C1 , and ! 0 has the same meaning for the other torus. We have therefore reached a transcendental determination of the conformal invariant through topological additions.

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6. Let us now consider a, say, piecewise analytically bordered ring domain on which two boundary points are distinguished. This ring domain can be mapped conformally onto 1 < jwj < R so that one boundary point goes to w D 1. The number R and the image of the other boundary point form a characteristic pair of conformal invariants. We must however distinguish whether both boundary points lie on the same boundary curve or on different boundary curves of the ring domain. In the first case, w D 1 and w D ei # are the two image boundary points on 1 < jwj < R and R; # (1 < R < 1, 0 < # < 2) are the invariants. In the second case, the boundary points w D 1 and w D Rei # of 1 < jwj < R are distinguished, but now # is only well-defined modulo 2. Using a continuous curve connecting w D 1 with w D Rei # and defined only up to continuous deformation, # (1 < # < C1) can be defined unambiguously. In the first case, such an approach would serve no purpose, because the space of those .R; #/ would then be no more connected, since in the first case # cannot be a multiple of 2. But in the second case, after decorating the ring domain with an additional curve connecting both boundary points and defined only up to deformation, we obtain a characteristic pair of conformal invariants R; # (1 < R < 1, 1 < # < 1).

3 Non-orientable regions 7. Up to now, we have been considering direct conformal mappings only, but not indirect conformal ones, which become conformal after a mirror symmetry. It would be easy to decide for the examples treated before whether indirect conformal mappings should be taken into consideration and then exclude them again by fixing an orientation. For the sake of completeness and generality, we now talk about the case where picking a definite orientation is impossible. The best known example of non-orientable region is the Möbius band. It arises from the round ring 1 < jwj < R when one identifies w with  Rw 1 ; or also when one cuts it up along the real axis and, for one piece, identifies all the boundary points t and  Rt (t D 1; : : : ; R). Every Möbius band (e.g. given in three dimensional space, singular-free) can be mapped conformally onto this Möbius band normal form, where of course it is no longer a question of making a distinction between direct or indirect conformal mappings. A map is conformal in the sense that the size of all angles remains unchanged and the ratio of linear dilatations is independent of direction. Each Möbius band can be covered by a two-fold orientable relatively unramified covering band and the latter can be mapped onto a ring domain 1 < jwj < R; this map readily transfers the given Möbius band to the special one constructed above. R is therefore the characteristic conformal invariant of the Möbius band. 8. The elliptic plane is another example of a non-orientable surface. It arises from the sphere when diametral points (i.e., both ends of a diameter) are identified and is

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topologically equivalent to the projective plane. If we suppose the radius of the sphere equal to 1, then the distance between two points of the elliptic plane is interpreted as the smallest length of an arc of a great circle of the sphere joining them; hence it is  2 since amongst two diametral points at least one has a distance  2 from one given point on the upper sphere. Conformal mappings from the elliptic plane to itself only come from sphere rotations. The characteristic conformal invariant of two points of the elliptic plane is therefore their distance. This distance increases from 0 to 2 and the pairs of points at distance 2 , which correspond to pairs of orthogonal diameters of the sphere or which correspond to four harmonically distributed points on the sphere, appear as a limit case. But now any two points of the elliptic plane can be joined by exactly two paths which cannot be deformed onto each other. If such a path is chosen and if the distance considered above is replaced by the smallest length of an arc of a circle connecting the points, into which the distinguished path can be continuously deformed, then we now have at our disposal the open interval from 0 to  and the exceptional position of the pairs of points at distance 2 disappears. n  3 points of the elliptic plane have 2n  3 invariants.

4 Finite Riemann surfaces 9. We now put the character common to all our examples in a more general form. First of all, we have always been dealing with Riemann surfaces;3 they could be simply-connected or not, bordered or not, planar or not, orientable or not. A Riemann surface is a connected topological space, every point P of which has a neighbourhood which is one-to-one and in both directions continuously (homeomorphically) mapped onto an open set of the z-plane; z is then called a local uniformizer. If the neighbourhood of P we have just mentioned contains a point Q, a neighbourhood of which is mapped onto an open set of the z 0 -plane, then the mapping z $ z 0 , which is bijective and continuous in the neighbourhood of Q, must be direct or indirect conformal. Since we allow indirect conformal parameter transformations, we also capture nonorientable surfaces. A mapping is called conformal when it becomes conformal after passing to local uniformizers. On an orientable Riemann surface there are two classes of local uniformizers that are connected among each other through direct conformal mappings; usually one class is chosen and the other local uniformizers are not used anymore. In some of our examples, the surface was bordered. By a boundary element of a Riemann surface we mean an open subset which is conformally mapped onto a domain of the open upper half-plane =z > 0, the boundary of which shares a segment with the real axis; moreover, this segment must correspond to the boundary of the surface in the sense that to each z-valued sequence which converges to the segment 3 (Translator’s note) We translate Riemannsche Mannigfaltigkeit by Riemann surface instead of Riemannian manifold, according to the definition given below by Teichmüller. However, let us emphasize that in the definition given here, coordinate changes can be antiholomorphic (indirect conformal).

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corresponds a sequence of points which eventually escapes every compact4 subset of the surface. To the points of our segment then correspond nothing but (ideal) boundary points of the surface; z is also said to be a local uniformizer for every such boundary point. This notion of boundary point is shown to be independent of any particular choice of local uniformizer z: instead of z one may also take any z 0 which 0 is (locally) real for real z and satisfies dz > 0. By including these boundary points dz and introducing the notions of neighbourhoods and local uniformizers for them, we obtain bordered Riemann surfaces. For example, for the ring domain 1 < jwj < R, z D i log w is a local uniformizer R of every boundary point w D ei # and z D i log w is a local uniformizer of every i# boundary point w D Re . It hardly needs to be mentioned that our notion of boundary element is from the start conformally invariant and should not be confused with the notion of accessible boundary points of planar domains that is needed in the study of the metric behaviour of conformal mappings. 10. By a finite Riemann surface we mean a (possibly bordered) Riemann surface which can be topologically seen as a simplicial complex. This means that it can be decomposed using some Jordan curves into finitely many pieces, each being simplyconnected and having three boundary points – the vertices – which cut its boundary curve in three sides, and every two of these pieces must have at most one vertex or one side in common with any other. These prerequisites are satisfied for every closed surface, but also for example for planar domains bordered by n Jordan curves. Each finite Riemann surface is compact;5 it is bordered by n  0 (ideal) closed curves. In topology, it is shown that all finite Riemann surfaces can be thought of as being obtained in the following way: one starts with the sphere, cuts out n  0 holes and places outside g  0 handles and   0 crosscaps.6 Of course, the handles and the crosscaps have nothing to do with the n holes: the resulting surface must still have n boundary curves. It is orientable or non-orientable, depending on whether  D 0 or  > 0. For orientable surfaces, g is called the genus (also in the case n > 0). For non-orientable surfaces, 2g C  is called the genus; but since in the case  > 0, one handle can be replaced by two crosscaps (and conversely, as long as  remains positive), we can restrict ourselves to the case g D 0,  > 0: the genus is then  . Finite Riemann surfaces have therefore three topological invariants: orientability or non-orientability, the genus g, respectively  , and the number n of boundary curves. These are also the only ones. All infinitely-connected domains are hence excluded and only those surfaces are studied that can be made planar through finitely many cuts. Also, punctures are forbidden: the unit disc is a finite Riemann surface, but the punctured plane is not. The latter has indeed no ideal boundary curve in the above sense and, even before trying to bound it at infinity, it is not compact. 4A

set is called compact if every sequence of points in the set has a cluster point in the set. the footnote in 9. 6 See for instance Seifert–Threlfall Lehrbuch der Topologie, Leipzig und Berlin 1934. 5 See

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The following division of possible cases is particularly well suited to our study: I. Closed orientable surfaces ( D 0; n D 0). II. Bordered orientable surfaces ( D 0; n > 0). III. Non-orientable surfaces ( > 0).

5 The notion of principal region 11. Two finite Riemann surfaces with the same g; ; n can be mapped upon each other topologically, but in general not conformally; rather, there are in general finitely many conformal invariants whose matching will be necessary and sufficient for conformal equivalence. One might for instance think, in the examples treated above, of the ring domain, of the torus and of the Möbius band. But other conformal invariants also appeared above that referred to a number of points of the surface, for example to two inner points or to four boundary points of a simply-connected domain. So the following definition seems appropriate: A principal region is a finite Riemann surface in which h  0 inner points and k  0 boundary points are distinguished. The surface itself is called the support of the principal region. A mapping from a principal region to another is a mapping from the support of the principal region to the support of the other sending the distinguished points on distinguished points. For instance, the simply-connected domain with two distinguished inner points is in this sense a principal region whose characteristic conformal invariant is the Green function. Of course, the number k of distinguished boundary points can be positive only if the number n of boundary curves of our surface is positive. Above, we had to exclude finite Riemann surfaces with punctures. Now, if for example a closed Riemann surface with one puncture is given, we will close it in order to obtain a finite Riemann surface and then distinguish the point where the puncture used to be; we thus obtain a principal region whose conformal invariants clearly coincide with those of the initial surface. So in general, punctures can be replaced by distinguished inner points and conversely; and it would not be wise to proceed without distinguishing inner points and allow punctures because there is nothing alike for distinguished boundary points. For this reason, we shall prefer the first way of doing and keep punctures out. 12. This definition of principal region is only temporary. In a later work, cases shall also be studied where for example two distinguished inner points are moved infinitely close; in the present work, such a situation is excluded. But the fact that this notion is to be revised must be mentioned right here. For the simply-connected domain with four distinguished boundary points, we have numbered these points to a certain extent arbitrarily, but in a way which could

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not be changed afterwards. For the torus, we have fixed a canonical cut by curves C1 , C2 which were well-defined only up to deformation, and so on. So in general we shall consider that a principal region is fixed only after adding certain topological determinations. We do not want to be more precise now, referring rather to the examples and to 49 where we shall come back to this subject. In the present work, which chiefly aims at giving a preliminary orientation, topological issues must step aside. 13. In the simple examples treated above, the principal region was always conformally mapped onto some normalized principal region of the same type which only depended on finitely many parameters; they then provided the conformal invariants. We admit the following statement without proof: when conformally equivalent principal regions are identified, principal regions of a fixed topological type form a topological manifold which is locally homeomorphic to the .  0/-dimensional Euclidean space and shall be, for this reason, denoted as the space R . The conformal invariants of a principal region are then precisely the functions on R . Hence, locally, there are precisely  independent conformal invariants. According to which notion of neighbourhoods conformal classes of principal regions form, we do not specify. We shall naturally work with mappings onto normalized regions. In the example of the elliptic plane with two distinguished points (or, better to say, the closed non-orientable surface of genus  D 1 with two distinguished inner points, since the latter can always be mapped conformally onto the former), the set of all classes of conformally equivalent principal regions was first bijectively mapped onto the set of distances A with 0 < A  2 . Thus, only a half-neighbourhood in R ( D 1!) corresponds to A D 2 (i.e. to the elliptic plane with two distinguished points between which there are two shortest paths), although we notice at first sight no singular character for this principal region. We nevertheless managed to establish regularity in 8 by distinguishing one of the two possible topological paths between both distinguished points. It is thus to reckon that such and even nastier occurrences might also arise for higher principal regions.

6 Characteristic numbers 14. We list right now a few non-negative integers which are characteristic of a principal region and for which we introduce notation once and for all. These are: g, the number of handles;  , the number of crosscaps; n, the number of boundary curves;

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h, the number of distinguished inner points; k, the number of distinguished boundary points. In case  > 0, it is possible to replace simultaneously g and  by g  1 and  C 2, by g  2 and  C 4, . . . , by 0 and  C 2g. From n D 0 follows k D 0; otherwise, g; ; n; h; k can be arbitrarily prescribed. Let further  be the number of parameters for the continuous group of conformal self-mappings of a principal region,  be the dimension of the space of all classes of conformally equivalent principal regions which are topologically equivalent to a given principal region. Hence, if a principal region has only finitely many conformal self-mappings, then necessarily  D 0. For the ring domain, we have  D 1 and  D 1; for the torus,  D 2 and  D 2; for simply-connected domains,  D 3 and  D 0; for the simplyconnected domain with one distinguished inner point and two distinguished boundary points,  D 0 and  D 1, and so on. We will see later that  and  are univoquely defined functions of g; ; n; h and k. The integer  is positive only in a few special cases which can be completely listed. The dimension formula    D 6 C 6g C 3 C 3n C 2h C k holds. One remarks a formal analogy with the Riemann–Roch theorem, which also displays the difference of two numbers of parameters in a closed expression. Later we shall justify the dimension formula and directly relate it with the Riemann–Roch theorem. Finally, we will consider later two quantities which have for supports of principal regions, resp. for principal regions themselves, a fundamental meaning similar to that of genus for orientable closed surfaces, namely, the algebraic genus ( g GD 2g C  C n  1

for closed orientable surfaces, for bordered or non-orientable surfaces.

and the reduced dimension ( for closed orientable surfaces, D 2  for bordered or non-orientable surfaces. The number  is always an integer.

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7 Statement of the problem 15. Given a mapping from a z D x C iy-domain to a w D u C iv-domain which is locally one-to-one and in both directions continuously differentiable, let us set 1 u2x C u2y C vx2 C vy2 I 2 ux vy  vx uy p D D jKj C K 2  1 D earccos jKj : KD

The number D  1 is the ratio of the axes of the infinitely small ellipse in the w-plane that corresponds under the mapping to an infinitely small circle of the z-plane and it is called the dilatation quotient. For conformal mappings, D D 1, otherwise D > 1; log D therefore measures the deviation of a mapping from conformality. Mapping conformally the z-plane or the w-plane does not change the dilatation quotient D D Dzjw . The dilatation quotient of a mapping from some principal region to another will be everywhere computed by passing to local uniformizers, which is possible thanks to conformal invariance. A one-to-one mapping with certain differentiability properties whose dilatation quotient is bounded is called quasiconformal.7 16. Since any conformal invariant J of a principal region — i.e., a quantity which can be thought of as a function of the point of R representing the principal region — remains absolutely unchanged under conformal mappings, it is then expected that it undergoes little changes under quasiconformal mappings if the supremum of the dilatation quotient is sufficiently close to 1. We are therefore led to the following Problem: Let a conformal invariant J , seen as a function on R , a fixed principal region and a number C > 1 be given. What are the values that J takes at those principal regions onto which the given principal region can be quasiconformally mapped so that the dilatation quotient remains everywhere  C ? We shall not discuss now the meaning of this problem for function theory.8 This problem has to be generalized in that the dilatation quotient is not just given a single bound C , but a bound which depends upon the points of the principal region. Getting a precise bound for J in this more general case is hard, though. Here, we shall deal with constant bounds only. 17. This problem can be placed in a broader context in two ways. Sometimes, it might be possible to construct the invariant J for many types of principal regions following a consistent rule (for example, for planar principal regions, the supremum of all the values of some geometrically defined quantity for all realizations in the plane of the principal region only defined up to conformal mapping) and to tackle the 7 Following Ahlfors, Zur Theorie des Überlagerungsflächen, Acta Math. 65. For another definition of quasiconformal mappings, see the reference given in the footnote of 161 8 See 141ff. and the reference in the footnote of 161

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problem simultaneously for all these different principal regions; one will then also think about taking limits toward no longer finite Riemann surfaces. Here we shall proceed the other way round and assume the type of principal regions fixed (i.e., their topological nature and especially the numbers listed in 14), hence also R , and then state the problem simultaneously for all invariants J . So given a principal region represented by a point P in R and a number C > 1, what are, for every function J defined over R , the values of J at those points Q for which there is a quasiconformal mapping from the principal region corresponding to P to the principal region corresponding to Q with dilatation quotient  C ? This general question is clearly equivalent to the following Problem: Let a point P of R and a number C > 1 be given. Let the principal region represented by P be mapped quasiconformally in any possible way onto another principal region of the same type so that the supremum of the dilatation quotient is  C , respectively < C , and let us represent the so-obtained principal region by a point Q in R again. What is the set UC .P /, respectively UC .P /, of the points Q obtained this way? Through the way it is written, it is already apparent that UC .P / should be thought of as an open neighbourhood and UC .P / as a closed neighbourhood of P in R . That this is correct will be made likely through the following explanation. 18. Let us consider two principal regions of the same type represented in R by the points P and Q. Given any quasiconformal mapping between these principal regions, let C denote the supremum of its dilatation quotient. We define the distance ŒPQ between the two points, or between the two principal regions, as the logarithm of the infimum of all these C . We have to check whether the axioms of a distance are satisfied. Surely we have ŒPQ  0. ŒPQ D ŒQP follows from the fact that the dilatation quotients of a mapping and its inverse are at each point equal. The triangle inequality ŒPQ C ŒQR  ŒPR follows from the proposition that under composition of two mappings, the dilatation quotient of the composite mapping is at each point at most equal to the product of the dilatation quotients of the two composed mappings.9 Now can P D Q be deduced from ŒPQ D 0? If two principal regions can be mapped upon each other quasiconformally with dilatation quotients whose difference with 1 becomes arbitrarily small, do we necessarily have a (direct or indirect) conformal mapping between them as well? This is not sure, because we do not know whether there always exists between two principal regions a quasiconformal mapping with the smallest maximum for dilatation quotients. But this is nevertheless likely since, according to the basic idea given in 16 already, we expect that two principal regions which cannot be mapped upon each other conformally cannot be mapped quasiconformally either so that the difference between the supremum of the dilatation quotient and 1 becomes arbitrarily small. We shall assume this without proof, and even that given any neighbourhood U of a point P in R there exists some C > 1 such that UC .P /  U. But it must be 9

O. Teichmüller, Eine Anwendung quasikonformer Abbildungen auf das Typenproblem. Deutsche Math. 2.

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explicitly emphasized that we are thus anticipating the results of a study that is not carried out yet. We further assume without proof that for each point P in R and each C > 1 there should be a neighbourhood U of P which is entirely contained in UC .P /. This assumption seems fully reasonable: if we think of all principal regions of a fixed type mapped conformally onto normalized principal regions with only finitely many parameters, then it should be possible to guarantee only a slight modification of these parameters using a mapping which only somewhat deviates from the identity and whose dilatation quotient only somewhat differs from 1. Both assumptions together imply that ŒPQ indeed satisfies the axioms of a metric and that the sets ŒPQ < const form a system of neighbourhoods in R equivalent to the initial one. 19. Thus, all spaces R that appeared before as topological spaces only (apart from a few trivial exceptions) are now provided with a metric in a uniform way. It will be our task to study this metric and particularly to find its geodesics. We will later be led to the reasonable hypothesis that R is, with regard to our metric, a Finsler space. But now our problem moves on to the following: Let two principal regions of the same type be given, represented in R by the points P and Q. What is the distance ŒPQ ? In particular, can we determine all extremal quasiconformal mappings whose dilatation quotients are everywhere  eŒPQ ? It is expected that there always exists an extremal quasiconformal mapping and that it is unique up to conformal self-mappings of the given principal regions. If this version of our problem is solved for some R , then the problems of 16 and 17 are also solved up to eliminations. Our extremal problem may look at first sight like something unfamiliar. One might rather expect to search for a mapping from a given principal region onto another which minimizes some integral related to the dilatation quotient. Here instead, the maximum of the dilatation quotient must be the smallest. However, this problem turns out to be accessible through analysis. But before we proceed any further with any general discussion, some examples should be presented for which a complete solution of our problem is possible. These examples are just simple cases given one after the other; we will later find that the character they all share is having reduced dimension  D 1.

8 Examples of bordered orientable principal regions10 20. We begin with the quadrilateral, i.e. the simply-connected domain with four distinguished boundary points. In 4, we mapped the quadrilateral, whose distinguished boundary points are to be thought of as numbered in the positive circular 10

H. Grötzsch, Über möglichst konforme Abbildungen von schlichten Bereichen, Sächs. Ber. 84.

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direction, conformally onto the upper half-plane =z > 0 so that the distinguished points were sent to 0; 1; ; 1, and we called  (1 <  < 1) the characteristic conformal invariant. This time, we map the quadrilateral onto a rectangle 0 < 0 ; =e ! >0;

0

and the mapping z 7! z has the property z 0 .z C e ! 1 / D z 0 .z/ C e ! 01 ; z 0 .z C e ! 2 / D z 0 .z/ C e ! 02 : Consequently, exactly like in 25, 1 ! =e !  =e ! 0  C =e C holds also in general. Equality holds only for an affine mapping composed of a translation, a dilatation in the direction of e ! 1 and a direct similarity. In case b D 0, these inequalities are not new, but for b ¤ 0, they say that ! 0 belongs to the zeroangled bigon made up of two arcs of circles of the !-plane that is sent to the strip 1 !  =e ! 0  C =e ! by the transformation e ! D cCd! C =e aCb! . It is advisable to endow the half-plane =! > 0 with the non-Euclidean metric with curvature 1 whose line element is given by jd!j : ds D =! ! and C =e ! from the real e ! -axis are then The parallels at a Euclidean distance C1 =e simply the two circles that touch the circle at infinity =e ! D 0 at e ! D 1 lying at a non-Euclidean distance log C from the point e ! . Getting back to ! through ! D cCae ! 0 we then see that ! lies within the two circles that touch the real axis in the d be ! a rational point ! D  b and are at a non-Euclidean distance log C from !. And if we had required ad  bc D 1 instead of ad  bc > 0, we would have then arrived to the same system of inequalities, for then  ab also runs over all rational numbers including 1. The range of values that ! 0 can take for given ! and C therefore lies in the intersection of infinitely many zero-angled bigons made up of arcs of circles. We now claim that this intersection is the disc with non-Euclidean center ! and non-Euclidean radius log C , that is, the set of all ! 0 at a non-Euclidean distance  log C from !. The reason is that the rational numbers  ab are everywhere dense in the real axis. If  is an intersection point between the real axis and the disc passing through ! and

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! 0 and symmetric with respect to the real axis, then  can be approximated by a sequence  abnn ; ! 0 must then lie within the two circles that touch the real axis in  and are at a non-Euclidean distance log C from !; but these circles just cut out from our orthogonal circle the segment of those points that are at most log C distant from !.  Cı! This conclusion can be grasped analytically as follows: each transformation ˛Cˇ! with real coefficients and with ˛ı  ˇ > 0 can be approximated through similar transformations with integer coefficients, from which we get in general  C ı! 1  C ı!  C ı! 0  C= = = I 0 C ˛ C ˇ! ˛ C ˇ! ˛ C ˇ! 0

 Cı!  Cı! and ˛Cˇ! ˛; ˇ; ; ı can be chosen so that ˛Cˇ! 0 have the same real part. But this disc is also the exact range of the values of ! 0 . For, the mapping

z 0 D Tk

z z z D K< C i= ; ˛!1 C ˇ!1 ˛!1 C ˇ!1 ˛!1 C ˇ!1

which is, up to some direct similarity of the z 0 -plane, a dilatation in the direction ˛!1 C ˇ!2 (˛; ˇ real), has dilatation quotient K and sends !1 ; !2 to the periods !0 !10 ; !20 , whose quotient ! 0 D !20 lies on the opposite side from  ˇ˛ on the orthogonal 1 circle through ! and  ˇ˛ and has non-Euclidean distance log K from !. This is clear from what precedes when ˛ and ˇ are integers and also follows for non simultaneously vanishing reals ˛; ˇ by taking the limit. The infimum of the logarithms of all upper bounds for dilatation quotients of mappings from a torus with ratio of periods ! onto a torus with ratio of periods ! 0 is therefore equal to the non-Euclidean distance between ! and ! 0 in the upper half-plane. This result has a great fundamental significance: for the torus case, the problem set in 19 would have inevitably led, however the conformal invariants of the torus may have been normalized, to the non-Euclidean metric of the !-half-plane which has already proved its worth so well in modular function theory. The modular functions turn out to be the analytic functions on the manifold R2 of the classes of conformally non-equivalent tori in the sense of Klein.16 We hope this way that the general problem stated in 19 should promote the theory of moduli (i.e., conformal invariants) of algebraic function fields of higher genus which has been barely investigated until now. So far, we have assigned a unique value ! through a fixed canonical cut of the torus. If we cancel this normalization, then ! is only defined up to the modular substitutions cCd! , where a; b; c; d are integers with ad  bc D 1; moreover, if aCb! ! come in addition. We the torus is not given any orientation, the substitutions cd ab! then have to draw the non-Euclidean circles with non-Euclidean radius log C about all points conjugate to ! with respect to this group and from the intersection of all these circles with a fundamental domain of the group; this intersection is obtained by carrying, with the substitutions of the group, all points of the circle about ! within the fundamental domain. 16 F.

Klein, Riemmannsche Flächen I und II, Vorlesungen Göttingen W.-S. 1891/92 und S.-S. 1892.

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27. In order to apply our result, let us deal with the closed surface of genus 0 with four distinguished points. It can be mapped conformally onto the sphere so that three distinguished points are sent to 0; 1; 1; the fourth point then goes to  and  is the characteristic conformal invariant. We thus think of the distinguished points as numbered and we also assume that an orientation of the surface has been fixed; without the latter,  and  would not be distinguishable and there would be in R2 only halfneighbourhoods about principal regions with real cross ratio  (cf. the end of 13). Now we consider the genus 1 two-fold covering of the sphere whose ramification points lie over 0; 1; ; 1; it has ratio of periods !, which is defined by  only up to modular substitutions cCd! aCb! , but which depends on  analytically;  is a six-valued function of ! which decomposes into six single-valued branches, each of which rewith even b and c. Now, mains fixed precisely under the modular substitutions cCd! aCb! with the adjunction of a topological ingredient, we are able to choose a definite value for !: we draw a continuous, double-point free curve from 0 to  passing through 1, well-defined only up to continuous deformation; we must also pay attention not to introduce double-points while we deform the curve. We choose on the covering torus, in the sense of 5, for C1 the twice-passed-through piece of our curve connecting 0 and 1 and, for C2 , the twice-passed-through piece of the curve connecting 1 and  (this yields a canonical cut indeed); through this, !1 and !2 are determined up to a 2 with =! > 0 is uniquely defined.  is common factor and up to sign, and so ! D ! !1 a single-valued function of ! which maps the upper half-plane =! > 0 conformally onto the well-known simply-connected covering of the -plane punctured in 0; 1; 1, the so-called moduli surface. The different sheets of this surface are distinguished from each other by the different topological types of the double-point free curve from 0 to  passing through 1. Now let a quasiconformal mapping of the sphere onto itself be given which maps 0; 1; 1 to themselves and  to 0 ; let its dilatation quotient be  C . We fix, using a curve 01, a specific point  on the infinitely-sheeted moduli surface. This curve is then sent by the mapping to a definite curve 010 which also determines a specific point over 0 of the moduli surface. According to 26, the non-Euclidean distance between the associated definite values !./ and !.0 / is  log C . This is also the only bound that the points lying over  and 0 are subject to, since any extremal affine mapping from 26 satisfies the symmetry property that is necessary for also providing, after passing to the two-fold covering of the sphere, a quasiconformal mapping from the sphere to itself. The problem stated in 19 therefore leads to the curvature 1 hyperbolic metric on the 0; 1; 1-punctured plane. The -image of the non-Euclidean disc about !./ with non-Euclidean radius log C is, for fixed  and for small log C , planar; for large C , it covers parts of the -sphere several times. If we only want to pay attention to  but not to the sheet of the moduli surface, it is advisable to construct in the !-plane the normal region about some fixed !./ which is made up of all points whose non-Euclidean distance from !./ is smaller (or equal) to the distance from cCd!./ with even b and c and odd aCb!./ 0 a and d . Each point ! whose non-Euclidean distance from one of the latest points is  log C also surely lies, if it belongs to the normal region, in the non-Euclidean

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disc about !./ with non-Euclidean radius log C . Hence, for fixed , one has only to transfer to the -sphere the intersection of this disc with the normal region in order to obtain the exact range of values of 0 . Should the principal region degenerate, i.e., should 0 converge to 0; 1 or 1, then, for fixed , the upper bound C of the dilatation quotient would become infinitely large. R2 is therefore a complete differential geometric space. The corresponding fact is also to be remarked in all other examples. 28. In 26 we found that, for fixed ! and C , the range of values of ! 0 is a welldefined closed disc. If ! 0 lies on the boundary of this disc and if the orthogonal circle through ! and ! 0 meets the real axis in at least one rational point, then it can be sent with a modular substitution to 1 and it follows from the proof given in 25 that the associated quasiconformal mapping from the z-plane to the z 0 -plane is affine. If, however, this orthogonal circle cuts the real axis in two irrational points, we also get an extremal affine mapping, but we still do not know in this case whether there are in addition non-affine extremal quasiconformal mappings. We are going to refute this possibility now. Since this does not depend on a rotation of the z-plane and the z 0 -plane, it is enough to show the following: Let TK z D K 0 and !10 D TK !1 ;

!20 D TK !2 :

Let the z D x C iy-plane be quasiconformally mapped with bound C onto the z 0 plane and thus let z 0 .z C !1 / D z 0 .z/ C !10 ;

z 0 .z C !2 / D z 0 .z/ C !20 :

Then we have C  K and equality holds only for the affine mapping z 0 D TK z C const : We can already deduce from 26 that C  K; so it just remains to find the condition under which C D K. Proof. Let QL denote the square 0  x  L, 0  y  L; L is a number likely to increase. z 0  TK z is bounded: jz 0  TK zj  M : The length of the z 0 -image of the segment =z D y D const, 0  x  L, is at least KL  2M , since its endpoints are at least this distance apart: Z Lˇ 0ˇ ˇ @z ˇ ˇ ˇ dx : KL  2M  ˇ @x ˇ 0

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Integration over y, from 0 to L, gives ˇ 0ˇ ˇ @z ˇ ˇ ˇ dx dy : ˇ ˇ QL @x

“ KL  2LM  2

Now let P be the period parallelogram associated to !1 ; !2 whose area is =!2 !1 and let be its diameter ( D maxfj!1 C !2 j; j!1  !2 jg). The union of period parallelograms covering QL has a total area  .L C 2 /2 , their number is thus 2 .LC2/2  .LC2/ =!2 !1 . The integral over QL is therefore at most equal to =!2 !1 -times the integral over one period parallelogram: “ ˇ 0ˇ “ ˇ 0ˇ ˇ @z ˇ ˇ @z ˇ .L C 2 /2 2 ˇ ˇ ˇ dz : ˇ KL  2LM  ˇ @x ˇ dz  =! ! ˇ ˇ 2 1 QL P @x Passing to the limit as L ! 1 yields “ ˇ 0ˇ ˇ @z ˇ ˇ ˇ dz : K=!2 !1  ˇ ˇ P @x Now we have “ ˇ 0 ˇ 2 “ “ ˇ @z ˇ ˇ ˇ dz  dz

ˇ ˇ P @x P P

ˇ 0 ˇ2 “ ˇ @z ˇ ˇ ˇ dz  =!2 !1 C dz : ˇ @x ˇ P

ˇ 0 ˇ2 ˇ @z ˇ dz 0 Anyhow, equality can hold only if ˇˇ ˇˇ D C . If we insert here the estimate @x dz “ ˇ 0ˇ ˇ @z ˇ ˇ ˇ dz , we get from below we found above for ˇ @x ˇ P

“ K 2 =!2 !1 

dz 0 D C =!20 !10 D C K =!2 !1

P

or

KC: ˇ 0 ˇ2 ˇ @z ˇ dz 0 holds identically, while the dilatation quoEquality holds only if ˇˇ ˇˇ D C @x dz tient is  C D K. Both equalities agree only if every infinitely small circle of the z-plane is sent to an infinitely small ellipse of the z 0 -plane with quotient of axes K and if moreover the great axis of this ellipse corresponds to a parallel to the real axis in the z-plane. But then z 0 is an analytic function of TK z, hence z 0 D aTK z C b, and moreover, a D 1 necessarily. 

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29. We can reduce all the results we obtained for bordered orientable principal regions to our new result relative to the torus. They were all reducible (and without using any lemma about conformal mappings) to the ring domain. Now if the ring domain 0 < log jwj < M is mapped quasiconformally with bound C onto the ring domain 0 < log jw 0 j < M 0 , then we extend the mapping by mirroring through the circles log jwj D M , log jwj D M 0 to a quasiconformal mapping from 0 < log jwj < 2M to 0 < log jw 0 j < 2M 0 . We then identify w D ei # 0 0 0 with w D e2M Ci # as well as w 0 D ei # with w 0 D e2M Ci # . We thus get two 0 0 tori which are mapped via z D log w, resp. z D log w , onto the punctured plane with pair of periods 2M , 2 i , resp. 2M 0 , 2 i , and which are quasiconformally mapped upon each other with dilatation quotient  C . The non-Euclidean i distance between ! D i and ! 0 D M 0 should therefore be at most log C , i.e. M 0 j log M  log M j  log C , as we already found in 21. The various examples treated at that time were reduced to the ring domain but they can also be reduced directly to the torus or to the sphere with four distinguished inner points: the quadrilateral will be either mapped onto a rectangle and extended through mirroring along its sides, or it will be mapped onto a disc with four distinguished boundary points and the sphere with four distinguished points is then obtained by mirroring. The simply-connected domain with two distinguished inner points is sent, by mapping it onto a disc and then mirroring, to the sphere with four points. Also, the simply-connected domain with one inner point and two boundary points is sent to the sphere with four distinguished points, if it is either mapped onto a disc and mirrored or mapped onto a half-disc so that the distinguished boundary points are sent to the vertices, then twice mirrored and if the vertices of the half-disc are no longer counted as distinguished points. Now it can also be understood why in 23 and 24 d  appeared as an elliptic differential of the first type. Similar observations also hold for the three forthcoming examples, so the result of 26 together with the lemma from 28 comprises all estimates we have managed to reach at this stage.

10 Examples of non-orientable principal regions 30 : A Möbius band is a non-orientable finite Riemann surface of genus 1 with one boundary curve and without distinguished points. It is hence described in the sense of 14 through gD0;

 D1;

nD1;

hD0;

k D0:

Each Möbius band can be conformally mapped onto the round ring 0 < log jwj < M which is turned into a Möbius band by identifying w with eM w 1 ; M is the characteristic conformal invariant (thus  D 1; incidentally  D 1 also) (cf. 7). To each mapping from a Möbius band with invariant M onto one with invariant M 0 corresponds a mapping from the round ring with modulus M onto one with modulus M 0 whose dilatation quotient has the same supremum, and to the extremal quasiconfor-

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mal mapping of the round ring corresponds the extremal quasiconformal mapping of the Möbius band. The sharp estimate j log M  log M 0 j  log C therefore holds. 31 : Now let us consider, as we did in 8, the elliptic plane E with two distinguished points A; B: g D0;  D1; nD0; hD2; k D0: E arises from the sphere K by identifying diametral points. The points A and B correspond to pairs of points A1 ; A2 and B1 ; B2 on K. We choose a path on E from A to B, well-defined up to deformation; two paths A1 B1 and A2 B2 on K correspond to it. The points A1 ; B1 ; A2 ; B2 lie, in this order, on a directed great circle  of K. In the sense of 27, we now draw on the oriented sphere with the four distinguished points A1 ; B1 ; A2 ; B2 a continuous double-point free curve A1 B1 A2 : it can simply be a half of . According to 27, this uniquely defines a ratio of periods ! and, in our case, ! D iv, v > 0. v does not change when A1 is swapped with A2 and, at the same time, B1 is swapped with B2 , and it is equal to the ratio of sides of the rectangle onto which the quadrilateral A1 B1 A2 B2 bordered by  can be mapped conformally. v and the spherical distance A1 B1 D A2 B2 (which is equal to the distance AB defined in 8) depend monotonically on each other. According to 27 (and with the preceding notation), the non-Euclidean distance between ! and ! 0 is under a quasiconformal mapping at most equal to log C , i.e. j log v 0  log vj  log C : The estimate is sharp; equality holds only for an easily given extremal mapping. In order to show this, we actually only need to show the following: if E is mapped topologically (quasiconformally) onto a second elliptic plane E0 with covering sphere K0 or if K is mapped onto K0 so that diametral points are sent to diametral points – we may assume that orientation is preserved –, and if A1 ; B1 ; A2 ; B2 are sent to A01 ; B10 ; A02 ; B20 , then the arc A1 B1 A2 of  is sent to a continuous double-point free curve t0 from A01 to A02 passing through B10 ; we have to show that t0 can be deformed to the half great circle  0 from A01 to A02 and passing through B10 in the sense of 27; in particular ! D iv and ! 0 D iv 0 can thus be compared, according to 27. Now this follows easily, which will not be carried out here, from the topological lemma:17 Any topological mapping of the elliptic plane onto itself can be deformed to the identity. 32 : For the last example we consider the Klein bottle. This is a closed nonorientable Riemann surface of genus 2: gD0;

 D2;

nD0;

hD0;

k D0:

17 Formulated by W. Mangler, Die Klassen von topologischen Abbildungen einer geschlossenen Fläche auf sich, M. Z. 44.

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A Klein bottle is obtained when we consider in the punctured u-plane the group of transformations   1 u 7! u C m C niv ; u 7! u C C m C niv ; 2 with freely chosen fixed v > 0, and when the points that are equivalent with respect to this group (i.e., that are obtained from each other through the transformations of the group) are identified. Any Klein bottle can be obtained this way (up to conformal mapping). We conclude that the Klein bottle has a two-fold orientable relatively unramified covering which is a torus and which can be, like in 5, unfolded over the plane. v is the characteristic conformal invariant of the Klein bottle. Applying a quasiconformal mapping yields j log v 0  log vj  log C : Again, we only have to check whether, under a topological mapping from a Klein bottle to another, and after developing the covering tori over the u-, resp. u0 -plane, the pair of periods 1; iv 0 really corresponds to the pair of periods 1; iv. But stated in this form, this cannot be true whatsoever; what we show instead is that the pair of mappings u 7! u ˙ 1 and u 7! u ˙ iv are sent to the corresponding mappings u0 7! u0 ˙ 1 and u0 7! u0 ˙ iv 0 ; this is enough, since with it and according to 26 (or even 25), the ratio of periods iv and iv 0 can be compared. u 7! u ˙ 1 is distinguished in that it generates the center of the group of transformations mentioned above; u 7! u ˙ iv 0 generates the (likewise cyclic) group of transformations that are sent to their inverses under a transformation with some group element reversing the orientation of the u-plane. The elliptic plane E with two points A; B can be cut along a continuous doublepoint free curve from B to A and two copies of E can be attached crosswise along this cut; we then get a Klein bottle. So the result of 31 can be reduced to the theorem we have just proved and the topological lemma applied in 31 can be in this way avoided.

11 A wrong track 33. This method cannot be applied as it stands to higher principal regions. As an easy example, let us consider like in 6 the ring domain with two distinguished boundary points (i.e. g D 0;  D 0; n D 2; h D 0; k D 2) and distinguish the two boundary points on different boundary components. We may first try the following: we map such a principal region conformally onto 1 < jzj < R with distinguished points 1 and Rei # and another one onto 1 < jz 0 j < R0 with distinguished points 1 0 and R0 ei # . The extremal quasiconformal mapping between the two principal regions (i.e., for which the maximum of dilatation quotient is the smallest) may carry log z to

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log z 0 affinely so that log R C i # is sent to log R0 C i # 0 and 2 i is sent to 2 i : log z 0 D

log R0 C i.# 0  #/ < log z C i = log z : log R

However, if we try to prove this using the methods applied before, we completely fail 0 at catching the additional condition that the mapping sends 1 to 1 and Rei # to R0 ei # . We shall see later that this family is not at all the family of extremal quasiconformal mappings. Let us now try to learn from the examples carried out before what extremal quasiconformal mappings look like! In these examples, we invariably mapped conformally the given principal region onto a normalized principal region from which we measured the conformal invariants and we mapped it, or a covering of it, conformally on a -plane, or on a part of it. The extremal quasiconformal mappings were constantly affine transformations in the -plane. When boundary curves occurred, like in 21 for ring domains, they were sent to straight lines of the -plane through which we mirrored (cf. 29). When some boundary points were distinguished, like in 20 for quadrilaterals, then they corresponded in the -plane to right-angled vertices. And when one inner point was distinguished, like in 27 for the sphere with four distinguished inner points, then the mapping had a two-fold ramification point in the -plane; if  D 0 corresponded to this point, then  and 20   were associated to the same point of the principal region. If z denotes some local uniformizer about the distinguished point,  2 we can say, as opposed to , that ddz is a local well-defined function of z, with a first-order pole at the distinguished point. This holds for distinguished inner points as well as for distinguished boundary points. All examples we treated before can be grasped within this point of view so it may be useful to think about them once again. 34 : So what can we do about the ring domain with two distinguished boundary points? Since no inner points are distinguished,  should be a locally univoquely defined function without any two-fold ramification points; the boundary components should become straight lines and each distinguished boundary point should give rise to a right-angled vertex. This is too much a demand. The idea suggests itself of replacing, in higher cases, the Euclidean -plane by the non-Euclidean plane – the unit disc jj < 1 –. The simply-connected relatively unramified covering of the ring domain can be readily mapped onto a domain of jj < 1 bordered by orthogonal arcs of circles, each meeting the adjacent ones perpendicularly, and whose vertices correspond to the distinguished boundary points. So the ring domain is mapped conformally onto 1 < jwj < R, is reflected at jwj D R and ei # is identified with R2 ei # , like in 29: we obtain a torus for which our distinguished boundary points now become distinguished inner points;  maps conformally the simply-connected covering, quadratically ramified over the distinguished inner points in all sheets, onto jj < 1. We would proceed similarly in all cases. For closed orientable surfaces without any distinguished points we would only have to map conformally the relatively unramified simply-connected covering on jj < 1.

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The principal region should accordingly be characterized through a group of conformal self-mappings of jj < 1. This group contains also indirect conformal mappings when the principal region is bordered or non-orientable. Any quasiconformal mapping from such a principal region onto another yields a quasiconformal self-mapping of jj < 1 which carries any group of conformal selfmappings of jj < 1 into an isomorphic group of such conformal mappings. 35 : We now ask the following question: if A is a quasiconformal mapping and S a conformal self-mapping of jj < 1, and if we know that S 0 D ASA1 is conformal, what conclusion can we draw about S 0 from that given S and from a given bound C for the dilatation quotient of A? We can clearly restrict to the case of a direct, hyperbolic, conformal mapping S . Since this situation does not depend on applying some conformal mapping, we replace jj < 1 by the upper half-plane = > 0 and assume that S  D F  ; S 0 D F 0 ; where F and F 0 denote some factors > 1. From S 0 D ASA1 and the assumption that the dilatation quotient of A is  C , we shall therefore deduce that F and F 0 cannot differ more than that (in case C D 1 we would get F D F 0 ). Identifying  with F  in = > 0 yields an ideal connected ring domain, which is conformally mapped through 2 w D ei log F log  onto a round ring with modulus M D

2 2 : log F

The mapping A carries this ring domain over the ideal connected ring domain that 2 2 . From arises by identifying  and F 0  (= > 0) and whose modulus is M 0 D log F0 0 the inequality j log M  log M j  log C proved in 21 we obtain j log log F  log log F 0 j  log C : Now this inequality holds for all direct conformal hyperbolic substitutions S arising in the group. For a given principal region and a bound C for the dilatation quotient, each of these inequalities reduces the set of possible image principal regions. It can be asked whether the neighbourhoods UC .P / defined in 17 are characterized by the totality of these inequalities. I am not able to answer this question. But two points are not in favour of pursuing this direction: Extremal quasiconformal mappings cannot be obtained this way. If, for some quasiconformal mapping A and some direct conformal hyperbolic self-mapping S of jj < 1, equality holds in j log log F  log log F 0 j  log C , then A only transforms such a conformal self-mapping of jj < 1 into a mapping of the same type which commutes with S . But in general the whole group will not commute with S .

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For genuine extremal quasiconformal mappings of the principal region, j log log F  log log F 0 j < log C always holds. Moreover, the orientable closed surface of genus 1 with one distinguished point does not lead, with this method, to the punctured -plane, but rather to jj < 1, although the conformal invariants of this principal region agree with those of its support and their behaviour under quasiconformal mapping was already exhaustively treated in 25–28. This method of covering uniformization would therefore be in this case unnecessarily complicated. The same holds for the ring domain and the Möbius band, each with one distinguished boundary point.

12 Extremal quasiconformal mappings 36. We therefore leave the path taken up in 34 and try to find properties of extremal quasiconformal mappings; we shall also use what we learned and gathered in 33. Let a principal region H be mapped extremal quasiconformally onto a principal region H0 , i.e. the maximum K of the dilatation quotient D is as small as possible. This amounts to assuming that this maximum K is not only attained at some points or lines, but everywhere, i.e. that the dilatation quotient is constant, equal to K. For, if D were only D K at some point and < K elsewhere, then one could introduce a suitable self-mapping of H0 differing from the identity only in a neighbourhood of the point, which would reduce there, after composing it with the given mapping from H to H0 , the dilatation quotient; the maximum of the dilatation quotient for the composite mapping would be < K. One would proceed similarly (possibly step by step) if the dilatation quotient were D K over some lines or pieces of surface. Of course, this should not be considered as a proof whatsoever, but only as heuristic considerations. We assume without proof that the dilatation quotient of an extremal quasiconformal mapping is constant. 37. But we should not expect that, conversely, every mapping with constant dilatation quotient is extremal quasiconformal. After passing to local uniformizers, a oneto-one, in both directions continuously differentiable mapping carries an infinitely small circle into an infinitely small ellipse whose ratio of axes D  1 is called the dilatation quotient. In case D > 1, we furthermore consider the circle’s diameter that goes under the mapping to the ellipse’s greatest axis, or rather its direction at the point whose neighbourhood is under study. Thus, a direction, or more precisely, a pair of opposite directions – a line element – is associated to every point. We get a direction field and we can integrate it to get the family of those curves that have in each point the stipulated direction (of maximal dilatation). If not only the dilatation quotient at each point of H is provided but moreover this direction field, then H0 is determined up to conformal mapping. The direction fields can be chosen quite arbitrarily.18 18

Details in 53ff. The idea seems to have been expressed first by Lawrentieff.

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We now ask the question: what direction fields are associated to extremal quasiconformal mappings? The property of a direction field or of a curve field to be associated to an extremal quasiconformal mapping will split into local and global conditions. 38. Again, let an extremal quasiconformal mapping from a principal region H onto a principal region H0 be given. Let its dilatation quotient be constant D K. We pick a simply-connected neighbourhood of some point p on H (a half-neighbourhood, of course, if p is a boundary point); the fact that p might be a distinguished point of the principal region does not play any role. It is then impossible to modify the mapping in this neighbourhood so that the dilatation quotient of the new mapping becomes  K and is < K at some point, for otherwise this would give an extremal quasiconformal mapping from H to H0 with non-constant dilatation quotient. Let us now give a definition. A quasiconformal mapping is called locally extremal if its dilatation quotient is constant D K and if there is, for each point p in the neighbourhood where the function is defined, a simply-connected neighbourhood U with the following property: the mapping is defined in U and on the boundary of U and any mapping which agrees outside U with the given one and whose dilatation quotient is  K has constant dilatation quotient K. Of course, if p is a boundary point, the modified mapping must map the entire piece of boundary of U that contains p onto the same curve as the original mapping; if p is a distinguished point, the image of p should not change. According to this definition, each extremal quasiconformal mapping is also locally extremal; a proof of this fact uses the assumption of 36, namely, that each extremal quasiconformal mapping has constant dilatation quotient. But now we can try to determine the set of locally extremal quasiconformal mappings, and afterwards, to look within for the subset of globally extremal quasiconformal mappings. This approach still needs some modifications. 39. Let us consider an extremal quasiconformal mapping from H to H0 with constant dilatation quotient K. Given some 1 < K ? < K, we construct a mapping from H onto a principal region H? of the same type for which the dilatation quotient is everywhere equal to K ? and to which is associated the same direction field, in the sense of 37, as to the given mapping; as this was already mentioned in 37 and as we shall see it even more precisely, this is always possible. The dilatation quotient of the mapping H? ! H ! H0 is then just KK? . We claim that the mapping H ! H? is extremal quasiconformal. Otherwise, there would be a mapping from H onto H? for which the maximum of the dilatation quotient would be < K ? , and it would give, together with the mapping H? ! H whose dilatation quotient is KK? , a mapping from H to H0 with maximum of dilatation quotients < K: contradiction. Likewise, the mapping H? ! H0 is extremal quasiconformal. As K ? grows from 1 to K, the point representing H? in the space R of all classes of conformally equivalent principal regions traces a curve in R which we will later call a geodesic with respect to the metric introduced in 18.

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We just draw here the following consequence: if the quasiconformal mapping associated to some direction field and to some constant dilatation quotient K is extremal, then it remains extremal when K is shrunk. The same statement obviously holds for locally extremal quasiconformal mappings. We shall therefore look for those direction fields for which the quasiconformal mappings associated to them and to dilatation quotients whose difference with 1 is sufficiently small are extremal, resp. locally extremal. And we shall soon modify this goal by requiring that the dilatation quotient differs from 1 only by an infinitely small quantity. Thereupon, some remodelings are still necessary.

13 The Riemann–Roch theorem 40. In order to better highlight the essence and not to let the basic ideas step back behind a lot of different cases, we now restrict ourselves for a moment to closed orientable surfaces without distinguished points, hence to principal regions with characteristic numbers g D 0; 1; 2; : : : I

 D0;

nD0;

hD0;

k D0:

For the sake of completeness only, we include the uninteresting case g D 0; the case g D 1 has already been carried out. In the notation of 14, following Riemann’s theory, we have for g D 0W  D 6;  D 0; for g D 1W  D 6;  D 2; for g > 1W  D 0;  D 6.g  1/. Hence, in all the cases, the dimension formula    D 6.g  1/ given in 14 holds. 41. There are on the closed oriented surface F non-constant functions which are in a neighbourhood of each point, considered as functions of local uniformizers, rational; they are called functions on the surface F for short. For each function on the surface, the sum of the multiplicities of its a-point is independent of a and is called the degree of the function. We assign to each point of the surface a “prime divisor” p bi-univoquely and take all these p as generators of a free abelian group, the group of divisors. A divisor is Q therefore an expression of the form p p˛p , where ˛p is an integer associated to each p (or to each point of the surface) which is different from 0 only for finitely many p.

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In particular, if z is a function on the surface, then at a zero of z, ˛p is set to be equal to the degree of this zero; at a pole of z, ˛p is set to be Qequal to minus the degree of the pole; and ˛p is equal to 0 elsewhere. The divisor p p˛p thus obtained is called the principal divisor .z/. The unit element of the group of divisors is also called a principal divisor; it is associated to the constants ¤ 0; 1. The principal divisors form a subgroup. The congruence classes of the group of all divisors quotiented by the normal subgroup of all principal divisors are called the divisor classes and form the divisor class group. 42. By an n-dimensional differential d  n we mean a rule which associates to each local uniformizer t on a neighbourhood of an arbitrary point of the surface a function g.t/ which is meromorphic in this neighbourhood; if t 0 is another local uniformizer about some point of this neighbourhood and if the differential assigns to it the function g 0 .t 0 /, then  0 n dt 0 0 g.t/ D g .t / dt must hold. The expression g.t/dt n is therefore invariant under parameter changes and is regular analytic over the whole surface except at poles; one writes g.t/dt n D n d  n and g.t/ D ddtn , although g.t/ is not an n-th power in general. Of course, n is  aninteger. If z is a function on the surface, then we can for example set g.t/ D dz dt

n

and observe that dz n is an n-dimensional differential. If d  n is an arbitrary dn dz n

g.t / n is a (possibly constant) function ' on . dz dt / n the surface; therefore, any d  is of the form 'dz n . Q We also assign a divisor .d  n / to every d  n ¤ 0, namely, .d  n / D p p˛p , where ˛p is the multiplicity of the zero, respectively the opposite of the multiplicity n of the pole of g.t/ D ddz n , or 0, depending on whether g.t/ has a zero, a pole or none at the point of the surface associated to p; t is a local uniformizer. (Only finitely many ˛p are different from 0, since neither poles nor zeroes can accumulate on a n/ is a principal divisor: all .d  n / lie in the same divisor class which is the point.) .d .dz/n n-th power of the divisor class of all 1-dimensional differentials; the latter is called the differential class W and contains among others .dz/.

n-dimensional differential, then

D

P Q 43. For each divisor d D p p˛p , deg d D p ˛p is called the degree of d. The degree of a principal divisor is 0. All divisors of a divisor class D have the same degree deg d calledQthe degree of the class. A divisor d D p p˛p is called an integral divisor whenever all ˛p  0. If d is a divisor, then the set of all (possibly constant) functions z on the surface for which .z/d is an integral divisor form, together with 0, a module with respect to complex numbers. The rank of this module, i.e. the maximal number of linearly independent such z with respect to the field of complex numbers, depends only on the divisor class D of d and is called the dimension of D, dimD. From deg D < 0 we get dimD D 0.

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The Riemann–Roch theorem says: dimD is always finite, and dimD  dim

W D deg D  g C 1 : D

Here, g is the genus of the surface and W is the differential class. In particular, deg W D 2.g  1/, dimW D g. If d is a divisor of the divisor class D, then dimDWn equals the rank of the module of all n-dimensional differentials d  n for which d.d  n / is an integral divisor; of course 0 has to be counted in this module. The Riemann–Roch theorem gives, when we replace D by DWn , dimDWn  dim

W1n D deg D C .2n  1/.g  1/ : D

44. In particular, by the Riemann–Roch theorem, dimW2  dim

1 D 3.g  1/ : W

Just as 0-dimensional differentials are functions and first dimensional ones are differentials in the usual sense, we call the two-dimensional differentials quadratic differentials and the .1/-dimensional ones inverse differentials. We say that a differential of the n-th order d  n is everywhere finite when it is equal to 0 or when the corresponding divisor .d  n / is integral. Hence, with this terminology, the difference between the maximal number dimW2 of linearly independent, everywhere finite, quadratic differ1 entials and the maximal number dim W of linearly independent, everywhere finite, inverse differentials is equal to 3.g  1/. If g D 0, then deg W2 D 4 < 0, hence dimW2 D 0 ;

dim

1 D3 W

.g D 0/ :

If g D 1, then W is the principal class – the unit element of the divisor class group –, hence 1 D1 .g D 1/ : dimW2 D 1 ; dim W 1 1 D 2.1  g/ < 0, hence dim W D 0: If finally g > 1, then deg W

dimW2 D 3.g  1/ ;

dim

1 D0 W

.g > 1/ :

We observe that for all closed oriented surfaces F without distinguished points, 2dimW2 D  ;

2dim

1 D W

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(cf. 40). It is noteworthy here that  and  count real parameters, whereas dimW2 1 and dim W count complex parameters: the maximal number of linearly independent, everywhere finite, quadratic differentials (respectively, inverse differentials) is equal 1 to 2dimW2 (respectively, 2dim W ), as soon as linear dependence is considered with respect to the field of real numbers. As soon as I knew the values of  given in 40, I noticed that they were also equal to 2dimW2 for small genera. I then conjectured some connection; this is what the following discussion is going to be about.

14 A conjecture 45. The supposition that the equality  D 2dimW2 is no coincidence gains credit 1 a sound meaning. through our being able to give to the equality  D 2dim W  denotes the number of parameters for the continuous group of all conformal selfmappings of our surface F. According to Lie theory, we shall therefore consider  as being the maximal number of (real) linearly independent infinitesimally conformal mappings from F onto itself; this can be verified by listing cases according to genus. An infinitesimally conformal mapping from F to itself moves every point only infinitesimally: if z is a local uniformizer on a neighbourhood of a point on the surface, then the point with parameter value z is sent to the point with parameter value z C g.z/; here  is a constant, infinitely small quantity.19 Since the infinitesimal mapping is conformal, g.z/ is regular analytic. Passing to another local uniformizer z 0 .z/ yields, if g 0 .z 0 / has a meaning analogous to that of g.z/, z 0 C g 0 .z 0 / D z 0 .z C g.z// D z 0 C  or g 0 .z 0 / D

dz 0 g.z/ dz

dz 0 g.z/ : dz

According to the definition given in 42, d  1 D g.z/=dz is a .1/-dimensional differential or an inverse differential, of course everywhere finite. Infinitesimal conformal mappings from F to itself correspond bijectively to everywhere finite inverse differentials. But now the maximal number of real linearly independent, everywhere 1 finite, inverse differentials is 2dim W . 46. We therefore conjecture that there is a connection between everywhere finite quadratic differentials and extremal quasiconformal mappings. According to 37, any extremal quasiconformal mapping is described by its constant dilatation quotient K and by its direction field. Now how can we link together direction fields, which lead 19 Computation

with infinitely small quantities has never been firmly grounded to the extent that we need it. But every mathematician knows what is meant. Our computations can not be justified by passing to the limit. Hence they have for the present heuristic value only.

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to extremal quasiconformal mappings, and everywhere finite quadratic differentials d 2? During a night of 1938, I came up with the following conjecture: Let d  2 be an everywhere finite quadratic differential on F, different from 0. Assign to every point of F the direction where d  2 is positive. Extremal quasiconformal mappings are described through the direction fields obtained this way and through arbitrary constant dilatation quotients K  1. Therefore, if z is a local uniformizer on a neighbourhood of some point, then writing d  2 D g.z/dz 2 , the condition d  2 > 0 means arg g.z/ C 2 arg dz D arg d  2 0 mod 2 or

1 arg dz  arg g.z/ mod  : 2

Hence, in the parameter plane, the direction determined by d  2 > 0 forms an angle  12 arg g.z/ with the direction of the positive z-axis and therefore defines a line element, i.e. a pair of opposite directions. This computation fails in the zeroes of d  2 . The sum of the multiplicities of these zeroes is deg W2 D 4.g  1/. The definition of quasiconformal mappings through dilatation quotients and direction fields might still appear as something not completely clear; it would be insightful to have the image surface before our eyes. For this reason, we formulate the conjecture once more but with other words. Let d  2 ¤ 0 be an everywhere finite quadratic differential on F. We set Z r 2 Z p d d 2 D dz : D dz 2 In general,  is a multivalued function; the various branches of  that correspond to the same point of F are related by substitutions of the form  ! ˙ C const :  maps a certain covering surface b F of F bijectively and conformally on a certain Riemann surface Z over the -plane. Let G be the group of deck transformations of b F over F; identifying the points of b F that are equivalent under G yields the former surface F (because for a turn along which some branch of  does not change, the other branches do not change as well). The group G is carried by the mapping from b F to Z to a group G of conformal self-mappings of Z which all have the analytic form  ! ˙ C const; identifying the points of Z that are equivalent under G gives the closed surface F back, up to conformal mapping. The mapping  0 D TK  D K 0. It is therefore conjectured that the mappings F ! F0 thus constructed from arbitrary d  2 and arbitrary K are the extremal quasiconformal mappings. Hence, first, these mappings should be extremal20 and second, F0 should, for fixed F, run through all surfaces of the same genus g. Let us count the parameters: 2dimW2 real parameters for d  2 and another one for K; however, things are unchanged if d  2 is multiplied by any positive factor; thus F0 depends on 2dimW2 parameters. Now general closed orientable surfaces of genus g defined only up to conformal mapping depend on  parameters, and we have already noticed above that  D 2dimW2 . In case g D 1, let the covering surface over F be mapped onto the z-plane, like in 25–28: the only everywhere finite quadratic differentials are then d  2 D a dz 2 and we come up exactly with the mappings constructed in 28. 47 : Let us now review the previous conjecture, which is not at all grounded yet. The reason why we just consider quadratic differentials is probably clear: through a quasiconformal mapping, a line element is assigned to every point, but without any well-defined direction; for this reason, d  is not single-valued, but d  2 is. This is where the tensorial character of the dilatation steps in. We now show that the mappings constructed in 46 are locally extremal quasiconformal in the sense of 38. We shall have to be particularly careful about the zeroes of d 2. For the moment,Zlet us consider a neighbourhood of a point where d  2 does not p vanish. Here,  D d  2 can be chosen as a local uniformizer. We take for a neighbourhood U of the point considered a rectangle parallel to the axes in the plane. It is affinely mapped with constant dilatation quotient K onto a rectangle parallel to the axes of the  0 -plane with ratio of edges K-times smaller. We have to show that any mapping of a rectangle onto another which agrees on the boundary with this given affine mapping and whose dilatation quotient is everywhere  K agrees with the given mapping and consequently has constant dilatation quotient K. But this follows from what was said in 20 about equality in the estimates given there. Now let us consider a -th order zero of d  2 . If we set again Z p Z r 2 d D d 2 D dz dz 2 and determine the integration constant so that  vanishes at the singular point, then 2

z D  C2 20 This

is made precise in 49ff. and proved in 132ff.

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is a local uniformizer. For even , a rectangle R, covered C2 -times, parallel to the 2 axes, containing 0, over the -plane is mapped conformally onto a neighbourhood U of z D 0. For odd , a rectangle R, covered . C 2/-times, parallel to the axes, containing 0, over the -plane is mapped conformally onto the two-fold covering of a neighbourhood U of z D 0 ramified at z D 0. Under the mapping constructed in 46 from F to some surface F0 , U appears to be mapped onto a corresponding neighbourhood U0 and R onto a rectangle R0 covered several times; the mapping from R to R0 is affine. In order to show that any mapping from U onto U0 which coincides on the boundary with the mapping we constructed and whose dilatation quotient is  K coincides with it in the interior as well, we prove the corresponding statement for the (not necessarily one-to-one) mapping R ! R0 . The proof agrees with that given in 20. The mapping was there supposed to be only vertex-preserving, whereas now the boundary behavior is prescribed exactly. There, the rectangles were planar, whereas they are now winding C2 2 -times, respectively . C 2/-times; under the mappings introduced to compare, ramification points need not be sent to ramification points, but this does not cause any trouble. The image on a sheet of R of a line parallel to an axis has in R0 a length which can be bounded from below; of course, we must integrate over all sheets. It is helpful to have a clear representation of the curves d  2 > 0 in the neighbourhood of a simple zero of d  2 . 48 : If d  2 is allowed to have zeroes, why couldn’t it Rhave p any poles then? We d  2 still converges. If consider on a trial basis a simple pole of d  2 . Here,  D  vanishes in the pole, z D 2 is a local uniformizer. A two-fold covering of a neighbourhood U of z D 0 is conformally mapped onto a rectangle of the -plane; the latter is affinely mapped onto a rectangle of the  0 -plane and it is mapped again conformally through z 0 D  02 onto a two-fold covering of a neighbourhood U0 of z 0 D 0. But if we want to prove extremality of this mapping U ! U0 like we did in 47, we fail in one point: we cannot bound from below the length of the  0 -image of the curves d  2 > 0 under the mappings introduced to compare. This can be done only if we introduce to compare only those mappings U ! U0 that carry z D 0 to z 0 D 0. Our mapping is therefore locally extremal quasiconformal in the vicinity of a pole of d  2 only if the pole is seen as a distinguished point of the principal region. This agrees with the observations we collected in 33.

15 Topological determination of principal regions 49. Of course, mathematical problems cannot be addressed the way they have just been carried out by having a guess at the solution. We are now looking for a way – heuristic, though – by which we will be able to find the extremal quasiconformal mappings systematically. Like before, our interest focuses for the moment exclusively on

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closed orientable surfaces without distinguished points. However, the considerations to follow hold unchanged for arbitrary principal regions as well, so for this reason, we proceed with them straight away in full generality. Two principal regions are called similar or of the same type when one can be mapped onto the other topologically (homeomorphically), that is, when their supports can be mapped onto each other bijectively and in both directions continuously so that distinguished points go to distinguished points. Let H0 be a principal region, H a principal region of the same type and H a topological mapping from H0 to H. We put in the same class as H all topological mappings H 0 from H0 to H that are obtained from H through deformation, i.e., those H 0 for which the mapping H 01 H from H0 onto itself can be deformed to the identity (cf. 139). Thus, the various classes of mappings from H0 onto H bijectively correspond to the congruence classes of the group of all topological self-mappings of H0 quotiented by the normal subgroup of those mappings deformable to the identity. H was defined only up to conformal mappings. If K is a conformal mapping from H to H, then H D KH is a topological mapping from H0 onto H. We declare that H should be defined only up to deformation and up to additional conformal mapping. The classes of conformally equivalent principal regions H form a space R in which two “close” principal regions are carried upon each other by a “small” deformation; we are not going to specify the meaning of this statement, referring instead to the examples from 2 to 8. If we compose H with such a small deformation D from H to the close principal region H? , we get a topological mapping H ? D DH from H0 to H? . We will accept without proof that two small deformations D and D 0 from H0 to H? can always be deformed onto each other; then the class of H corresponds bijectively to the class of H ? because H ? D DH . In this way, the classes of such H form a space R . This space R appears to be mapped on R in a well-defined manner since one can associate to any H its accompanying H D H H0 . We shall deal with this space R in what follows. We can also describe how to pass from the space R to the covering space R as follows: we equip each principal region H with some topological mapping H from H0 onto H defined only up to deformation. Here, composition by conformal mappings of H is ignored. We obtain in this way the points of R . Hence, in general, to the principal region H, which corresponds to only one point in R , correspond several points in R , namely, as many as the index of the group of those topological self-mappings of H that are deformable to conformal mappings in the group of all topological mappings from H onto itself (cf. 143). 50. As an example, we choose the torus, already considered in 5. Let us choose an orientation on a fixed torus H0 and a pair of closed curves C10 ; C20 (a canonical cut) like in 5. If an arbitrary torus H and a topological mapping from H0 onto H are given, this mapping instantly determines an orientation on H as well as a pair of curves C1 ; C2 , where Ci is simply the H-image of Ci 0 . We thus get a well-defined value for the period ratio ! which is however not uniquely defined by H only. Therefore, it is not R2 ( D 2!), but rather the covering space R2 that turns out to be mapped bijectively onto the upper !-half-space.

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Therefore, instead of determining ! uniquely, as we did before, by distinguishing an orientation and a pair of curves on H, we can also determine ! in a unique way by distinguishing a topological mapping from H onto H0 defined up to deformation. And the latter is more likely to be generalisable. Also, the topological decorations for the other examples presented in 2 to 8 can always be carried out by using a topological mapping determined up to deformation on a fixed principal region of the type in question. We notice through the examples that R is always homeomorphic to the  -dimensional Euclidean space. We shall have reasons to suppose this holds true in general. 51. From now on our interest focuses on the points of R only, or, to put it briefly, on the topologically determined principal regions. More precisely, a topologically determined principal region is a pair composed of a principal region and a topological mapping, determined only up to deformation, from this principal region onto a fixed principal region of the same type. The considerations from 15 to 19 now receive their correct meaning if they are applied on topologically determined principal regions and their space R . We just have to define what quasiconformal mappings from one topologically determined principal region onto another look like. Let H1 and H2 be two similar principal regions which are topologically determined by some mappings H1 ; H2 going from a fixed principal region H0 of the same type to H1 ; H2 . Then H2 H11 is a topological mapping from H1 to H2 ; we shall only consider the mappings from H1 to H2 obtained by deforming this mapping H2 H11 , i.e., only these mappings should be considered as mappings from some topologically determined principal region onto another and should be used for comparison in the problem of extremal quasiconformal mappings from a topologically determined principal region onto another. 52. The problem stated in 19 now takes the following form: Let a principal region of a certain type be fixed. One shall give a set of quasiconformal mappings, each with constant dilatation quotient, from some principal region of this type onto another. One shall prove that every mapping of this set is extremal quasiconformal, in the sense that for any mapping between the same principal region obtained by deforming it, the maximum of its dilatation quotient is bigger, or equal only if the new comparison mapping is obtained by composing the former mapping which comes from our set with a conformal mapping. Finally, one shall show that any topological mapping from one principal region of the type considered onto another can be deformed to a mapping from this set. Then, in order to compute the distance defined in 18 between two points in R representing the classes of conformally equivalent topologically determined principal regions, we will deform any topological mapping from a topologically determined principal region onto another to a mapping of that set; this will then give an extremal quasiconformal mapping and the logarithm of its constant dilatation quotient will be the sought-after distance.

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It is especially conjectured – for the time being, still with a huge uncertainty – that in the case of closed orientable surfaces without distinguished points, the set of mappings we constructed in 46 solves the problem we have just stated here.

16 Definition of principal regions through metrics 53. Again, let H0 be a fixed principal region, let H be an arbitrary principal region of the same type and let some topological mapping from H0 to H be given. We assume that this mapping is sufficiently regular. If z D x C iy is a local uniformizer about a point of H0 and if z 0 D x 0 C iy 0 is a local uniformizer for the corresponding point of H, then z 0 is locally a function of z which in general does not satisfy the Cauchy–Riemann differential equations, but anyway behaves sufficiently regularly. We have ˇ 0 ˇ2 ˇdz ˇ D dx 02 C dy 02 D Edx 2 C 2F dxdy C Gdy 2 with  ED

@x 0 @x

2



@y 0 C @x

2 I

@x 0 @x 0 @y 0 @y 0 F D C I @x @y @x @y

 GD

@x 0 @y

2



@y 0 C @y

2 :

A new metric ds 2 , well-defined up to some positive factor, is introduced on H0 through ˇ ˇ2   ds 2 D  ˇdz 0 ˇ D  Edx 2 C 2F dxdy C Gdy 2 : The metric ds 2 remains invariant if another local uniformizer z on H0 is used; modifying the local uniformizer z 0 on H may only change the unimportant factor in ds 2 . The topologically determined principal region H is uniquely determined by the H is another metric ds 2 on H0 which is well-defined up to some factor. Namely, if e  jde z 0 j2 similar principal region which, after mapping it onto H0 , gives rise to a metric e (e z 0 is a local uniformizer on e H) which differs from ds 2 D  jdz 0 j2 only by a factor, 0 2 z 0 from H to e H locally prethen we have jde z j D  jdz 0 j2 ; the mapping z 0 $ e serves lines and is consequently direct or indirect conformal. But a (topologically determined) principal region is defined only up to conformal mapping. If H0 is “arbitrarily” endowed with some positive definite metric ds 2 D Edx 2 C 2F dxdy C Gdy 2 ; well-defined up to some positive factor, then there always exist some principal region H and some mapping from H0 to H associated to it. For a proof, we go back to the notion of Riemann surfaces from 9. It was required there that there exists in every neighbourhood on such a surface a local uniformizer well-defined up to conformal

363

9 Extremal quasiconformal mappings

mappings. One and the same two-dimensional manifold can therefore be turned into a Riemann surface in completely different ways by choosing different systems of local uniformizers. H0 itself is already a Riemann surface; we determine some functions  > 0; x 0 ; y 0 of x; y through (  0 2 )

0 0   @y @x @x @y 0 @y 0 @x 0 2 C I F D C I ED @x @x @x @y @x @y (  0 2 )  @y @x 0 2 : C GD @y @y Since

EG  F D  2

2

@x 0 @y 0 @x 0 @y 0  @x @y @y @x

2 ;

the Jacobi determinant of x 0 ; y 0 with respect to x; y is different from 0; the neighbourhood of the point where z D x C iy is a local uniformizer is mapped bijectively onto a piece of the z 0 D x 0 C iy 0 -plane and we have ˇ ˇ2 ds 2 D  ˇdz 0 ˇ : We see from this last formula that z 0 is exactly determined up to conformal mappings; z 0 defines a new local uniformizer and H0 is thus turned this way, with the help of our metric ds 2 , into a new Riemann surface. That the latter can be seen as a principal region is clear: it can be triangulated like H0 and it has the same distinguished points as H0 ; we will also accept without proof that the boundary components remain fixed since there is no reason why z 0 should be necessarily singular at the boundary; but we can also make this clear through the doubling method that will be introduced in 92ff. Now, one and the same point set H0 has been turned into a Riemann surface in two ways and consequently represents two conceptually distinct principal regions: on the one hand, the former H0 and, on the other hand, if z 0 D x 0 C iy 0 is used as a local uniformizer, a principal region that will be denoted by H. Both are mapped upon each other bijectively, simply with the identity; seen as point sets (not as principal regions), H0 and H are indeed identical and we associate every point of H0 to itself, seen as an H-point. This yields a mapping from H0 to H which clearly defines the given metric ds 2 (only defined up to some factor) directly on H0 . Of course, for the sake of clarity, H may also be mapped conformally onto a principal region which differs from H0 . Here, it has tacitly been assumed that any topological mapping from one principal region onto another can be deformed to some sufficiently regular mapping. Moreover, we have not given the conditions under which ; x 0 ; y 0 can be deduced from E; F; G. 54. We have turned all at once all principal regions of a fixed type into a single point set H0 endowed with various metrics. In this way, a difficulty that has bothered us all the way through since 13 ceases immediately, namely, the never sharply apprehended

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Oswald Teichmüller

notion of neighbourhood in R , respectively in R : we shall now say that two principal regions are close in R , respectively in R when the corresponding metrics on H0 only slightly differ. But there is still something to bear in mind. The principal region H as such does not determine the metric on H0 yet; it much more determines it, only up to some factor, if a definite mapping H from H0 to H is underlaid. If H is replaced by another mapping H 0 , then H 0 D AH , where A is a self-mapping of H0 . Hence, if ds 2 .p/ is associated to H (ds 2 is a “function” of the point p of H0 ), then ds 2 .Ap/, seen as a “function” of p, is associated to H 0 . Therefore, if we want to describe principal regions H through metrics ds 2 on H0 , which are naturally defined only up to some factor, two metrics are to be put in the same class if and only if one is carried on the other by the rule ds 2 .p/ ! ds 2 .Ap/; here, A is a (sufficiently regular) topological self-mapping of H0 . We are particularly interested in topologically determined principal regions H for which we take into account, besides the mapping H from H0 to H, only those mappings H 0 that can be deformed to H . Therefore, writing H 0 D HA, only the mappings A that can be deformed to the identity are taken into account. So now, we let two metrics ds 2 ; ds 02 on H0 be in the same class if and only if ds 02 .p/ D .p/ds 2 .Ap/ ; where  is a positive function and A a self-mapping onto H0 deformable to the identity. These classes of metrics ds 2 defined over H0 are in one-to-one correspondence with the classes of conformally equivalent topologically determined principal regions, i.e., with the points of R . 55. Let H1 and H2 be two topologically determined principal regions defined through the metrics ds12 ; ds22 on H0 . What corresponds to quasiconformal mappings from H1 to H2 ? As H1 and H2 are mapped onto H0 , to each mapping from H1 onto H2 corresponds a mapping A from H0 onto itself. Since we are dealing with topologically determined principal regions, according to 51, only enter into consideration those mappings A that can be deformed to the identity. But we can replace ds22 .p/ by ds22 .Ap/; hence, now, we have to compare the two metrics ds12 .p/ and ds22 .Ap/, which are both to be thought of as relative to the same point p of H0 . If they are proportional (i.e., ds12 .p/ D .p/ds22.Ap/), then the mapping is conformal. In the general case, the local uniformizers z10 .p/; z20 .p/ of H1 ; H2 are thought of as determined, like in 53, through ds12 .p/ D 1 jdz10 j2 I

ds22 .Ap/ D 2 jdz20 j2 :

The dilatation quotient D of the locally one-to-one mapping z10 $ z20 depends, as a function of p, only on the metrics ds12 .p/ and ds22 .Ap/ but not on the particular choice of local uniformizers z10 ; z20 . It must be possible to compute it algebraically from the coefficients E; F; G of both metrics, as we shall do it soon in a special case.

9 Extremal quasiconformal mappings

365

The problem of the extremal quasiconformal mapping from H1 onto H2 is now replaced by the following problem: Let two “arbitrary” metrics ds12 and ds22 be given on the principal region H0 . One shall find a mapping A from H0 onto itself, which can be deformed to the identity, such that the maximum of the dilatation quotient D for the pair of metrics ds12 .p/; ds22 .Ap/ is the smallest. The dilatation quotient is here an algebraic function of the coefficients E; F; G of both metrics which depends only upon the ratios EW F W G, which is unchanged if the independent variables x; y are simultaneously replaced by some others in both quadratic forms ds12 .p/; ds22.Ap/, which is D 1 if these metrics are proportional and is > 1 otherwise, and which measures the deviation from proportionality and whose logarithm satisfies the triangle inequality. Little thinking will easily bring the problem in a form corresponding to that of 52. 56. We introduce a simplification. We do not compare two arbitrary metrics on H0 , but we rather compare the initial metric ds 2 D jdzj2 D .dx 2 C dy 2 / of H0 , where z is a local uniformizer of H0 , with another one, i.e., we identify H0 with H1 . This can be done either by setting H0 , in itself arbitrary, to be simply equal to H1 – equal to one of the principal regions to be compared –, or otherwise by describing with z a local uniformizer on H0 not associated to H0 but to H1 . Hence now, the metric jdzj2 D .dx 2 C dy 2 / is given on the principal region that was before called H0 D H1 and is now called H but, in addition, a second metric ds 2 D Edx 2 C 2F dxdy C Gdy 2 is given, which pinpoints a principal region that was previously called H2 and which is now called H0 . We are going to compute the dilatation quotient. We think of  > 0 and z 0 D x 0 C iy 0 as determined by jdz 0 j2 D ds 2 D Edx 2 C 2F dxdy C Gdy 2 : If ƒ1 and ƒ2 with ƒ1  ƒ2 are the (positive) eigenvalues of the form ds 2 , then ƒ2 jdzj2  jdz 0 j2  ƒ1 jdzj2 ; and the bounds are attained. We thus have ˇ 0ˇ r ˇ 0ˇ r ˇ dz ˇ ˇ dz ˇ ƒ1 ˇ ˇ ˇ ˇ D ƒ2 max ˇ D I min ˇ ˇ dz  dz ˇ  and the dilatation quotient

ˇ 0ˇ s ˇ ˇ max ˇ dz dz ˇ ƒ1 ˇ ˇ D : DD ˇ dz 0 ˇ ƒ2 min ˇ dz ˇ

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Oswald Teichmüller

For more comfortable computations, we introduce K  1 by   p 1 1 DC I D D K C K2  1 D earccos K : KD 2 D s

s

Then 1 KD 2

ƒ1 C ƒ2

ƒ2 ƒ1

!

ƒ1 C ƒ2 D p : 2 ƒ1 ƒ2

But, as it is well-known, ƒ1 C ƒ2 D E C G I hence

ƒ1 ƒ2 D 2 D EG  F 2 I

p E CG I D D K C K2  1 : 2 We also come up in this way with the formula presented at the beginning of 15. D

p

EG  F 2 I

KD

57. If we now summarize 52, 55 and the final formulas in 56, we obtain the following statement of the problem: Let a principal region H be given. One shall find a set of metrics on H, ds 2 D Edx 2 C 2F dxdy C Gdy 2

.z D x C iy local uniform. on H/ ;

with the following properties: (1) If one computes, for an arbitrary metric ds 2 D Edx 2 C 2F dxdy C Gdy 2 , the “dilatation quotient” D from the formulas D

p EG  F 2 I

KD

E CG I 2

D DKC

p

K2  1 ;

then the metrics of the set have constant dilatation quotient D D K D const. (2) If for a metric ds 2 there exists a mapping A from H onto itself which can be deformed to the identity, for which ds 2 .Ap/ belongs to the set and which has constant dilatation quotient K, then the maximum of the dilatation quotients of ds 2 is itself  K and equal to K only if ds 2 .p/ D ds 2 .Ap/. (3) Such an A exists for any “arbitrary” metric ds 2 . In particular when H is a closed orientable surface without distinguished points, then the conjecture stated in 46 translates into the following: let one choose some everywhere finite quadratic differential d  2 different from 0 as well as some K  1 and set jd  2 j  0, d  2 ¤ 0 regular) and the vectors k associated to them. A vector k ¤ 0 is associated 2 to any c jdd  2 j because if

d  2 D a1 d 12 C C a d  2 ¤ 0 then k D c

.a real/ ;

“ ˇ 2ˇ ˇ d  ˇ d 2 ˇ ˇ ˇ 2 ˇ < d  2 dz F dz

and therefore

ˇ 2ˇ ˇd ˇ ˇ ˇ ˇ dz 2 ˇ dz > 0 : M

“ a1 k1 C C a k D c

Exactly as shown in 89, it is also seen that different k are associated to different 2 c jdd  2 j : c is uniquely defined by k and d  2 is uniquely defined up to some positive factor. Hence, the map d 2 0; c 7! k jd  2 j is one-to-one. Each d  2 ¤ 0, defined only up to some positive factor, is attached by this map to a k ¤ 0, defined only up to some positive factor. Hence, if again one sets d  2 D a1 d 12 C C a d  2 , then to any .a1 ; : : : ; a / is associated some .k1 ; : : : ; k / ( and  are the arbitrary positive factors). But .a1 ; : : : ; a /, respectively .k1 ; : : : ; k /, are rays emanating from the origin of the real  -dimensional linear space. Therefore, for each ray in the .a1 ; : : : ; a /-space there is some associated ray in the .k1 ; : : : ; k /-space; like in 90 this map is also one-to-one. (Rays were described there by their intersection with the unit sphere which was introduced completely arbitrarily.) According to the topological lemma mentioned in 90, under this association, each ray of the k-space is indeed associated to a ray of the a-space. We conclude like in 90 that 2 Each vector k ¤ 0 corresponds to some suitable c jdd  2 j (d  2 D a1 d 12 C C a d  2 ); c is uniquely defined by k while .a1 ; : : : ; a / is only defined by it up to some positive factor. d2 dz 2 This means that within the class of every B jdzj 2 , either 0 or some c jd  2 j is to be found which is, according to 101, extremal: its norm is constant, equal to 0, dz 2 respectively to c, and the norm of any B jdzj 2 in its class but different from it has greater maximum. We set kkk D 0, respectively kkk D c, if k D 0, respectively if k corresponds to d2 dz 2 c jd  2 j . If now k represents the class of B jdzj 2 , then according to 66 we set likewise ( dz 2 0 B jdzj2 D kkk D c

:

423

9 Extremal quasiconformal mappings

Like in 91, we have for this metric kakk D jaj kkk .a real!/ and the triangle inequality

kk C k0 k  kkk C kk0 k

holds with equality only in case k0 D  k (  0) or k D 0. Vectors k therefore form a  -dimensional linear metric space L . Its metric is determined by the convex ovoid kkk D 1. Let A be the space of vectors a D .a1 ; : : : ; a /; we set d  2 D a1 d 12 C C a d  2 and a k D a1 k1 C C a k . One could associate to each point a of A the hyperplane a k D 1 of L , and to each point k of L the hyperplane a k D 1 of A. This contact transformation (polarity) transfers the convex ovoid kkk D 1 in the convex surface “ ˇ 2ˇ ˇd ˇ ˇ ˇ ˇ 2 ˇ dz D 1W M dz to the point a of A with

ˇ 2ˇ ˇd ˇ ˇ ˇ ˇ 2 ˇ dz D 1 M dz

“

corresponds the tangent hyperplane to kkk D 1 in the point k associated to The convex ovoid is an ellipsoid if and only if “ ˇ 2ˇ ˇd ˇ ˇ ˇ ˇ 2 ˇ dz D 1 M dz

d  2 32 . jd  2 j

is an ellipsoid in A. 110. If the support M of the principal region H is closed and orientable, we let d 12 ; : : : ; d 2 be a basis of the regular quadratic differentials on H with respect to complex coefficients. It only needs to be set 2 d C D

1 2 d i 

. D 1; : : : ;  / ;

2 in order to obtain a basis d 12 ; : : : ; d 2 with respect to real coefficients. Hence

 D 2 I 32 For

0 d  2 ¤ 0 is another regular quadratic differential, then if d  02 D a10 d 12 C    C a  ˇ “ ˇ 02 ˇ “ ˇ 2 ˇ ˇ ˇ d2 ˇ d  ˇ ˇ d ˇ ˇ a  k0 D ˇ dz 2 ˇ < d  02 dz ˇ dz 2 ˇ dz D a  k D 1 ; M M

i.e., for all boundary points k0 of the convex ovoid body, a  k 1 holds and D 1 only for k0 D k.

424

Oswald Teichmüller

 is an even integer. Furthermore, ( “ k C i kC D

< B M

“ D

B M

d 2

)

dz 2

d 2 dz 2

(

“ dz C i

1 d 2 < B 2 i dz M

) dz

dz :

These k Ci kC , for  D 1; : : : ;  , can therefore be used as complex coordinates in L D L2 which becomes this way a space of complex dimension  . Then everything could be expressed as we did in 88 to 91 when the special case of no distinguished points was discussed. In particular, a one-parameter group of rotations likewise acts upon the convex ovoid. The integer  D 2  0 is called the reduced dimension. However, for bordered or non-orientable principal regions we set the reduced dimension  to be equal to  . Then in all cases  D dimDW2 : According to 106, a bordered or non-orientable region has the same reduced dimension as its double. 111. If H is a closed and orientable principal region, hence arising from distinguishing h  0 points on a closed orientable Riemann surface of genus g  0, then according to 108 and 99,  D 2dimDW2 I

 D 2dim

1 DW

and, as this was already observed in 98, according to 43, dimDW2  dim This gives

1 D 3 C 3g C h : DW

   D 6 C 6g C 2h :

Now if H is a bordered or non-orientable principal region with characteristic numbers g; ; n; h; k (see 14), then according to 108 and 99  D dimDW2 I  D dim

1 DW

and according to 98, dimDW2  dim This gives

1 D 6 C 6g C 3 C 3n C 2h C k : DW

   D 6 C 6g C 3 C 3n C 2h C k :

9 Extremal quasiconformal mappings

425

This formula thus holds in case  C n > 0 and remains true if  D n D k D 0, yielding back the formula    D 6 C 6g C 2h valid for closed orientable principal regions. Hence, in all cases we have the Dimension formula    D 6 C 6g C 3 C 3n C 2h C k : It should be noticed here that from degDW > 0 follows dim1=DW D 0, therefore  D 0. Hence,  is positive at best when degDW  0, which just corresponds to finitely many cases that will be completely listed later on. The dimension formula enables in each case to compute  from g; ; n; h; k. 112. We will now give another, heuristic, justification for the dimension formula which does not rely upon the Riemann–Roch theorem nor upon the theory of infinitesimal quasiconformal mappings. It was already known to Klein33 that when H is a surface of algebraic genus G without distinguished points, ( 6.G  1/ for  C n D 0  D 3.G  1/ for  C n > 0 : Introducing herein the value ( g GD 2g C  C n  1

for  C n D 0 ; for  C n > 0

we get

   D 6 C 6g C 3 C 3n : So it only remains to show that    increases by 2 each time one inner point is distinguished and by 1 each time one boundary point is distinguished. Let the principal region H? arise from the principal region H by distinguishing the still undistinguished inner or boundary point p on H. Let a (a D 0; 1 or 2) be the dimension of the set of points in which p can be carried by conformal self-mappings of the topologically determined principal region H (that is, by conformal mappings from H onto itself, which, seen as topological mappings, can be deformed to the identity). Let  ? and ? have the same meaning for H? as  and  for H. The conformal self-mappings of the topologically determined principal region H? are the conformal self-mappings of H that fix p inside this a-dimensional set, hence ? D   a : H provides  parameters; then H? arises by determining the a-dimensional set containing p, which gives 2  a or 1  a parameters. Therefore, (  C 2  a .p inner point/ ?  D  C 1  a .p boundary point/ : 33

See the footnote in 26.

426

Oswald Teichmüller

By subtracting, a disappears: ( 2   D C 1 ?

?

.p inner point/ .p boundary point/ ;

which was to be proved.

25 Passing to finite mappings. R  as a Finsler space 113. We have been occupied since 61 with infinitesimal quasiconformal mappings and we solved in 109 the problem of extremal infinitesimal quasiconformal mappings. It is now time to go back to finite quasiconformal mappings between two principal regions of the same determined topological type. Let an extremal quasiconformal mapping from the principal region H onto the principal region H0 with constant dilatation quotient K > 1 be given. We get from 39 a one-parameter family of principal regions H.K ? / (1  K ?  K) with H.1/ D H and H.K/ D H0 as well as definite extremal quasiconformal mappings from H to H.K ? / by requiring that the mapping H ! H.K ? / has the same dilatation quotient K ? and the same direction field as the mapping H ! H0 . This can also be described with the help of metrics. Let ds 2 D Edx 2 C 2F dxdy C Gdy 2 D ƒjdzj2 C 0 : We now consider the family of metrics ƒjdzj2 C t 0

9 Extremal quasiconformal mappings

427

(except for t D 0 of course) but the dilatation quotient K ? is computed from tjH j K2  1 K ?2  1 D D t : K ?2 C 1 ƒ K2 C 1 When t increases from 0 to 1, K ? increases from 1 to K. Our mapping H ! H.K ? /, known since 39, is therefore described through the metric ƒjdzj2 C

K ?2  1 K 2 C 1 0 is constant and d  2 is a regular quadratic differential of H, because these are the extremal infinitesimal quasiconformal mappings. Hence the metric ds 2 associated to the limit extremal quasiconformal mapping H ! H0 should be   d 2 D .jd  2 j C c 0 and Yƒ is its image under the mapping Y , then Z Z jd j  jd j  2M : Yƒ

ƒ

For the proof, we naturally introduce once more the metric ˇ ˇ ds 2 D ˇd  2 ˇ on the support M of H. Let us consider, about any point p of M, the neighbourhoods U.p; / defined for sufficiently small  as follows: R U.p; / is the set of all points that can be connected with p by a curve of “length” jd j < . The number  is always chosen so small that the neighbourhood lying on M is planar and simply-connected and furthermore contains no distinguished points of H nor zeroes of d  2 (except possibly for p itself). In U.p; /, set Z p D d 2 I p

then,

8 ˆ 0 line ƒ and with ends P; Q. Since by assumption the distinguished points were avoided as Q ƒ can be deformed on E in the same way as on M to a line ƒ was deformed to ƒ, Q which connects the endpoints P; Q of ƒ; we equally denote it by ƒ. Q lying over ƒ Should we have now Z Z jd j < jd j ; Q ƒ

ƒ

then we would construct the shortest geodesic line connecting P and Q on E. Its length would all the more be smaller than the length of ƒ. But ƒ is also geodesic. The points P and Q on E would then be connected by two different geodesic lines. There would therefore be on the simply-connected surface E a geodesic two-gon, that is, a simply-connected domain whose boundary consists of two geodesic arcs with common ends R; S . We shall prove that this fact leads to a contradiction. Let T1 ; : : : ; Ti be boundary points different from R and S and let U1 ; : : : ; Ul be the inner points of the geodesic two-gon that are zeroes of d  2 (with respect to the local uniformizers of E! Hence poles do not arise). We shall denote the multiplicity of these zeroes by  for short, omitting indices, although they almost always have the same value. R p We surround R; S; T1 ; : : : ; Ti ; U1 ; : : : ; Ul by small circles jj D const ( D p d  2 , cf. 137) and connect with curves the circles about U1 ; : : : ; Ul with the boundary of the two-gon. The Gauß-Bonnet formula holds inside the simplyconnected domain that remains: Z X d arg d  C a D 2 ; «

R where a is the sum of the external angles. Now d arg d  vanishes by integrating over the geodesics arg d  2 D const and over the twice-traversed lines connecting the circles about U1 ; : : : ; Ul with the two-gon’s boundary. Let the angle of Rthe two-gon be equal to ˇ in R, to  in S and to ı in T . An easy computation of « d arg d  P

454

Oswald Teichmüller

over the circle arcs and of the sum of the external angles gives     X  i  C2 C2 C2  ˇ C   C ı  2 2 2 D1 C

l  X

D1

2 

C2 2 2



D 2 :

Of course the value  depends on the point. We have ˇ > 0 and  > 0. Hence the first two summands are, taken together, smaller than 2. Further, we always 2 , since we are considering geodesic arcs (see above). Consequently, have ı  C2 the third sum contains nothing but non-positive summands. The fourth sum contains only negative summands. So the left-hand side cannot be equal to 2; contradiction.  139. In 137 we started, as a prerequisite, with a self-mapping Y of the principal region H which could be obtained from the identity through deformation. That is, there should be a family of topological self-mappings of H p ! q.p;  / ; depending continuously upon a parameter  (0    1), which should be the identity for  D 0 and equal to Y for  D 1. Therefore, a point q.p;  / is associated to each point p of H and to each  (0    1). This map is continuous and is, for fixed  , a homeomorphism from the support M of H onto itself. The distinguished points remain fixed, i.e., q.ps ;  / D ps for the h distinguished points ps . Finally, q.p; 0/ D pI

q.p; 1/ D Y p :

For a fixed p, when  runs from 0 to 1, the point q.p;  / moves along a continuous curve which connects p to Y p; this curve will be called the train curve of p. Second lemma: there exists a positive number M with the property that every train curve can be deformed to a curve with length  M while keeping the endpoints fixed and avoiding the distinguished points. Proof. Because of the uniform continuity of the function q.p;  /, there exists ı > 0 with the following property: If j2  1 j  ı and if q.p; 1 / lies in U.pn ; n /, then q.p; 2/ lies in U.pn ; 2n /. We cut the interval 0    1 into v intervals   C1 whose lengths are  ı. Let p be fixed on H. Let q.p;  / lie in U.pn ; n /. Then the piece     C1 of the train curve q.p;  / lies in U.pn ; 2n / and can be deformed within to a curve of length < 3n  3 max.1 ; : : : ; N /. If pn is a distinguished point of H, this can also be done without passing through pn . We proceed this way with all the v pieces.  We can therefore take M D 3v max.1 ; : : : ; N /.

9 Extremal quasiconformal mappings

455

140. Proof of the metric-topological lemma (137) Let p.t/ (0  t  1) be the parametric representation of the line ƒ with length L. We set q.p.t/;  / D r.t;  / .0  t  1; 0    1/ : Q with length LQ has therefore the parametric representation The deformed line ƒ r.t; 1/ .0  t  1/. Now, on the square 0  t  1; 0    1, the segment  D 0 can clearly be deformed to the concatenation of segments t D 0;  D 1; t D 1. If the continuous mapping .t;  / ! r.t;  / is applied to this deformation process then it is seen that ƒ can be deformed to a curve which consists of the train curve of p.0/ D P , Q and of the train curve of p.1/ D Q followed backwards. According to the second in ƒ lemma, each of the two train curves can be further deformed to a line connecting its endpoints with length  M . Thus, ƒ is deformed to a curve with length  2M C LQ without having covered any distinguished point. According to the first lemma, Q L 2M C L or

LQ  L  2M :



28 Conformal mappings of a principal region onto itself 141. So far, our study has been based upon the space R of all classes of conformally equivalent topologically determined principal regions of a fixed type. The space R , first introduced in 13, is derived from R by dropping the topological determination. According to 112, it turns out that R and R have the same dimension  . We will now have a closer look at this relationship which is essentially of topological nature. In the thirteen cases with  > 0 listed in 125, we always find a definite connected, -parameter continuous group C of conformal self-mappings of the topologically determined principal region H; this group is a normal subgroup of the group K of all conformal self-mappings of H. We also explicitly put into K those mappings that cannot be deformed to the identity, even though, according to 51, these mappings actually transport H into another topologically determined principal region which just corresponds to the same point in R . C is made up of those conformal self-mappings of H that can be deformed as conformal mappings to the identity, i.e. that can be connected to 1 within K. In all the other cases, where  D 0, we set C D 1. K is a subgroup of the group G of all topological self-mappings of H. Mappings from G also map H onto itself, but only as a principal region and not as a topologically determined principal region, that is, it might not be possible to deform them to the identity. The group A of those topological self-mappings of H that can be deformed to the identity is a normal subgroup of G. Clearly, C  A and consequently, C  K \ A (\ means intersection). We now claim that CD K\A:

456

Oswald Teichmüller

In words: Every conformal mapping from H onto itself which, considered as a topological mapping, can be deformed to the identity, can also be deformed to the identity while keeping conformality. Proof. All cases with

degDW  0

were listed in 125;  was in these cases always positive. The claim is easily checked in these thirteen cases (be careful for the Klein bottle!). In the cases where degDW > 0 the claim reads

K\AD1:

When the support of the principal region is not closed nor orientable, we can easily pass to the doubles. It is therefore sufficient to prove the following: Let H be a closed orientable surface of genus g with h distinguished points; let degDW D 2.g  1/ C h > 0 : Then any direct conformal self-mapping of H which can be deformed to the identity as a topological mapping is necessarily the identity. We prove this by enumerating cases according to genus. The condition 2.g  1/ C h > 0 splits namely into 1. g D 0, h  3; 2. g D 1, h  1; 3. g > 1. I. g D 0, h  3. A direct conformal mapping of the sphere onto itself which fixes at least three points is the identity. II. g D 1, h  1. Let the torus be conformally developed like in 6 over the punctured u-plane; let !1 ; !2 be a pair of primitive periods. Each direct conformal mapping of the torus onto itself corresponds to a direct conformal self-mapping u0 .u/ of the u-plane, hence u0 D au C b. Should the mapping be deformed to the identity, then we must have u0 .u C !i / D u0 .u/ C !i (i D 1; 2); therefore a D 1, u0 D u C b. Moreover, if at least one point must be fixed, then b D m!1 C n!2 and the mapping of the torus onto itself is the identity. III. g > 1. A conformal self-mapping of a closed oriented surface F which can be deformed to the identity as a topological mapping carries any everywhere finite one-dimensional differential du into another and also carries every cut of the surface into a topologically equivalent (homologous, even homotopic) cut. But since the differential is uniquely defined by its periods for a cut which is only determined up

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to deformation, any everywhere finite one-dimensional differential du is carried to 1 of two such differentials are also invariant. But itself. Hence, the quotients du du2 these quotients generate the function field of the surface; they generate an invariant subfield of relative degree 2 only in the hyperelliptic case; but in the hyperelliptic case as well, all functions on the surface are invariant, since not only the quotients du1 but also du itself are invariant. Any conformal self-mapping of a closed oriented du2 surface for which all functions on the surface are invariant is but the identity.  142. Our result

C DK\A

implies that the quotient group KjC, which is by the way always finite, can be mapped isomorphically in a uniquely defined manner on a subgroup of the quotient group GjA: KjC ' KAjA  GjA : KA is the group of the topological self-mappings of the principal region H that can be deformed to conformal mappings and A is the normal subgroup of KA. Let G0 and A0 have the same meaning for the topologically determined principal region H0 as G and A have for the topologically determined principal region H. Then G0 jA0 ' GjA and this isomorphism is uniquely defined. For, if H is any topological mapping from H to H0 , then simply G0 D H GH 1 ;

A0 D H AH 1 I

G0 jA0 simply arises from GjA by transforming it using H . But it is still possible to replace H by HA, where A is any self-mapping of H which lies, because we are dealing with topologically determined principal regions, in A. The isomorphism G 0 D HGH 1

.G 2 G ; G 0 2 G0 /

between G and G0 that carries A to A0 can therefore be replaced by the isomorphism G 0 D HAGA1 H 1 : Now this isomorphism is in general different from the first one, but since A lies in A, it provides the same isomorphism as the previous one, GjA $ G0 jA0 , between the quotient groups. The elements of the quotient group GjA, the mapping classes, hence have a meaning which is independent of any particular principal region. We set GjA D F and call F the mapping class group of our type of topologically determined principal regions. For each topologically determined principal region of a given type, GjA

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is therefore isomorphically mapped in a uniquely defined way on the mapping class group F which is fixed once and for all. Also, KjC is isomorphically mapped in a uniquely defined way on a subgroup of F. We shall always tacitly identify in what follows GjA with F and also see KjC as a subgroup of F. 143. Any topologically determined principal region H is now carried through an element G of the corresponding group G to a principal region GH which differs from H only by its topological determination: if H0 is some fixed principal region of the same type as H and if H is topologically determined, following 49, by some topological mapping H from H0 to H which is well-defined only up to deformation, then GH is the same principal region and it is topologically determined by the mapping GH from H0 to H. As a result, this clearly depends only on the congruence class of G mod A, hence on the elements of GjA D F. Each element GA of F carries each topologically determined principal region H to the topologically determined principal region GH which differs from H only by topological determination. Here, of course, conformally equivalent, topologically determined principal regions are transferred to conformally equivalent ones. We therefore get a presentation of F by self-mappings of the space R of classes of conformally equivalent, topologically determined principal regions. This means that such mappings are homomorphically associated to the elements of F. The metric of R is invariant under these mappings. For let E be some extremal quasiconformal mapping from the topologically determined principal region H onto the topologically determined principal region H0 ; let F be any element of F and let G; G 0 be the mappings from H; H0 that correspond to F (F D GA, F D G 0 A0 ); then E is at the same time an extremal quasiconformal mapping from GH to G 0 H0 , hence ŒH; H0 D ŒGH; G 0 H0 : Any topologically determined principal region H is carried precisely under the mappings of KA to a conformally equivalent one. To the non-topologically determined principal region H hence correspond as many points of R as the index .GW KA/ is (cf. 49). Let N0 be the group of the elements of F that fix each point of R ; let N be the subgroup of K containing C with NjC D N0 : The subgroup N0 is normal in F and N is normal in K. For each H, NjC can be viewed as a subgroup of F and N0 is the intersection of all these subgroups. R arises from R when the points of R that are equivalent under F (or under FjN0 ) are identified. We naturally conjecture that FjN0 acts on R properly discontinuously.

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144. Together with KjC, N0 is always finite for sure, but N0 can also contain other elements than 1. For example, this is the case if H is the torus, which develops over the u-plane with pairs of primitive periods   !2 !1 ; !2 = >0 : !1 To each topological mapping G from H onto itself corresponds an integral linear substitution of periods !10 D a!1 C b!2 ; with determinant

!20 D c!1 C d!2

ad  bc D ˙1 :

It is defined in the following way: G carries some (hence any) path on H along which u increases up to !i over a path along which u increases up to !i0 . G keeps orientation or reverses it depending on whether is C1 or 1. G can be deformed   the determinant  to the identity if and only if ac db D 10 01 .D 1/. GjA D F is therefore isomorphic to the group of all integer matrices with determinant ˙1. Every conformal mapping from H onto itself yields, by passing to the u-plane, the mapping u0 D u C  or u0 D uN C  ; that is subject to the only condition !1 D a!1 C b!2 !2 D c!1 C d!2

(

 or

!1 D a!1 C b!2 ; !2 D c!1 C d!2

where a; b; c; d are integers with determinant ad  bc D ˙1. C is made up of the mappings u0 D u C  : For this reason, KjC consists of 2; 4; 8 or 12 elements: of 8 elements when a quadratic lattice appears; of 12 elements when a lattice corresponding to an equilateral triangle appears; of 4 elements for a symmetric lattice (rectangle or rhombus) and of 2 elements in general cases. For, the mapping u0 D u ;

 1 0  which lies in K but not in C, is always available  1 0 and the matrix 0 1 corresponds to it. This matrix generates with the unit 0 1 the subgroup of matrix group corresponding to N0 . The quotient group FjN0 is isomorphic  to a group of motions of R2 which is the upper half-plane =! > 0: the matrix ac db corresponds to the nonEuclidean motion !0 D

c C d! ; a C b!

respectively ! 0 D

depending on whether ad  bc D ˙1.

c C d !N ; a C b !N

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145. Another example: Every closed orientable surface H of genus 2 is hyperelliptic and, as such, has a distinguished order-2 conformal self-mapping yielding, when each point is identified with its image, a closed surface of genus 0; this mapping corresponds to an order-2 element in F. But we must first show that the same element of F always arises here, whatever the topologically determined surface of genus 2 we started with; then it will correspond to the group N0 . It is therefore to be shown: Let H and H0 be two closed orientable surfaces of genus 2, let N and N 0 be the self-mappings distinguished above and let H be a topological mapping from H to H0 . Then HNH 1 can be deformed to N 0 : HNH 1 D N 0 A0 ; A0 in A0 : The claim remains unchanged if H is deformed. We deform H to an extremal quasiconformal mapping and claim: If H is extremal quasiconformal, then HNH 1 D N 0 :

Proof. H can be thought of as the two-fold covering of the whole z-sphere with six ramification points r1 ; : : : ; r6 . The most general everywhere finite quadratic differential is   2 az C bz C c dz 2 2 d D : .z  r1 / .z  r6 / But this is at the same time the most general regular quadratic differential of the z-sphere with six distinguished points r1 ; : : : ; r6 . Therefore, if the most general extremal quasiconformal mapping from the z-sphere with distinguished points r1 ; : : : ; r6 to the z 0 -sphere with distinguished points r10 ; : : : ; r60 from which topological determination follows is constructed, then it arises from d  2 in the way given in 114 and if, using this mapping, the two-fold covering of the z-sphere with ramification points r1 ; : : : ; r6 is mapped onto the two-fold covering of the z 0 -sphere with ramification points r10 ; : : : ; r60 , then this is the most general extremal quasiconformal mapping H from H onto another topologically determined, closed, oriented surface H0 of genus 2. But here, N and N 0 simply swap sheets and, for these mappings H , the meaning of HNH 1 D N 0 is clearly understood.  This would not have been established so quickly without the theory of extremal quasiconformal mappings. A planar domain with three boundary curves admits, as this is well-known, exactly one indirect conformal self-mapping which transports each boundary curve on itself as a whole. Here also, the corresponding element of F is independent of the domain directly chosen and lies for this reason in N0 .

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146. Let K and K0 be conformal self-mappings of the principal region H, resp. H0 (hence K in K, K0 in K0 ). The congruence class of K mod C agrees with the congruence class of K0 mod C0 , seen as elements of F. Let E be the extremal quasiconformal mapping from the topologically determined principal region H onto H0 that arises, according to 114, from the regular quadratic differential d  2 and from the dilatation quotient K. Then,

K01 E K is a mapping from H to H0 which can be deformed to E and with the same dilatation quotient. Consequently, K01 E K D C 0 E or

E K D .K0 C 0 /E ;

where C 0 is an element of C0 , i.e., a conformal mapping from H on H0 which can be deformed to the identity. In particular, E and E K have the same direction field d  2 > 0 on H, that is, d  2 is multiplied under K by a positive factor. This factor must be equal to 1 since the convergent integral “ d H

is invariant by K. When H is orientable and K is indirect conformal, we must of course pass immediately after applying K on d  2 to the complex conjugate. This is obvious and will not be further mentioned. We therefore see that: d  2 is invariant by K. (This holds in particular for all K in C because we can take here K0 D 1; in 135 and 136, we saw that d  2 is only multiplied by a constant of norm 1 under K in C.) Now let K be in K and d  2 be a non zero, regular quadratic differential which remains invariant under K. Let E be the extremal quasiconformal mapping from H onto H0 which, according to 114, corresponds to d  2 and to the dilatation quotient K. Then

K0 D E KE 1 is a self-mapping of H0 which, as this is easily seen, is conformal; the congruence class of K mod C is, as an element of F, equal to the congruence class of K0 mod C0 . This is a converse of the above result, and it should be noticed that this conclusion holds for any dilatation quotient K  1. d  2 is invariant under K if and only if there is in the mapping class defined by K in F a conformal mapping K0 from H0 D EH onto itself. If one element of F lies in KjC for two different topologically determined principal regions H; H0 , then this also holds for all principal regions on the geodesic passing through the points representing H and H0 in R .

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This can also be seen directly as follows: according to 143, to any element of F is associated an isometric mapping from R to itself which carries geodesics to geodesics; if this mapping leaves two distinct points of R fixed, then the uniquely defined geodesic going through these two points is fixed as well and, because lengths are preserved, this geodesic is even carried to itself pointwise. We can reason in the same way for the case of closed orientable principal regions and complex geodesics: they are carried pointwise to themselves when K is direct conformal and undergo a reflexion when K is indirect conformal. 147. By a “geodesic manifold” in R we mean a nonempty subset of R which, given any two distinct points in it, also contains the whole geodesic running through these two points. Whenever an element F of F fixes some points of R , this builds a geodesic manifold whose points correspond to the topologically determined principal regions H for which the quotient group KjC, seen as a subgroup of F, contains the given element F of F. The intersection of geodesic manifolds is again a geodesic manifold. This is the reason why the points of R that remain invariant under a subgroup U of F always form a geodesic manifold (or the empty set) whose points correspond to the topologically determined principal regions for which KjC contains the given subgroup U. When a point of such a geodesic manifold is known, then all the others can be given and in particular the dimension of this manifold can be computed. Given the principal region H, suppose U is contained in KjC; then according to 146, it suffices to determine all regular quadratic differentials of H that are invariant by U; from them and from any arbitrary dilatation quotient K  1, the construction given in 114 then yields all topologically fixed principal regions H0 of the sought-after manifold. In particular, the dimension of this manifold is equal to the maximal number of reallinearly independent regular quadratic differentials of H that are invariant by U. An algebraic proof of this rule for computing the dimension should be looked for. A conformal mapping from a principal region H onto itself leaves all regular quadratic differentials of H fixed if and only if it lies in N (i.e. if and only if its congruence class mod C lies in N0 ). For only then this geodesic manifold is the whole R . This is a simpler way (more abstract though) to the result of 145. It must be noticed here as in general that a quadratic differential of an oriented principal region is said to be invariant under an indirect conformal mapping if it is carried by it to its complex conjugate. It has also been frequently referred to “invariance only up to complex-conjugate” before, though at that time this referred to nonoriented principal regions. The chief point is always the direction field d  2 > 0. 148 . Given some finite subgroup U of F, we now ask whether there might exist some principal regions H for which U is contained in KjC. (We already mentioned before that KjC is always finite; this is why only finite subgroups U will enter this discussion). We require that there exist for the principal region H0 of a given type representatives of the congruence classes in U of G mod A which form a subgroup

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U? of G. Hence, U? must be a subgroup of G so that U? \ A D 1 I

U? AjA D U

.hence U? ' U/ :

Furthermore, the elements of U? should be sufficiently regular. Then we take the support M0 of H0 and identify points of M0 that are equivalent with respect to U? (i.e. that can be carried to each other by elements in U? ); this still yields a surface which can be topologically mapped onto some finite Riemann surface M? (we shall not prove it!). H0 is topologically mapped onto a principal region H whose support is a finite covering of M? , and U? goes under this mapping to a group of conformal self-mappings of H, as required. The requirement that U? exists is necessary. This follows from the following theorem: Given any principal region H, there exists a subgroup K? of K with K? \ C D 1 I

K? C D K

.hence K? ' KjC/ :

(This can also be expressed by saying that K decomposes as the crossed product of C with KjC.) Proof. When  D 0, C D 1 and K? D K. We must consider the thirteen cases with  > 0 given in 125. In each of these thirteen cases, there are some geometric configurations … on the principal region so that, given any …1 and …2 , there always exists one and only one C in C with C …1 D …2 . (For the sphere, … can be taken to be the ordered triple of points; for the Möbius band, … can be taken to be the boundary point, and so on.) Now choose some … and take for K? the group of elements of K that fix ….  149 .

Given any S in G for which S 2 lies in U, there exists some U in G with S U.mod A/; U 2 D 1 :

We prove this purely topological theorem with the help of the theory of extremal quasiconformal mappings (which is itself so far unestablished). Let S be deformed to an extremal quasiconformal mapping E. Because S 2 lies in A, S 1 can be deformed to S , hence into E. On the other hand, S 1 can be deformed to E 1 . Both extremal quasiconformal mappings E and E 1 can therefore be deformed to each other. Consequently, E 1 D CE; C in C : Now if  D 0, then necessarily C D 1 and we can take U D E. But if  > 0, we consider again the geometric configurations … of the previous proof: we are allowed to modify E (replacing it by C 0 E, C 0 in C) so that E carries a definite … onto itself; then this … remains fixed under C and again C D 1, and we can set U D E. Let H0 be a general principal region and U be a finite subgroup of F D GjA. Is there a multiplicative representative system U? of U in G, that is, a subgroup U? of G with U? \ A D 1; U? AjA D U .hence U ' U? /‹

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If U? exists, there is, according to 148, a principal region of the same type as H0 with U?  K. Hence, if U (resp. UjU \ N0 ) is, according to 143, represented through isometric transformations of R , the point of R corresponding to H is a fixed point of this finite subgroup. (148 is vague!) If conversely P is a fixed point of the group U (resp. UjU \ N0 ) in R and if H is the corresponding principal region, then U  KjC and U? exists: simply take the corresponding subgroup of K? constructed in 148. So the question about existence of a multiplicative representative system for U in G as stated by Nielsen is equivalent to the question about existence of a fixed point for the finite group of isometric transformations UjU \ N0 of R . 150. In order to illustrate the preceding general discussion, we consider the hyperelliptic surface. Let H0 be a closed orientable surface of genus g > 1 without distinguished points. A topological mapping G from H0 onto itself is called distinguished if it has the following properties: G2 D 1 G preserves orientation. Identifying the points p and Gp of H0 yields a closed surface of genus 0. Such distinguished mappings can be obtained in the following way: take a twofold covering H of the sphere with 2g C2 ( 6) ramification points; sheet swapping is a distinguished mapping onto it with 2g C2 fixed points; then map H topologically on H0 ; this gives a distinguished mapping also on H0 which corresponds to exchanging sheets on H. Every distinguished mapping G of H0 arises this way. For, identifying p and Q of genus 0 and H0 is a two-fold covering of M Q which must Gp yields a surface M Q naturally have 2g C 2 ramification points. If M is topologically mapped on a sphere, then H0 goes to a two-fold (hyperelliptic) Riemann surface H over this sphere and G is carried to the sheet exchanging. To the distinguished mappings G from H0 onto itself correspond certain congruence classes F in the factor group F D GjA with F 2 D 1. I don’t know how many classes there are. As soon as F D 1 is out of the matter, then exchanging sheets on a (hyperelliptic) two-fold covering of the sphere of genus g > 1 is a conformal mapping which according to 141 cannot be deformed to the identity (because it reverses the sign of one-dimensional everywhere finite differentials). Let F be such a class; let H be a closed orientable Riemann surface of genus g which admits a conformal distinguished mapping G (i.e. H is hyperelliptic and lies as a two-fold covering over the sphere obtained by identifying p and Gp) and let the congruence class of G mod C, resp. mod A, be just the element F of F. In order to find the geodesic manifold of those H0 with the same properties, we have to find which everywhere finite quadratic differentials remain unchanged under G. If r1 ; : : : ; r2gC2

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are the ramification points of the two-fold surface H over the z-plane and if we set q .z  r1 / .z  r2gC2 / D w ; then the most general everywhere finite quadratic differential is d 2 D

P .z/ C Q.z/w 2 dz ; w2

where P .z/ is a polynomial of degree at most 2g  2 and Q.z/ is a polynomial of degree at most g  3. (2g  1 complex constants are found in P and further g  2 in Q so that d  2 depends upon 3g  3 constants, as it should be according to 44.) Now, under G, z is sent to itself and w to w; hence, d  2 remains invariant by G if and only if Q.z/ D 0 or d 2 D

a0 C a1 z C C a2g2 z 2g2 2 dz : w2

The geodesic manifold to be studied is therefore .4g  2/-dimensional. This dimension 4g  2 is equal to the dimension  D 6g  6 of the whole R only in the case g D 2 (cf. 145). Hence, to each of our classes of distinguished mappings F in F corresponds a .4g  2/-dimensional manifold in R6.g1/ which, given any two distinct points in it, also contains the whole complex geodesic through them. These manifolds are disjoint (the case there are several of them occurs only for g  3) for, on a hyperelliptic surface H, the conformal distinguished mapping is uniquely defined. As in 145, the extremal quasiconformal mappings between two topologically determined H lying on the same geodesic manifold of ours arises simply by setting H as a two-fold covering over the z-plane, by distinguishing the ramification points r1 ; : : : ; r2gC2 , by mapping extremal quasiconformally the z-plane with these distinguished points and by then going back to the two-fold covering. All our geodesic manifolds can be transferred onto each other through the geometric transformations of R coming from F (resp. FjN0 ), since two H, each endowed with one distinguished mapping, can always be carried onto each other topologically so that the distinguished mappings correspond. All “distinguished classes” F , as we will put it in short, (i.e., the F in F D GjA that contain a distinguished G), are conjugate in F for this very reason. The number of such F is equal to the index of the normalizer of F in F. 151. If G0 and G00 are two distinguished mappings of H0 and if G0 G00 .mod A/ ; then there is some A in A with G00 D A G0 A1 :

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If G0 can be deformed to G00 , then G0 can also be connected to G00 within the set of distinguished mappings (as A is allowed to approach 1 continuously.) A purely topological proof of this proposition is to be hoped for. Proof. Let the surface H0 be mapped on H by H and on H0 by H 0 so that G D HG0 H 1

and G 0 D H 0 G00 H 01

are conformal (this is always possible according to 150). We deform the mapping H 0 H 1 from H to H0 to an extremal quasiconformal mapping E: E D H 0 AH 1 ; A in A : Any mapping which comes from H 0 H 1 through deformation can be written so. Now from G0 G00 .mod A/ follows that

G 01 EG D H 0 G001 AG0 H 1

can be deformed to E. Since the extremal quasiconformal mapping is uniquely defined (indeed we have C D 1), it follows that G 01 EG D E or or

H 0 G001 AG0 H 1 D H 0 AH 1 AG0 A1 D G00 :



152. Let H be a closed orientable surface of genus g > 1 and let G be a distinguished mapping from H to itself. If F is the congruence class of G mod A then we showed in 150 that the number of distinguished classes is equal to the index .FW NF /, where NF is the normalizer of F in F (i.e. the group of all elements of F commuting with F ). If now P lies in G and the congruence class of P mod A lies in NF , then G 0 D P GP 1 is a distinguished mapping G (mod A). Consequently and according to 151, P GP 1 D AGA1

.A in A/ ;

that is, A1 P commutes with G. We denote by Z the group of elements of G commuting with G and obtain NF D ZAjA : Hence

.FW NF / D .GW ZA/ :

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Z has the following geometric meaning: if H is seen as the two-fold covering of the Q and G acts by exchanging sheets, then every topological mapping of the sphere M Q onto itself which permutes the .2g C 2/ ramification points only can be sphere M carried in two ways to a topological mapping of H onto itself; Z consists in these mappings. The number of distinguished classes is therefore equal to the index in the group G of all topological self-mappings of H of the group ZA of all topological selfmappings of H that can be deformed to those mappings that carry points of H “lying Q to points on top of the others” (i.e. those points that lie over the same point of M) lying on top of the others too. We thus state three topological problems: I. Prove the result of 151 in a purely topological way. II. Prove in a purely topological way that in the case g D 2, ZA D G, as it should be from 145. III. Compute the index .GW ZA/ for each g. 153. We could likewise show that symmetric surfaces build  -dimensional geodesic manifolds in R2 which can intersect each other, though. Instead, we prefer talking quickly about the general notion of covering. Let two finite Riemann surfaces M? and M be given. The surface M is called an m-fold covering of M? if to each point of M is associated a “trace point” of M? and if the following conditions are satisfied: Each inner or boundary point of M? is a trace point of at least one and at most m points of M; of less than m points only for finitely many points and pieces of boundary curves. The mapping from M to M? is continuous and, except on the boundary and at the finitely many ramification points, conformal. If less than m points of M lie over an inner point of M? , then these are algebraic ramification points and the sum of the numbers of sheets is m. The boundary curves of M are mapped to the boundary of M? . Still, some fault lines of M can lie over some pieces of the boundary curves of M? ; these are analytic curves for which two analytically mirrored points of M have the same image point in M? . Some ramification points might also lie over the boundary of M? ; they are then either inner points of M, where fault lines cross, or boundary points of M from which fault lines emanate. When M? is orientable, M is orientable. When M? is closed, M is closed. The covering is said to be normal if there is a group S of m conformal selfmappings of M such that two points of M have the same trace point if and only if they are equivalent under S. Functions on M? can also be seen as functions on M. The function field of M has degree m over the function field of M? . It is Galois (normal) over the latter field if and only if the covering is normal; then the Galois group can be isomorphically mapped onto S in a well-defined manner. An n-dimensional differential d  n on M

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is an n-dimensional differential on M? if and only if it is invariant by S (up to passing to the complex conjugate). For normal coverings there are at most two-fold ramification points on the boundary of M. Let M be an arbitrary finite Riemann surface and let S be any finite group of conformal mappings from M onto itself. Identifying the points that are equivalent under S yields a finite Riemann surface M? and M is then a normal covering of M? with group S. 154. Let H and H? be principal regions with supports M, M? . We call H a normal covering of H? if the following conditions are satisfied:  M is a normal covering of M? with group S;  Distinguished points of H are sent to distinguished points by S;  The trace point of each distinguished point and each ramification point of H is a distinguished point of H? ;  H has distinguished points or ramification points only over every distinguished point of H? . It should be noticed that a ramification point of H lying over a distinguished point of H? need not be distinguished. Our definition turns out to be justified through the following proposition: The regular quadratic differentials on H? are exactly the regular quadratic differentials on H that are invariant by S. Proof. The quadratic differentials on M? are exactly the S-invariant quadratic differentials on M. We must only check that for the quadratic differentials on M? regularity for H and for H? has the same meaning. For ordinary inner and boundary points, even distinguished, this is obvious; it remains to look at ramification points. Let p be for instance a -fold ramification point of M lying over the interior of M? and let z ? be a local uniformizer of M? vanishing at p such that p z D z? is a local uniformizer for M. Let d  2 D f .z ? /dz ?2 be some quadratic differential on M? . In order for d  2 to be regular on H? , f .z ? / is allowed to have at most one first-order pole, i.e. z ? f .z ? / must be regular, since the trace point of p is a distinguished point of M? . In order for d  2 to be regular on H, the factor of dz 2 in  ? 2 z? 2 dz 2 ? d  D f .z / dz 2 D 2 2 f .z ? /dz 2 dz z

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must either be finite or, if p is a distinguished point of M, must be infinite of at most ?2 ?2 the first order (in z), i.e. zz 2 f .z ? /, resp. z z f .z ? /, must be finite. These conditions actually have the same meaning because f .z ? / is a single-valued function of z ? . The same holds for ramification points lying over the boundary of M? .  155. Let all extremal quasiconformal mappings corresponding to regular quadratic differentials of H? and to arbitrary dilatation quotients  1 be applied to H; this gives rise to a submanifold R? of the space R of all classes of conformally equivalent topologically determined principal regions of the same type as H and it is isometrically mapped to the space R ? of all classes of conformally equivalent topologically determined principal regions of the same type as H? . R? is a geodesic manifold and it corresponds in the sense of 147 to the subgroup U D SCjC of F. Each extremal quasiconformal mapping carries the group S to a group of conformal self-mappings of some principal region H0 (represented by a point of R? in R ) which is however well-defined only up to transformation by a conformal selfmapping of H0 deformable to the identity. We met several times examples of normal coverings of principal regions: the first time was in 22 where the ring domain appeared as the two-fold normal covering of the quadrilateral. Similar occurrences then appeared in 23, 24, 27, 29 and 30–32. The double of a bordered or non-orientable surface (92 and forward) is a normal twofold covering, just like the principal region obtained in 106 by doubling. Finally, just recall 128. If the problem of extremal quasiconformal mappings is solved for the type H, then, because R ? ' R?  R ; it is also solved for the type H? . This was already used in 136. 156. It is said in 63: “In particular we compare the metric   ds 2 D . C L/ jdzj2 C  0 cases beforehand as we did in 125, then proceed directly and, like in 141, shown that C D 1 elsewhere. I did not point this out before in order to avoid confusion. Actually, this was already used tacitly in 63 when topologically determined principal regions were considered. For, conformal mappings from K (or their congruence classes mod N) transform d  2 linearly homogeneically and also form a group of linear homogeneous isometric transformations from the infinitesimal space L onto itself. (KjN is just the subgroup of transformations from FjN0 that fix the point P of R corresponding to H and, for this reason, map in first-order approximation a neighbourhood of it linearly onto itself.) If we hadn’t fixed H topologically, we would have had identified the points of L that are equivalent under KjN and in the case K is bigger than N we would not have had any linear metric space at our disposal anymore.

29 Remarks on the torus 157. In 73 and 100 we quickly put aside the quadratic differentials d  2 that were defined only up to positive factors and regular elsewhere. We conjectured that they provided mappings which were not globally but only locally extremal quasiconformal. We will have a look at such mappings more closely in a simple case. Let a torus be developed over the u-plane with a pair of primitive periods !1 ; !2 . We are looking for everywhere finite quadratic differentials d  2 which are multiplied by positive numbers as we go around the surface. If we set d  2 D f .u/du2 , then f .u/ has to be an entire function for which f .u C !1 /=f .u/ and f .u C !2 /=f .u/ are positive constants. As an elliptic function of the second type without poles, f .u/ has the form f .u/ D aeu : Here, e!1 and e!2 must be positive: =! 0 .mod 2/

. D 1; 2/ :

The numbers  satisfying this condition form a lattice. Let  be any non zero number of this lattice. We had d  2 D aeu du2 ;

9 Extremal quasiconformal mappings

hence D

Z p

d 2

471

p 2 a u

D e 2 D be 2 u : 

 maps the finite u-plane conformally onto the logarithmic surface which ramifies infinitely over  D 0; 1. The mapping u 7! uC! translates as the multiplication of 

by e< 2 ! and the rotation in positive direction by the angle = 2 ! 0 .mod /. Our former torus arises again (up to a conformal mapping) from the logarithmic surface over the -plane through a suitable identification. Now we must find a quasiconformal mapping of the torus whose dilatation quotient is constant equal to K  1 and whose direction field of maximal dilatation is d  2 > 0. We simply apply the affine mapping  0 D K 1, this mapping is never extremal quasiconformal; according to 47, it is locally extremal quasiconformal only. Completely similar considerations can be carried out for the ring domain by considering the lines d  2 > 0. 158. A singularity free torus, lying in a tridimensional Euclidean space, endowed with the metric induced by the surrounding space gives rise, if it is symmetric (as for exemple the usual torus which arises by rotating a circle about a line not intersecting it and contained in its plane), by conformally developing it on the plane, to a period ratio that is purely imaginary under suitable normalization. Are there also tori with non purely imaginary period ratios under any normalization? We will find an answer based on heuristic arguments. Let a torus be conformally mapped onto the w D u C iv-plane mod .!1 ; !2 /. The position vector r is therefore a mod .!1 ; !2 / doubly periodic function of w D u C iv. We have ru  rv ¤ 0 and

ds 2 D d r d r D jdwj2 ;

with .w/ > 0. (Hence, E D ru ru D , F D ru rv D 0, G D rv rv D .) Let  be the normal vector. We pass to an infinitely close surface r C ın  ;

472

Oswald Teichmüller

where ın is hence a first-order infinitely small function of w. If infinitely small quantities of higher degree than the first are neglected, the close surface is endowed with the metric ds 2 C ıds 2 D d.r C ın / d.r C ın / D d r d r C 2ın d r d  or, if it is set as usual d r d  D L du2 C 2Mdu dv C N dv 2 ; ds 2 C ıds 2 D jdwj2  2 ın .L du2 C 2Mdu dv C N dv 2 / : In accordance with 62, we set  2 ın

M LN i 2 

 D B ;

where  is an infinitely small positive constant quantity (the former ıE, ıF , ıG must be replaced by 2Lın; 2M ın; 2N ın!). Then, B

dw 2 jdwj2

again gives the infinitesimal variation of the conformally invariant part of the metric. According to 88, as we pass to the close surface r C ın  conformally developed on the w 0 -plane, the variation of the period ratio is determined alone by “ B dw ; kD P

where integration is taken over a period parallelogram of the w-plane, or by   “ M LN ın Ci dw : 2 2  P Now we never have L  N D M D 0 identically, for then, the surface would have only ombilic points and would be a plane or a sphere, but not a torus. On the M other hand, it is possible that all the values of LN 2  i  fall on a single line passing through zero, i.e. that the ratio .L  N /=M is constant. Then by rotating the w-plane we can get M D0: The lines u D const, v D const are then the curvature lines of the torus. This is for example the case for the usual torus. The period ratio can then be changed in first-order approximation by r C ın  not arbitrarily but in just one direction. For example, the period ratio which is purely imaginary for the usual torus, remains in first-order approximation purely imaginary under r 7! d r d .

9 Extremal quasiconformal mappings

473

On the other hand, there are tori for which, already because of their shape, curvature lines cannot have the analytic form u D const, v D const. Period ratios can nevertheless be purely imaginary. But they can then be freely modified and be carried through r C ın  on close ones which will not be purely imaginary anymore. Based on these considerations, we expect spatial tori without purely imaginary period ratios. It might even be possible to find out spatial tori with arbitrarily prescribed period ratios !.

30 Generalization 159. Consider a sufficiently regular topological mapping from the circle jzj D 1 onto itself. One shall extend it to a quasiconformal topological mapping from the disc jzj  1 onto itself so that the maximum of the dilatation quotient is as small as possible. Unlike what precedes, not only must the boundary be mapped onto itself as a whole, but each boundary point has a prescribed image point. This is as if all boundary points were distinguished. The extremal quasiconformal mappings are easily guessed: they are obtained, like in 114, from a constant dilatation quotient K and a quadratic differential d  2 D f .z/dz 2 ; where f .z/ is regular in jzj < 1. But now we are not allowed to require for instance that f .z/dz 2 is real along the boundary jzj D 1: this would correspond to the problem of the boundary being carried to itself only as a whole. On the other hand, f .z/ cannot also have poles at all the boundary points at the same time because they are all distinguished. We will much more let the boundary behaviour of f .z/ be arbitrary and obtain this way as many extremal mappings as needed. Certain regularity assumptions on the boundary must definitely be made whose appropriate formulation is a problem in itself. One would at best expect some proposition of the following type: “Whenever a given boundary mapping can be extended to a quasiconformal mapping, then it can also be extended to an extremal quasiconformal mapping. It has the indicated form and “ “ d D jf .z/j dz “ converges. If

jzj 0 and some regular quadratic differential d  2 on .H; p/ with a pole at p such that

.wx C iwy /

dz 2 dz 2 d2 D .w1x C iw1y / Cc : jdzj2 jdzj2 jd  2 j

For any non-zero complex number a, the expressions w; w1 ; c and d  2 in this formula can be simultaneously replaced by aw; aw1 ; jajc and ad  2 . We hence get at the point p: ˇ w ˇ2 ˇ ˇ ds 2 D c 2 ˇ : ˇ dz

9 Extremal quasiconformal mappings

475

32 An estimate 161. As we are now turning toward the simplest example, we still have to deal with a question only researchers working in the field should find some interest in. It concerns the proper development of the Grötzsch–Ahlfors method. Similarly to 129, we consider a hexagon S in the -plane which is bordered by the sequence of edges  D 0 a a C qi pa C qi pa C i i 0 ; and a hexagon S0 in the  0 -plane which is bordered by the sequence of segments  0 D 0 a0 a0 C q 0 i p0 a0 C q 0 i p0 a0 C i i 0 : Of course 0 < pa < a, 0 < q < 1, 0 < p0 a0 < a0 , 0 < q 0 < 1. Let a quasiconformal mapping from S onto S0 be given which sends the pointing-out vertices 0; a; a C qi; pa C i; i to 0; a0 ; a0 C q 0 i; p0 a0 C i; i and whose dilatation quotient is  C . We saw in 129 that whenever p D p0 and q D q 0 then a0 C: a But how can we bound C from below if we don’t simultaneously have p D p0 and q D q 0 ? We know the answer: map conformally S and S0 onto hexagons so that pointing-out vertices go to corresponding pointing-out vertices and so that the condition p D p0 and q D q 0 which was formerly not satisfied is now fulfilled; then apply 129. Computational difficulties arise against this method. For this reason, we ask how far we can go without such an auxiliary mapping. The estimation method we followed in 129 (integration over the lines = D const) can be applied and it gives a0 02 .p .1  q/ C pq/  Cp.p0 C q 0  p0 q 0 / : a But it is a general rule of Grötzsch that integrating over the lines = D const and integrating over the lines