Braids, Conformal Module, Entropy, and Gromov's Oka Principle 9783031672873, 9783031672880

Table of contents :
Preface
Acknowledgments
Introduction
Contents
1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory
1.1 Free Homotopy Classes of Mappings and Conjugacy Classes of Group Elements
1.2 Riemann Surfaces
1.3 Braids
1.4 Mapping Class Groups
1.5 The Relation Between Braids and Mapping Classes
1.6 Beltrami Differentials and Quadratic Differentials
1.7 Elements of Teichmüller Theory
1.8 Thurston's Classification of Mapping Classes
2 The Entropy of Surface Homeomorphisms
2.1 The Topological Entropy of Mapping Classes
2.2 Absolutely Extremal Homeomorphisms and the Trajectories of the Associated Quadratic Differentials
2.3 The Entropy of Pseudo-Anosov Self-homeomorphisms of Surfaces with Distinguished Points
2.4 Pseudo-Anosov Self-homeomorphisms of Closed Surfaces Are Entropy Minimizing
2.5 The Entropy of Mapping Classes on Second Kind Riemann Surfaces
3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem
3.1 Ahlfors' Extremal Length and Conformal Module
3.2 Another Notion of Extremal Length
3.3 Invariants of Conjugacy Classes of Braids: Statement of the Main Theorem
3.4 Main Theorem. The Upper Bound for the Conformal Module. The Irreducible Case
3.5 The Lower Bound for the Conformal Module. The Irreducible Case
4 Reducible Pure Braids. Irreducible Nodal Components, Irreducible Braid Components, and the Proofof the Main Theorem
4.1 Irreducible Nodal Components. Pure Braids
4.2 The Irreducible Braid Components. Pure Braids
4.3 The Relation Between Irreducible Nodal Components and Irreducible Braid Components
4.4 Recovery of the Conjugacy Class of a Pure Braid from the Irreducible Braid Components
4.5 Pure Braids, the Reducible Case. Proof of the Main Theorem
5 The General Case. Irreducible Nodal Components, Irreducible Braid Components, and the Proofof the Main Theorem
5.1 Irreducible Nodal Components. The General Case
5.2 Irreducible Braid Components. The General Case
5.3 The Building Block of the Recovery. The General Case
5.4 Recovery of Conjugacy Classes of Braids from the Irreducible Braid Components. The General Case
5.5 Proof of the Main Theorem for Reducible Braids. The General Case
6 The Conformal Module and Holomorphic Familiesof Polynomials
6.1 Historical Remarks
6.2 Families of Polynomials, Solvability and Reducibility
7 Gromov's Oka Principle and Conformal Module
7.1 Gromov's Oka Principle and the Conformal Module of Conjugacy Classes of Elements of the Fundamental Group
7.2 Description of Gromov-Oka Mappings from Open Riemann Surfaces to C{-1,1}
7.3 Gromov-Oka Mappings from Tori with a Hole to P3
8 Gromov's Oka Principle for (g,m)-Fiber Bundles
8.1 (g,m)-Fiber Bundles. Statement of the Problem
8.2 The Monodromy of (g,m)-Bundles. Isotopy and Isomorphism of Bundles
8.3 (0,4)-Bundles Over Genus 1 Surfaces with a Hole
8.4 (0,4)-Bundles Over Tori with a Hole. Proof of Theorem 7.4
8.5 Smooth Elliptic Fiber Bundles and Differentiable Families of Complex Manifolds
8.6 Complex Analytic Families of Canonical Tori
8.7 Special (0,4)-Bundles and Double Branched Coverings
8.8 Lattices and Double Branched Coverings
8.9 The Gromov-Oka Principle for (1,1)-Bundles over Tori with a Hole
9 Fundamental Groups and Bounds for the Extremal Length
9.1 The Fundamental Group and Extremal Length. Two Theorems
9.2 Coverings of C {-1,1} and Slalom Curves
9.3 Building Blocks and Their Extremal Length
9.4 The Extremal Length of Words in π1. The Lower Bound
9.5 The Upper Bound of the Extremal Length for Words Whose Syllables Have Degree 2
9.6 The Extremal Length of Arbitrary Words in π1. The Upper Bound
9.7 The Extremal Length of 3-Braids
10 Counting Functions
11 Riemann Surfaces of Second Kind and Finiteness Theorems
11.1 Riemann Surfaces of First and of Second Kind and Finiteness Theorems
11.2 Some Further Preliminaries on Mappings, Coverings, and Extremal Length
11.3 Holomorphic Mappings into the Twice Punctured Plane
11.4 (g,m)-Bundles over Riemann Surfaces
Appendix A
A.1 Several Complex Variables
A.2 A Lemma on Conjugation
A.3 Koebe's Theorem
References
Index

Citation preview

Lecture Notes in Mathematics  2360

Burglind Jöricke

Braids, Conformal Module, Entropy, and Gromov’s Oka Principle

Lecture Notes in Mathematics Volume 2360

Editors-in-Chief Jean-Michel Morel, City University of Hong Kong, Kowloon Tong, China Bernard Teissier, IMJ-PRG, Paris, France Series Editors Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Rupert Frank, LMU, Munich, Germany Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth

, University of Cologne, Cologne, Germany

Ariane Mézard, IMJ-PRG, Paris, France Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg Mark Policott, Mathematics Institute, University of Warwick, Coventry, UK László Székelyhidi

, MPI for Mathematics in the Sciences, Leipzig, Germany

Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, MPI for Mathematics in the Sciences, Leipzig, Germany

This series reports on new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. The type of material considered for publication includes: 1. Research monographs 2. Lectures on a new field or presentations of a new angle in a classical field 3. Summer schools and intensive courses on topics of current research. Texts which are out of print but still in demand may also be considered if they fall within these categories. The timeliness of a manuscript is sometimes more important than its form, which may be preliminary or tentative. Please visit the LNM Editorial Policy (https://drive.google.com/file/d/ 1MOg4TbwOSokRnFJ3ZR3ciEeKs9hOnNX_/view?usp=sharing) Titles from this series are indexed by Scopus, Web of Science, Mathematical Reviews, and zbMATH.

Burglind Jöricke

Braids, Conformal Module, Entropy, and Gromov’s Oka Principle

Burglind Jöricke Berlin, Germany

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-031-67287-3 ISBN 978-3-031-67288-0 (eBook) https://doi.org/10.1007/978-3-031-67288-0 Mathematics Subject Classification: 20F36, 30F60, 32E10, 32G05, 32G08, 32G15, 32L05, 32Gxx, 32Q56, 37B40 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland If disposing of this product, please recycle the paper.

Preface

This book takes the reader on an excursion through a wide range of topics that are focused on the treatment of conformal invariants and dynamical invariants. The collection of conformal invariants of complex manifolds, considered in the book, was proposed by Gromov for the case of the twice punctured complex plane. They are called here the conformal modules of conjugacy classes of elements of the fundamental group of complex manifolds and play a role in understanding the obstructions to Gromov’s Oka Principle. The fundamental group of the space of monic polynomials of degree n without multiple zeros is the braid group on n strands. The respective conformal invariants are also called the conformal modules of conjugacy classes of n-braids. They appeared much earlier in connection with the interest in the Thirteen’s Hilbert Problem. The interesting point is that the conformal invariant of a conjugacy class of braids, that is related to Gromov’s Oka principle, is inversely proportional to a dynamical invariant, the entropy, which was studied in connection with Thurston’s celebrated theory of surface homeomorphisms. This result can be considered as another instance of the numerous manifestations of the unity of mathematics. The relation has applications. Known results concerning the entropy can be applied to problems that use the concept of the conformal module, and vice versa; methods of complex analysis and quasiconformal mappings used for the study of the conformal invariants imply results on the entropy. After prerequisites on Riemann surfaces, braids, mapping classes, and elements of Teichmüller theory, a detailed introduction to the entropy of braids and mapping classes is given with thorough, sometimes new proofs. The book contains the first published proof of the inverse proportionality of the conformal and the dynamical invariants of conjugacy classes of braids, and a number of applications of the concept of conformal module to Gromov’s Oka Principle as well as to an older problem that appeared in connection with the Thirteen’s Hilbert problem. Gromov’s conformal invariants of the twice punctured complex plane are estimated from above and from below. The upper and the lower bounds differ by universal multiplicative constants. The estimates imply estimates of the entropy of any pure three-braid as well as quantitative statements on the limitation of Gromov’s Oka Principle in the v

vi

Preface

sense of finiteness theorems. For the finiteness theorems slightly more powerful conformal invariants related to elements of the fundamental group (not merely to conjugacy classes) are used. Including detailed proofs of all the main results and proposing several research problems, the book is aimed at graduate students and researchers.

Acknowledgments While working on the project of this book the author had the opportunity to gain from the wonderful working conditions at IHES, Max-Planck-Institute for Mathematics, Weizmann Institute, CRM Barcelona, and at Humboldt-University Berlin, in particular at the SFB “Raum-Zeit-Materie.” Other parts of the work were done while visiting the École Normale Supérieure Paris, the Universities of Grenoble, Bern, Calais, Toulouse, Lille, Bloomington, and Canberra. Talks, or series of lectures, given on the topic of the book at various places encouraged research and helped to improve the explanation. A talk given at the Algebraic Geometry seminar at Courant Institute and a stimulating discussion with F. Bogomolov and M. Gromov had a special impact. The author is indebted to many other mathematicians for interesting and fruitful discussions and for information concerning references to the literature. Among them are B. Berndtsson, R. Bryant, D. Calegari, F. Dahmani, P. Eyssidieux, B. Farb, P. Kirk, M. Reimann, S. Orevkov, L. Stout, O. Viro and M. Zaidenberg. O. Viro asked about a notion of conformal module of braids rather than of conjugacy classes of braids. M. Reimann gave references to the literature related to the computation of the conformal module of special doubly connected domains and of quadrilaterals. The author is grateful to B. Farb who suggested to use the concept of conformal module and extremal length for a proof of finiteness ¯ theorems, and to B. Berndtsson for proposing the kernel for solving the .∂-problem that arises in the proof of Proposition 11.1. Special thanks go to Alexander Weiße for teaching how to draw figures and for producing the essential parts of some difficult figures. The author is much obliged to F. Dufour and Marie-Claude Vergne for drawing some other figures, and to Cécile Gourgues for typing a preliminary version of some chapters. Berlin, Germany Bures-sur-Yvette, France June 2024

Burglind Jöricke

Introduction

I am the spirit that always denies! And rightly so. Mephistopheles, Faust I, verse 1338/1339

The book treats invariants from different topics of mathematics, namely conformal invariants and dynamical invariants. While the dynamical invariant, the entropy, is very popular and intensively studied, the conformal invariants were undeservedly almost forgotten.

The conformal invariants are important in connection with the heuristic principle, which is widely known as Oka’s Principle. The Oka Principle goes back to J.P. Serre, who phrased it in 1951 as follows: In the absence of topological obstructions analytic problems on Stein manifolds admit analytic solutions. The Cartan-Serre Theorems A and B, and the equivalence between topological and holomorphic classification of principal fiber bundles over Stein manifolds are important examples of this principle. In [35] M. Gromov offered an interpretation of the Oka Principle in terms of the h-principle for holomorphic mappings: “Oka Principle is an expression of an optimistic expectation with regard to the validity of the h-principle for holomorphic maps in the situation where the source manifold is Stein”. The h-principle is said to hold for holomorphic mappings from a complex manifold X to a complex manifold Y , if each continuous mapping .X → Y is homotopic to a holomorphic one. Gromov’s Oka Principle became the subject of intensive and fruitful research on the “affirmative side”, i.e. on situations, when the principle holds. For a comprehensive account see [29]. On the other side, what about the “negating side”? How interesting is a look at situations when Gromov’s Oka Principle is violated? Are there interesting structures behind? A basic problem concerning the “negating side” of Gromov’s Oka Problem is the following. vii

viii

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Problem Understand obstructions for Gromov’s Oka Principle. Obstructions for Gromov’s Oka Principle are based on the relation between conformal invariants of the source and the target, in case the relation between topological invariants does not completely describe the situation. Gromov himself referred in his seminal paper [35] from 1989 to mappings from annuli into the twice punctured complex plane as the simplest example, where this principle fails and pointed out the special role of mappings from annuli for analyzing the failure of the principle. The occurring conformal invariants of the twice punctured complex plane (called here the conformal modules of conjugacy classes of elements of the fundamental group) are supposed to capture certain “conformal rigidity”. The conformal module of a conjugacy class of elements of the fundamental group .π1 (Y, y0 ) of a complex manifold Y is defined as follows. The conjugacy class .e ˆ of an element .e ∈ π1 (Y, y0 ) can be identified with the free homotopy class of curves containing the curves of the class e. A continuous map g from an annulus .A = {z ∈ C : r < |z| < R} into .X is said to represent the conjugacy class .eˆ if for some (and hence for any) .ρ ∈ (r, R) the map .g : {|z| = ρ} → X represents .e. ˆ The definition of the conformal module of .eˆ is based on Ahlfors’ concept of conformal module and extremal length. According to Ahlfors the conformal module 1 of an annulus . A = {z ∈ C : r < |z| < R} is equal to . m(A) = 2π log( Rr ) 1 and its extremal length equals .λ(A) = m(A) . For any complex manifold .X and any conjugacy class .eˆ of elements of the fundamental group of .X the conformal ˆ of .eˆ is defined as the supremum of the conformal modules of annuli module .M(e) that admit a holomorphic mapping into .X representing .e. ˆ The extremal length .Λ(e) ˆ equals . M1(e) . ˆ An obstruction to Gromov’s Oka principle is given by the following observation. For a holomorphic mapping .f : X → Y from a complex manifold X to a complex manifold Y and any element .e ∈ π1 (X, x0 ) the conformal module of the free homotopy class of its image .f ∗ (e) is not smaller than the conformal module of the free homotopy class .eˆ of the element itself. It was an encouraging observation, that for the twice punctured complex plane it is possible to give good estimates of the conformal invariants proposed by Gromov. More precisely, in Chap. 9 we give upper and lower bounds for the conformal module of conjugacy classes of elements of the fundamental group of the twice punctured complex plane. The upper and lower bound differ by multiplicative constants not depending on the conjugacy class. Interestingly, the notion of the conformal module of conjugacy classes of elements of the fundamental group appeared already 1974 (without name) much before Gromov’s work. It was used in the paper [33] which was motivated by the interest of the authors in Hilbert’s Thirteen’s Problem for algebraic functions. In Chap. 6 we present some related historical remarks. The complex manifold in this case was the space .Pn , the space of monic polynomials of degree n without multiple zeros. (The adjective “monic” refers to the property that the coefficient by

Introduction

ix

the highest power of the variable equals 1). The elements of the fundamental group of this space are braids on n strands. Thus, the invariant that was considered in [33] is the conformal module of conjugacy classes of braids. It was applied to the question of global reducibility of families of polynomials .Pn that depend holomorphically on a parameter varying over a connected open (i.e. non-compact) Riemann surface (in particular over an annulus). Braids play a role in several mathematical fields. It is therefore not surprising that invariants of conjugacy classes of braids occurred in different areas of mathematics. There is a dynamical invariant, the entropy. The dynamical concept was considered in 1979 in [25] in the Asterisque volume dedicated to Thurston’s theory of surface homeomorphisms. Thurston’s theory of surface homeomorphisms was motivated by his celebrated geometrization conjecture. For the dynamical concept one assigns to each braid on n strands an isotopy class of self-homeomorphisms of the unit disc all elements of which fix the unit circle pointwise and permute a set .En of n chosen points in the unit disc. The entropy of the braid is the infimum of the entropy of the mappings in the class. The entropy does not change under conjugation, in other words it is an invariant of conjugacy classes of braids. The entropy of braids, more generally, of classes of surface homeomorphisms has received a lot of attention. It turns out that the two invariants related to the dynamical aspect and to the conformal aspect of braids are related. The following theorem holds. Main Theorem For each conjugacy class of braids .bˆ ∈ Bˆn , .n ≥ 2, the following equality holds ˆ = M(b)

.

π 1 . ˆ 2 h(b)

ˆ denotes the entropy of the conjugacy class .b. ˆ Here .h(b) The relation between the invariants coming from different fields has applications. First, results related to the more intensively studied dynamical invariant entropy may be applied to problems that use the concept of conformal module. For instance, Chap. 6 contains a short conceptual proof (of a slightly stronger version) of the result of [33], based on a result [70] related to entropy. Vice versa, methods of complex analysis including quasi-conformal mappings, that can be used for the study of the conformal module, may give results related to the entropy. For example, the relation between the conformal and the dynamical invariants together with the effective estimates of the conformal module gives effective upper and lower bounds for the entropy of pure 3-braids. There is an invariant of braids (not merely an invariant of conjugacy classes of braids), the conformal module (or extremal length) of braids with totally real horizontal boundary values. It uses Ahlfors’ extremal length of rectangles instead of the extremal length of annuli. This notion is more powerful for applications to problems related to Gromov’s Oka Principle than the extremal length of conjugacy classes of braids. For instance, it is used to obtain quantitative statements concerning

x

Introduction

Gromov’s Oka Principle in the spirit of analogs of the Geometric Shafarevich Conjecture (see Chap. 11). We tried to make the book as self-contained as possible and coherent for graduate students. Chapter 1 provides prerequisites from several topics, like Riemann surfaces, braids, and mapping classes, as well as Teichmüller theory. In some cases proofs are included, for instance, in the case when the material is less standard and not extensively treated in the literature, or when the idea behind a statement will appear to be important in the following. Readers who are familiar with the material may skip this chapter and consult it later for notation and for some less standard facts that are included in this chapter. In Chap. 2 a self-contained thorough treatment of the entropy of braids and mapping classes is given. One of the theorems of [25] on irreducible mapping classes on closed surfaces is proved along the lines of [25]. The proof of another theorem is new. We include detailed prerequisites on foliations generated by quadratic differentials. A thorough treatment of the entropy of mapping classes on finite Riemann surfaces of second kind (i.e. on connected Riemann surfaces that are neither closed nor punctured and have finitely generated fundamental group) is presented. This is needed to treat the entropy of braids and the entropy of reducible mapping classes. The proof of the Main Theorem for irreducible braids is given in Chap. 3. The chapter starts with a short introduction to Ahlfors’ concept of the extremal length of families of curves in planar domains, and presents an extension of the concept to homotopy classes of curves in complex manifolds. The chapter contains explicit computations of the extremal length of some examples of conjugacy classes of 3braids (which means by the Main Theorem, to give another way of computing some entropies), and it contains explicit computations of the extremal length with totally real horizontal boundary values of some examples of 3-braids. In Chap. 4 we define the irreducible braid components of a conjugacy class of reducible pure braids and show that the conformal module of the conjugacy class of a pure braid equals the minimum of the conformal modules of the irreducible components. Similarly, we define the irreducible nodal components of the conjugacy class of mapping classes associated to the braid, and show that the entropy of the mapping class is equal to the maximum of the entropies of the irreducible nodal components. The Main Theorem for irreducible pure braids implies the Main Theorem for reducible pure braids. The conjugacy class of a reducible pure braid can be recovered from its irreducible components. The conjugacy class of the mapping class can be recovered up to products of commuting Dehn twists from the conjugacy classes of the irreducible nodal components. Similar statements for not necessarily pure braids are contained in Chap. 5. As mentioned above, Chap. 6 discusses global reducibility of holomorphic families of polynomials. Since the conformal invariants are not much studied we present deeper applications of the concept of the conformal module of braids or their conjugacy classes (more generally, of elements of fundamental groups or their conjugacy classes) in Chaps. 7–11.

Introduction

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In Chap. 7 we study situations when Gromov’s Oka Principle fails, using the concept of the conformal modules of conjugacy classes of elements of the fundamental group. For instance, we call a mapping .f : X → Y from a connected finite open Riemann surface X to a complex manifold Y a Gromov-Oka mapping, if for each conformal structure on X with only thick ends the mapping is homotopic to a holomorphic mapping. (A conformal structure on a connected finite open Riemann surface has only thick ends if it makes the manifold a closed Riemann surface with a finite number of closed discs but no point removed.) The Gromov-Oka mappings from finite open Riemann surfaces to the twice punctured Riemann surface are completely described. In Chap. 8 we prove theorems related to the failure of the Gromov-Oka Principle for bundles whose fibers are Riemann surfaces of type .(1, 1) (in other words, the fibers are once punctured tori). The problem can be reduced to the respective problem for .(0, 4)-bundles (and, hence, for mappings into .P3 ) by considering double branched coverings. Chapter 9 is devoted to effective upper and lower bounds for the extremal length with totally real horizontal boundary values of any element of the fundamental group of the twice punctured complex plane (not merely of the extremal length of conjugacy classes). The bounds differ by a multiplicative constant not depending on the element. This is done in terms of a natural syllable decomposition of the word representing the element. In the last section (Sect. 9.7) we give respective estimates for 3-braids. The extremal length with totally real horizontal boundary values is capable of giving more subtle information regarding Gromov’s Oka Principle and limitations of its validity than the respective invariant of conjugacy classes. Moreover, the estimates for the extremal length with totally real horizontal boundary values of pure braids imply estimates of the extremal length of conjugacy classes of these braids, and hence, of their entropy. In Chap. 10 we give effective estimates for the growth of the number of those elements of the braid group modulo center .B3 Z3 , whose extremal length with totally real boundary values is positive and does not exceed a positive number Y . As a corollary, we give an alternative proof of the result of Veech [80] on the exponential growth of the number of conjugacy classes of elements of .B3 Z3 that have positive entropy not exceeding Y . The proof does not use deep techniques from Teichmüller theory. In Chap. 11 we apply the concept of conformal module and extremal length to obtain quantitative statements on the limitation of Gromov’s Oka Principle. More specifically, for any finite open Riemann surface X (maybe, of second kind) we give an effective upper bound for the number of irreducible holomorphic mappings up to homotopy from X to the twice punctured complex plane, and an effective upper bound for the number of irreducible holomorphic torus bundles up to isotopy on such a Riemann surface. These statements are analogs for certain objects on Riemann surfaces of second kind of the Geometric Shafarevich Conjecture and the Theorem of de Franchis, that state the finiteness of the number of certain holomorphic objects on closed or punctured Riemann surfaces.

Contents

1

2

3

Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Free Homotopy Classes of Mappings and Conjugacy Classes of Group Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Mapping Class Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Relation Between Braids and Mapping Classes . . . . . . . . . . . . . . . 1.6 Beltrami Differentials and Quadratic Differentials . . . . . . . . . . . . . . . . . 1.7 Elements of Teichmüller Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Thurston’s Classification of Mapping Classes . . . . . . . . . . . . . . . . . . . . . . The Entropy of Surface Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Topological Entropy of Mapping Classes . . . . . . . . . . . . . . . . . . . . . . 2.2 Absolutely Extremal Homeomorphisms and the Trajectories of the Associated Quadratic Differentials . . . . . . . . . . . . . 2.3 The Entropy of Pseudo-Anosov Self-homeomorphisms of Surfaces with Distinguished Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Pseudo-Anosov Self-homeomorphisms of Closed Surfaces Are Entropy Minimizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Entropy of Mapping Classes on Second Kind Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conformal Invariants of Homotopy Classes of Curves. The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Ahlfors’ Extremal Length and Conformal Module . . . . . . . . . . . . . . . . . 3.2 Another Notion of Extremal Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Invariants of Conjugacy Classes of Braids: Statement of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Main Theorem. The Upper Bound for the Conformal Module. The Irreducible Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 12 16 19 23 28 41 55 55 66 71 80 95 109 109 113 127 129

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3.5 4

5

The Lower Bound for the Conformal Module. The Irreducible Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Reducible Pure Braids. Irreducible Nodal Components, Irreducible Braid Components, and the Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Irreducible Nodal Components. Pure Braids . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Irreducible Braid Components. Pure Braids . . . . . . . . . . . . . . . . . . . 4.3 The Relation Between Irreducible Nodal Components and Irreducible Braid Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Recovery of the Conjugacy Class of a Pure Braid from the Irreducible Braid Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Pure Braids, the Reducible Case. Proof of the Main Theorem . . . . . The General Case. Irreducible Nodal Components, Irreducible Braid Components, and the Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Irreducible Nodal Components. The General Case . . . . . . . . . . . . . . . . . 5.2 Irreducible Braid Components. The General Case . . . . . . . . . . . . . . . . . 5.3 The Building Block of the Recovery. The General Case . . . . . . . . . . . 5.4 Recovery of Conjugacy Classes of Braids from the Irreducible Braid Components. The General Case . . . . . . . . . . . . . . . . . 5.5 Proof of the Main Theorem for Reducible Braids. The General Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143 146 150 152 156

161 161 165 171 174 179

6

The Conformal Module and Holomorphic Families of Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.1 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.2 Families of Polynomials, Solvability and Reducibility . . . . . . . . . . . . . 198

7

Gromov’s Oka Principle and Conformal Module. . . . . . . . . . . . . . . . . . . . . . . 7.1 Gromov’s Oka Principle and the Conformal Module of Conjugacy Classes of Elements of the Fundamental Group . . . . . . . 7.2 Description of Gromov-Oka Mappings from Open Riemann Surfaces to C \ {−1, 1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Gromov-Oka Mappings from Tori with a Hole to P3 . . . . . . . . . . . . . .

205

Gromov’s Oka Principle for .(g, m)-Fiber Bundles . . . . . . . . . . . . . . . . . . . . . 8.1 (g, m)-Fiber Bundles. Statement of the Problem . . . . . . . . . . . . . . . . . . . 8.2 The Monodromy of (g, m)-Bundles. Isotopy and Isomorphism of Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 (0, 4)-Bundles Over Genus 1 Surfaces with a Hole . . . . . . . . . . . . . . . . 8.4 (0, 4)-Bundles Over Tori with a Hole. Proof of Theorem 7.4 . . . . . . 8.5 Smooth Elliptic Fiber Bundles and Differentiable Families of Complex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Complex Analytic Families of Canonical Tori . . . . . . . . . . . . . . . . . . . . . . 8.7 Special (0, 4)-Bundles and Double Branched Coverings . . . . . . . . . .

235 235

8

205 216 226

238 247 252 258 262 269

Contents

xv

8.8 8.9

Lattices and Double Branched Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 The Gromov-Oka Principle for (1, 1)-Bundles over Tori with a Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

9

Fundamental Groups and Bounds for the Extremal Length . . . . . . . . . . 9.1 The Fundamental Group and Extremal Length. Two Theorems . . . 9.2 Coverings of C \ {−1, 1} and Slalom Curves . . . . . . . . . . . . . . . . . . . . . . . 9.3 Building Blocks and Their Extremal Length. . . . . . . . . . . . . . . . . . . . . . . . 9.4 The Extremal Length of Words in π1 . The Lower Bound . . . . . . . . . . 9.5 The Upper Bound of the Extremal Length for Words Whose Syllables Have Degree 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 The Extremal Length of Arbitrary Words in π1 . The Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 The Extremal Length of 3-Braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

289 289 292 300 305 315 321 338

10

Counting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

11

Riemann Surfaces of Second Kind and Finiteness Theorems . . . . . . . . . 11.1 Riemann Surfaces of First and of Second Kind and Finiteness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Some Further Preliminaries on Mappings, Coverings, and Extremal Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Holomorphic Mappings into the Twice Punctured Plane. . . . . . . . . . . 11.4 (g, m)-Bundles over Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

357

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Several Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 A Lemma on Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Koebe’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

411 411 412 414

A

358 363 368 386

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

Chapter 1

Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

In this chapter we provide prerequisites from several topics, like Riemann surfaces, braids and mapping classes, as well as Teichmüller theory. For completeness and convenience of the reader and for fixing notation and terminology we give here a recollection of basic facts, that will be used later frequently. References to the literature are given. In some cases proofs are included, for instance, in the case when the material is less standard and not extensively treated in the literature, or when the idea behind a statement will occur to be important in the following. Readers who are familiar with the material may skip this chapter and consult it later for notation and for some less standard facts that are include in this chapter.

1.1 Free Homotopy Classes of Mappings and Conjugacy Classes of Group Elements The Change of the Base Point Let .X be a connected smooth manifold with base point .x0 and non-trivial fundamental group .π1 (X , x0 ). Let .α be an arc in .X with initial point .x0 and terminal point x. Change the base point .x0 ∈ X along a curve .α to the point .x ∈ X . This leads to an isomorphism .Isα : π1 (X , x0 ) → π1 (X , x) of fundamental groups induced by the correspondence .γ → α −1 γ α for any loop .γ with base point .x0 and the arc .α with initial point .x0 and terminal point x. We will denote the correspondence .γ → α −1 γ α between curves also by .Isα . We call two homomorphisms hj : G1 → G2 , j = 1, 2, from a group G1 to a group G2 conjugate if there is an element g  ∈ G2 such that for each g ∈ G1 the equality h2 (g) = g  −1 h1 (g)g  holds. For two arcs α1 and α2 with initial point x0 and terminal point x we have α2−1 γ α2 = (α1−1 α2 )−1 α1−1 γ α1 (α1−1 α2 ). Hence, the two isomorphisms Isα2 and Isα1 differ by conjugation with the element of π1 (X , x) represented by α1−1 α2 . © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025 B. Jöricke, Braids, Conformal Module, Entropy, and Gromov’s Oka Principle, Lecture Notes in Mathematics 2360, https://doi.org/10.1007/978-3-031-67288-0_1

1

2

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

Free homotopic curves are related by homotopy with fixed base point and an application of a homomorphism Isα that is defined up to conjugation. Hence, free homotopy classes of curves can be identified with conjugacy classes of elements of the fundamental group π1 (X , x0 ) of X . Let X and Y be oriented smooth connected non-compact surfaces with finitely generated fundamental group, with base points x0 ∈ X and y0 ∈ Y. For a continuous mapping F : X → Y with F (x0 ) = y0 we denote by F∗ : π1 (X , x0 ) → π1 (Y, y0 ) the induced map on fundamental groups. For each element e0 ∈ π1 (X , x0 ) the image F∗ (e0 ) is called the monodromy along e0 , and the homomorphism F∗ is called the monodromy homomorphism corresponding to F . The mapping F → F∗ defines a one to one correspondence from the set of homotopy classes of continuous mappings X → Y with fixed base point x0 in the source and fixed value y0 at the base point to the set of homomorphisms π1 (X , x0 ) → π1 (Y, y0 ). Consider a free homotopy Ft , t ∈ (0, 1), of continuous mappings from X to def

Y. Consider the curve α(t) = Ft (x0 ), t ∈ [0, 1]. Suppose F0 (x0 ) = y0 and F1 (x0 ) = y1 . Each curve β in Y with initial point y0 and terminal point y1 defines an isomorphism π1 (Y, y0 ) → π1 (Y, y1 ) by considering the compositions of curves β −1 γβ for loops γ representing elements of π1 (Y, y0 ). For different curves β the isomorphisms differ by conjugation. Identifying fundamental groups with different base point by a chosen isomorphism of the described kind we obtain a mapping that assigns to each free homotopy class of maps X → Y a conjugacy class of homomorphisms π1 (X , x0 ) → π1 (Y, y0 ). The mapping is surjective. It is also injective by the following reason. Let Fj : X → Y, j = 0, 1, be continuous mappings, Fj (x0 ) = y0 , such that for an element e ∈ π1 (Y, y0 ) the equality (F1 )∗ = e−1 (F0 )∗ e holds. Let α be a smooth curve in Y that represents e. There exists a free homotopy F t , t ∈ [0, 1], of mappings such that F 0 = F0 and F t (x0 ) = α(t), t ∈ [0, 1]. Then (F 1 )∗ = e−1 (F 0 )∗ e. Hence, F1 and F 1 define the same homomorphism π1 (X , x0 ) → π1 (Y, y0 ), and hence, they are homotopic. We obtain the following theorem (see also [36],[76], [75].) Theorem 1.1 The free homotopy classes of continuous mappings from X to Y are in one-to-one correspondence to the set of conjugacy classes of homomorphisms between the fundamental groups of X and Y.

1.2 Riemann Surfaces We will use the common definition of a surface, of a smooth surface, of a surface with boundary, and of a Riemann surface. A homeomorphic mapping .ω : S → X from a surface S onto a Riemann surface X is called a conformal structure or a complex structure, having in mind, that the pull back of a complex analytic atlas on X provides a complex analytic atlas on S.

1.2 Riemann Surfaces

3

If the interior of a smooth surface with boundary is equipped with a complex structure, the surface with boundary is called (following Ahlfors) a bordered Riemann surface. A surface (not a surface with boundary) is called open if no connected component is compact. A compact surface (without boundary) is called a closed surface. A closed surface with finitely many points removed is called a punctured surface. The removed points are called punctures. A connected closed oriented surface or a connected closed Riemann surface is called of type .(g, m), if it has genus .g and is equipped with .m distinguished points. The intersection number of an ordered pair of smooth loops with transversal intersection in a smooth oriented surface X is the sum of the intersection numbers over all intersection points. The intersection number at an intersection point equals .+1 if the orientation determined by the tangent vector to the first curve followed by the tangent vector to the second curve is the orientation of X, and equals .−1 otherwise. The intersection number does not change under homotopy, i.e. the intersection number of a homotopic pair of smooth curves with transversal intersection is equal to the intersection number of the original pair. If the loops are not smooth or do not intersect transversally, their intersection number is defined as intersection number of a homotopic pair of smooth loops with transversal intersection. We will consider only oriented surfaces, often without further mentioning. If not mentioned otherwise, the considered surfaces will be connected. A surface is called finite if its fundamental group is finitely generated. A finite surface is obtained from a closed surface by removing finitely many disjoint simply connected closed sets (called holes). The fundamental group of a surface of genus g with .m > 0 holes is a free group in .2g + m − 1 generators. Each finite connected open Riemann surface X is conformally equivalent to a domain (denoted again by X) on a closed Riemann surface .Xc such that each connected component of the complement .Xc \ X is either a point or a closed topological disc with smooth boundary [77]. A connected Riemann surface is said to be of first kind, if it is a closed or a punctured Riemann surface, otherwise it is said to be of second kind. If all holes of a finite open Riemann surface are closed topological discs, the Riemann surface is said to have only thick ends. Standard Bouquets of Circles Let X be an oriented smooth open surface of genus g ≥ 0 with .m > 0 holes, equipped with a base point .q0 . The union B of noncontractible circles in X with base point .q0 is called a bouquet of circles in X, if .q0 is the only common point of any pair of circles in B, and different circles in B with base point .q0 represent different elements of .π1 (X, q0 ). Suppose B is the union of simple closed oriented curves .αj , βj , .j = 1, . . . , g  , and .γk , .k = 1, . . . , m , with base point .q0 with the following property. Labeling the rays of the loops emerging from the base point .q0 by .αj− , βj− .γj− , and the incoming rays by .αj+ , βj+ .γj+ , and

.

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1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

Fig. 1.1 A standard bouquet of circles for a connected finite open Riemann surface

moving in counterclockwise direction along a small circle around .q0 we meet the rays in the order .

. . . , αj− , βj− , αj+ , βj+ , . . . , γk− , γk+ , . . . .

Then B is called a standard bouquet of circles in X. Let B be a standard bouquet of circles in X with base point .q0 . If the collection E of elements of the fundamental group .π1 (X, q0 ) represented by the collection of curves in B is a system of generators of .π1 (X, q0 ) (then in particular, .g  = g,  .m = m), we call B a standard bouquet of circles for X, and say that the system .E of generators is associated to a standard bouquet of circles for X. See also Fig. 1.1. Label the generators .E ⊂ π1 (X, q0 ) associated to a standard bouquet of circles for X as follows. The elements .e2j −1,0 ∈ π1 (X, q0 ), .j = 1, . . . , g, are represented by .αj , the elements .e2j,0 ∈ π1 (X, q0 ), j = 1, . . . , g, are represented by .βj , and the elements .e2g +k,0 ∈ π1 (X, q0 ), k = 1, . . . , m, of .π1 (X, q0 ) are represented by 2g +m−1 .γk . The system .E = {ej } labeled in this way is called a standard system of j =1 generators of .π1 (X, q0 ). A standard bouquet of circles for a connected finite open Riemann surface can be obtained as follows. Recall that the Riemann surface X is conformally equivalent to a domain in a closed Riemann surface .Xc of genus g. Take a fundamental polygon of the compact Riemann surface .Xc , so that the projection of its sides to .Xc do not meet the holes of X. The projections of the sides provide a collection of simple .

1.2 Riemann Surfaces

5

closed curves .αj , .βj , .j = 1, . . . , g, with base point .q0 , that are pairwise disjoint outside .q0 , and represent the elements of the fundamental group .π1 (Xc , q0 ). Each pair .(αj , βj ) corresponds to a handle of .Xc . Cut .Xc along the curves .αj , .βj , ˚ Denote the connected .j = 1, . . . , g. We obtain a simply connected domain .X. c components of .X \X by .C1 , C2 , . . . , Cm . (The .C are also the connected components    ˚\ X of .X j (αj ∪ βj ) .) Each component is either a point or a closed disc. We of .m − 1 simple associate to the components .C ,  = 1, . . . , m − 1, a collection  closed curves .γ in X with base point .q0 that meet the union . j (αj ∪ βj ) exactly at the point .q0 and are pairwise disjoint outside .q0 . Moreover, each .γ is contractible in .X ∪ C and divides X into two connected components, one of them containing .C . We will say that .γ surrounds .C . We orient .γ so that .C is on the left when walking along the with this orientation. We may choose the .γ so that the g curve .γ equipped m−1 γ is a standard bouquet of circles for X. union . j =1 (αj ∪ βj ) ∪ =1 Let X be a connected finite open smooth surface with base point .q0 , and B a standard bouquet of circles in X. Choose a neighbourhood .N(B) of B in X with the following properties. .N(B) contains a smoothly bounded open disc D in X around .q0 , such that the intersection of each circle in the bouquet with .X \ D is a closed arc with endpoints on the boundary .∂D. Denote the arcs by .sj . Moreover, .N(B) is equal 2g +m−1 Vj of D with disjoint sets .Vj , where each .Vj has the to the union .D ∪ j =1 following properties. There exists a diffeomorphism .ϕj from a half-open rectangle .{x + iy ∈ C : |x| < a, |y| ≤ b} in the plane with sides parallel to the axes onto .Vj , that maps the horizontal sides into .∂D, and the rest of the rectangle into .X \ D. Moreover, its restriction to the vertical symmetry axis of the rectangle (with the direction of the imaginary axis) parameterizes the arc .sj (with the orientation of the respective circle of the bouquet). We call the .Vj bands attached to D representing .sj , and we call a neighbourhood of the described form a standard neighbourhood of the standard bouquet B. Lemma 1.1 Suppose a standard bouquet B of circles for a connected open Riemann surface X consists of g pairs of curves .αj ,.βj , and .m − 1 curves .γk as above, that represent a system of generators of .π1 (X, q0 ), and let .N(B) be a standard neighbourhood of S. Then there exists a continuous family of mappings .ϕt : X → X, t ∈ [0, 1], that map X homeomorphically onto a domain .Xt in X such that .ϕ0 is the identity and .X1 = N(B). In particular, X has genus g and m holes. Proof For each j the pair of loops .αj , .βj in .N(B) has non-vanishing intersection number. Each of these pairs corresponds to a handle of .N(S). For any other pair of loops the intersection number equals zero. Each loop .γk , k = 1, . . . , m − 1, divides .N (B). One of the connected components of .N(B) \ γk is an annulus .Ak whose second boundary component .∂k is a boundary component of .N(B). The loop .γk also divides X. Otherwise there would be a simple closed curve .γ in X with base point .q0 whose only intersection point with .γk is .q0 , and that has points on .Ak and in .N(B) \ Ak . Then the intersection number of the curves .γ and .γk would be different from zero. But all elements of the fundamental group

6

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

of .N (B), and hence all elements of the fundamental group of X, have vanishing intersection number with .γk . Let .Ωk be the component of .X \ γk that contains .∂k . Put .Xk = Ωk \ Ak . The last boundary component .∂m of .N(B) also divides X. Otherwise there would be an arc in X that joins two points that are close to a point in .∂m and locally on different sides of .∂m . Then there would exist a simple closed loop in X with base point .q0 that has non-zero intersection number with .∂m . This is impossible, since all elements of the fundamental group .π1 (X, q0 ) can be represented by loops that are contained in .N(B) and hence have zero intersection number with .∂m . Let .Xm be the connected component of .X \ ∂m , that contains the part of a neighbourhood of .∂m which is contained in .X \ N(B). Each of the .Xk is an open Riemann surface of some genus with a number of holes. Since .αj , .βj , and .γk represent generators of the fundamental group of both, X and .N (B), each .Xk has genus zero. No .Xk is a disc. If some .Xk , k ≤ m − 1, was a disc, .γk would be contractible in X, which is a contradiction. .Xm is not a disc, because .∂m is free homotopic to .γ1 . . . γm−1 α1 β1 α1−1 β1−1 . . . αg βg αg−1 βg−1 which is not the identity in the fundamental group of X. Since the fundamental groups of X and .N (B) are isomorphic, each .Xk is an annulus bounded be a boundary component of X and .∂k . We proved that X has genus g and m holes. Moreover, there exists a continuous family of mappings .ϕt : X → X, t ∈ [0, 1], that map X diffeomorphically onto a domain .Xt in X such that .ϕ0 is the identity and .X1 = 

N (B). Coverings For the following facts see [23, 27]. By a covering .P : Y → X we mean a continuous map P from a connected topological space .Y to a connected topological space .X such that for each point .x ∈ X there is a neighbourhood .V (x) of x such that the mapping P maps each connected component of the preimage of .V (x) homeomorphically onto .V (x). (Note that in function theory sometimes these objects are called unlimited unramified coverings to reserve the notion “covering” for more general objects.) A covering .P : Y → X is called a universal covering if for each covering .PZ : Z → X of .X by a connected topological space .Z and for each pair of points .y ∈ Y and .z ∈ Z with .P (y) = PZ (z) there exists a unique fiber preserving continuous mapping .f : Y → Z that takes y to z. A connected locally simply connected topological space has up to isomorphism at most one universal covering. If .X and .Y are connected manifolds, .Y is simply connected, and .P : Y → X is a covering, then P is the universal covering. For each connected manifold .X the universal covering exists and is denoted by P : X˜ → X . It is constructed as follows. Fix a base point .x0 in .X . For each .x ∈ X we denote by .π1 (X ; x0 , x) the set of homotopy classes of arcs in X with initial point .x0 and terminal point x. Consider the set that consists of the union of all elements .ex0 ,x ∈ π1 (X ; x0 , x) for all .x ∈ X . Consider the mapping that assigns to each element .ex0 ,x ∈ π1 (X ; x0 , x) the point .x ∈ X . We define a system of fundamental neighbourhoods of elements of the defined set as follows. For each element .ex0 ,x ∈ π1 (X ; x0 , x) we take any simply connected neighbourhood .U ⊂ X .

1.2 Riemann Surfaces

7

and consider the set of all products .ex0 ,x ex,y ∈ π1 (X ; x0 , y) for which .ex,y can be represented by curves that are contained in U and join x with y. We obtain a simply connected manifold .X˜ and a covering .P : X˜ → X . (For a proof see [27].) Consider the universal covering .P : X˜ → X of a manifold X. A covering transformation is a homeomorphism .ϕ : Y ⊃  such that .P ◦ ϕ = P. The covering ˜ X). By transformations form a group under composition, denoted by .Deck(X, the definition of the universal covering for each pair of points .z1 , z2 of .X˜ with ˜ X) with .z2 = ϕ(z1 ). .P(z1 ) = P(z2 ) there exists a unique element .ϕ ∈ Deck(X, Equip X with a Riemannian metric d, and denote by the same letter d the lifted ˜ In other words, the length of a curve in the metric d on .X˜ is the length metric on .X. ˜ The Dirichlet region containing .z0 is of its projection to X. Take any point .z0 ∈ X. defined as ˜ X) } . Dz0 = {z ∈ X˜ : d(z, z0 ) < d(z, g(z0 )) for all g ∈ Deck(X, def

.

(1.1)

The Dirichlet region is an open set and .g1 (Dz0 ) ∩ g2 (Dz0 ) = ∅ if .g1 and .g2 are different covering transformations. The union of the closures .g(Dz0 ) over all covering transformations g is equal ˜ Hence, the union of .Dz0 with a suitable subset of its boundary is a to the whole .X. fundamental region. A subset U of .X˜ is called a fundamental region for the covering ˜ → X if each point in X has exactly one preimage under the mapping .P : .P : X U → X. Further, for each point z of the boundary of .Dz0 (taken in the topological ˜ there is a covering transformation g such that .d(z, z0 ) = d(z, g(z0 )). space .X) We saw that for each point .z0 ∈ X˜ there exists a neighbourhood U such that .U ∩ g(U ) = ∅ for all covering transformations except the identity. In particular, no element of this group except the identity has a fixed point, and the group acts ˜ i.e. for two compact sets A and B in .X˜ the set .A ∩ g(B) is discontinuously on .X, non-empty for at most finitely many elements g of the group. Let X be a connected finite open Riemann surface with base point .q0 and let .P : X˜ → X be the universal covering map. We fix a base point .q0 ∈ X and a base point ˜ The group of covering transformations of .X˜ can be identified .q ˜0 ∈ P−1 (q0 ) ⊂ X. with the fundamental group .π1 (X, q0 ) of X by the following correspondence (See ˜ X). Let .γ˜0 be an arc in e.g. [27]). Take a covering transformation .σ ∈ Deck(X, ˜ .X with initial point .q ˜0 and terminal point .σ (q˜0 ). Denote by .Isq˜0 (σ ) the element of .π1 (X, q0 ) represented by the loop .P(γ˜0 ). By the convention to write products of curves from left to right and compositions ˜ X)  σ → Isq˜0 (σ ) ∈ of mappings from right to left for the mapping .Deck(X, π1 (X, q0 ) the following relation holds Isq˜0 (σ1 σ2 ) = Isq˜0 (σ2 )Isq˜0 (σ1 ) .

.

Hence, the mapping .σ → Isq˜0 (σ −1 ) is a group homomorphism. This homomorphism is injective and surjective, hence it is a group isomorphism. The inverse q˜ −1 .(Is 0 ) of the mapping .Isq˜0 is obtained as follows. Represent an element .e0 ∈

8

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory π1 (X, q0 )

Fig. 1.2 A commutative diagram related to the change of the base point

Isq˜0

˜ X) Deck(X,

Isα

Isq˜

π1 (X, q)

π1 (X, q0 ) by a loop .γ0 . Consider the lift .γ˜0 of .γ0 to .X˜ that has initial point .q˜0 . Then q˜ −1 .(Is 0 ) (e0 ) is the covering transformation that maps .q˜0 to the terminal point of .γ˜0 . def ˜ ∈ X the isomorphism .Isq˜ : For another point .q˜ of .X˜ and the point .q = P(q) ˜ ˜ X) the element of .π1 (X, q) Deck(X, X) → π1 (X, q) assigns to each .σ ∈ Deck(X, ˜ that is represented by .P(γ˜ ) for a curve .γ˜ in .X that joins .q˜ with .σ (q). ˜ .Isq˜ is related q˜0 ˜ to .Is as follows. Let .α˜ be an arc in .X with initial point .q˜0 and terminal point .q. ˜ Put .α = P(α). ˜ Then for the isomorphism .Isα : π1 (X, q0 ) → π1 (X, q) the equation ˜ X), Isq˜ (σ ) = Isα ◦ Isq˜0 (σ ), σ ∈ Deck(X,

.

(1.2)

holds, i.e. the diagram Fig. 1.2 is commutative. Indeed, let .α˜ −1 denote the curve that is obtained from .α˜ by inverting the direction on .α, ˜ i.e. moving from .q˜ to .q˜0 . For a ˜ in .X˜ has initial point curve .γ˜0 in .X˜ that joins .q˜0 with .σ (q˜0 ), the curve .α˜ −1 γ˜0 σ (α) −1 .q ˜ and terminal point .σ (q). ˜ Therefore .P(α˜ γ˜0 σ (α)) ˜ represents .Isq˜ (σ ). On the other hand P(α˜ −1 γ˜0 σ (α)) ˜ = P(α˜ −1 ) P(γ˜0 ) P(σ (α)) ˜ = α −1 γ0 α

.

(1.3)

represents .Isα (e0 ) with .e0 = Isq˜0 (σ ). In particular, if .q˜0 ∈ P−1 (q0 ) is another preimage of the base point .q0 under the projection .P, then the associated isomorphisms to  the fundamental group .π1 (X, q0 ) are conjugate, i.e. .Isq˜0 (e0 ) = (e0 )−1 Isq˜0 (e0 )e0 for each .e0 ∈ π1 (X, q0 ). The element .e0 is represented by the projection of an arc in .X˜ with initial point .q˜0 and terminal point .q˜0 . Keeping fixed .q˜0 and .q0 we will say that a point .q˜ ∈ X˜ and a curve .α in X are compatible if the diagram Fig. 1.2 is commutative, equivalently, if Eq. (1.2) holds. We may also start with choosing a curve .α in X with initial point .q0 and terminal point q. Then there is a point .q˜ = q(α), ˜ such that .q˜ and .α are compatible. Indeed, let .α˜ be the lift of .α, that has initial point .q˜0 . Denote the terminal point of .α˜ by .q(α), ˜ and repeat the previous arguments. For two Riemann surfaces .X and .Xˆ and a non-constant holomorphic mapping ˆ → X a point .xˆ ∈ Xˆ is called a branch point or ramification point if there is .p : X no neighbourhood of .xˆ such the restriction of p to it is injective. For each branch point .xˆ there exists a neighbourhood V of .xˆ and an integer number .k ≥ 2, such that k ˆ near .xˆ and on .X .p | V is equal to the mapping .z → z in suitable coordinates on .X near .x = p(x). ˆ

1.2 Riemann Surfaces

9

A non-constant holomorphic mapping .p : Xˆ → X is called a branched covering, ˆ if the image E of the set .Eˆ of branch points is discrete and the restriction .p | (X \ E) is a covering. The set E is called the branch locus. The branched covering is called simple if for each point .x ∈ E the fiber .p−1 (x) contains exactly one branch point and in a neighbourhood of this branch point the mapping is a double branched covering (i.e. in suitable coordinates it equals .z → z2 ). Hyperbolic Riemann Surfaces For the following collection of well-known results we refer e.g. to [58], and [59]. By the Uniformisation Theorem the universal covering space .X˜ of each connected Riemann surface X (equipped with the complex structure that makes the projection .P : X˜ → X holomorphic) is conformally equivalent to one of the following: the Riemann sphere .P1 , or the complex plane def

C, or the upper half-plane .C+ = {z ∈ C : Imz > 0}. The upper half-plane is equipped with the hyperbolic metric which is defined by the infinitesimal length element . |dz| 2y . If the upper half-plane is replaced by the unit disc .D, the hyperbolic

.

|dz| metric (that is induced by a conformal mapping .C+ → D) is defined by . 1−|z 2| . A Riemann surface whose universal covering is (conformally equivalent to) the upper half-plane will be called hyperbolic.

Let X be a finite Riemann surface with .X˜ being the upper half-plane. Each covering transformation is a conformal mapping of the upper half-plane onto itself. Such a mapping extends to Möbius transformation on the Riemann sphere. Hence, the group of covering transformations is a subgroup of the projective special linear group .P SL2 (R) = SL2 (R)−Id, where .SL2 (R) is the group of mappings .z → az+b cz+d with .ad − bc = 1, for which .a, b, c, d are real numbers, equivalently, that map the real axis to itself, and .−Id is the subgroup generated by .−Id for the identity .Id. The group of covering transformations acts properly discontinuously. Subgroups of .P SL2 (R) that act properly discontinuously are called Fuchsian groups. For each fixed point free Fuchsian group .Γ the quotient .C+ Γ is a Riemann surface. The limit set of a Fuchsian group .Γ is the set of accumulation points of .{g(z0 ) : g ∈ Γ }, where .z0 is any point of the upper half-plane. The set is independent of the choice of the point .z0 . The limit set of a fixed point free Fuchsian group is either the real axis .R or a nowhere dense subset of the real axis without isolated points. A Fuchsian group is said to be of first kind if the limit set is the real axis, and is said to be of second kind if the limit set is a nowhere dense subset of the real axis without isolated points. The following theorem motivates our definition of Riemann surfaces of first and second kind. Theorem 1.2 A connected Riemann surface is of first kind iff it is the quotient C+ Γ of the upper half-plane by a Fuchsian group .Γ of first kind.

.

For a proof (in a more general situation) see e.g. [56], II Theorem 3.2 and [2].

10

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

Fig. 1.3 The universal covering of the twice punctured complex plane

The Universal Covering of the Twice Punctured Complex Plane We recall the well-known explicit description of the universal covering of .C \ {0, 1}. We consider the domain .D0 = {z ∈ C+ : 0 < Rez < 1, |z − 12 | > 12 } in the upper half-plane, which is bounded by a half-circle and two half-lines. These curves are geodesic curves in the hyperbolic metric of the upper half-plane. We call .D0 the geodesic triangle with vertices .0, 1, and .∞. Denote by .P the conformal mapping from .D0 onto the upper half-plane, whose continuous extension to the boundary takes 0 to def

0, 1 to 1 and .∞ to .∞. The extension of .P to the boundary takes . = (0, i∞) to def

def

(−∞, 0), .C = {|z − 12 | = 12 } ∩ C+ to .(0, 1) and .r = 1 + (0, +i∞) to .(1, ∞).

.

We reflect .D0 in the circle .|z − 12 | = 12 . By symmetry reasons the reflected domain is the geodesic triangle .D1 bounded by .C (the reflection of itself), and the half-circles in .C+ with diameter .(0, 12 ) (the reflection of .), and diameter .( 12 , 1) (the reflection of r). In the same way we reflect .D0 in the half-lines in the boundary (see Fig. 1.3). We reflect now .D1 in the circle .|z − 34 | = 14 with diameter .( 12 , 1). We obtain the geodesic triangle .D2 with vertices . 12 , . 23 , and 1. This can be seen by the following calculation. The mapping .z → ζ (z) = 4(z − 34 ) maps the circle .|z − 34 | = 14 to the unit circle .|ζ | = 1. The reflection map in .|ζ | = 1 is .ζ → ζ1 , hence, the reflection in the circle .|z − 34 | = z=

.

1 4

is defined by the mapping

1 3¯z − 2 1 1 1 1 + 3) = . (ζ + 3) → ( + 3) = ( 4 4¯z − 3 4¯z − 3 4 4 ζ

The mapping takes 1 to 1, 0 to . 23 , and . 12 to . 12 . We get further geodesic triangles by continuing reflecting in each geodesic arc of each already obtained geodesic triangle. We get a collection of open geodesic triangles whose closures in .C+ fill the whole .C+ .

1.2 Riemann Surfaces

11

Extend the mapping .P continuously to the closure of .D0 , and by Schwarz’ Reflection Principle through the geodesic arcs in the boundary of .D0 . Extending .P repeatedly through the geodesic boundary arcs of the images under repeated reflection, we obtain a holomorphic mapping form .C+ to .C \ {0, 1}, denoted also by .P. The extended .P takes each geodesic triangle, obtained from .D0 by an even number of successive reflections, conformally onto the upper half-plane, and maps each geodesic triangle, obtained from .D0 by an odd number of successive reflections, conformally onto the lower half-plane. It is easy to see that .P : C+ → C \ {−1, 1} is a covering, in fact it is the universal covering of the twice punctured complex plane. Each reflection in a geodesic side of one of the geodesic triangles obtained from .D0 by reflection is an anti-holomorphic self-homeomorphism of .C+ . The composition .ϕ of an even number of successive reflections is a conformal mapping of .C+ onto itself, for which .P◦ϕ = P, in other words, it is a covering transformation. The covering transformation obtained by reflection in the imaginary axis followed z by reflection in one of the half-circles .{z ∈ C+ : |z ± 12 | = 12 } equals .z → 1∓2z . 1 1 ∓¯z Indeed, the reflection in .{z ∈ C+ : |z ± 2 | = 2 } is given by the mapping .z → 2¯z±1 , the reflection in the imaginary axis is given by .x + iy → −x + iy. Reflection in the imaginary axis followed by reflection in .{z ∈ C+ : Rez = 1} equals .z → z + 2. These covering transformation generate the group of covering transformations. In particular, each covering transformation is an element of .P SL2 (Z). Here .SL2 (Z) is the group of .2 × 2 matrices with integer entries and determinant 1, and .P SL2 (Z) is the quotient .SL2 (Z)−Id, for the subgroup .−Id where .Id is the unit matrix. Properties of Elements of .P SL2 (Z) Let .T (z) = az+b cz+d represent an element of .P SL2 (Z). The fixed points of this mapping satisfy the equation . az+b cz+d = z, d−a b 2 hence, if .c = 0, .z + c z − c = 0. This implies that the fixed points are √ (a+d)2 −4 d−a .z± = − ± . We used the fact that the determinant of the matrix 2c  2c ab .A = equals .ad − bc = 1. The eigenvalues .t± of the matrix A satisfy the cd √ (a+d)2 −4 equation .t 2 − (a + d)t + 1 = 0, hence, .t± = a+d ± . We used again that 2 2 the determinant of the matrix equals 1. If .|a + d| > 2 there are two different real fixed points. The matrix A has two different   real eigenvalues .t± , and can be conjugated by a real matrix to the matrix t+ 0 . . Hence, T can be conjugated by a holomorphic self-homeomorphism of 0 t− t+ z  2 .C+ to the mapping .T (z) = t− = t+ z. If .|a + d| = 2, T has a double fixed point .z0 . This point is real. Conjugating this fixed point to .∞ (for instance by the conformal self-mapping .z → z01−z of .C+ ) we arrive at a mapping of the form .z → z + b for a real number .b. The eigenvalues are equal to .t± = 1. If .c = 0 then .a = d = ±1, hence .|a + d| = 2.

12

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

If .|a + √d| < 2 the eigenvalues are equal to .t± = ±i, if .|a + d| = 0, and to √ t± = 12 ± 23 i or .t± = − 12 ± 23 i, if .|a + d| = 1. In the first case the square of the mapping .T 2 is the identity. In the second case .T 3 is the identity.

.

1.3 Braids In this section we outline basic facts on braids. For details the reader may consult [14] or [52]. Most intuitively, braids are described in terms of geometric braids. We will use here the complex plane .C though the complex structure will play a role only later. Let .En be a subset of .C containing exactly n points. A geometric braid in .[0, 1] × C with base point .En is a collection of n mutually disjoint arcs in the cylinder .[0, 1]×C which joins the set .{0} × En in the bottom with the set .{1} × En in the top of the cylinder and intersects each fiber .{t}×C along an unordered n-tuple .En (t) of points. The arcs are called the strands of the geometric braid. There is a more conceptual point of view. We consider points .(z1 , . . . , zn ) ∈ Cn as ordered tuples of points in .C. Usually the points .zj ∈ C will be required to be pairwise distinct, in other words, we require that .(z1 , . . . , zn ) belongs to the configuration space .Cn (C) = {(z1 , . . . , zn ) ∈ Cn : zi = zj for .i = j } of n particles moving in the plane without collision. Denote by .Sn the symmetric group. Each permutation in .Sn acts on .Cn (C) by permuting the coordinates. The quotient .Cn (C)Sn is called the symmetrized configuration space. We denote its elements by .{z1 , . . . , zn } and consider them as unordered n-tuples of points in .C or as subset .En of .C consisting of exactly n points. For a subset A of .C we will put .Cn (A) = Cn (C) ∩ An . The configuration space inherits the topology and complex structure from .Cn . The symmetrized configuration space is given the quotient topology and quotient complex structure. Note that .Sn acts freely and properly discontinuously on .Cn (C). The canonical projection .Psym : Cn (C) → Cn (C)Sn is holomorphic. Identifying subsets of .C consisting of n points with elements of .Cn (C)Sn , we may regard a geometric braid with base point .En as a set of the form {(t, f (t)), t ∈ [0, 1]} ⊂ [0, 1] × C

.

f

(1.4)

for a continuous mapping .[0, 1]  t −→ Cn (C)Sn with .f (0) = f (1) = En . We will often identify the geometric braid (1.4) with the mapping f defining it. Two geometric braids with base point .En are called isotopic if there is a continuous family of geometric braids with base point .En joining them. A braid on n strands with base point .En is an isotopy class of geometric braids with base point .En . Isotopy classes of geometric braids with base point .En form a group. The operation is obtained by putting one geometric braid on the top of another.

1.3 Braids

13

Take a geometric braid with base point .En . Assigning to each point in .{0} × En the point in .{1} × En which belongs to the same strand we obtain a permutation of the points of .En . It depends only on the isotopy class which we denote by b. This permutation is denoted by .τn (b). The braid b is called pure if .τn (b) is the identity. The set of pure n-braids forms a group which we denote by .PBn . Algebraically, n-braids are represented as elements of the Artin braid group .Bn . This is the group with generators denoted by .σ1 , . . . , σn−1 and (if .n ≥ 3) with relations .σi σj = σj σi for .|i − j | ≥ 2, and .σi σi+1 σi = σi+1 σi σi+1 for .i = 1, . . . , n − 2. An isomorphism between .Bn and isotopy classes of geometric braids with given base point .En can be obtained as follows. Choose a projection .Pr of .C onto the real line .R which is injective on .En . Label the points in .En so that .En = {z1 , . . . , zn } with .Pr(zj ) < Pr(zj +1 ). The strand of a (geometric) braid with initial point .zj is called the j -th strand. The projection .[0, 1] × C  (t, z) → (t, Pr z) ∈ [0, 1]×R assigns to each geometric braid a braid diagram (at intersection points of images of strands it is indicated which strands is “over” and which strand is “under”). Isotopy classes of geometric braids with base point .En correspond to equivalence classes of diagrams. The equivalence classes of diagrams can be interpreted as elements of the Artin braid group. The braid diagram corresponding to the generator .σj of the Artin braid group is the diagram with one intersection point, at this point the j -th strand is over the .j + 1-st strand. The isomorphism from the group of isotopy classes of geometric braids to the Artin group depends on the base point .En and the projection .Pr. A change of the base point and a change of the projection lead to conjugation with an element of the Artin group. Indeed, let .α be a curve in .Cn (C)Sn with initial point .En and endpoint .En . A change of base point from .En to .En along .α assigns to each curve .γ in .Cn (C)Sn with initial point and endpoint equal to .En (i.e. to each geometric braid with base point .En ) the curve .α −1 γ α. The latter curve is obtained by travelling first along the arc .α −1 in the symmetrized configuration space, that joins the new base point .En with .En , then along the curve .γ , and then along .α. Looking at the associated braid diagram using a fixed projection (provided it is injective on both,  .En and .En ) and considering equivalence classes we obtain the statement concerning the change of base point. A change of the projection can be made by fixing the projection and considering an isotopy of the space .C. This leads to an isotopy of geometric braids, hence we obtain the statement for the change of projections. The center .Zn of the Artin braid group .Bn is generated by the element .Δ2n , where def

Δn = (σ1 σ2 . . . σn−1 )(σ1 σ2 . . . σn−2 ) . . . (σ1 σ2 )σ1 is called the Garside element. The braid .Δn is represented by the following geometric braid. Choose a set .En ⊂ R that is invariant under reflection in the imaginary axis. The representing geometric braid is obtained from the trivial geometric braid with base point .En by a half-twist of the cylinder .C × [0, 1], i.e. by fixing the bottom of the cylinder and turning the top by the angle .π . We will say that the braid .Δn corresponds to a half-twist. The braid .Δ2n corresponds to a full twist. For an element e of a group we denote by .e the subgroup that is generated by e. We put .Zn = Δ2n .

.

14

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

The pure braid group for .n = 3 is related to a free group, more precisely the following lemma holds. Lemma 1.2 The group .PB3 Δ23  is isomorphic to the fundamental group of .C \ {−1, 1} with base point 0. The generators .σj2 Δ23  of .PB3 Δ23  correspond to the generators .aj of .π1 (C \ {−1, 1}, 0). Here .a1 is represented by a loop that surrounds .−1 counterclockwise, and .a2 is represented by a loop that surrounds 1 counterclockwise. Proof For a point .z = (z1 , z2 , z3 ) ∈ C3 (C) we denote by .Mz = M(z1 ,z2 ,z3 ) the Möbius transformation that maps .z1 to 0, .z3 to 1 and fixes .∞. Then .Mz (z2 ) omits 1 0, 1 and .∞. Notice that .Mz (z2 ) is equal to the cross ratio .(z2 , z3 ; z1 , ∞) = zz23 −z −z1 · z3 −∞ z2 −∞

=

def z2 −z1 z3 −z1 . We define .C(z) =

2Mz (z2 ) − 1 with .z = (z1 , z2 , z3 ) ∈ C3 (C). The def

mapping .C takes .C3 (C) to .C \ {−1, 1} and .C3 (R)0 = {(x1 , x2 , x3 ) ∈ R3 : x1 < x2 < x3 } to .(−1, 1). Associate to a curve .γ˜ (t) = (γ˜1 (t), γ˜2 (t), γ˜3 (t)), t ∈ [0, 1], γ˜1 (t) in .C3 (C) the curve .C(γ˜ )(t) = 2 γγ˜˜23 (t)− (t)−γ˜1 (t) − 1 , t ∈ [0, 1], in .C which omits the points .−1 and 1. The mapping .γ˜ → C(γ˜ ) defines a homomorphism .C∗ from the fundamental group .π1 (C3 (C), (−1, 0, 1)) of .C3 (C) with base point .(−1, 0, 1) to def

def

the fundamental group .π1 = π1 (C \ {−1, 1}, 0) of the twice punctured complex plane with base point 0. Identify the pure braid group .PB3 with a subgroup of the fundamental group of .C3 (C)S3 with base point .{−1, 0, 1}. For curves .γ representing elements of this subgroup we consider the lift .γ˜ under .Psym with base point .(−1, 0, 1) ∈ C3 (C). This gives an isomorphism from the pure braid group .PB3 onto the fundamental group of .C3 (C) with base point .(−1, 0, 1). def

γ , .˚ γ (t) = (−1, C(γ˜ )(t), 1) =  The braids represented by the two loops .γ˜ and .˚ 2Mγ˜ (t) (γ˜1 (t))−1 , 2Mγ˜ (t) (γ˜2 (t))−1 , 2Mγ˜3 (t) (γ˜1 (t))−1) , t ∈ [0, 1], differ by 2 γ we act for each t by a complex a power .Δ2N 3 of the full twist .Δ3 . Indeed, to obtain .˚ linear mapping on the point .γ˜ (t). The number N can be interpreted as the linking number of the first and the third strands of the geometric braid .γ˜ (t), t ∈ [0, 1]. This linking number is obtained as follows. Discard the second strand. The resulting braid γ equals .σ 2N where N is the mentioned linking number. For the geometric braid .˚ the linking number of the first and third strand is zero. It follows that .C∗ is surjective and its kernel equals .Δ23 . A word in the generators of a free group is called reduced, if neighbouring terms are powers of different generators. We saw that for each element .b of the pure braid group modulo its center .PB3 Z3 there is a unique element .b ∈ PB3 that represents 2 2 .b and can be written as reduced word in .σ and .σ . Indeed, this representative b of .b 1 2 is determined by the property that the linking number between the first and the third strand equals zero. Assigning to each element .b ∈ PB3 Z3 the mentioned element we obtain the isomorphism to the free group in two generators .σ12 Zn and .σ22 Zn , or, equivalently to the fundamental group .π1 (C \ {−1, 1}, 0) of the twice punctured complex plane with base point 0 with generators .a1 and .a2 , respectively. 

1.3 Braids

15

Geometric braids with base point .En were interpreted as paths in .Cn (C)Sn with initial and terminal point equal to .En , in other words, as loops in this space with base point .En . Isotopy classes of geometric braids with base point .En correspond to homotopy classes of loops in .Cn (C)Sn with base point .En , in other words, braids with base point .En correspond to elements of the fundamental group .π1 (Cn (C)Sn , En ) of .Cn (C)Sn with base point .En . Thus the Artin braid group .Bn is isomorphic to .π1 (Cn (C)Sn , En ). We saw that a change of the base point leads to an automorphism of .Bn defined by conjugation with an element of .Bn . n the set of conjugacy classes of .Bn . Its elements .bˆ ∈ B n can be Denote by .B interpreted as free homotopy classes of loops in .Cn (C)Sn . In other words, two geometric braids .f0 : [0, 1] → Cn (C)Sn , f1 : [0, 1] → Cn (C)Sn , represent n , iff there is a free homotopy joining them, i.e. if there exists the same class .B a continuous mapping .h : [0, 1] × [0, 1] → Cn (C)Sn such that .h(t, 0) = f0 (t), h(t, 1) = f1 (t), for .t ∈ [0, 1], and .h(s, 0) = h(s, 1) for .s ∈ [0, 1]. We may consider the continuous family of braids .fs : [0, 1] → Cn (C)Sn with variable base point as a free isotopy of braids. We will also use the following terminology. A loop in .Cn (C)Sn (i.e. a continuous map of the circle into .Cn (C)Sn ) is called a closed geometric braid. A free homotopy class of loops in .Cn (C)Sn is called a closed braid. Closed braids n . correspond to elements of .B Arnold interpreted the symmetrized configuration space .Cn (C)Sn as the space of monic polynomials .Pn of degree n without multiple zeros. (”Monic” refers to the property that the coefficient by the highest power of the variable equals 1.) Denote by .Pn the set of all monic polynomials of degree n. If we assign to each unordered n-tuple .En = {z1 , . . . , zn } (this time the .zj are not necessarily n  pairwise distinct) the monic polynomial . (z − zj ) = a0 + a1 z + . . . zn j =1

whose set of roots equal .En , we obtain a bijection onto the set .Pn . The set of monic polynomials of degree n can also be parameterized by the ordered n-tuple n .(a0 , a1 , . . . , an−1 ) ∈ C of the coefficients. Hence, we obtain a bijection between n .Cn (C)Sn and the set of points .(a0 , a1 , . . . , an−1 ) ∈ C which are coefficients of polynomials without multiple zeros. There is a polynomial .Dn in the variables .a0 , . . . , an−1 , called the discriminant, which vanishes exactly if the polynomial with these coefficients has multiple zeros. Hence, the symmetrized configuration space def

Cn (C)Sn can be identified with .Cn \VDn where .VDn = {(a0 , . . . , an−1 ) ∈ Cn : Dn (a0 , . . . , an−1 ) = 0}. The identification is actually a biholomorphic map. Thus, n .Cn (C)Sn is biholomorphic to a pseudoconvex domain in .C , even stronger, it is biholomorphic to the complement of an algebraic hypersurface in .Cn . The space .Pn ∼ = Cn (C)Sn received much attention in connection with problems of algebraic geometry. For instance, motivated by his interest in the Thirteen’s Hilbert problem, Arnold [7] computed its topological invariants. .

16

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

1.4 Mapping Class Groups Mapping class groups are defined as follows. Let A be a topological space, not necessarily compact, but paracompact. Let .A1 and .A2 be disjoint closed subsets of A. Denote by .Hom(A; A1 , A2 ) the set of self-homeomorphisms of A that fix .A1 pointwise and .A2 setwise. We will also write .Hom(A; A1 ) for .Hom(A; A1 , ∅), but in case we do not require that some points are fixed we write .Hom(A; ∅, A2 ). We will also write .Hom(A) for .Hom(A; ∅, ∅). Equip the set .Hom(A; A1 , A2 ) with compact open topology. Let A be an oriented manifold. By .Hom+ (A; A1 , A2 ) we denote the set of orientation preserving self-homeomorphisms in .Hom(A; A1 , A2 ). It forms a group with respect to composition. The set of connected components of + .Hom (A; A1 , A2 ) equipped with the inherited group structure is the mapping class group .M(A; A1 , A2 ) corresponding to the triple .(A, A1 , A2 ). ¯ be the closed unit disc in the complex plane .C with boundary .∂D and Let .D interior .D . At this point we may forget the complex structure of .D, but we will need it later.

Let .En0 = 0, n1 , . . . , n−1 be the “standard” subset of .D containing n points. n We also identify .En0 with the respective unordered n-tuple of points. The set ¯ ∂D, En0 ) is commonly known as the mapping class group of the n-punctured .M(D; ¯ ∂D, En0 ) is isomorphic to .M(D\E ¯ n0 ; ∂D) since each element disc. Note that .M(D; 0 ¯ n ; ∂D) extends continuously to each point of .En0 (the “punctures”). of .M(D\E The points of .En0 are also called “distinguished points”. The homeomorphisms ¯ ∂D, En0 ) are called homeomorphisms of .D ¯ with distinguished points in .Hom+ (D; 0 .En . By an abuse of language we will also call them homeomorphisms of the npunctured disc, having in mind the identification with their restrictions to .D \ En0 . ¯ ∂D, En0 ) of .Hom+ (D; ¯ ∂D, En0 ) containing the The connected component .Hom0 (D; ¯ which can be joined to the identity consists of the self-homeomorphisms of .D ¯ ∂D, En0 ). In other identity by a continuous family of homeomorphisms in .Hom+ (D; + ¯ 0 words, it consists of homeomorphisms in .Hom (D; ∂D, En ) which are isotopic to the identity through homeomorphisms fixing .∂D and .En0 pointwise. We have ¯ ∂D, En0 ) = Hom+ (D, ¯ ∂D, En0 )Hom0 (D; ∂D, En0 ) . M(D;

.

(1.5)

¯ ∂D, En ) can be defined for any The respective mapping class group .M(D; unordered n-tuple .En of points in .D. ¯ ∂D, En ) we consider its restriction For a homeomorphism .ϕ ∈ Hom+ (D; + .ϕ | D ∈ Hom (D; ∅, En ) and the corresponding mapping class in .M(D; ∅, En ). This mapping class is denoted by .mfree ϕ and is called the free isotopy class of .ϕ. We ¯ also call the elements of .M(D; ∂D, En ) the relative mapping classes. The restriction ¯ ∂D, En ) to .M(D; ∅, En ). map .ϕ → ϕ | D defines a surjective mapping from .M(D;

1.4 Mapping Class Groups

17

Indeed, each element of .M(D; ∅, En ) contains representatives which extend to the boundary .∂D as the identity mapping on .∂D. There is a short exact sequence ¯ ∂D, En ) → M(D; ∅, En ) → 0 . 0 → K → M(D;

.

(1.6)

¯ ∂D, En ) is in the kernel .K iff each representative is isotopic An element .m ∈ M(D; to the identity through homeomorphisms of .D which fix .En setwise (and, hence, pointwise). Let X be an oriented surface. A Dehn twist about a simple closed curve .γ in an oriented surface S is a mapping that is isotopic to the following one. Take a neighbourhood of .γ that can be parameterized as a round annulus .A = {e−ε < |z| < ε 1} so that .γ corresponds to .|z| = e− 2 . The mapping is an orientation preserving selfhomeomorphism of S which is the identity outside A and is equal to the mapping −εs+2π it → e−εs+2π i(t+s) for .e−εs+2π it ∈ A, i.e. .s ∈ (0, 1). Here .ε is a small .e positive number. For the following theorem see e.g. [42]. Theorem 1.3 The kernel .K is generated by a Dehn twist about a simple closed curve .γ in .D which is homologous to .∂D in .D\En . Let again .En ⊂ D be a set that contains exactly n points. The inclusion .D → P1 induces a surjective homomorphism .H∞ ¯ ∂ D, En ) → M(P1 ; ∞, En ), H∞ : M(D;

.

(1.7)

which is described in detail as follows. Take a mapping class .m ∈ M(D; ∂ D, En ). Represent it by a homeomorphism .ϕ. Extend .ϕ to a self-homeomorphism .ϕ∞ of .P1 by putting .ϕ∞ = ϕ on .D and .ϕ∞ = id outside .D. Let .m∞ ∈ M(P1 ; ∞, En ) be the mapping class of .ϕ∞ . It depends only on the mapping class .m of .ϕ in ¯ ∂ D, En ), not on the choice of the representative .ϕ. Put .M(D; def

H∞ (m) = m∞ ∈ M(P1 ; ∞, En ).

.

(1.8)

The homomorphism is surjective since each element of .M(P1 ; ∞, En ) can be represented by a homeomorphism that is equal to the identity on .P1 \ D. By Theorem 1.3 the kernel of the homomophism (1.8) consists of the class of powers of Dehn twists about a simple closed curve that is homologous to .∂D. Let A and .A be oriented manifolds, let .A1 and .A2 be subsets of A, and let      .A and .A be subsets of .A . Take a homeomorphism .ψ : A → A that maps .A j 1 2 −1 to .Aj for .j = 1, 2. The mapping .ϕ → ψ ◦ ϕ ◦ ψ is a one-to-one mapping from .Hom+ (A; A1 , A2 ) onto .Hom+ (A ; A1 , A2 ). It defines an isomorphism .isψ from the mapping class .M(A; A1 , A2 ) onto the mapping class .M(A ; A1 , A2 ). The isomorphism is determined by .ψ up to conjugation by an element of .M(A; A1 , A2 ).

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1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

Indeed, let .ψ1 : A → A be another homeomorphism that maps .A1 to .A1 and .A2 to .A2 . Then (ψ1 )−1 ◦ ϕ ◦ ψ1 = (ψ −1 ◦ ψ1 )−1 ◦ ψ −1 ◦ ϕ ◦ ψ ◦ (ψ −1 ◦ ψ1 ) .

.

 of an element .m of the mapping class group .M(A; A1 , A2 ) The conjugacy class .m consists of all elements .g−1 mg with .g ∈ M(A; A1 , A2 ).The set of conjugacy classes  A1 , A2 ). We will identify the conjugacy class of an element is denoted by .M(A; .m ∈ M(A; A1 , A2 ) with the set  .

ψ −1 ϕ ◦ ψ : ϕ ∈ m, ψ : A → A a homeomorphism that maps A1 to A2 and A2 to A2 ,

having in mind the isomorphisms .is−1 ψ that are determined up to conjugation by elements of .M(A; A1 , A2 ) .  1 ; ∞, En ). Any homeomorphism .h : Consider the set of conjugacy classes .M(P D \ En → C \ En that maps points close to .∂D to points close to .∞ defines an isomorphism M(D; ∅, En ) → M(P1 ; ∞, En )

.

by associating to each orientation preserving self-homeomorphism .ϕ : D \ En ⊃ the self-homeomorphism .h ◦ ϕ ◦ h−1 of .C \ En . The isomorphisms corresponding to different homomorphisms .D \ En → C \ En differ by conjugation by a selfhomeomorphism of .C \ En whose extension to .P1 fixes .∞. We obtain a one-to-one  correspondence between the set of conjugacy classes .M(D \ En ) and the set of  1 \ En ; ∞) ∼  1 ; ∞, En ). conjugacy classes .M(P = M(P More generally, if X is a closed oriented surface and .En a subset of X consisting of n points, the space .M(X; ∅, En ) ∼ = M(X\En ) is called the mapping class group of the n-punctured surface. Let X be a compact connected oriented surface with boundary and let .En be a subset of X that contains exactly n points. The set .M(X; ∂X, .En ) .∼ = .M(X\En ; ∂X) is called the mapping class group of the npunctured surface with boundary. Consider the mapping .ϕ → ϕ|IntX that assigns to each self-homeomorphism .ϕ of X the restriction to the interior .IntX of X. This mapping defines a homomorphism .M(X; ∂X, En ) → M(IntX; ∅, En ), whose kernel is generated by Dehn twists about closed curves in X that are homologous to a boundary component of X. Let X be a bordered Riemann surface with smooth boundary. There is a compact Riemann surface .Xc and a diffeomorphism from X onto the closure of a smoothly bounded domain in .Xc with the following properties. The diffeomorphism is conformal on .Int X. The domain in .Xc is obtained by removing from .Xc a finite number of smoothly bounded topological discs. This follows from the fact, that the interior of X admits a conformal mapping onto a smoothly bounded domain on a closed Riemann surface ([77], see also Sect. 1.2), and the smooth

1.5 The Relation Between Braids and Mapping Classes

19

analog of Caratheodory’s Theorem (Theorem 4 Chapter 2, Section 3, in [32]) that states smooth extension to the boundary of conformal mappings between smoothly bounded domains (see e.g. [8]). Let .∂1 , . . . , ∂N be the boundary components of X and let .δ1 , . . . , δN be the open discs on .Xc bounded by the .∂j . Let .En ⊂ Int X be a finite set. For each .j = 1, . . . , N we pick a point .ζj ∈ δj . Put .ζ = {ζ1 , . . . , ζN }. There is a homomorphism Hζ : M(X; ∂X, En ) → M(Xc ; {ζ1 , . . . , ζN }, En )

.

(1.9)

which can be described as follows. Take .m ∈ M(X; ∂X, En ). Let .ϕ ∈ Hom+ (X; ∂X, En ) be a representing homeomorphism. Let .ϕ c be the extension of .ϕ to .Xc which is the identity outside X. Then .Hζ (m) = mζ where .mζ is the class of .ϕ c in .M(Xc ; {ζ1 , . . . , ζN }, En ). The homomorphism (1.9) is surjective. Its kernel is generated by Dehn twists about closed curves that are homologous to a boundary component of X. We remove now the requirement that the elements of a mapping class fix the boundary of a bordered Riemann surface X pointwise (or fixes each boundary component setwise), in other words we consider mapping classes in .M(X; ∅, En ). We obtain a homomorphism M(IntX; ∅, En ) → M(Xc \ {ζ1 , . . . , ζN }; ∅, En ) .

.

We may identify the conjugacy classes   c \ {ζ1 , . . . , ζN }; ∅, En ) . M(IntX; ∅, En ) ∼ = M(X

.

1.5 The Relation Between Braids and Mapping Classes In this section we describe the isomorphism between the braid group .Bn and the mapping class group of the n-punctured disc. We use complex notation. For any subset E of the unit disc and any .ψ in

self-homeomorphism 1 n−1 + 0 .Hom (D, ∅, ∅) we put .evE ψ = ψ(E). For .En = 0, , . . . , ( considered as n n unordered tuple of n points or as set) we obtain  

  n−1 1 ,...,ψ . evEn0 ψ = ψ(0), ψ n n

.

(1.10)

We define the mapping .en ,  en (ψ) = ψ(0), . . . , ψ

.



n−1 n

 (1.11)

20

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

that assigns to .ψ the ordered n-tuple of values of .ψ at the points .0, n1 , . . . , n−1 n . Note that .en (ψ) ∈ Cn (C), and .Psym en (ψ) = evEn0 ψ. The isomorphism between the mapping class group .M(D, ∂ D, En0 ) and the group of isotopy classes of geometric braids with base point .En0 is obtained as follows. Let .ϕ ∈ Hom+ (D; ∂ D, En0 ). Consider a path .ϕt ∈ Hom+ (D, ∂ D), .t ∈ [0, 1], which joins .ϕ with the identity. In other words, .ϕt is a continuous family of selfhomeomorphisms of .D which fix the boundary .∂ D pointwise, such that .ϕ0 = id and 0 .ϕ1 = ϕ. Notice that we do not require that .ϕt maps .En to itself. By the AlexanderTietze Theorem (see e.g. [52], Section 1.6.1) such a family exists for each .ϕ ∈ Hom+ (D; ∂ D, En0 ). Consider the evaluation map  

n−1 ∈ Cn (C)Sn . .[0, 1]  t → evE 0 ϕt = ϕt (0), . . . , ϕt n n

(1.12)

This map defines a geometric braid in the cylinder .[0, 1] × D (i.e. .evEn0 ϕt ∈ Cn (D)Sn for each t). Its base point is En0 = ϕ0 (En0 ) = ϕ1 (En0 ) .

.

Notice that the isotopy class of the obtained geometric braid depends only on the class of .ϕ in .M(D; ∂ D, En0 ). The obtained mapping from .M(D; ∂ D, En0 ) to the group of braids with base point .En0 , hence to .Bn , is a homomorphism. It is, in fact, an isomorphism. This is a consequence of Proposition 1.1 below. Let .En be an arbitrary unordered n-tuple of points in .D. A continuous family of homeomorphisms .ϕt ∈ Hom+ (D; ∂ D), .t ∈ [0, 1], is called a parameterizing isotopy of a geometric braid .f : [0, 1] → Cn (D)Sn with base point .En if .ϕ0 = id and .evEn ϕt = f (t), .t ∈ [0, 1]. Proposition 1.1 Let .f : [0, 1] → Cn (D)Sn be a smooth geometric braid in [0, 1] × D with any base point .En ∈ Cn (D)Sn . Then there exists a smooth parameterizing isotopy .ϕt for f . The mappings .ϕt can be chosen so that the map .[0, 1] × D  (t, z) → (t, ϕt (z)) ∈ [0, 1] × D is a diffeomorphism. .

For the continuous version see e.g. [52]. Proof of Proposition 1.1 Lift the mapping f to a mapping .f˜ : [0, 1] → Cn (D). Denote the coordinate functions of .f˜ by .fj : [0, 1] → D. Choose .δ > 0 so that for each t the discs .Uj (t) of radius .δ around the points .fj (t), .j = 1, . . . , n, are pairwise disjoint subsets of .D.  (t, Uj (t)). Define a smooth mapping v Consider for each j the tube .Tj = t∈[0,1]  from the union . Tj of the tubes to the complex plane .C by putting j def

v(t, ζ ) =

.

∂ fj (t), (t, ζ ) ∈ Tj . ∂t

(1.13)

1.5 The Relation Between Braids and Mapping Classes

21

Notice that for .(t, ζ ) ∈ Tj the point .ζ is in the .δ-neighbourhood .Uj (t) of .fj (t) in .D. For each t the mapping v is constant on each .Uj (t) as a function of .ζ . The function .fj satisfies the differential equation .

∂ fj (t) = v(t, fj (t)), t ∈ [0, 1] . ∂t

(1.14)

Indeed, .(t, fj (t)) is contained in .Tj , hence, .v(t, fj (t)) = ∂t∂ fj (t). Let .T0j  Tj be the . 2δ -tubes in .[0, 1] × D around the graphs of .fj . Let .χ0 be a δ ∞ .C -function on .C with values in .[0, 1] that equals 1 on .|ζ | ≤ 2 and equals 0 outside .|ζ | < δ. Define a mapping .χ on .[0, 1] × D by the equations .χ (t, ζ ) = χ0 (ζ − fj (t)) ∞ for .(t, ζ ) ∈ Tj , .j = 1, . . . , n, and .χ (t, ζ ) = 0 else. Then  .χ 0is a .C -function on .[0, 1] × D with values in .[0, 1] that equals 1 in the union . Tj of the smaller tubes j  and equals 0 outside the union of the bigger tubes . Tj . Put j

V (t, ζ ) =

.

⎧ ⎨v(t, ζ ) · χ (t, ζ ) ⎩ 0

if (t, ζ ) ∈



Tj (1.15)

j

otherwise

For .ζ ∈ D we denote by .ϕt (ζ ) the solution of the initial value problem .

∂ ϕt (ζ ) = V (t, ϕt (ζ )), ϕ0 (ζ ) = ζ , ∂t

(1.16)

on the maximal interval of existence. By the smooth version of the Picard-Lindelöf Theorem the initial value problem has a unique solution on a maximal interval of existence and the solution depends smoothly on all parameters (see e.g. [37], Theorem V, 3.1, Corollary V, 3.1 and the remark after it, and also Theorem V, 4.1). Near the ends of the maximal interval of existence the solution curve approaches the boundary of the domain. Since .V = 0 in a neighbourhood of .[0, 1]×∂D, the solution curve approaches .{1} × D, in other words the maximal interval of existence equals .[0, 1]. Moreover, for each t the mapping .ζ → ϕt (ζ ) is a local diffeomorphism, and hence by the uniqueness theorem for solutions of initial value  problems, this mapping is a global diffeomorphism. Since .V = v on the union . T0j of the smaller j

tubes, Eq. (1.14) yields .ϕt (fj (0)) = fj (t) for all j and .t ∈ [0, 1]. Since V vanishes in a neighbouhood of .[0, 1]×∂D, .ϕt (ζ ) = ζ for .ζ close to .∂D. Hence, .ϕt , t ∈ [0, 1], is a parameterizing isotopy. 

Remark 1.1 Let .f : [0, 1] × [0, 1] → Cn (D)Sn be a smooth isotopy of braids with fixed base point .En , that joins the braids .t → f (t, 0) and .t → f (t, 1). Then ϕt,s (ζ )

there exists a diffeomorphism .[0, 1] × [0, 1] × D  (t, s, ζ ) −→ [0, 1] × [0, 1] × D such that .ϕt,s (En ) = f (t, s), (t, s) ∈ [0, 1] × [0, 1], and .ϕt,s is the identity on 0 .[0, 1] × [0, 1] × ∂D. If .ϕt is a given smooth parameterizing isotopy for .t → f (t, 0) 1 and .ϕt is a given smooth parameterizing isotopy for .t → f (t, 1), then the family 0 1 .ϕt,s may be chosen so that .ϕt,0 = ϕt , .ϕt,1 = ϕt , and .ϕ0,s = Id.

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1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

The proof follows along the same lines as the proof of Proposition 1.1. We construct a vector field V on .[0, 1] × [0, 1] × D, i.e. a vector field that depends on .(ζ, t) ∈ D × [0, 1] and the additional parameter .s ∈ [0, 1], such that V (t, s, fj (t, s)) =

.

∂ fj (t, s) for (t, s) ∈ [0, 1] × [0, 1] ∂t

for the strands .fj of f , and solve the initial value problem .

∂ ϕt,s (ζ ) = V (t, s, ϕt,s (ζ )), ϕ0,s (ζ ) = ζ , (s, ζ ) ∈ [0, 1] × D . ∂t

The families .ϕt,0 and .ϕt,1 define vector fields . ∂t∂ ϕt,0 and . ∂t∂ ϕt,1 for .(t, s, ζ ) in .[0, 1] × {0} × D and .[0, 1] × {1} × D. The vector field V can be chosen to coincide with these vector fields on the respective sets. We omit the details. By the proposition the inverse of the mapping .M(D; ∂ D, En0 ) → Bn is obtained as follows. Take a braid .b ∈ Bn and choose a representing smooth geometric braid in .[0, 1]×D. Consider a parameterizing isotopy .ϕt . Associate to b the mapping class of the homeomorphism .ϕ1 . Explicitly, the inverse mapping assigns to each generator .σj ∈ Bn the class of the following homeomorphism which is called a half-twist around the interval   j −1 j . , . Take two open discs .D1 and .D2 centered at the midpoint of the segment  n n   j −1 j j −1 j ⊂ D1 , D¯ 1 ⊂ D2 , .D¯ 2 does not contain points . n , n , such that . n , n j of .En0 other than . j −1 n and . n . Define .ϕσj to be the identity on .D\D2 and to be counterclockwise rotation by the angle .π on .D¯ 1 . Extend this mapping by a homeomorphism of .D¯ 2 \D1 which changes the argument of each point by a nonnegative value at most equal to .π (see Fig. 1.4). We will denote the mapping 0 .Bn → M(D; ∂ D, En ) by .Θn .

Fig. 1.4 The braid .σ1 ∈ B3 and the corresponding mapping class .mσ1

1.6 Beltrami Differentials and Quadratic Differentials

23

The mapping class .mΔ2n corresponding to .Δ2n is the Dehn twist about a curve in .D that is homologous to .∂D in .D \ En0 . Notice that the elements of the set of conjugacy n of braids are in one-to-one correspondence with the elements of the set classes .B  of conjugacy classes .M(D; ∂ D, En0 ) of mapping classes.

1.6 Beltrami Differentials and Quadratic Differentials In this section we collect some prerequisites from Teichmüller theory, in particular we recall some facts related to Beltrami differentials and quadratic differentials. For more details we refer the reader for example to [4, 59, 67, 78]. Let X be a Riemann surface. A Beltrami differential .μ on X assigns to each chart on X with holomorphic coordinates z an essentially bounded measurable function dz .μ(z) so that .μ(z) dz is invariant under holomorphic coordinate changes. In other words, the functions .μ1 (z) and .μ2 (ζ ) associated to local coordinates .z = z(ζ ) and .ζ are related by the equation  μ1 (z(ζ ))

.

dz dζ

−1 

dz dζ

 = μ2 (ζ ) .

(1.17)

We may interpret Beltrami differentials as sections in the bundle .κ −1 .⊗ .κ¯ for the cotangent bundle .κ of X. By an abuse of notation we will denote the Beltrami differential on X by .μ and write .μ = μ(z) ddzz¯ , where the left hand side denotes the globally defined Beltrami differential and .μ(z) on the right hand side is a representing function in local coordinates z. The value .|μ(z)| is invariant under holomorphic coordinate change. def

Put .μ∞ = sup |μ|. X

Let X and Y be Riemann surfaces and let .ϕ : X → Y be a smooth orientation preserving homeomorphism. Let z be coordinates near a point of X and let .ζ be coordinates on Y near its image under .ϕ. We write .ζ = ϕ(z) in these coordinates. ¯

∂

ϕ

Consider the function . ∂∂ ϕϕ = ∂∂z¯ in these coordinates. Since for the Jacobian .J (z) ϕ  2 ∂z 2 ∂  ∂  2 of .ϕ(z), .J (z) =  ∂z ϕ(z) −  ∂ z¯ ϕ(z) > 0, the denominator .∂ϕ does not vanish   ∂ ϕ(z) d z¯ ¯  and . ∂∂ ϕϕ  < 1. The expression . ∂∂z¯ is invariant under holomorphic coordinate ϕ(z) dz ∂z

changes on X and on Y . It defines a Beltrami differential .μϕ on X. The mapping .ϕ is called quasiconformal if .μϕ ∞ < 1. If X and Y are compact this is automatically so. The condition that .ϕ is differentiable can be weakened.

24

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

For the Beltrami differential of superpositions of quasiconformal mappings the following formulas hold. μg ◦ f =

.

∂z f μg ◦f − μf . ∂z f 1 − μf μg ◦f

(1.18)

If g is conformal, then .μg = 0 and .μg ◦f = μf . If f is conformal, then .μf = 0 and μg ◦ f =

.

 f  2 μg ◦f . |f  |

(1.19)

For a detailed account on quasiconformal mappings we refer to [4]. 1+μ  The quasiconformal dilatation of the mapping .ϕ is defined as .K(ϕ) = 1−μϕϕ ∞ ∞ if .ϕ is quasiconformal and is defined to be equal to .∞ otherwise. A meromorphic (respectively, holomorphic) quadratic differential .φ on X assigns to each chart on X with holomorphic coordinates z a meromorphic (respectively, holomorphic) function which by abusing notation we denote by .φ(z), such that .φ(z)(dz)2 is invariant under holomorphic changes of coordinates. In other words, the functions .φ1 (z) and .φ2 (ζ ) associated to local coordinates .ζ and .z(ζ ) are related by the transformation rule  φ1 (z(ζ ))

.

dz dζ

2 = φ2 (ζ ) .

(1.20)

Holomorphic quadratic differentials on X can be regarded as holomorphic sections of the bundle .κ 2 . By an abuse of notation we will write .φ = φ(z) dz2 , where the left hand side denotes the quadratic differential and .φ(z) on the right hand side denotes the meromorphic function that represents .φ in local coordinates z. If the function .φ1 (ζ ) corresponding to local coordinates .ζ vanishes or equals .∞ at a point .ζ0 the function .φ2 (z) corresponding to local coordinates .z(ζ ) vanishes at the point .z(ζ0 ) or equals .∞ there. The set of points on X where this happens is called the set of singular points of the quadratic differential .φ. The other points are called regular points of the quadratic differential. Suppose on a simply connected domain U with coordinates .ζ the quadratic differential equals .φ(ζ )(dζ )2 where .φ = 0 is a nowhere vanishing analytic function √ (in particular .φ has no pole on U ). Consider the differential form . φ(ζ )dζ on U for a branch of the square root of .φ. There are local holomorphic coordinates z of the quadratic differential .φ(ζ ) dζ 2 on U in which the quadratic differential has the form .dz2 . They are called flat coordinates. Such coordinates can be obtained as an integral of the mentioned differential form,  z(ζ ) =

ζ

.

ζ0

 φ(ζ  )dζ  .

(1.21)

1.6 Beltrami Differentials and Quadratic Differentials

25

√ Together with the complex differential form . φ(ζ )dζ on √ √ U we will consider the two real forms .dx(ζ ) = Re( φ(ζ )dζ ) and .dy(ζ ) = Im( φ(ζ )dζ ) on U . The coordinates (1.21) vanish at .ζ0 . Flat coordinates that vanish at the point .ζ0 are called distinguished coordinates at .ζ0 . Different flat coordinates on U differ from each other by sign (since we may choose the other branch of the square root) and an additive constant (as the integral of the form is defined up to an additive constant). Denote by .E  the set of singular points of the quadratic differential on X. The .φregular set .X \E  has a flat structure. This means that there is an atlas with transition functions being .z → ±z + b for complex numbers b. The flat structure defines foliations on the .φ-regular part of X as follows. Cover  .X \ E by simply connected open sets .Uj each of which is equipped with flat coordinates .zj . Then the forms .dxj on .Uj define a foliation on .X \ E  which is called horizontal foliation. The plaques (the local parts of leaves in .Uj ) are the connected components of the sets .xj = const in .Uj . The transition functions map plaques to plaques. In the same way the forms .dyj on .Uj define a foliation on .X \ E  which is called the vertical foliation. Actually, for each real .θ the forms .Re(eiθ dzj ) define a foliation on .X \ E  . The horizontal foliation on each .Uj is generated by a vector field .vh (ζ ), ζ ∈ Uj . In other words, the leaves of this foliation are the integral curves of this vector field. In arbitrary coordinates .ζ on .Uj the vector field is defined by the conditions   Re( φ(ζ ) dζ )(vh (ζ )) = 1 , Im( φ(ζ ) dζ )(vh (ζ )) = 0 .

.

(1.22)

Notice that if .φ(ζ ) = |φ(ζ )|eiθ in coordinates .ζ = ξ + iη, then in these coordinates   θ θ Re( φ(ζ ) dζ ) = |φ(ζ )|(cos dξ − sin dη) , 2 2   θ θ Im( φ(ζ ) dζ ) = |φ(ζ )|(sin dξ + cos dη) . 2 2

.

In distinguished coordinates .zj the conditions (1.22) become .dxj (vh ) = 1 , dyj (vh ) = 0 . In other words, in these coordinates the vector field points in the direction of the positive real axis (or in the direction of the negative real axis if the other branch of the square root is chosen). Identifying a vector with a point in the complex plane we obtain the following. If .φ(ζ ) = |φ(ζ )|eiθ in coordinates  cos θ2  −1 .ζ then in these coordinates .vh (ζ ) = |φ(ζ )| 2 which we identify with the − sin θ2 1 1 1  sin θ  2 = point .|φ(ζ )|− 2 e− 2 iθ in the complex plane. Moreover, .vv (ζ ) = |φ(ζ )|− 2 cos θ2 θ−π  1  cos − 21 − 12 i(θ−π ) 2 e in the |φ(ζ )|− 2 θ−π which can be identified with the point .|φ(ζ )| .

− sin

2

complex plane. In general the .vh (the .vv , respectively) in the coordinate patches do not define a global vector field on .X \ E  but merely a global line field. Indeed,

26

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

the horizontal foliation is in general not orientable. The leaves of the horizontal foliation (of the vertical foliation, respectively) are also called horizontal trajectories (vertical trajectories, respectively). Arcs that are contained in horizontal (vertical, respectively) trajectories are called horizontal (vertical, respectively) arcs. In a neighbourhood U of a singular point .p0 there are holomorphic coordinates z vanishing at the point in which the quadratic differential has the form  φ(z) dz =

.

2

a+2 2

2 za dz2

(1.23)

for some integer a. The coordinates in which .φ has the form (1.23) are uniquely defined up to multiplication by an .(a + 2)-nd root of unity and are called distinguished coordinates at the singular points. The number a is called the order of the point. For distinguished coordinates near regular points .φ has the form (1.23) with .a = 0. We will consider quadratic differentials with at worst poles of order one, i.e. .a ≥ −1. Equip a simply connected subset .U  of .U \ {p0 } with coordinates z satisfying (1.23). We assume that in these coordinates U has the form .U = {|z| < r0 } for a positive number .r0 . 2 − a2 − iaθ The vector field that equals . a+2 r e 2 at the point .z = eiθ in .U  in the mentioned coordinates generates the horizontal foliation on .U  . A ray .{reiθ : 0 < r < r0 } (in these coordinates) is a subset of a horizontal trajectory if at each point the unit tangent vector to the ray is equal to the unit vector in a horizontal iaθ 2kπ , .k = 0, . . . , a + 1. The direction, i.e .eiθ = ±e− 2 , or equivalently, .θ = a+2 (2k+1)π i

rays that are subsets of the vertical trajectories are .{re a+2 : r > 0 small}, for .k = 0, . . . , a + 1 (we use the same coordinates satisfying (1.23)). We call these rays horizontal bisectrices (also horizontal separatrices), or vertical bisectrices (also vertical separatrices), respectively. Any other horizontal leaf in U is a subset of a sector between two neighbouring horizontal bisectrices and avoids a neighbourhood (depending on the leaf) of the singular point. The respective facts hold for the vertical leaves. The same facts hold in a neighbourhood of each singular point. We obtained a singular foliation on X. On a sector between two consecutive horizontal bisectrices a+2 a branch of the power .z → z 2 provides flat coordinates. The image of the sector under the latter mapping is a subset of the upper or lower half-plane with horizontal leaves being the sets .x = const. A sector between two consecutive vertical bisectrices is mapped to a subset of the right or left half-plane with vertical leaves being the sets .y = const. We want to associate a singular metric to .φ. Consider again a simply connected domain U in .X \ E  equipped with coordinates .ζ . The quadratic differential in these coordinates has the form .φ(ζ )dζ 2 . Let z be flat coordinates on U . Let .γ : [0, 1] → U be a piecewise smooth curve. We define the length .φ (γ ) with respect to .φ of the curve .γ as its Euclidean length in some flat coordinates. This value is independent

1.6 Beltrami Differentials and Quadratic Differentials

27

on the choice of the flat coordinates, and therefore it is well defined. If we denote by .γ1 the curve .t → z(γ (t)) in some flat coordinates z then 



φ (γ ) =

0

γ1

  d  (γ1 (t)dt . dt

1

|dz| =

.

In arbitrary coordinates .ζ we obtain with .dz = 

  | φ(ζ )dζ | =

φ (γ ) =

.

1

0

γ

(1.24)

√ φ(ζ )dζ

d    |φ(γ (t))|  (γ (t))dt. dt

(1.25)

√ We call .|dz| = | φ(ζ )dζ | the length element in the .φ-metric and .φ (γ ) the .φ-length of the curve .γ . An arbitrary piecewise smooth curve in the regular part of .φ can be subdivided into pieces contained in small discs around regular points. The .φ-length of the curve is the sum of the .φ-length of these pieces. The invariant form (1.25) makes sense also for curves passing through singular points. As the singular points of .φ are at worst poles of order one, the .φ-length of any piecewise smooth compact curve .γ : [0, 1] → X is finite. In the same way we define the horizontal .φ-variation .φ,h (γ ) of a curve .γ (the vertical .φ-variation .φ,v (γ ), respectively). For a curve .γ in a simply connected set U with flat coordinates z this is the total variation of the x coordinate (the y coordinate, respectively) along the curve .γ , 



   d Re (γ (t)dt , dt γ 0   1    d φ,v (γ ) = |dy| = Im (γ (t)dt . dt γ 0

φ,h (γ ) =

.

|dx| =

1

(1.26)

An arbitrary curve is divided into pieces contained in such sets and the variations of the pieces are added. The area element in the .φ-metric is defined as .dz ∧ d z¯ in flat coordinates z, or, √ √ equivalently, . φ(ζ )dζ ∧ φ(ζ )dζ = |φ(ζ )|dζ ∧ dζ in arbitrary  local coordinates .ζ . For a small open subset .U ⊂ X the integral . |φ| = 12 |φ(ζ )| dζ ∧ d ζ¯ is U

U

def

invariant under holomorphic coordinate changes. Hence, .φ1 =



|φ| is well

X

defined. .φ is called integrable on X if this integral is finite. The singularities of integrable meromorphic quadratic differentials are zeros or simple poles. The set of integrable meromorphic quadratic differentials on X with norm . · 1 forms a complex Banach space. The following fact is an immediate corollary of the definitions. For each meromorphic quadratic differential .φ and any number .k ∈ (0, 1) the object .k |φ| φ =

28

k

φ¯ |φ|

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory ¯ φ(z) |φ(z)| 1+k 1−k .

(given in local coordinates z by .k

a Beltrami differential of norm .K =

with a meromorphic function .φ(z)) is

1.7 Elements of Teichmüller Theory Teichmüller Theorem (Closed Surfaces) The following celebrated theorem was the key point for solving Riemann’s problem on describing the classes of conformally equivalent Riemann surfaces. It was Teichmüller’s vision that isotopy classes of conformally equivalent Riemann surfaces are easier to handle and their description provides a solution of Riemann’s problem. For more details, see [2], [9]. Theorem 1.4 Let X and Y be closed Riemann surfaces of genus g ≥ 2, and let ϕ : X → Y be a homeomorphism. Then there is a unique homeomorphism isotopic to ϕ with smallest quasiconformal dilatation. This homeomorphism is either conformal or its Beltrami differential has the form k· |φ| φ for a holomorphic quadratic differential φ on X and a constant k ∈ (0, 1). φ is unique up to multiplication by a positive constant. A homeomorphism with Beltrami differential k · |φ| φ for a holomorphic quadratic differential φ on X and a constant k ∈ (0, 1) is called a Teichmüller mapping and φ is called its quadratic differential. Teichmüller mappings are characterized by their local description in distinguished coordinates. Let X and Y be closed Riemann surfaces. Suppose ϕ : X → Y is a Teichmüller mapping with quadratic differential φ and constant k. Then the inverse mapping ϕ −1 is again a Teichmüller mapping with quadratic differential denoted by −ψ and constant k. The order of φ at a point z is the same as the order of −ψ at the image ϕ(z). The quadratic differential ψ is called the terminal quadratic differential of ϕ. The quadratic differential φ of ϕ(z) is also called the initial quadratic differential of ϕ(z). For more insight see [11], or Theorem 8.1, V.8, of [59]. There are distinguished coordinates z for φ near a point z0 ∈ X that vanish at z0 and distinguished coordinates ζ for −ψ near the image point ϕ(z0 ) ∈ Y which vanish at ϕ(z0 ) so that the mapping ϕ has the form  ζ =

.

za+2 + 2 k|z|a+2 + k 2 z¯ a+2 1 − k2

1  a+2

(1.27)

with ζ > 0 for z > 0. If a = 0 this is equivalent to 1

1

ζ = ξ + iη = K 2 x + i K − 2 y

.

with z = x + iy and K =

1+k 1−k

being the quasiconformal dilatation.

(1.28)

1.7 Elements of Teichmüller Theory

29

We will describe now an idea of Ahlfors ([2], section 4, pp. 19–20) which allows to reduce questions concerning self-homeomorphisms of punctured Riemann surfaces (self-homeomorphisms of closed Riemann surfaces with distinguished points, respectively) to the related questions concerning closed Riemann surfaces. Ahlfors used it for a proof of Teichmüller’s theorem for punctured Riemann surfaces. It can also be used for reducing the study of the entropy of self-homeomorphisms of closed Riemann surfaces with distinguished points to the study of the entropy of self-homeomorphisms of closed Riemann surfaces. Ahlfors’ idea is the following. Let X and Y be closed Riemann surfaces, both with a set of m distinguished points. (Equivalently, we may think about two mpunctured Riemann surfaces by using the identification of Hom+ (X, ∅, E) with Hom+ (X\E, ∅, ∅).) Assume 2g − 2 + m > 0, so that the universal covering of the m-punctured surfaces equals C+ . Except for g = 0 and m ≤ 4 Ahlfors associates to X and Y closed Riemann surfaces Xˆ and Yˆ , which are holomorphic simple branched coverings pX : Xˆ → X of X and pY : Yˆ → Y of Y with branch locus being the set of distinguished points, and have genus at least two. Such coverings can be obtained as follows. If the number of punctures is even (and not zero) and either g ≥ 1, or g = 0 and m ≥ 6, then one can take a double branched covering with branch locus being the set of distinguished points. If g > 0 and the number of points in the set E of distinguished points is odd and bigger than 1, one first considers an intermediate double branched covering with branch locus being a non-empty set E  ⊂ E that contains an even number of points,and takes then a double branched cover of the intermediate branched covering, this time with branch locus being the preimage of E \ E  in the intermediate branched covering (which consists of an even number of points). If the number of distinguished points equals 1 and g > 0, we take an intermediate unramified covering and proceed further as before. If g = 0 and m ≥ 5 is odd we consider an intermediate double branched covering with branch locus being m−1 of the punctures, and then a double branched covering over the immediate covering branched at the preimages of the remaining puncture of X. This treats all cases except g = 0 and m ≤ 4. If m ≤ 3 there exist holomorphic self-mappings of P1 that map each distinguished point in X to the respective distinguished point in Y . If m = 4, the double branched covering with the required branch locus is a torus (i.e. a closed Riemann surface of genus 1), in which case the extremal mappings are well described, and also problems related to entropy are well-known in this case. Consider a homeomorphism ϕ : X → Y which maps the set of distinguished points of X to the set of distinguished points of Y . There is a lift ϕˆ of ϕ, ϕˆ : Xˆ → Yˆ , i.e. a homeomorphism between closed surfaces such that pY ◦ ϕˆ = ϕ ◦ pX . The homeomorphism ϕˆ has some “additional symmetries”. Vice versa, a homeomorphism between Xˆ and Yˆ with such symmetries is a lift of a homeomorphism between X and Y which maps the set of distinguished points in X to the set of distinguished points in Y . Notice that a self-homeomorphism ϕ of a Riemann surface X which preserves a set E consisting of an even number of points, lifts to the double branched covering

30

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

with branch locus E as follows. Cut X along a set Γ of disjoint arcs that join pairs of points of E. Consider two copies of X \ Γ . ϕ maps each copy to a copy of X \ ϕ(Γ ). Glue the copies in the source and the copies in the target crosswise along respective strands of the cuts and extend the mapping. Teichmüller’s theorem also applies in the situation of homeomorphisms with “additional symmetry” between closed Riemann surfaces. One obtains quadratic differentials on Xˆ with a “symmetry”, and one obtains quadratic differentials on ˆ This implies that the X which lift to the mentioned quadratic differentials on X. quadratic differentials on X have at most simple poles at the branch locus. This is a simple calculation using the behaviour of quadratic differentials under coordinate changes. Indeed, let zˆ 0 be a branch point in Xˆ and let z0 be its image under the projection Xˆ → X. Let ζ be distinguished coordinates on Xˆ at zˆ 0 , and let z = z(ζ ) = ζ 2 be coordinates near z0 on X. If φ is the quadratic differential on X and φˆ its lift to Xˆ then by (1.20) ˆ ) = φ(z(ζ ))z (ζ )2 = φ(z(ζ ))(2ζ )2 = 4φ(z(ζ ))z(ζ ) = 4φ(z)z . φ(ζ

.

(1.29)

Since φˆ does not have poles, φ has at most a simple pole at z0 . The arguments imply that Teichmüller’s theorem remains true for homeomorphisms between closed surfaces with distinguished points, if instead of holomorphic quadratic differentials one considers meromorphic quadratic differentials on closed surfaces with at most simple poles at distinguished points. These are exactly the integrable meromorphic quadratic differentials. In a similar way Ahlfors treats finite Riemann surfaces of the second kind (i.e. Riemann surfaces with finitely many boundary components, some of which may be points, some are continua). He uses an extension of the homeomorphism to a homeomorphism between the doubles of the Riemann surfaces. We do not need this case here. For details see [2, 3]. The local characterization of Teichmüller maps implies the following useful fact. Let X, X1 and X2 be closed or punctured Riemann surfaces, and let ϕ1 : X → X1 and ϕ2 : X1 → X2 be Teichmüller maps with quadratic initial differentials φ1 and φ2 , respectively. Suppose φ2 is equal to the terminal quadratic differential of ϕ1 . Then ϕ2 ◦ ϕ1 is a Teichmüller map with quadratic differential φ1 . Teichmüller Spaces Let X be a connected Riemann surface of genus g, with  boundary continua and m punctures. We always assume that the universal covering of X equals C+ . (This excludes the Riemann sphere, C, C∗ = C\{0}, and tori (compact Riemann surfaces of genus 1).) Let wj : X → Yj , j = 1, 2, be two quasiconformal homeomorphisms onto Riemann surfaces (considered as conformal structures on X). They are called (Teichmüller) equivalent if there exists a conformal mapping c : Y1 → Y2 such that w2−1 ◦ c ◦ w1 is isotopic to the identity by an isotopy which fixes the set of boundary continua pointwise (if any). We denote the equivalence class of a quasiconformal complex structure w : X → Y on X by [w]. The set of equivalence classes is the

1.7 Elements of Teichmüller Theory

31

Teichmüller space T (X). Equip the Teichmüller space with the Teichmüller metric dT ,

def

dT ([w1 ], [w2 ]) = inf

.

 1 log K(v2 ◦ v1−1 ) : v1 ∈ [w1 ] , v2 ∈ [w2 ] . 2

(1.30)

Note that the infimum is equal to the following

1 log K(g) : g : Y1 → Y2 such that w2−1 ◦ g ◦ w1 : X → X is isotopic .inf 2  (1.31) to the identity fixing the boundary continua pointwise . For a quasi-conformal homeomorphism w with [w] ∈ T (X) the space T (w(X)) is canonically isometric to T (X). We choose a reference Riemann surface X with m punctures and  boundary continua and write T (g, m, ) for the respective Teichmüller space. We also need the Fuchsian model of the Teichmüller space. (For more details see [4], VI B.) Consider Riemann surfaces X and Y of first kind whose universal coverings equal C+ . Represent X and Y as quotients of the upper half-plane C+ by the action of Fuchsian groups Γ and Γ1 of first kind, i.e. X ∼ = C+ Γ , Y ∼ = C+ Γ1 . Then each homeomorphism w from X to Y lifts to a self-homeomorphism w˜ of the upper half-plane such that for each γ ∈ Γ there is γ1w˜ ∈ Γ1 such that w˜ ◦ γ = γ1w˜ ◦ w˜ ,

.

(1.32)

and each self-homeomorphisms of C+ with this property projects to a homeomorphisms from X onto Y . Indeed, a mapping w˜ 1 also lifts w if and only it differs from w˜ by precomposition with a covering transformation for the covering C+ → Y . Beltrami differentials and quadratic differentials on X lift to Beltrami differentials μ˜ and quadratic differentials φ˜ on C+ with the invariance property μ˜ ◦ γ = μ˜

.

γ γ

for γ ∈ Γ

2 φ˜ ◦ γ · γ  = φ˜

.

for γ ∈ Γ

(automorphic (−1, 1)-forms) , (automorphic 2-forms) .

(1.33) (1.34)

¯ + (a self-homeomorphism A self-homeomorphism of the closed upper half-plane C 1 of the Riemann sphere P , respectively) is called normalized if it maps 0 to 0, 1 to 1 and ∞ to ∞. Note that each quasiconformal self-homeomorphism of C+ extends ¯ + [2]. to a self-homeomorphism of C Let Γ be a Fuchsian group. Denote by Qnorm (Γ ) the set of normalized quasiconformal self-homeomorphisms of C+ that satisfy (1.32) for another Fuchsian

32

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

group Γ1 . The Beltrami differentials on C+ that satisfy (1.33) are in one-toone correspondence to elements of Qnorm (Γ ). Indeed, associate to each Beltrami differential μ on C+ the Beltrami differential μˆ on C, for which μ(z) ˆ = μ(z), z ∈ C+ , μ(z) ˆ = μ(¯ ¯ z), z ∈ C− (C− denotes the lower half-plane). There is a unique normalized solution w of the equation wz¯ = μ(z) ˆ wz on the complex plane. It maps C+ onto itself. Its restriction to C+ is denoted by w μ . Let Γ1 = Γ μ be the group wμ ◦ γ ◦ (w μ )−1 , γ ∈ Γ . This is a Fuchsian group. If μ satisfies (1.33) then wμ satisfies (1.32) for Γ1 = Γ μ . Hence, w μ induces a quasiconformal mapping W μ : C+ Γ → C+ Γ1 . Two elements of Qnorm (Γ ) are called equivalent iff their restrictions to the real axis coincide. The Teichmüller space T (Γ ) is defined as set of equivalence classes of elements of Qnorm (Γ ). Two mappings w μ , w ν ∈ Qnorm (Γ ) are equivalent iff the mappings wμ , w ν of C+ induce Teichmüller equivalent mappings W μ , W ν on X = C+ Γ ([4], VI B, Lemma 2). Let μ be a Beltrami differential on X, let μ˜ be its lift to C+ and let w μ˜ be the ˜ The projection of w μ˜ to normalized solution of the Beltrami equation on C+ for μ. μ X is denoted by W . For later use we give the following definition. Definition 1.1 For each Beltrami differential μ on X the homeomorphism W μ is called the normalized solution of the Beltrami equation on X for the Beltrami differential μ. Also, we assign to μ the element [W μ ] of the Teichmüller space T (X). We use the notation {μ} for [W μ ]. The obtained mapping from T (Γ ) to T (X) is a bijection. Let now X be a Riemann surface of genus g with m punctures and no boundary continuum, i.e. X = Xc \ E for a closed Riemann surface Xc with set of distinguished points E. We assume that the universal covering of X equals C+ . The Teichmüller space is denoted by T (X) ∼ = T (g, m, 0). Instead of T (g, m, 0) we will write T (g, m). Teichmüller’s theorem implies the following. Denote by QC(X) the set of quasiconformal homeomorphisms of X onto []

another Riemann surface. The mapping QC(X) −→ T (X) assigns to each element w ∈ QC(X) its class [w] in the Teichmüller space T (X). The Teichmüller space is equipped with the Teichmüller metric dT . Associate to each non-trivial class [w] ∈ T (X) a (unique up to composition with conformal mappings) extremal quasiconformal homeomorphism in this class. We obtain a bijection from non-trivial elements of the Teichmüller space to Beltrami differentials of the form k |φ| φ on X, where k is a constant in (0, 1) and φ is a holomorphic quadratic differential on X, that extends to a meromorphic quadratic differential on Xc with at most simple poles at the points of E. Notice that a meromorphic quadratic differential on Xc is integrable iff it has at most simple poles. Assign to the trivial class in the Teichmüller space (corresponding to a conformal extremal mapping) the zero quadratic differential. We obtain a bijection of the Teichmüller space to the space of integrable holomorphic quadratic differentials of norm less than 1. The real dimension of the space of integrable holomorphic quadratic differentials on

1.7 Elements of Teichmüller Theory

33

X is equal to 6g − 6 + 2m. It can be proved that the Teichmüller space T (X) is homeomorphic to the unit ball in the Banach space of such quadratic differentials. There is a unique conformal structure on T (g, m) with the following property. Take any family of complex structures in QC(X) whose Beltrami differentials depend holomorphically on certain complex parameters. More precisely, for almost all x ∈ X the Beltrami differentials at x of the members of the family depend holomorphically on the complex parameters. Then the equivalence classes in T (X) depend holomorphically on the complex parameters. The explicit construction uses the Fuchsian model (see [10]). Write X = C+ Γ . Let μ be a Beltrami differential on C+ satisfying (1.33). Let wμ be the unique normalized solution of the Beltrami equation on the Riemann sphere with Beltrami coefficient equal to μ on C+ and equal to 0 on C− . The mapping wμ does not map C+ onto C+ , but if the Beltrami differentials depend holomorphically on complex parameters then the mappings wμ depend holomorphically on them. The mappings wμ are conformal on the lower half-plane C− . Moreover, for two Beltrami differentials ν and μ the equality wμ = w ν holds on R iff the equality wμ = wν holds on R (and hence the latter equality holds on C− ) ([4], VI B, Lemma 1). Consider the Schwarzian derivative S(wμ | C− ). (For a locally conformal mapping f on an open set in P1 the Schwarzian derivative is defined   2    as S(f ) = ff  − 12 ff  .) The Schwarzian derivatives of Möbius transformations equal zero and S(f ◦ g) = S(f ) ◦ g · (g  )2 + S(g) for locally conformal mappings f and g. This implies that φμ = S(wμ | C− ) is a quadratic differential on C− which satisfies (1.34) (with respect to Γ acting on C− ). If μ depends holomorphically on parameters then so does φμ . We obtained a map μ → φμ = S(wμ | C− ) from the set of Beltrami differentials on C+ satisfying (1.33) to the space of holomorphic quadratic differentials on C− . The mapping satisfies the condition φμ = φν

.

if w μ , w ν ∈ Qnorm (Γ ) are equivalent.

(1.35)

Note that for each holomorphic function f on a simply connected domain in the complex plane there is a meromorphic function w in the domain, unique up to a Möbius transformation for which S(w) = f . There is an explicit way to find such a function. Consider the Banach space B(C− , Γ ) of holomorphic quadratic differentials in C− with norm ϕ = sup |y 2 ϕ(z)|. By the condition (1.35) the mapping T (X) ∼ = T (Γ )  {μ} → φμ

.

(1.36)

is well defined. It can be proved that it defines a homeomorphism of T (X) onto an open subset of the unit ball of B(C− , Γ ) [4]. This homeomorphism is called Bers embedding. The complex structure on T (X) induced by this homeomorphism from B(C− , Γ ) is the desired one. For Beltrami differentials μ on X depending holomorphically on parameters, the Teichmüller classes [W μ ] = {μ} also depend

34

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

holomorphically on parameters, but the homeomorphisms W μ do possibly not have this property. Teichmüller Discs Let X be a Riemann surface of genus g with m punctures with universal covering C+ . Write X = C+ Γ for a Fuchsian group Γ . Let φ be a meromorphic quadratic differential on X which is holomorphic on Xc except, maybe, at some punctures, where it may have simple poles. For each |e−iθ φ| z = r eiθ ∈ D we consider the Beltrami differential μz = z |φ| φ = r e−iθ φ . Notice that the absolute value |μz | of μz is the constant function |z| on C+ . Each Beltrami differential μz defines a unique element w μz ∈ Qnorm (Γ ), equivalently a unique normalized solution W μz ∈ QC(X) associated to μz , and a unique Teichmüller class [W μz ] = {μz } ∈ T (X) ∼ = T (Γ ). The mapping D  z → {μz } ∈ T (X)

(1.37)

.

is a holomorphic embedding. For each z = 0 the homeomorphism W μz is a Teichmüller map. Let z1 , z2 ∈ D, z1 = z2 . Up to positive multiplicative μ constants the quadratic differential of W zj equals e−iθj φ, j = 1, 2, i.e. the quadratic differentials differ by a constant factor. Hence, by Lemma 9.1 of [59] the composition W μz2 ◦ (W μz1 )−1 is a Teichmüller mapping of the form W μz for a point z ∈ D. By formula (9.5) of [59] the point z is real, if z1 and z2 are real. One can show that the absolute value of the Beltrami differential of W μz2 ◦ (W μz1 )−1 is a constant function on C+ that equals k = |μW

.

Hence, with K =

1+k 1−k

.

μ z2

◦(W μz1 )−1

  z2 − z1 | =  1 − z z¯

2 1

  . 

(1.38)

the value 1 |1 − z2 z1 | + |z2 − z1 | 1 log K = log 2 2 |1 − z2 z1 | − |z2 − z1 |

is equal to the distance of z1 and z2 in the Poincaré metric on the unit disc. (For details see, e.g. [59].) Thus the mapping D  z → {μz } ∈ T (X) is an isometric proper holomorphic embedding of the disc with Poincaré metric into Teichmüller space with Teichmüller metric. It is called the Teichmüller disc associated to φ. We will denote its image in T (X) by DX,φ . The Modular Group Consider a closed Riemann surface of genus g with m punctures, denoted by X = Xc \ E. Here Xc is a closed Riemann surface with a set E of m distinguished points. A quasiconformal self-homeomorphism ϕ of X induces a mapping ϕ ∗ of the Teichmüller space T (X) ∼ = T (g, m) to itself. It is defined as follows. For each homeomorphism w ∈ QC(X), w : X → Y the composition w ◦ ϕ : X → Y is another quasiconformal homeomorphism. Its class [w ◦ ϕ] ∈ T (g, m) depends only on the class [w] of w. Put ϕ ∗ ([w]) = [w ◦ ϕ]. For

1.7 Elements of Teichmüller Theory

35

each quasiconformal self-homeomorphism ϕ of X the mapping ϕ ∗ is an isometry on the Teichmüller space T (g, m). Moreover, it maps T (g, m) biholomorphically onto itself. The mapping ϕ ∗ is called the modular transformation of ϕ. Notice that ϕ ∗ depends only on the mapping class of ϕ in the mapping class group M(X; ∅) = M(Xc , ∅, E), in other words, isotopic self-homeomorphisms of X have the same modular transformation. The modular group is isomorphic to the mapping class group M(X) = M(Xc , ∅, E). For two quasiconformal self-homeomorphism ϕ1 and ϕ2 of the punctured Riemann surface X the equality (ϕ1 ◦ϕ2 )∗ = ϕ1∗ ϕ2∗ holds, because [w ◦ (ϕ1 ◦ ϕ2 )] = ϕ2∗ ([w ◦ ϕ1 ]) = ϕ1∗ ϕ2∗ ([w]). Hence, the set of modular transformations forms a group, called the modular group. It is often denoted by Mod(g, m). The quotient T (g, m)Mod(g, m) can be identified with the Riemann space of conformally equivalent complex structures, i.e. with the moduli space of Riemann surfaces of genus g with m punctures. Royden’s Theorem The following deep theorem of Royden [71] has many applications. Theorem 1.5 (Royden) The Teichmüller metric on the Teichmüller space T (g, m) is equal to the Kobayashi metric. In other words, a holomorphic mapping from the |dz| unit disc D with Poincaré metric 1−|z| 2 into T (g, m) with Teichmüller metric is a contraction. (Equivalently, a holomorphic map from C+ with hyperbolic metric into T (g, m) is a contraction.)

|dz| 2y

Configuration Space and Teichmüller Space We will relate geometric n-braids to paths in the Teichmüller space T (0, n + 1) of the Riemann sphere with n + 0 1 punctures. The reference Riemann surface

will be denoted by X0 = C\En = . Let En1 ⊂ C be another set containing P1 \({∞}∪En0 ), where En0 = 0, n1 , . . . n−1 n

exactly n points. For a homeomorphism from C\En0 onto C\En1 we will use the same notation for this homeomorphism and for the extension of this homeomorphism to a self-homeomorphism of P1 which maps En0 to En1 and ∞ to ∞. Consider the set QC∞ (0, n + 1) of orientation preserving quasiconformal homeomorphisms w : C \ En0 → C \ En1 whose extensions to P1 fix ∞. We equip this set with the topology for which a neighbourhood basis of an element w0 is given by the sets  Nε (w0 ) = w ∈ QC∞ (0, n + 1) :

.

1 1 μw − μw0 ∞ < ε, |w(0) − w0 (0)| < ε, |w( ) − w0 ( )| < ε . n n Convergence in this topology implies uniform convergence on compact subsets of C (of the extensions of the mappings across the punctures). Indeed, let aw,w0 : C ⊃  be the complex affine mapping that takes w(0) to w0 (0) and w( n1 ) to w0 ( n1 ). Then aw,w0 ◦ w takes 0 to w0 (0) and n1 to w0 ( n1 ), and for any sequence of

36

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

elements wn ∈ Nεn (w0 ) with εn → 0 the sequence awn ,w0 converges to the identity uniformly on compacts. By formulas (1.18) and (1.19) the L∞ -norm of the Beltrami coefficient of the mapping awn ,w0 ◦ wn ◦ w0−1 converges to 0 for any sequence wn ∈ Nεn (w0 ), εn → 0. The mappings awn ,w0 ◦ wn ◦ w0−1 fix the points w0 (0), w0 ( n1 ), and ∞, i.e. after conjugating by a complex linear mapping they are normalized solutions of Beltrami equations on C with Beltrami coefficients converging to 0 in the L∞ -norm. Hence, these mappings converge to the identity uniformly on compacts (see [4], Ch. V, B, Lemma 1 and the proof of Theorem 3). Each self-homeomorphism ψ of C acts diagonally on the configuration space Cn (C). Denote this action again by ψ: ψ : (z1 , . . . , zn ) → (ψ(z1 ), . . . , ψ(zn )), (z1 , . . . , zn ) ∈ Cn (C) .

.

(1.39)

The action descends to an action on the symmetrized configuration space Cn (C)Sn . Let A be the set of complex affine mappings on the complex plane (equivalently, the set of Möbius transformations on the Riemann sphere that fix ∞). Each element a ∈ A has the form a(z) = az + b, z ∈ C. Here b ∈ C and a ∈ C∗ = C\{0} are constants. A has the complex structure of C∗ × C. It forms a group under composition. 1 For z = (z1 , z2 , . . . , zn ) we put az (ζ ) = n1 zζ2−z −z1 . Then  1 1 z3 − z1 1 zn − z1  az (ζ )((z1 , z2 , . . . , zn )) = 0, , , ..., n n z 2 − z1 n z 2 − z1

.

1 1 ∈{0} × { } × Cn−2 (C\{0, }) . n n Recall that   1   1 n−2 Cn−2 C\{0, } = (z3 , . . . , zn ) ∈ C \{0, } : zi = zj for i = j . n n

.

The mapping  1 z3 − z1 1 zn − z1  (z1 , z2 , . . . , zn ) → z1 , z2 − z1 , , ..., n z 2 − z1 n z 2 − z1

.

is a holomorphic isomorphism from Cn (C) onto C × C∗ × Cn−2 (C\{0, n1 }). We denote by PA the projection from Cn (C) onto the quotient Cn (C)A by the action of A. The quotient Cn (C)A with the inherited complex structure is isomorphic to {0} × { n1 } × Cn−2 (C\{0, n1 }). Denote by Isn the isomorphism Isn : Cn (C)A → {0} × { n1 } × Cn−2 (C\{0, n1 }) that assigns to each element of the quotient Cn (C)A the normalized element of Cn (C) whose first two coordinates are 0 and n1 . The

1.7 Elements of Teichmüller Theory

37 Isn ◦PA

def

composition A,n = Isn ◦ PA is equal to the mapping (z1 , z2 , . . . , zn ) −−−−→  1 1 z3 −zP −z1 . 0, n , n z2 −z11 , . . . , n1 zzn2 −z 1 Recall that for each w of P1 the map en (w) is defined by    self-homeomorphism . Recall also that two complex structures en (w) = w(0), w n1 , . . . , w n−1 n w1 , w2 on C\En0 are Teichmüller equivalent if there is a conformal mapping c : w1 (C\En0 ) → w2 (C\En0 ) such that w2−1 ◦ c ◦ w1 : C\En0 → C\En0 is isotopic to the identity on C\En0 . Denote by the same letters c, w1 , w2 the extensions of the previous mappings to P1 . The mapping c is a Möbius transformation that fixes ∞, hence c ∈ A. This implies that for two Teichmüller equivalent conformal structures w1 and w2 on C\En0 there exists a mapping c ∈ A such that en (w2 ) = c(en (w1 )). Indeed, if w2−1 ◦ c ◦ w1 is isotopic to the identity on C\En0 its extension to C fixes each point in En0 , hence w2 and c ◦ w2 take the same values at each point of En0 . The arguments imply that to each Teichmüller class [w] ∈ T (0, n + 1) corresponds a unique class in Cn (C)A. It is obtained as follows. Take an element τ ∈ T (0, n + 1). Represent τ by a self-homeomorphism w ∈ τ . The element en (w) ∈ Cn (C) is defined by τ modulo a diagonal action of the group A of complex affine self-mappings of C. We denote by PT (τ ) the unique element of the class in Cn (C)A corresponding to τ , that has the form (0, n1 , z3 , . . . , zn ). In the following simple but useful lemma we again identify homomorphisms between punctured surfaces and their extensions to P1 . Lemma 1.3 Let En1 and En2 be subsets of C, each containing exactly n points. Let w1 : C\En0 → C\En1 and w2 : C\En0 → C\En2 be Teichmüller equivalent homeomorphisms. If w1 and w2 take equal values on two different points jn1 and jn2         of En0 , i.e. w1 jn1 = w2 jn1 and w1 jn2 = w2 jn2 , then En1 = En2 and w1 , w2 are isotopic as mappings from C\En0 onto C\En1 (i.e. w1 ◦ w2−1 : C \ En2 ⊃ is isotopic to the identity.) In particular, en ◦ w1 = en ◦ w2 . Proof Under the conditions of the lemma there is an affine map c ∈ A such that the mapping w2−1 ◦ c ◦ w1 (see Fig. 1.5) is isotopic to the identity through self0 homeomorphisms of CEn0 . Then w2−1 ◦ c ◦ w1 fixes   En pointwise,   i.e. c ◦ w1 | En0 = w2 | En0 . Since w1

j1 n

= w2

j1 n

and w1

j2 n

= w2

j2 n

, c fixes two

points in C, hence c is the identity. Thus = = = En2 and w2−1 ◦ w1 is isotopic to the identity through self-homeomorphisms of C\En0 . In other words, there is a continuous family ϕ t , t ∈ [0, 1], of self-homeomorphisms of En1

w1 (En0 )

w2 (En0 )

X0

Fig. 1.5 Teichmüller equivalent mappings

w1 1 \ En

w2

c

2 \ En

38

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

C | En0 with ϕ 0 = w2−1 ◦ w1 and ϕ 1 = id. The required isotopy is w2 ◦ ϕ t , t ∈ [0, 1]. 

The equality en ◦ w1 = en ◦ w2 follows. The following theorem was proved by Kaliman [49, 50] and later independently by Bers and Royden [12]. Our proof follows the approach of Kaliman. Theorem 1.6 The mapping PT : T (0, n + 1) → {0} × { n1 } × Cn−2 (C\{0, n1 }) is a holomorphic covering. Hence, the Teichmüller space T (0, n+1) is the holomorphic universal covering space of the space {0} × { n1 } × Cn−2 (C\{0, n1 }). Remark Theorem 1.6 implies in particular that the Teichmüller space T (0, 4) is isomorphic to C+ . For the case n = 3 it will be convenient to consider instead of the covering PT : T (0, 4) → {0} × { 13 } × C\{0, 13 } the covering PT : T (0, 4) → C\{0, 1} that is obtained as follows. Consider the mapping C3 (C) → {0} × (C \ {0, 1}) × {1} that assigns to each class (z1 , z2 , z3 )A the unique element of C3 (C) whose firstentry equals 0, and whose last entry equals 1. This element equals Mz (z1 , z2 , z3 ) , where for z = (z1 , z2 , z3 ) the mapping Mz   1 is the Möbius transformation Mz (ζ ) = zζ3−z −z1 , ζ ∈ C, and Mz (z1 , z2 , z3 ) =    Mz (z1 ), Mz (z2 ), Mz (z3 ) . Instead of PA,n we consider the mapping PA ,n :     C3 (C) → C \ {0, 1}, PA,n (z1 , z2 , z3 ) = Mz (z2 ). The mapping PT : T (0, 4) → C \ {0, 1} is defined by choosing for each τ ∈ T (0, 4) a self-homeomorphism w of  (e (w)) = z2 −z1 . P1 representing τ and putting PT (τ ) = PA z3 −z1 ,n 3 The covering PT : T (0, 4) → C\{0, 1} and the covering P : C+ → C\{0, 1} described in Sect. 1.2 are canonically identified as follows. Consider the points 1 [Id] ∈ T (0, 4) and 12 ∈ C\{0, 1}. Then PT ([Id]) = P( 1+i 2 ) = 2 . Hence, there is a holomorphic isomorphism ω : T (0, 4) → C+ that takes [Id] to 1+i 2 such that  −1 P = PT ◦ ω . Proof of Theorem 1.6. The theorem is a consequence of the following lemma.



In the Lemma 1.4 we consider {0} × { n1 } × Cn−2 (C\{0, n1 }) as subspace of Cn (C) and denote points of this space by E˜ n . Lemma 1.4 For any point E˜ n∗ ∈ {0} × { n1 } × Cn−2 (C\{0, n1 }) there exists a neighbourhood U in {0} × { n1 } × Cn−2 (C\{0, n1 }), such that for any preimage τ0 ∈ PT−1 (E˜ n∗ ) there exists a holomorphic mapping gU : U → T (0, n + 1) that takes the value τ0 at the point E˜ n∗ and is a local inverse of PT . Moreover, for any given homeomorphism w0˜ ∗ ∈ QC∞ (0, n + 1), such that En

en (w 0˜ ∗ ) = E˜ n∗ and [w 0˜ ∗ ] = τ0 , there exists a smooth mapping En

En

gU1

U  E˜ n −→ wE˜ n ∈ QC∞ (0, n+1) with wE˜ ∗ = w 0˜ ∗ and en (wE˜ n ) = E˜ n , such n

En

that gU (E˜ n ) = [wE˜ n ] = [gU1 (E˜ n )], and hence, PT ([wE˜ n ]) = PT (gU (E˜ n )) = E˜ n .

1.7 Elements of Teichmüller Theory

39

Proof We consider the family vzD : D⊃ of self-homeomorphisms of the unit disc, vzD (ζ ) =

.

ρ(ζ )z + ζ

, ζ ∈ D,

1 + ρ(ζ )zζ

(1.40)

where the parameter z runs over a disc εD with center 0 and small radius ε. Here ζ → ρ(ζ ) is a (real) non-negative smooth function on D, that vanishes near the unit circle, equals 1 in a neighbourhood of the closed disc εD, and satisfies the inequality z+ζ |ρ(ζ )| ≤ 1 on D. For each z ∈ εD the restriction vzD |εD equals ζ → 1+zζ . This mapping takes 0 to z. The mapping vzD is equal to the identity on the unit circle. For each ζ ∈ D the mapping z → vzD (ζ ), z ∈ D, is holomorphic. The following equalities hold. vzD (ζ ) − ζ =

.

∂ζ (vzD (ζ ) − ζ ) =

ρ(ζ )z(1 − |ζ |2 ) 1 + zρ(ζ )ζ

z∂ζ ρ(ζ )(1 − |ζ |2 ) − ρ(ζ )zζ 1 + zρ(ζ )ζ −

∂ζ (vzD (ζ ) − ζ ) =

,

ρ(ζ )z(1 − |ζ |2 )(z∂ζ ρ(ζ )ζ + zρ(ζ )) (1 + zρ(ζ )ζ )2

z∂ζ ρ(ζ )(1 − |ζ |2 ) − ρ(ζ )zζ 1 + zρ(ζ )ζ

−

,

ρ(ζ )z(1 − |ζ |)2 z(∂ζ ρ)(ζ )ζ (1 + zρ(ζ )ζ )2

.

(1.41) def

Put C = max|ζ |≤1 |(∂ζ ρ)(ζ )| = max|ζ |≤1 |(∂ζ ρ)(ζ )|. For each z ∈ εD we extend vzD to P1 by the identity outside the unit disc. Denote the extended function again by vzD . By Eq. (1.41) for each z ∈ εD the mapping vzD (ζ ) − ζ, ζ ∈ P1 , is smooth and vanishes outside the unit disc. Its supremum norm and the supremum norm of its differential do not exceed C  ε for a constant C  depending on C. These facts imply, that if ε is small, for each z ∈ εD the mapping vzD : P1 ⊃  is a diffeomorphism. Indeed, vzD is a local diffeomorphism and it is C  ε-close to the identity in the supremum norm, hence it is injective, if ε is small. Hence, vzD is a diffeomorphism from P1 onto its image. Since it is the identity outside the unit disc, it is a diffeomorphism from P1 onto itself, and it maps the unit disc onto itself. Moreover, the Beltrami coefficient μvzD (ζ ) =

∂ζ vzD (ζ ) ∂ζ vzD (ζ )

of vzD for |ζ | < 1 satisfies the

 inequality supζ ∈D |μD z (ζ )| ≤ C ε, and, hence, is small for each z ∈ εD if ε small. Moreover, the mapping z → μvzD (ζ ), z ∈ εD, is holomorphic for each ζ ∈ D. Write E˜ n∗ = (0, 1 , z0 , . . . , zn0 ), and let Dj be disjoint discs in C \ {0, 1 } n

3

n

def

with center zj0 , j = 3, . . . , n. The set U  = {0} × {1} × D3 × . . . × Dn ⊂

40

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

{0} × { n1 } × Cn−2 (C\{0, n1 }) is a neighbourhood of (0, n1 , z30 , . . . , zn0 ) in {0} × { n1 } × Cn−2 (C\{0, n1 }). Let Djε , j = 3, . . . , n, be the disc with the same center as Dj and radius equal to the radius of Dj multiplied by the small number ε def

chosen above. We consider the neighbourhood U = {0} × {1} × D3ε × . . . × Dnε , U ⊂ U  , of (0, n1 , z30 , . . . , zn0 ). Take any τ0 ∈ T (0, n + 1) for which PT (τ0 ) = (0, n1 , z30 , . . . , zn0 ) = E˜ n∗ and any w 0˜ ∗ with [w 0˜ ∗ ] = τ0 . We will define a continuous En

En

family wE˜ n ∈ QC∞ (0, n + 1), E˜ n ∈ U, of quasiconformal homeomorphisms of P1 with the following properties. en (wE˜ n ) = E˜ n , gU (E˜ n ) = [wE˜ n ], and PT ([wE˜ n ]) = E˜ n . Moreover, wE˜ ∗ = w 0˜ ∗ , and for all ζ ∈ P1 the Beltrami coefficient μwE˜ (ζ ) En

n

n

depends holomorphically on E˜ n ∈ U . For j = 3, . . . , n we denote by aj ∈ A a complex affine mapping that maps the unit disc onto Dj and maps 0 to zj0 . Let E˜ n = (0, n1 , z3 , . . . , zn ) ∈ U . Put vE˜ n (ζ ) equal to aj ◦ v D−1

◦ a−1 j (ζ ) if ζ ∈ Dj , j = 3, . . . , n, and equal to ζ on the rest

aj (zj ) D vz is the

of P1 . Since identity near ∂D, for each E˜ n = (1, n1 , z3 , . . . , zn ) ∈ U the mapping (ζ, E˜ n ) → vE˜ n (ζ ) is smooth on P1 , takes zj0 to zj , j = 3, . . . , n, and fixes 0, n1 , and ∞. vE˜ ∗ is equal to the identity. Moreover, for each ζ ∈ P1 each vE˜ n in the n family is differentiable at ζ . By Eqs. (1.18) and (1.19) (see also [4], 1D, equations (7) and (8)) for the Beltrami differential of compositions, the absolute value of its Beltrami coefficient is bounded by C  ε. Put gU1 (E˜n ) = wE˜ n = vE˜ n ◦ w 0˜ ∗ . En

Then wE˜ ∗ = w 0˜ ∗ . The mapping E˜ n → μwE˜ is uniformly continuous in E˜ n n

En

n

and ζ ∈ P1 (recall that each μwE˜ vanishes outside D). Hence, the mapping n E˜ n → vE˜ n ◦ w 0˜ ∗ ∈ QC∞ (0, n + 1) is continuous. Moreover, the mapping En

E˜ n → μwE˜ (ζ ) is holomorphic for every ζ ∈ P1 . By the choice of the complex n structure on the Teichmüller space T (0, n + 1) the mapping E˜ n → [wE˜ n ] ∈ T (0, n + 1) is holomorphic. It is clear that PT ([g 1 (E˜n )]) = PT ([w ˜ ]) = E˜ n = U

(0, n1 , z3 , . . . , zn ). The lemma is proved. For later use we state the following tautological lemma. Lemma 1.5 The following diagram is commutative.

.

All mappings in the diagram are continuous.

En



1.8 Thurston’s Classification of Mapping Classes

41

The Modular Transformation Corresponding to Braids For each n-braid b ∈ Bn we consider the mapping class mb,∞ = H∞ (mb ) ∈ M(P1 ; {∞}, En0 ) ⊂ M(P1 \ (En0 ∪ {∞})) .

.

∗ We associate to mb,∞ the modular transformation ϕb,∞ on the Teichmüller space T (n + 1, 0), that is induced by a homeomorphism ϕb,∞ that represents the mapping class mb,∞ . The modular transformation does not depend on the choice of the representing homeomorphism ϕb,∞ . It is convenient to choose for ϕb,∞ the mapping ϕ1 of a parameterizing isotopy for a geometric braid representing b. We denote the ∗ mapping ϕb,∞ on T (n + 1, 0) by Tb and call it the modular transformation Tb of b (though the mapping Tb depends only on mb,∞ , i.e. on bZn ). Since braids are composed as elements of a fundamental group and modular transformations are composed as mappings, the relation

Tb 1 b 2 = Tb 2 Tb 1

.

(1.42)

holds for two braids b1 and b2 . Vice versa, take a modular transformation T on the Teichmüller space T (n + 1, 0), i.e. a mapping T : T (n + 1, 0)⊃ such that for a representative ϕ of a class in M(P1 ; ∞, En0 ) for each element [w] ∈ T (n + 1, 0) the equality T ([w]) = [w ◦ ϕ] holds. We associate to T an element bZn as follows. Let q˜0 = [Id] ∈ T (n + 1, 0). Then T ([Id]) = [ϕ]. Let ϕ be a representative of [ϕ] and let ϕt be a continuous family of self-homeomorphisms of P1 that fix ∞ such that ϕ0 = Id and ϕ1 = ϕ. Then en (ϕ0 ) = en (Id) and en (ϕ1 ) = en (ϕ) = Sϕ (en (Id)) for a permutation Sϕ of the set En0 , since ϕ maps En0 onto itself permuting the points of En0 . Then the curve Psym en (ϕt ), t ∈ [0, 1], in Cn (C)Sn defines a geometric braid. Let b be the braid represented by it. Then T = Tb . Indeed, ϕt , t ∈ [0, 1], is a parameterizing isotopy for the geometric braid Psym en (ϕt ), t ∈ [0, 1], and ϕ1 = ϕ = ϕb,∞ . We proved that the mapping Bn Zn  bZn → Tb ∈ T (n + 1, 0) is a bijection that satisfies (1.42).

1.8 Thurston’s Classification of Mapping Classes Thurston’s interest in surface homeomorphisms was motivated by his celebrated geometrization conjecture. He considered a closed surface S and a selfhomeomorphism .ϕ of S. The mapping torus ([0, 1] × S)(0, x) ∼ (1, ϕ(x))

.

42

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

is obtained by gluing the fiber over the point 0 of the cylinder .[0, 1] × S to the fiber over 1 using the homeomorphism .ϕ. Thurston observed that for a class of homeomorphisms, which he called pseudo-Anosov, the mapping torus admits a complete hyperbolic metric of finite volume. This was one of the eight geometric structures. Moreover, Thurston gave a classification of mapping classes of surface homeomorphisms. Here is Thurston’s theorem on classification of mapping classes. Theorem 1.7 (Thurston [79]) A self-homeomorphism of a closed surface which is not isotopic to a periodic one is either isotopic to a pseudo-Anosov homeomorphism or is reducible, but not both. We explain now Thurston’s notion of pseudo-Anosov homeomorphisms. Let S be a connected finite smooth oriented surface. It is either closed or homeomorphic to a surface with a finite number of punctures. We will assume from the beginning that S is either closed or punctured. A finite non-empty set of mutually disjoint Jordan curves .{C1 , . . . , Cα } on a connected closed or punctured oriented surface S is called admissible if no .Ci is homotopic to a point in X, or to a puncture, or to a .Cj with .i = j . Thurston calls an isotopy class .m of self-homeomorphisms of S (in other words, a mapping class on S) reducible if there is an admissible system of curves .{C1 , . . . , Cα } on S such that some (and, hence, each) element in .m maps the system to an isotopic system. In this case we say that the system .{C1 , . . . , Cα } reduces .m. A mapping class which is not reducible is called irreducible. A conjugacy class is called reducible if the representing mapping classes are reducible. Thurston calls an individual self-homeomorphism .ϕ of a closed or punctured surface S reduced by the set .{C1 , . . . , Cα }, if this set is admissible and ϕ(C1 ∪ C2 ∪ . . . ∪ Cα ) = C1 ∪ C2 ∪ . . . Cα .

.

A self-homeomorphism of S is reduced by an admissible system of curves if and only if the mapping class of .ϕ is reducible. Let S be a closed or punctured surface with set E of distinguished points. We say that .ϕ is a self-homeomorphism of S with distinguished points E, if .ϕ is a self-homeomorphism of S that maps the set of distinguished points E to itself. Notice that each self-homeomorphism of the punctured surface .S \ E extends to a self-homeomorphism of the surface S with set of distinguished points E. We will sometimes identify self-homeomorphisms of .S \ E and self-homeomorphism of S with set E of distinguished points. For a (connected oriented closed or punctured) surface S and a finite subset E of S a finite non-empty set of mutually disjoint Jordan curves .{C1 , . . . , Cα } in .S \ E is called admissible for S with set of distinguished points E if it is admissible for .S \ E. An admissible system of curves for S with set of distinguished points E is said to reduce a mapping class .m on S with set of distinguished points E, if the induced mapping class on .S \ E is reduced by this system of curves. Similarly, a braid .b ∈ Bn is called reducible, if its associated mapping class 0 .mb = Θn (b) ∈ M(D; ∂ D, En ) is reducible and is called irreducible otherwise.

1.8 Thurston’s Classification of Mapping Classes

43

Bers [11] gave a proof of Thurston’s Theorem from the point of view of Teichmüller theory and obtained a description of reducible mappings. The proof of our Main Theorem makes explicit use of the technique developed by Bers. We will outline now the results of Bers’ approach to Thurston’s theory which we need. Consider a smooth closed or punctured surface S. We do not require that it is connected. We allow S to be the union of more than one, but at most finitely many, connected components. A conformal structure on S is a homeomorphism w of S onto a Riemann surface. Let .ϕ be a self-homeomorphism of S. Consider the extremal problem to find the following infimum I (ϕ) = inf {K(w ◦ ϕ˜ ◦ w −1 ) : w : S → w(S) is a conformal structure on S,

.

def

ϕ˜ is free isotopic toϕ}.

.

(1.43)

This extremal problem differs from Teichmüller’s extremal problem by varying also the conformal structure. Notice that in (1.43) we do not required that the conformal structures are quasiconformal. If the infimum is realized on a pair .(w0 , ϕ0 ), then −1 .w0 ◦ ϕ0 ◦ w 0 is called absolutely extremal and .w0 is called a .ϕ0 -minimal conformal structure. ϕ its conjugacy class, Denote by .mϕ the mapping class of .ϕ and by .m ϕ = {ψ = w ◦ ϕ˜ ◦ w −1 : w is a conformal structure on S, ϕ˜ is isotopic to ϕ}. m

.

Then .I (ϕ) can be written as follows ϕ }. I (ϕ) = inf{K(ψ) : ψ ∈ m

.

(1.44)

In the following we will again consider connected surfaces S unless said otherwise. A self-homeomorphism .ϕ of S is periodic if .ϕ n is the identity on S for a natural number n. For mappings that are isotopic to periodic self-homeomorphisms the infimum is attained. More precisely, the following theorem holds. Theorem 1.8 (Bers, see [11]) A self-homeomorphism .ϕ of a surface S is (free) isotopic to a periodic self-homeomorphism iff there is a conformal structure w on S and a self-homeomorphism .ϕ˜ of S such that .w ◦ ϕ˜ ◦ w −1 is conformal (thus .K(w ◦ ϕ˜ ◦ w −1 ) = 0). Moreover, if .ϕ is free isotopic to a periodic self-homeomorphism then there is a .ϕ-minimal conformal structure of first kind. For self-homeomorphisms of S that are not isotopic to periodic ones the following theorem holds. Theorem 1.9 (Bers, see [11]) A conformal structure w of second kind on a surface S cannot be .ϕ-minimal for a self-homeomorphism .ϕ of S that is not free isotopic to a periodic one. Moreover, there exists a conformal structure .w1 of first kind and a self-homeomorphism .ϕ˜ of S which is isotopic to .ϕ and such that .K(w1 ◦ ϕ˜ ◦ w1−1 ) < K(w ◦ ϕ ◦ w −1 ).

44

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

In the light of the two theorems it is sufficient to consider the extremal problem only for conformal structures of first kind. Moreover, we may fix a reference conformal structure of first kind and replace S by the obtained Riemann surface X. Composing the mappings w in (1.43) with the inverse of the reference conformal structure we may consider conformal structures on X rather than on S. Assume that X is of genus g with m punctures and .3g −3+m > 0. (We require that the universal covering of X is .C+ and, in case of genus 0, that the number of punctures is at least 4, to avoid trivial cases.) The Teichmüller space of a Riemann surface .T (X) of first kind can equivalently be described as follows. Consider the set of all homeomorphisms of X onto another Riemann surface Y of first kind (instead of quasiconformal homeomorphisms). Call two (not necessarily quasiconformal) homeomorphisms .wj : X → Yj , j = 1, 2, Teichmüller equivalent if there is a conformal mapping .c : Y1 → Y2 such that −1 .w 2 ◦ c ◦ w1 is isotopic to the identity. The thus obtained equivalence classes are the same as the Teichmüller classes. Indeed, if the Riemann surfaces X and Y are of first kind and w is an arbitrary homeomorphism from X onto Y then w can be extended to a homeomorphism between closed Riemann surfaces. The extended homeomorphism can be uniformly approximated by smooth, and thus, quasiconformal homeomorphisms. For Riemann surfaces of first kind this implies also that the Teichmüller distance (1.30) can be computed by letting all homeomorphisms in (1.30) be arbitrary homeomorphisms instead of quasiconformal homeomorphisms. In particular, for a Riemann surface X of first kind and for homeomorphisms .w1 : X → X1 , .w2 : X → X2 , the Teichmüller distance .dT ([w1 ], [w2 ]) can also be written as follows:

dT ([w1 ], [w2 ]) = inf

.

1 log K(g) : g : X1 → X2 a surjective homeomorphism 2

 which is isotopic to w2 ◦ w1−1 through such homeomorphisms .

.

(1.45)

Let X be a Riemann surface of first kind and let .ϕ be a self-homeomorphism of X. Denote by .ϕ ∗ the modular transformation induced by (the mapping class of) .ϕ on .T (g, m). Put L(ϕ ∗ ) =

.

def

inf

τ ∈T (g ,m)

dT (τ, ϕ ∗ (τ )) .

(1.46)

The quantity .L(ϕ ∗ ) is called the translation length of .ϕ ∗ . Write .τ = [w] . Then ∗ .ϕ ([w]) = [w ◦ ϕ], and by (1.30), (1.43) and (1.45) .

1 log I (ϕ) = inf dT (τ, ϕ ∗ (τ )). 2 τ ∈T (g ,m)

(1.47)

1.8 Thurston’s Classification of Mapping Classes

45

Bers uses the following terminology in analogy to the classification of elements of .PSL(2, Z). Recall that .PSL(2, Z) is the quotient of the group .SL(2, Z) of quadratic matrices of determinant 1 with integer entries by the subgroup consisting of the identity .Id and .−Id. Consider a modular transformation .ϕ ∗ , i.e. an element of the modular group .Mod(g, m). A point .τ ∈ T (g, m) is called .ϕ ∗ -minimal, if ∗ ∗ ∗ ∈ Mod(g, m) is elliptic, if .dT (τ, ϕ (τ )) = L(ϕ ). A modular transformation .ϕ it has a fixed point in .T (g, m), parabolic, if it has no fixed point but .L(ϕ ∗ ) = 0, hyperbolic, if .L(ϕ ∗ ) > 0 and .L(ϕ ∗ ) is attained, and pseudohyperbolic, if .L(ϕ ∗ ) > 0 but .dT (τ, ϕ ∗ (τ )) > L(ϕ ∗ ) for all .τ ∈ T (g, m). A conformal structure w is .ϕ-minimal for a self-homeomorphism .ϕ of X iff .[w] is .ϕ ∗ -minimal. Theorem 1.10 (Bers, see [11]) An element .ϕ ∗ ∈ Mod(g, m) is elliptic iff it is periodic. This happens iff the absolutely extremal map in the isotopy class of selfhomeomorphisms containing .ϕ is conformal. The following theorem is a reformulation of Thurston’s result. Theorem 1.11 (See [11]) For an irreducible self-homeomorphism .ϕ of a Riemann surface X of first kind the modular transformation .ϕ ∗ of .ϕ is either elliptic or hyperbolic. Corollary 1.1 An irreducible self-homeomorphism .ϕ of a Riemann surface X of first kind leads to an absolutely extremal self-homeomorphism .ϕ˜ of a Riemann surface Y by isotopy and conjugation with a homeomorphism. For the quasiconformal dilatation .K(ϕ) ˜ of the absolutely extremal mapping .ϕ˜ we have .

1 1 log K(ϕ) ˜ = log I (ϕ) = L(ϕ ∗ ). 2 2

(1.48)

The following theorem characterizes the absolutely extremal maps with hyperbolic modular transformation in terms of Teichmüller mappings and quadratic differentials. Theorem 1.12 (Bers, see [11]) Let X be a Riemann surface of genus g with m punctures, .3g − 3 + m > 0. A mapping .ϕ : X → X is absolutely extremal, iff it is either conformal or a Teichmüller mapping satisfying the following two equivalent conditions (i) the mapping .ϕ ◦ ϕ is also a Teichmüller mapping with .K(ϕ ◦ ϕ) = (K(ϕ))2 , (ii) the initial and terminal quadratic differentials of .ϕ coincide. Remark 1.2 Suppose .ϕ is an absolutely extremal self-homeomorphism of X with quadratic differential .φ. Then its modular transformation maps the Teichmüller disc .DX,φ homeomorphically onto itself. Indeed, the mapping .ϕ has Beltrami differential .k |φ| φ , and hence it has initial

quadratic differential .φ. The inverse .ϕ −1 has Beltrami differential .−k |ψ| ψ , where .ψ is the terminal quadratic differential of .ϕ. Since .ϕ is absolutely extremal, .ψ = φ.

46

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory ∗ μz For each .z ∈ D and .μz = z |φ| φ the element .ϕ ([W ]) ∈ T (X) is represented

φ for a number .k ∈ (0, 1), by .W μz ◦ ϕ. Since .ϕ ±1 has Beltrami differential .±k |φ| ±1 μ ±k the homeomorphism .ϕ has the form .W . Hence the composition .W μz ◦ ϕ = μ μ μ −1  z −k W ◦ (W ) equals .W z for a number .z ∈ D. Since also for the inverse .ϕ −1 the composition .W μz ◦ ϕ −1 = W μz ◦ (W μk )−1 equals .W μz for a number .z ∈ D, .ϕ maps the Teichmüller disc .DX,φ homeomorphically onto itself. In case .z = x ∈ R is a positive real number, the composition .W μz ◦ (ϕ)±1 is equal to .W μz± for real numbers .z± . In other words, .ϕ ∗ maps real points .{μz }, z ∈ R, in the Teichmüller disc to real points in the Teichmüller disc. The following definition of pseudo-Anosov mappings is equivalent to Thurston’s definition (see [79], [25]).

Definition 1.2 Let S be an oriented smooth connected surface. A selfhomeomorphism .ϕ of S is called a pseudo-Anosov mapping, if there exists a homeomorphism .w : S → X onto a Riemann surface X of first kind such that −1 is an absolutely extremal self-homeomorphism of X with hyperbolic .w ◦ ϕ ◦ w modular transformation (equivalently, a non-periodic absolutely extremal selfhomeomorphism of X ). The absolutely extremal self-homeomorphisms .ϕ of a Riemann surface of first kind with hyperbolic modular transformation (equivalently, the non-periodic absolutely extremal self-homeomorphisms .ϕ of a Riemann surface of first kind) are characterized by the following properties. There is a quadratic differential .φ such that .ϕ maps singular points to singular points, it maps the leaves of the horizontal foliation to leaves of the horizontal foliation and leaves of the vertical foliation to leaves of the vertical foliation. These two foliations are orthogonal outside the common singularity set and are measured foliations in the sense of Thurston by 1 using the metric .ds = |φ| 2 to measure the distance between leaves. The mapping −1 .ϕ decreases the distance between horizontal trajectories by the factor .K 2 and 1 increases the distance between vertical trajectories by the factor .K 2 . Here K is the quasiconformal dilatation of .ϕ. dx The image of .(−1, 1) under an isometry from .(−1, 1) with metric . 1−x 2 into the Teichmüller space .T (g, m) with Teichmüller metric is called a geodesic line. Each pair of distinct points .τ1 , τ2 ∈ T (g, m) lies on a unique geodesic line. This geodesic line contains each point .τ for which .dT (τ1 , τ ) + dT (τ, τ2 ) = dT (τ1 , τ2 ) (see [11], [57]). Self-homeomorphisms of Riemann surfaces with parabolic or pseudohyperbolic modular transformation are reducible. Notice that self-homeomorphisms with elliptic modular transformation also may be reducible but those with hyperbolic modular transformation are irreducible (see Theorem 1.13, statement (3) below). Following Bers [11] we consider now the extremal problem for the quasiconformal dilatation in the case of reducible self-homeomorphisms of Riemann surfaces. Suppose again X is a Riemann surface of first kind with .3g − 3 + m > 0 and a self-homeomorphism .ϕ : X → X is reduced by a non-empty admissible system

1.8 Thurston’s Classification of Mapping Classes

47

of curves .{C1 , . . . , Cα }. .ϕ is called maximally reduced by this system if there is no admissible system of more than .α curves which reduces a mapping which is isotopic to .ϕ. .ϕ is called completely reduced by this system of curves if for each connected α  component .Xj of the complement .X\ C and the smallest positive integer .Nj , =1

for which .ϕ Nj (Xj ) = Xj , the map .ϕ Nj | Xj is irreducible. Lemma 1.6 ([11]) A reducible mapping is isotopic to a maximally reduced mapping. Lemma 1.7 ([11]) If .ϕ is maximally reduced by a system of curves .{C1 , . . . , Cα } it is completely reduced by this system. Let .ϕ be a self-homeomorphism of a Riemann surface X of first kind, .3g − 3 + m > 0, which is completely reduced by a non-empty admissible system .{C1 , . . . , Cα } of curves. If .ϕ is not periodic then an absolutely extremal selfmapping of a Riemann surface related to .ϕ by isotopy and conjugation does not exist (see Theorem 1.13 below). But there exists an absolutely extremal self-mapping .ψ of a nodal Riemann surface Y . The nodal Riemann surface Y is the image of X by a α  continuous mapping w which collapses each curve .Cj to a point and maps .X \ Cj 1

homeomorphically onto its image. The absolutely extremal mapping .ψ is related to .ϕ by isotopy on X and semi-conjugation with w. The nodal Riemann surface Y and the absolutely extremal mapping .ψ on it can be regarded as a “limit” of a sequence of non-singular Riemann surfaces .Xj and self-homeomorphisms .ϕj of .Xj . For the .Xj we have .Xj = wj (X) for a quasiconformal complex structure .wj on X. The .ϕj are related to .ϕ by isotopy and conjugation: .wj−1 ◦ ϕj ◦ wj is isotopic to .ϕ on X. The quasiconformal dilatations .K(ϕj ) converge to the infimum inf {K(w ◦ ϕ˜ ◦ w −1 ) : w ∈ QC(X), ϕ˜ isotopic to ϕ} = e2L(ϕ

.

∗)

and are strictly larger than the infimum. We will need the details later and give them here. A nodal Riemann surface X (or Riemann surface with nodes) is a one-dimensional complex space, each point of which has a neighbourhood which is either biholomorphic to the unit disc .D in the complex plane or to the set .{z = (z1 , z2 ) ∈ D2 : z1 z2 = 0}. (A mapping on the latter set is holomorphic if its restriction to either of the sets, .{(z1 , 0) : z1 ∈ D}, and .{(0, z2 ) : z2 ∈ D}, is holomorphic.) In the second case the point .(0, 0) is called a node. We assume that X is connected and has finitely many nodes. Let .N be the set of nodes. The connected components of .X\N are called the parts of the nodal Riemann surface. We do not require that the set .N of nodes is non-empty. If it is empty we also call the Riemann surface non-singular. Thus, non-singular Riemann surfaces are particular cases of nodal Riemann surfaces. We will say that a non-singular Riemann surface is of finite type and stable if it has no boundary continuum, has genus g and m punctures with .2g − 2 + m > 0 (hence the universal covering is .C+ ). A connected nodal Riemann surface with

48

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

finitely many parts, each of which is a stable Riemann surface of finite type, is called of finite type and stable. Let X and Y be stable nodal Riemann surfaces of finite type. A surjective homeomorphism .ϕ : X → Y is orientation preserving if its restriction to each part of X is so. Notice that .ϕ defines a bijection between the parts of X and the parts of Y . The quasiconformal dilatation of .ϕ is defined as K(ϕ) = max K(ϕ | Xj ) ,

(1.49)

.

Xj

where .Xj runs over all parts of X. Let .ϕ be an orientation preserving self-homeomorphism of a stable nodal Riemann surface X of first kind. The mapping .ϕ permutes the parts of X along cycles. Let .X0 be a part of X and let n be the smallest number for which .ϕ n (X0 ) = X0 . Put .Xj = ϕ j (X0 ) for .j = 1, . . . , n − 1,, and call .(X0 , X1 , ..., Xn−1 ) a .ϕ-cycle of length n. The mapping .ϕ is called absolutely extremal if for any .ϕcycle .(X0 , X1 , ..., Xn−1 ) the restriction .ϕ|X0 ∪ ... ∪ Xn−1 to the Riemann surface .X0 ∪ ... ∪ Xn−1 (which is not connected if .n > 1) is absolutely extremal. In other words the following holds. Let .w : X → Y be a homeomorphism onto another nodal Riemann surface Y (considered as conformal structure on X). Let .ϕˆ be a selfhomeomorphism of X which is isotopic to .ϕ. Let .(X0 , ..., Xn−1 ) be a cycle of parts for .ϕ. Then .

max

0≤j ≤n−1

K(ϕ | Xj ) ≤

max

0≤j ≤n−1

K(w ◦ ϕˆ ◦ w −1 | w(Xj )) .

(1.50)

If the mapping .ϕ|X0 ∪...∪Xn−1 is absolutely extremal, then the restriction .ϕ n |X0 is an absolutely extremal self-homeomorphism of the connected Riemann surface .X0 . Notice that if .ϕ fixes all parts of X then (1.50) is equivalent to the condition that .ϕ | Xj is an absolutely extremal self-homeomorphism of .Xj for each part .Xj of X. In this case Theorem 1.11 applied to the parts gives a description of the absolutely extremal self-homeomorphisms. Let .X0 , .X1 , . . . Xn be Riemann surfaces, and let .ϕˆ be a self-homeomorphism of def

the disjoint union .X = X0 ∪ ... ∪ Xn−1 that commutes the .Xj along the cycle ϕˆ

ϕˆ

ϕˆ

def

X0 −−−→ X1 −−−→ . . . −−−→ Xn = X0 .

.

(1.51)

Let w be a conformal structure on X, in other words, consider Riemann surfaces def n−1 .Yj , their disjoint union .Y = j =0 Yj , and a conformal mapping .w : X → Y . We may assume that w maps .Xj conformally onto .Yj for each j . We define .ψ = w ◦ ϕˆ ◦ w−1 on .Y = w(X). Let .ϕˆ j = ϕ|X ˆ j , .ψj = ψ|Yj for .j = 0, . . . , n − 1, and .wj = w | Xj , j = 0, . . . , n − 1, .Xn = X0 , and .wn = w|Xn = w0 . Then we get the commutative diagram Fig. 1.6.

1.8 Thurston’s Classification of Mapping Classes  ϕ

X0 w0

Y0

 ϕ

X1

49  ϕ

...

ψ

Y1

Xn = X0 .

wn−1

w1 ψ

 ϕ

Xn−1

ψ

...

wn =w0 ψ

Yn−1

Yn = Y0

Fig. 1.6 A commutative diagram for self-homeomorphisms of non-connected surfaces

The commutativity of the diagram implies the following equations ψj

.

=wj +1 ◦ ϕˆj ◦ wj−1 , j = 0, . . . n − 1 ,

ψ j |Y0 =wj ◦ ϕˆ j ◦ w0−1 , j = 1, . . . n , ψ n |Y0 =w0 ◦ ϕˆ n ◦ w0−1 .

(1.52)

It will be convenient to have in mind the following lemma which describes the absolutely extremal self-homeomorphisms on the cycles of parts of the nodal surface X. def

Lemma 1.8 Let .X0 , X1 , ..., Xn−1 , Xn = X0 be non-singular stable Riemann surfaces of finite type. Suppose .ϕ is a self-homeomorphism of .X0 ∪ ... ∪ Xn−1 which permutes the .Xj along the n-cycle ϕ

ϕ

ϕ

def

X0 −−−→ X1 −−−→ . . . −−−→ Xn = X0 .

.

(1.53)

Then .ϕ is absolutely extremal iff the following two conditions hold. def

(1) The mapping .F = ϕ n |X0 is absolutely extremal. def

(2) The quasiconformal dilations .K(ϕj ) of the mappings .ϕj = ϕ|Xj satisfy the equality .

1 1 1 log K(ϕ) = max log K(ϕj ) = log K(ϕ n | X0 ) . j 2 2 2n

(1.54)

Moreover, if .ϕ is absolutely extremal then one of the following two situations occurs. (2a) The mapping . F is conformal and . ϕ is conformal. (2b) The mapping . F : X0 ⊃ is a Teichmüller mapping whose initial and terminal quadratic differentials are equal, and the Teichmüller classes . [ϕ j ] ∈ T (X0 ) , . j = 1, ..., n − 1, divide the bounded segment with endpoints .[id] and .[F ]

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1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

on the unique geodesic line through these two points into n segments, each of which has .dT -length equal to .

1 1 dT (X0 ) ([id], [F ]) = log K(F ) . 2n 2n def

Moreover, for each .j, j = 1, ..., n − 1, the mapping .ϕj = ϕ|Xj is a Teichmüller map from .ϕ j (X0 ) onto .ϕ j +1 (X0 ). Proof Notice first that for all homeomorphisms .ϕˆ that permute the .Xj along the cycle (1.53) the inequalities 1 1 1 log K(ϕˆ j ) ≥ log K(ϕˆ j ) ≥ log K(ϕˆ n |X0 ) 2 2 2 n−1

n max

.

j

(1.55)

0

hold. We prove first that .ϕ is absolutely extremal if .F = ϕ n |X0 is absolutely extremal and equality (1.54) holds. Let .ϕˆ be a self-homeomorphism of .X0 ∪ . . . ∪ Xn−1 that is isotopic to .ϕ, and let .ψ be related to .ϕ by the diagram Fig. 1.6. If .F = ϕ n |X0 is absolutely extremal, then .

1 1 log K(ϕ n |X0 ) ≤ log K(ψ n |X0 ) . 2 2

If also equality (1.54) holds, then .

1 n 1 1 n log K(ϕ) = K(ϕ n |X0 ) ≤ K(ψ n |X0 ) ≤ n max log K(ψj ) = log K(ψ) . j 2 2 2 2 2

We proved that .ϕ is absolutely extremal. Vice versa, suppose .ϕ is absolutely extremal. Then .F = ϕ n |X0 is absolutely extremal. It remains to show that condition (2) holds. If F is conformal, we associate to F the self-homeomorphism Id

Id

F

X0 −−−→ X0 −−−→ . . . −−−→ X0

.

of the formal disjoint union of n copies of .X0 . This homeomorphism is conformal, hence its quasiconformal dilation equals zero. By the Lemma on Conjugation (see Appendix A.2) this homeomorphism is conjugate to .ϕ. Since .ϕ is absolutely extremal, .ϕ must have vanishing quasiconformal dilation, hence .ϕ is conformal (see Eq. (1.50)). Consider the remaining case when F is a Teichmüller mapping whose initial and terminal quadratic differentials are equal. Let .φ be the quadratic differential of F and .DX0 ,φ the Teichmüller disc in .X0 corresponding to .φ. Suppose .μφ = k |φ| φ . The

1.8 Thurston’s Classification of Mapping Classes

51



set . {x |φ| } : x ∈ R , equipped with the metric .dT , is the unique geodesic line φ through .[F ] = {k |φ| = {0}. Consider the points .τj = {xj |φ| φ } and .[Id] φ } on this

|φ| line that divide the segment . {x φ } : x ∈ [0, k] on this line into n segments of 1 equal .dT -length . 2n log K(F ). The class .τj , j = 1, . . . , n − 1, is represented by μ the absolutely extremal self-homeomorphism .W xj of .X0 , .τ0 is represented by the μ μ x ˚n = X0 and ˚0 = X identity .Id and .τn is represented by .W n = W k = F . Put .X μxj ˚ .Xj = W (X0 ), j = 1, . . . , n − 1. By Lemma 9.1 of [59] for each .j = 0, . . . , n − def μ μ ˚j → X ˚j +1 is a Teichmüller ϕj = W xj +1 ◦ (W xj )−1 : X 1, the homeomorphism .˚ mapping with quadratic differential .φ. For its quasi-conformal dilatation the equality |φ| 1 1 . log K(˚ ϕj ) = dT ({xj |φ| 2 φ }, {xj +1 φ }) = 2n log K(F ) holds. n−1 ˚ def ˚j that equals .˚ ˚j . Then .˚ Consider the mapping .˚ ϕ on .X ϕj on .X ϕ0n | = X 0 1 1 n n n ˚0 = F , . log K(˚ ˚0 ). Since .X ˚0 = X0 and .˚ ϕ | X0 = ϕ | X0 X ϕ ) = 2n log K(˚ ϕ |X 2 ˚ ϕ on .X is conjugate to the mapping .ϕ on X. Since .ϕ is absolutely the mapping .˚ extremal, the inequality

.

1 1 1 ˚0 ) = 1 log K(ϕ n | X0 ) log K(ϕ) ≤ log K(˚ ϕ) = log K(˚ ϕn | X 2 2 2n 2n

holds. Condition .(2) follows from (1.55). Conditions (1) and (2) imply that .ϕ is conformal if F is conformal, and if F is a Teichmüller mapping, then equality (1.54) implies that the elements .τj of the Teichmüller space represented by the .ϕj divide the segment on the geodesic line between .[I d] and .[F ] into n segments of equal length. The lemma is proved. 

Now we describe in more detail the solution of the extremal problem (1.43) in the reducible case. Let X be a connected Riemann surface, which is closed or of first kind, with universal covering equal to .C+ . Let .m ∈ M(X) be an isotopy class of orientation preserving self-homeomorphisms. Let .C be an admissible system of curves which completely reduces an element .ϕ of .m. By an isotopy we may assume that .C is real analytic (or even geodesic). Associate to X and the system of curves .C a nodal surface Y and a continuous surjection .w : X → Y . This can be done as follows. Surround each connected component .Cj of .C by an annulus .A(Cj ), which admits a conformal mapping .cj onto a round annulus .Arj = { r1j < |z| < rj } for some . rj > 1, such that .cj maps the curve .Cj to the unit circle. For each j we define the def

set .Vj = {(z1 , z2 ) ∈ C2 : z1 z2 = 0, |zj | < rj for j = 1, 2} and the holomorphic mapping .ψj : Arj \ {|z| = 1} → Vj  ψj (z) =

.

(0, z)

if 1 < |z| < rj ,

( 1z , 0)

if

1 rj

< |z| < 1 .

52

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

Notice that .ψj (Arj \{|z| = 1}) = Vj \D2 . The nodal surface Y is obtained by gluing  each .Vj to .X \ Cj along the sets .Vj \ D2 and .A(Cj ) \ Cj , respectively, using the −1 2 holomorphic homeomorphism .c−1 j ◦ ψj : Vj \ D → A(Cj ) \ Cj . The set of nodes .N of Y is the set of nodes of the .Vj , hence, .N is in bijective correspondence to the set of connected components of .C. By construction the nodal surface Y contains the “copy” .Y \∪j Vj of .X\∪j A(Cj ). The restriction .w | X \ ∪j A(Cj ) is defined to be the canonical mapping from .X \ ∪j A(Cj ) onto its copy in Y . For all . j we let . χj : [1, rj ] → [0, 1] be a diffeomorphism such that . χj (1) = 0 and .χj (rj ) = 1. We define for each . j a continuous surjective mapping . wj : Arj \ Cj → Vj \ {(0, 0)} , wj (z) =

.

 χj (|z|)ψj (z) 1 )ψj (z) χj ( |z|

if 1 < |z| < rj , if r1j < |z| < 1 .

The mapping .wj ◦ cj maps .A(Cj ) \ Cj homeomorphically onto .Vj \ {(0, 0)}. It extends to a continuous surjection from .A(Cj ) onto .Vj that collapses .Cj to a point. The mapping w that is equal to .w | X \ ∪j A(Cj ) on .X \ ∪j A(Cj ) and is for each j equal to the extension of .wj ◦ cj to .A(C  j ) is the required surjection .w : X → Y . Since .ϕ maps the complement .X\ C∈C C of the set of curves homeomorphically onto itself, we may consider the mapping .w ◦ ϕ ◦ w −1 on .Y \ N . It extends continuously to the nodes. Denote the obtained mapping on Y by .ϕ and its isotopy class on Y by .m . Note that the nodal surface Y is defined up to homeomorphism by the isotopy class of .C. (We consider isotopies of the system of curves .C within the class of real analytic systems of curves). The class .m is determined by .m, and by the system of  curves .C. The conjugacy class .m  of .m is defined by the isotopy class of .C and . the class .m The mapping .ϕ , and, hence its isotopy class .m , permutes the parts .Yk of .Y \N ˚ j . Let .m,j be the restrictions of the class .m to the along cycles denoted by .cyc ˚ j . We call the conjugacy classes .m  cycles .cyc ,j of the restrictions to the cycles ˚ j the irreducible components of .m  related to the class of .C. Notice that the class .cyc   only modulo products of powers of Dehn twists around .m  determines the class .m curves which are homotopic to curves of the system .C. It is not hard to see that the  .m  are irreducible (see also Remark 5.1). Notice also that the isotopy class of a system of curves .C which completely reduces an element of .m is not uniquely determined by X and .m, even if we require that the system maximally reduces the homeomorphism. In particular, the type of the nodal surface Y is not uniquely determined. This may occur, for instance for reducible homeomorphisms with elliptic modular transformation. It is a remarkable fact that the extremal problem for the quasiconformal dilatation has a solution in terms of nodal Riemann surfaces and irreducible parts of mapping classes.

1.8 Thurston’s Classification of Mapping Classes

53

The following theorem is due to Bers. Theorem 1.13 ([11]) Let .m be a mapping class of orientation preserving selfhomeomorphisms of a Riemann surface X of first kind with universal covering .C+ . Let .C be an admissible system of real analytic curves which completely reduces an element .ϕ ∈ m. Choose a stable nodal Riemann surface Y of first kind with set of nodes .N and a continuous surjection .w : X → Y which contracts each curve of .C to a point and whose restriction to the complement of .C is a homeomorphism onto  .Y \ N . Denote by .m the mapping class on Y induced by .m and w, and by .m  its conjugacy class. Then the following holds. 1. There exists a conformal structure .w˜ on .Y, .w˜ : Y → w(Y ˜ ) = Y˜ , and an ˜  absolutely extremal self-homeomorphism .ϕ˜ of .Y , representing the class .m . The self-homeomorphism .ϕ˜ has the following stronger extremal properties. 2. The equality I (ϕ) = e2L(ϕ

.

∗)

= K(ϕ) ˜

(1.56)

holds for the modular transformation .ϕ ∗ of .ϕ. Moreover, for any continuous surjection .w  : X → Y  onto a nodal Riemann surface .Y  , such that the preimages of the nodes are disjoint Jordan curves of the system .C and the restriction of .w  to the complement of the curves is a homeomorphism, and any  induced by .m and .w  , we have  self-homeomorphism .ϕ  in the class .m  K(ϕ) ˜ ≤ K(ϕ  ).

.

3. If .m is reducible and not periodic then for each non-singular Riemann surface .Y  , each surjective homeomorphism .w  : X → Y  , and each self-homeomorphism    −1 ◦ ϕ  ◦ w  ∈ m , we have strict inequality .ϕ of .Y such that .(w ) K(ϕ) ˜ < K(ϕ  ).

.

4. If .m is reducible then there exists a sequence .Y (j ) of non-singular Riemann surfaces .Y (j ) , surjective homeomorphisms .w (j ) : X → Y (j ) , and selfhomeomorphisms .ϕ (j ) of .Y (j ) with the following property: (w (j ) )−1 ◦ ϕ (j ) ◦ w (j ) ∈ m and K(ϕ (j ) ) → K(ϕ). ˜

.

Recall that a self-homeomorphism .ϕ˜ of a nodal Riemann surface .Y˜ is abso˜ = maxY˜j K(ϕ|Y ˜ j ) is smallest among the lutely extremal if the quantity .K(ϕ) respective quantities for all nodal Riemann surfaces that are homeomorphic to Y and all self-homeomorphisms of Y that are obtained from .ϕ˜ by isotopy and conjugation. Here .Y˜j are the parts of .Y˜ . Recall also that the quantity .I (ϕ) for a self-homeomorphism .ϕ of a non-singular Riemann surface equals the infimum of

54

1 Riemann Surfaces, Braids, Mapping Classes, and Teichmüller Theory

K(ϕ  ) over all self-homeomorphisms .ϕ  of non-singular Riemann surfaces in the class of .ϕ. Equality (1.56) relates the value .I (ϕ) to the quasiconformal dilatation of the absolutely extremal self-homeomorphism of the associated nodal surface. The second part of Statement 2 is strictly stronger than statement 1. In Statement 1 the nodal surface up to homeomorphism and the class of .ϕ˜ are fixed. In statement 2 the type of the nodal surface .Y  is not prescribed. It may be different from the type of Y . We require only that .Y  is the image of X under a continuous surjection that maps some subcollection of the set of admissible curves to nodes. The deepest part of the theorem is its particular case concerning irreducible homeomorphisms and Statement 3.

.

Chapter 2

The Entropy of Surface Homeomorphisms

Here we give a self-contained proof of the Theorem of Fathi and Shub on the entropy minimizing property of pseudo-Anosov self-homeomorphisms of closed Riemann surfaces of genus at least two following mainly along the lines of the original proof. A proof of the theorem for the case of punctured surfaces is also included. The exposition contains a thorough account on the necessary prerequisites on the trajectories of quadratic differentials. The present proof of the Fathi and Shub Theorem on the relation between the entropy and the quasi-conformal dilatation is new. The chapter concludes with a thorough treatment of the entropy of selfhomeomorphisms of Riemann surfaces of second kind which is a prerequisite for the treatment of reducible braids.

2.1 The Topological Entropy of Mapping Classes The topological entropy of continuous surjective mappings of a compact topological space to itself is defined as follows. Let X be a compact topological space and .ϕ a continuous mapping from X onto itself. Let .A be a collection of open subsets of X which cover X (for short, .A is an open cover of X). For two open covers .A and .B we define A ∨ B = {A ∩ B : A ∈ A, B ∈ B}.

.

Let .N (A) be the minimal cardinality of a subset .A1 of .A which is a cover of X. The entropy .h(ϕ, A) of .ϕ with respect to .A is defined as def

h(ϕ, A) = limN →∞

.

1 log N (A ∨ ϕ −1 (A) ∨ . . . ∨ ϕ −N (A)) . N

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025 B. Jöricke, Braids, Conformal Module, Entropy, and Gromov’s Oka Principle, Lecture Notes in Mathematics 2360, https://doi.org/10.1007/978-3-031-67288-0_2

(2.1)

55

56

2 The Entropy of Surface Homeomorphisms

(Here .ϕ −1 (A) = {ϕ −1 (A) : A ∈ A}, .ϕ −k−1 (A) = ϕ −1 (ϕ −k (A)).) The entropy .h(ϕ) of .ϕ is defined as def

h(ϕ) = sup h(ϕ, A).

.

A

(2.2)

An open cover .A is a refinement of an open cover .B if each set .A ∈ A is contained in a set .B ∈ B. We write .A ≺ B. A sequence .{An }, n = 1, 2, . . . , of open covers is refining if for each n the cover .An+1 is a refinement of .An and each open cover .B has a refinement among the .An . For a refining sequence of open covers .An the equality h(ϕ) = lim h(ϕ, An )

.

n→∞

holds. (The sequence of numbers on the right is non-decreasing.) See [1], Proposition 12. An example of a refining sequence of covers is the following. Suppose X is equipped with a metric. Let .An be the cover consisting of all open balls of radius .εn . If .εn decreases to zero for .n → ∞ then the sequence is refining. The entropy is a conjugacy invariant (see Theorem 1 of [1]): If .ψ : X → Y is a homeomorphism of topological spaces, then .h(ψ ◦ϕ ◦ψ −1 ) = h(ϕ). Let .ϕ be a self-homeomorphism of X. The following relation is not difficult to prove (see Theorem 2 of [1]). For any non-zero integral number n, .h(ϕ n ) = |n| h(ϕ). Moreover, the following result of [1] (see Theorem 4 there) holds. Let .X1 and .X2 be two closed subsets of a topological space X such that .X = X1 ∪ X2 . If a self-homeomorphism .ϕ of X fixes each .Xj , .j =1, 2, setwise, then .h(ϕ) = maxj =1,2 h(ϕ | Xj ). For more details see [1]. We also need the following property of entropies (see Theorem 5 of [1]). Suppose X is a compact topological space and .∼ is an equivalence relation on X. Let .p : X → X ∼ be the projection of X to the quotient. If .ϕ˜ is a continuous ˜ ∼ ϕ(y) ˜ if .x ∼ y, then for the mapping .ϕ mapping from X into itself such that .ϕ(x) on the quotient that is defined by .ϕ ◦ p = p ◦ ϕ˜ the inequality .h(ϕ) ≤ h(ϕ) ˜ holds. We will be concerned with the topological entropy of self-homeomorphisms of compact surfaces (with or without boundary). Let X be a compact surface (with or without boundary) with a finite set .En of distinguished points.

2.1 The Topological Entropy of Mapping Classes

57

Let .m ∈ M(X; ∂X, En ) (or .m ∈ M(X; ∅, En ) if .∂X = ∅) be a mapping class. The entropy of the mapping class .m is defined as follows h(m) = inf {h(ϕ) : ϕ ∈ m} .

.

(2.3)

Since the topological entropy of a homeomorphism is invariant under conjugation the following holds: h(m) = h( m) = inf {h(ψ ◦ ϕ ◦ ψ −1 ) : ϕ ∈ m , ψ : X → Y a homeomorphism} . (2.4)

.

For a braid .b ∈ Bn with base point .En we define the entropy .h(b) as the entropy of ¯ ∂D, En ) corresponding to b: the mapping class .mb = Θn (b) ∈ M(D; def

h(b) = h(mb ).

.

(2.5)

By (2.4) the entropy .h(b) does not depend on the choice of the base point .En and is a conjugacy invariant. Hence, ˆ = h( b } . h(b) = h(b) mb ) = inf {h(ϕ) : ϕ ∈ m def

.

(2.6)

The entropy of self-homeomorphisms of closed surfaces was studied first in Exposé 10 of the Asterisque volume dedicated to Thurston’s work [25]. In Sects. 2.2 and 2.4 of this chapter we give the proof of the following two theorems. Theorem 2.1 Let X be a (closed connected) Riemann surface of genus g with a set Em of .m ≥ 0 distinguished points, .3g − 3 + m > 0. Let .ϕ0 ∈ Hom(X; ∅, Em ) be a non-periodic absolutely extremal self-homeomorphism of X with set of distinguished points .Em (by an abuse of language we speak about a non-periodic absolutely extremal self-homeomorphism of .X\Em ). Then

.

h(ϕ0 ) =

.

1 log K(ϕ0 ) = L(ϕ0∗ ) . 2

(2.7)

Here .K(ϕ0 ) is the quasiconformal dilatation of .ϕ0 , and .ϕ0∗ is the modular transformation on the Teichmüller space .T (g, m) induced by .ϕ0 and .L(ϕ0∗ ) is its translation length. The theorem was first proved for closed surfaces by Fathi and Shub in the exposé 10 of the volume [25]. The present proof of Theorem 2.1 (see Sect. 2.3) differs from that of Fathi and Shub and works also for punctured surfaces. Theorem 2.2 ([25]) Let X be a closed connected Riemann surface of genus .g ≥ 2. Then any pseudo-Anosov self-homeomorphism of X is entropy minimizing in its isotopy class.

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2 The Entropy of Surface Homeomorphisms

Theorem 2.2 is proved in [25] but the ingredients of the proof of Theorem 2.2 in [25] are spread over several chapters. In Sects. 2.2 and 2.4 we present a selfcontained proof of Theorem 2.2 following mainly the ideas of Fathi and Shub. Ahlfors’ trick (see [2], section 4, pp. 19–20), and the Lemma 2.1 below on the entropy of lifts to simple branched coverings are needed to prove the analog of Theorem 2.2 for punctured Riemann surfaces. Theorem 2.2 for punctured Riemann surfaces is proved in Sect. 2.3. The result of Sect. 2.5 on the entropy of selfhomeomorphisms of Riemann surfaces of second kind is needed to treat the entropy of reducible braids. Recall that a branched covering is called simple, if over each point there is at most one branch point and the mapping is a double branched covering in a neighbourhood of this branch point. Lemma 2.1 Let .ϕ be a self-homeomorphism of a compact Riemann surface X with p set of distinguished points E. Let .Xˆ −→ X be a simple branched covering of X with branch locus in E and let .ϕˆ : Xˆ → Xˆ be a self-homeomorphism of .Xˆ such that .p ◦ ϕ ˆ = ϕ ◦ p (for short, .ϕˆ is a lift of .ϕ). Then .h(ϕ) ˆ = h(ϕ). Proof of the Lemma 2.1 We assume that the covering multiplicity m is bigger than 1. I. The Inequality .h(ϕ) ≤ h(ϕ) ˆ follows from Theorem 5 [1]. Here is a proof for def

convenience of the reader. Let .A be an open cover of X. Put .p−1 (A) = {p−1 (A) : ˆ Since .p ◦ ϕˆ = ϕ ◦ p it follows that A ∈ A}. .p−1 (A) is an open cover of .X. −1 −1 −1 −1 .p (ϕ (A)) = ϕˆ (p (A)). Further, .p−1 (A ∨ B) = p−1 (A) ∨ p−1 (B) for two open covers .A and .B of X. Hence     N p−1 (A) ∨ . . . ∨ ϕˆ −N (p−1 (A)) = N p−1 (A) ∨ . . . ∨ p−1 (ϕ −N (A))     = N p−1 (A ∨ . . . ∨ ϕ −N (A)) = N A ∨ . . . ∨ ϕ −N (A) .

.

Hence h(ϕ) = sup h(ϕ, A) =

.

A

sup −1 (A) for ˆ A=p a cover A of X

ˆ ≤ h(ϕ, ˆ A)

sup A˜ an arbitrary cover of Xˆ

˜ = h(ϕ) h(ϕ, ˆ A) ˆ .

Notice that whatever sequence of covers .An of X we take, the sequence of covers p−1 (An ) is not refining if the covering multiplicity is bigger than 1.

.

II. The Opposite Inequality .h(ϕ) ≥ h(ϕ) ˆ We will choose a suitable refining sequence of coverings of X and associate to it a refining sequence of coverings ˆ Equip X with a metric d and denote by d also the induced metric on .Xˆ (which of .X. is defined by putting the length of any smooth curve in .Xˆ equal to the length of its projection). Let .σ > 0 be the minimal distance between two points in the branch locus of the covering.

2.1 The Topological Entropy of Mapping Classes

59

A connected open subset V of X is called nicely covered if .p−1 (V ) contains no more than one branch point. For a nicely covered connected open set V the preimage .p−1 (V ) consists either of m or .m − 1 connected components. In the first case p is a homeomorphism from each connected component of .p−1 (V ) onto V . In the second case this statement is true for .m − 2 of the connected components of −1 (V ). The restriction of p to the remaining component of .p −1 (V ) is a double .p branched covering of V with a single branch point. Each open set of diameter not exceeding .σ is nicely covered. Remarks Preceding the Proof of the Opposite Inequality Suppose we have three simply connected sets .A1 , .A2 , and .A3 belonging to a certain open cover .A of X. Suppose for integer numbers .k1 , k2 , and .k3 the set .A1 ∩ ϕ −k2 (A2 ) ∩ ϕ −k3 (A3 ) is not empty and the union .A1 ∪ ϕ −k2 (A2 ) ∪ ϕ −k3 (A3 ) is contained in a simply connected set B, that does not intersect the branch locus. Since .ϕ is a homeomorphism with set of distingiushed points E, none of the sets .Aj , j = 1, 2, 3, intersects the branch locus. For .k  = 1, 2, 3, we let .Alk  be the m connected components of .p−1 (Ak  ), and .ϕ ˆ a self-homeomorphism of .Xˆ for which .p ◦ ϕˆ = ϕ ◦ p. Then the intersection of j j j two or three sets among the .A11 , .ϕˆ −k2 (A22 ), and .ϕˆ −k3 (A23 ) is not empty if and only if the sets lie in the same of the m connected components of .p−1 (B). Hence, there j j j are exactly m non-empty sets of the form .A11 ∩ ϕˆ −k2 (A22 ) ∩ ϕˆ −k3 (A23 ). Consider the case when .A1 and .A3 do not meet the branch locus but .A2 contains a single point of the branch locus. Suppose the intersection .A1 ∩ϕ −k2 (A2 )∩ϕ −k3 (A3 ) is not empty and the union .A1 ∪ ϕ −k2 (A2 ) ∪ ϕ −k3 (A3 ) is contained in a simply connected set B with a single point in the branch locus. The previous arguments remain true for the .m − 2 connected components of .p−1 (B) that do not contain a branch point. The connected component .B m−1 , for which .p : B m−1 → B is a double branched covering contains two preimages of .A1 , and two preimages of .ϕ −k3 (A3 ), and we cannot exclude that each of the two components of .p−1 (A1 ) intersects each of the two components of .p−1 (ϕ −k3 (A3 )). We may get more than m, but no more j j j than .m − 2 + 4 non-empty sets of the form .A11 ∩ ϕˆ −k2 (A22 ) ∩ ϕˆ −k3 (A33 ). More −1 detailed, there are at most two of the connected components of .p (A1 ), such that j j j each of them may contain 2 different sets of the form .A11 ∩ ϕˆ −k2 (A22 ) ∩ ϕˆ −k3 (A33 ), and all other components contain exactly one set of this form. Equivalently, there are at most two of the connected components of .p−1 (ϕ −k3 (A3 )), such that each of j j j them may contain 2 sets of the form .A11 ∩ ϕˆ −k2 (A22 ) ∩ ϕˆ −k3 (A33 ), and all other components contain exactly one set of this form. Beginning of the Formal Proof We make the following choices of positive numbers by induction on the natural number .. For each . there exists a positive number .ε < ε−1 , .ε1 < σ2 , such that for any pair of points .z1 , z2 ∈ X the following implication holds d(z1 , z2 ) < ε ⇒ d(ϕ ±l (z1 ), ϕ ±l (z2 ))
1 and .k0 + k1 < n and .k1 ≥ . Notice that .Ak0 and .Ak0 +k1 +1 do not meet the branch locus. By the remark in the beginning of the proof there are no more than .m − 2 + 4 < 2m jk +k +1 jk different non-empty sets of the form .ϕˆ −k0 +1 (Aˆ 0 ) ∩ . . . ∩ ϕˆ −k0 −k1 (Aˆ 0 1 ), and .

k0 +k1 +1 jk0 +k1 +1 −1 −k −k 0 1 of .p (ϕ (Ak0 +k1 +1 )) for

k0

each set is contained in a connected component the set .Ak0 +k1 +1 that does not intersect the branch locus. Moreover, each connected component of .p−1 (ϕˆ −k0 −k1 (Ak0 +k1 +1 )) contains at most two sets of the form jk +k +1 jk .ϕ ˆ −k0 +1 (Aˆ 0 ) ∩ . . . ∩ ϕˆ −k0 −k1 (Aˆ 0 1 ). k0 +k1 +1

k0

Consider now any collection .A1 , A2 , . . . , An ∈ A . Divide it into maximal collections of consecutive sets, so that either all sets of a collection intersect the branch locus or all sets of the collection do not intersect the branch locus. For each integer number k we consider all the maximal collections of consecutive sets .Ak0 +1 , . . . , Ak0 +k1 , that intersect the branch locus and for which .k1 ≥  and .k0 + k1 ≤ k. Its number is denoted by .n(k). If .Ak0 +1 , . . . , Ak0 +k1 is a maximal collection of consecutive sets among the .A1 , A2 , . . . , An ∈ A , that do not intersect the branch locus, then each connected jk +k component of the preimage .p−1 (ϕ −k0 −k1 −1 (Ak00+k11 )) contains at most .2n(k0 ) = jk +k j 2n(k0 +k1 ) different sets of the form .Aˆ 11 ∩ . . . ∩ ϕˆ −k0 −k1 −1 (Aˆ k00+k11 ). If .Ak0 +1 , . . . , Ak0 +k1 is a maximal collection of consecutive sets that intersect the branch locus and .k0 + k1 ≤ n, then, if .k1 < , each connected component jk

+k

of .p−1 (ϕ −k0 −k1 +1 (Ak00+k11 )) contains at most .2n(k0 ) different sets of the form

jk +k j Aˆ 11 ∩ . . . ∩ ϕˆ −k0 −k1 −1 (Aˆ k00+k11 ), and contains at most .2 · 2n(k0 ) = 2n(k0 +k1 ) different

.

jk +k j sets of the form .Aˆ 11 ∩ . . . ∩ ϕˆ −k0 −k1 +1 (Aˆ k00+k11 ), if .k1 ≥ . Since .n(n) ≤ [ n ], we obtain

    n N Aˆ  ∨ . . . ∨ ϕˆ −n (Aˆ  ) ≤ 2[  ] · m · N A ∨ . . . ∨ ϕ −n (A ) .

.

Hence, h(Aˆ  , ϕ) ˆ ≤

.

log 2 + h(A , ϕ) 

(2.18)

2.1 The Topological Entropy of Mapping Classes

65

for each of the described covers .A of X. Since the sequences .A and .Aˆ  are refining we obtain h(ϕ) ˆ ≤ h(ϕ) .

.

(2.19)

The lemma is proved.

 

Theorem 2.2 and Lemma 2.1 together with Ahlfors’ trick (see Sect. 1.7) will imply the following. Theorem 2.3 Let X be a closed connected Riemann surface of genus g with a set Em of .m ≥ 0 distinguished points, .3g −3+m > 0. Then any non-periodic absolutely extremal self-homeomorphism of X with set of distinguished points .Em is entropy minimizing in its isotopy class.

.

Together with Corollary 1.1 we obtain the following statement, which includes homeomorphisms with elliptic or hyperbolic modular transformation. Corollary 2.1 Let .ϕ be an irreducible self-homeomorphism of a closed connected Riemann surface X with a set .Em of .m ≥ 0 distinguished points, .3g − 3 + m > 0. Let .ϕ0 be an absolutely extremal self-homeomorphism of a Riemann surface Y with m distinguished points, which is obtained from .ϕ by isotopy and conjugation. Then 1 m ϕ0 ) log K(ϕ0 ) = L(ϕ ∗ ) = h( 2 = inf {h(ϕ1 ) : ϕ1 is obtained from ϕ by isotopy and conjugation} .

h(ϕ0 ) =

.

Proof of the Implication Theorem 2.2 .⇒ Theorem 2.3 Let X be a connected closed Riemann surface with set of distinguished points .{z1 , . . . , zm }, .3g − 3 + m > 0. According to [2] (see also Sect. 1.7) there is a closed Riemann surface .Xˆ of genus at least two which is a simple branched covering of X so that the set of branch points projects onto the set .{z1 , . . . , zm }. The non-periodic absolutely extremal self-homeomorphism .ϕ0 lifts to a non-periodic absolutely extremal selfhomeomorphism .ϕˆ0 on .Xˆ with .K(ϕ0 ) = K(ϕˆ0 ). By Lemma 2.1 .

inf {h(ϕ1 ) : ϕ1 is a self-homeomorphism of Xwhich is isotopic to ϕ} = inf {h(ϕˆ 1 ) : ϕˆ1 is the lift of a self-homeomorphism of X which is isotopic to ϕ} ≥ inf {h(F1 ) : F1 is a self-homeomorphism of Xˆ which is isotopic to ϕ}. ˆ

Since .ϕˆ0 is absolutely extremal on .Xˆ and isotopic to .ϕ, ˆ the Fathi-Shub Theorem 2.2 implies that the last infimum equals .h(ϕˆ 0 ). Thus, the last infimum is attained on a homeomorphism of .Xˆ which is a lift. Hence the inequality between the second and the third infimum is an equality too. Since by Lemma 2.1 we have

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h(ϕ0 ) = h(ϕˆ 0 ), the first infimum equals .h(ϕ0 ). Also, .K(ϕ0 ) = K(ϕˆ0 ). It remains to use Corollary 1.1, and the invariance of the entropy under conjugation.  

.

2.2 Absolutely Extremal Homeomorphisms and the Trajectories of the Associated Quadratic Differentials We collect here some facts concerning trajectories of quadratic differentials that are related to non-periodic absolutely extremal self-homeomorphisms. For a more comprehensive exposition we refer the reader to the book of Strebel [78]. Let .φ be a quadratic differential associated to a non-periodic absolutely extremal self-homeomorphism .ϕ on a closed Riemann surface X possibly with set of distinguished points E. Recall that .φ is a meromorphic quadratic differential. It is analytic except, maybe, at some distinguished points where it has at worst first order poles. We need some information about the horizontal and vertical trajectories of .φ. For a horizontal trajectory .γ and a point p on it we call the union of .{p} with any of the connected components of .γ \ {p} a horizontal trajectory ray. If the set of limit points of a trajectory ray (that is the closure of the ray in the .φ-metric minus the ray itself) contains a regular point it contains a horizontal arc through this point. If the limit set of a trajectory ray consists of a single point this point must be critical and the trajectory ray is called critical. Proposition 2.1 For a quadratic differential .φ associated to a pseudo-Anosov selfhomeomorphism .ϕ of a closed Riemann surface (possibly with distinguished points) there are no leaf of the horizontal foliation and point on it with both trajectory rays critical (and no leaf of the vertical foliation and point on it with this property). Proof Suppose there is a horizontal leaf joining two critical points. There are only finitely many critical points and in a neighbourhood of each critical point there are only finitely many horizontal bisectrices. Since .ϕ maps the set of singular points of .φ onto itself (perhaps, permuting the points) and .ϕ maps horizontal trajectories to itself, a power .ϕ n of .ϕ maps the leaf onto itself. Adding the two endpoints to the leaf we obtain a compact arc which has finite .φ-length. But by (1.28) the n homeomorphism .ϕ n expands the length of the horizontal leaves by the factor .K 2 which is impossible.   Proposition 2.2 There is no closed leaf of the horizontal foliation and no closed leaf of the vertical foliation of a quadratic differential .φ associated to a pseudoAnosov homeomorphism .ϕ. Proof Suppose there is a closed horizontal leaf .γ . Choose an orientation for .γ . We claim that there is an annulus A around this leaf which is contained in the regular part of X and is the union of closed horizontal leaves. Indeed, each point x of the leaf .γ is regular. Hence, there is a neighbourhood V of each point x, that

2.2 Absolutely Extremal Homeomorphisms and the Trajectories of the. . .

67

is equipped with distinguished coordinates centered at the point, such that .V ∩ γ is the set of points in V that are real in distinguished coordinates, and the direction of the positive orientation of .γ corresponds to the positive direction of the real axis. We may assume that, perhaps after shrinking, the neighbourhood of each x is a square .R(x) with center x written in the distinguished coordinates centered at x as .{|Re z)| < ε(x), |Im z| < ε(x)}. Cover .γ by a finite number of such squares labeled by .Rj , j = 1, . . . , N, of side length .εj . We may do this so, that considering an infinite sequence of squares with .Rj +N = Rj for each j , any .Rj intersects exactly the squares .Rj ±1 . Let .ε be the minimum of the .εj . The squares .Rj cover an annulus A around .γ . By the choice of the distinguished coordinates on the .Rj each transition function is a translation by a real number. The arcs given in distinguished coordinates on .Rj by .(−εj , εj ) cover the closed def

horizontal curve .γ . Hence, for any real parameter t, .|t| < = min εj , the arcs .(−εj + it, εj + it) cover a closed horizontal curve. The union of these curves is an annulus A around .γ whose existence was claimed. Take any boundary point of the annulus A in the sense, that the point is a limit point of A in the .φ-metric and is not contained in A. If this point is a regular point then there is a horizontal arc contained in .∂A that contains the point. If all points of a boundary component of A are regular, then by the preceding arguments there is a bigger annulus that contains the previous annulus as well as the boundary component, and is the union of closed horizontal curves. Take the maximal annulus containing .γ that is the union of closed horizontal curves. Then each boundary component must contain a critical point and the complement in each boundary component of the critical points consists of horizontal leaves with both rays being critical. This is not possible by Proposition 2.1.   We call a trajectory ray divergent if its limit set contains more than one point. A trajectory ray with initial point p is called recurrent if p is contained in the limit set of the ray. We will consider trajectory rays parameterized so that the initial point has the smallest parameter and write .p2 > p1 for two points .p1 and .p2 on the ray if the parameter of .p1 is smaller than that of .p2 . The parametrization defines an orientation of the ray. Let as before X be a Riemann surface with quadratic differential .φ, and let R be an open rectangle (and let .R¯ be a closed rectangle, respectively) in the complex plane with sides parallel to the axes. A mapping .F : R → X is called an open .φrectangle (and a mapping .F : R¯ → X is called a closed .φ-rectangle, respectively,) if ¯ respectively) to .φ-horizontal arcs and takes .F takes horizontal segments in R (in .R, ¯ respectively) to .φ-vertical arcs, so that the Euclidean vertical segments in R (in .R, length of the horizontal (vertical arc, respectively) is equal to the .φ-length of its image. Recall that trajectories do not contain .φ-singular points, hence the images of .φrectangles are contained in the .φ-regular part of X. We call a .φ-rectangle embedded if the mapping .F is injective.

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2 The Entropy of Surface Homeomorphisms

Theorem 2.4 Every divergent trajectory ray is recurrent. More precisely, let .α + be a divergent horizontal trajectory ray with initial point .p0 . Let .β be any oriented open arc through .p0 contained in a vertical leaf, that is cut positively by .α + at .p0 . Then for each point .p1 on the ray .α + there exists a point .p2 > p1 on .α + , such that .α + cuts .β positively at .p2 . The .φ-length of the arc on .α + between .p1 and .p2 is bounded from below by a constant c that depends only on the quadratic differential .φ. Proof Let .β  ⊂ β be a vertical arc which has center .p0 and .φ-length .φ (β  ), such that the vertical arc with center .p0 and .φ-length .3φ (β  ) is contained in .β. We call a point .p ∈ β  exceptional, if the horizontal trajectory ray, which cuts .β  positively at p, is critical and does not cut .β  positively at a point greater than p. Notice that a point p is not exceptional, if the trajectory ray which cuts .β  positively at p is critical but cuts .β  positively at a point greater than p before running into the critical point. We claim that only finitely many points of .β  are exceptional. Indeed, each such point p and the respective critical ray are obtained as follows. Take a critical point, take any oriented trajectory emerging from this critical point, consider its part between the critical point and its first negative intersection point with .β  (if there is such), and invert orientation. There are only finitely many such rays, hence the claim. Recall that the ray .α + is not critical. We take a closed arc .β0 , that is contained in .β  , has midpoint .p0 , and does not contain exceptional points. We will prove now the following Claim There is a horizontal trajectory ray that emerges at a point .p ∈ β0 and intersects .β  positively. For positive numbers t we consider the closed rectangle .Rt ⊂ C with sides parallel to the axes, with horizontal Euclidean side length equal to t, vertical Euclidean side length .φ (β0 ), and with midpoint of the left side equal to the origin. For small t there is a closed .φ-rectangle .Fφ,t : Rt → X which maps the left side of .Rt onto .β0 . Such a .φ-rectangle can be defined for parameters t as long as each horizontal trajectory ray that emerges at a point .p ∈ β0 in positive direction does not run into a critical point before running along a closed arc of .φ-length t with initial point p. Assume by contradiction that the claim is not true. Since .β0 does not contain exceptional points, each trajectory ray that emerges at a point in .β0 in positive direction is non-critical (otherwise it would intersect .β  positively at a point after its initial point). Hence, a closed .φ-rectangle .Fφ,t : Rt → X which maps the left side of .Rt onto .β0 exists for all .t > 0. If .Fφ,t is injective on .Rt , the image .Fφ,t (Rt ) has .φ-area .t · φ (β0 ). Since the .φ-area .φ1 of X is finite, there exists a smallest number .t0 > 0 for which the .φ-rectangle .Fφ,t0 : Rt0 → X is not injective (see Fig. 2.1). Denote by .st0 the right side of .Rt0 . The mapping .Fφ,t0 is injective on .Rt0 \ st0 . Hence, .Fφ,t0 (st0 ) intersects the rest of the boundary of .Fφ,t0 (Rt0 ). We claim, that the vertical segment

2.2 Absolutely Extremal Homeomorphisms and the Trajectories of the. . .

69

Fig. 2.1 The smallest number .t0 with non-injective .φ-rectangle .Rt0 → Fφ,t0 (Rt0 )

Fφ,t0 (st0 ) cannot intersect the image of the open horizontal sides of .Rt0 . Suppose, in contrary, .Fφ,t0 (st0 ) intersects the image of an open horizontal side .sh of .Rt0 . Let .p be the intersection point. Then .p = Fφ,t0 (z ) = Fφ,t0 (z ), where .z is on an open horizontal side .sh of .Rt0 , and .z = t0 + iy is contained in .st0 . But then for all .t < t0 and close to .t0 the point .Fφ,t0 (t + iy) is also contained in the image .Fφ,t0 (sh ) of the open horizontal side .sh . But this means, that .Fφ,t0 is not injective on .Rt for .t < t0 and close to t, contrary to the definition of .t0 . Hence, .Fφ,t0 (st0 ) intersects the image of the left side of .Rt0 , i.e .Fφ,t0 (st0 ) intersects .β0 ⊂ β  . Since .Fφ,t0 is injective on .Rt0 \ st0 , the points on .Rt0 that are close to different vertical sides of the rectangle, but not on the vertical sides, are mapped to different (local) sides of the vertical arc containing .β0 . This means, that there is a horizontal trajectory that starts at a point .β0 and cuts .β0 ⊂ β  positively after running along a segment of length .t0 . This contradicts our assumption that the claim is not true. The claim is proved. Let .t0 be the infimum of positive numbers t for which a horizontal trajectory ray that emerges at a point of .β0 in positive direction intersects .β  positively after running along a segment of length t. Then, since .β0 does not contain exceptional points, the closed .φ-rectangle .Fφ,t0 : Rt0 → X exists. Moreover, the image of the right side .Fφ,t0 (st0 ) of .Rt0 intersects .β  . Then .Fφ,t0 (st0 ) is a vertical arc of .φ-length  .φ (β0 ) that intersects .β , hence, it is contained in .β. This means that each trajectory ray that emerges at a point of .β0 in positive direction, in particular .α+ , intersects .β positively after running along a segment of length .t0 . We proved that .α+ cuts .β positively after .p0 . Denote the intersection point by .p1 . The same argument applied to .p1 instead of .p0 shows that .α+ cuts .β positively at a point .p2 after .p1 . .

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2 The Entropy of Surface Homeomorphisms

There is a positive constant c such that the .φ-length of an arc on .α + between two distinct positive intersection points with .β exceeds c. Indeed, such an arc must leave a simply connected neighbourhood of the closure of .β, that depends only on the quadratic differential .φ. Repeating the argument for the already found intersection points with .β we obtain a sequence of positive intersection points .p0 < p1 < . . . < pn < . . . of .α+ with .β such that the .φ-length of the arc on .α + between .p0 and .pn tends to .∞. The theorem is proved.   Theorem 2.5 For each .φ-regular point .x0 ∈ X there exists a simple closed smooth curve in X that passes through .x0 , is contained in the .φ-regular part of X, and is transversal to the vertical foliation. The horizontal .φ-length .φ,h of this curve is positive. Proof By Proposition 2.1 one of the horizontal trajectory rays that start at .x0 is not critical and therefore recurrent. We consider this recurrent trajectory ray, denoted by .α + , oriented in the direction away from .x0 . Let .β be a small vertical arc through + intersects .β positively at .x . By Theorem 2.4 .α + .x0 that is oriented so that .α 0 intersects .β positively at a point .x1 after .x0 . Consider an open .Φ-rectangle V with small horizontal side length whose right open vertical side contains the closed arc ∗ .β ⊂ β that joins the points .x0 and .x1 by. Let .p1 be the last (before .x1 ) intersection point of .α+ with .∂V . We take the closed curve which is the union of the part of .α+ between .x0 and .p1 and a curve in V that joins .p1 with .x0 and is transversal to the vertical foliation (see Fig. 2.2). The curve can be chosen to be smooth and simple closed. Its horizontal length equals the length of the part of .α+ between .x0 and .x1 . The theorem is proved.   Fig. 2.2 A recurrent horizontal trajectory gives to a simple closed curve that is transversal to the horizontal foliation

β

V

• p1

x0

•

α+

• x1

2.3 The Entropy of Pseudo-Anosov Self-homeomorphisms of Surfaces with. . .

71

2.3 The Entropy of Pseudo-Anosov Self-homeomorphisms of Surfaces with Distinguished Points In this section we will prove Theorem 2.1. Our proof will be different from that in [25]. We begin with an especially simple example which is the entropy of a pseudoAnosov self-homeomorphism of a torus with a distinguished point. This example is considered in [1]. The arguments given here will be adapted later to give a proof of the general case of the theorem. Consider the standard torus .X = C(Z + iZ) with distinguished point .0(Z + iZ). Let .ϕ be a self-homeomorphism of X that lifts to a real linear self-homeomorphism of the complex plane which maps the integer lattice onto itself. Then .ϕ fixes the distinguished point .0(Z  + iZ) of X. The lift .ϕ˜ of .ϕ to ab with integer entries and the universal covering .C corresponds to a matrix . cd determinant equal to one, i.e. ϕ˜

.

     ξ ab ξ = η cd η

where .ζ = ξ + iη ∈ C are complex coordinates on .X˜ = C, and .ad − bc = 1. Suppose the eigenvalues of the matrix are positive and not equal to each other. Denote the bigger eigenvalue by .λ and the corresponding eigenvector by .v+ . The other eigenvalue equals . λ1 . Denote the corresponding eigenvector by .v− . Then .ϕ(xv ˜ + +yv− ) = λxv+ +λ−1 yv− . Consider the conformal structure .w : X → w(X) ˜ that takes the point .xv+ + yv− to the on X that lifts to a mapping .w˜ : X˜ → w( ˜ X) point .z = x + iy. Then w˜ ◦ ϕ˜ ◦ w˜ −1 (x + iy) = λx + iλ−1 y .

.

(2.20)

def

Consider the self-homeomorphism .ϕ0 = w◦ϕ◦w −1 of .w(X) with distinguished point .0w(Z ˜ + iZ). Take local flat coordinates z vanishing at a point .z0 and local flat coordinates .ζ vanishing at the point .ϕ0 (z0 ). The mapping .ϕ0 in these local coordinates becomes 1 ζ = w ◦ ϕ ◦ w −1 (x + iy) = λx + i y . λ

.

(2.21)

The Beltrami differential of the mapping is .

(λ − λ−1 ) d z¯ (λ − λ−1 ) d z¯ dz ∂ϕ0 (z) = = . ∂ϕ0 (z) (λ + λ−1 ) dz (λ + λ−1 ) (dz)2

(2.22)

The equation shows the following. Let the quadratic differential .φ be given in local d z¯ coordinates by .(dz)2 . The Beltrami differential of the mapping .ϕ0 equals .k |φ| φ = k dz

72

with .k =

2 The Entropy of Surface Homeomorphisms (λ−λ−1 ) . (λ+λ−1 )

Hence, the quadratic differential of .ϕ0 is the globally defined

quadratic differential .(dz)2 on .w(X), and .ϕ0 has quasiconformal dilatation .K = 1+k 2 1−k = λ . The quadratic differential of the inverse mapping .ξ + iη → λξ + iλη has Beltrami differential .

(λ−1 − λ)d z¯ dz . (λ−1 + λ)(dz)2

Hence, the quadratic differential of the inverse mapping is .−(dz)2 , and therefore the terminal quadratic differential of .ϕ0 is equal to its initial quadratic differential .φ. We proved that .ϕ0 is absolutely extremal on .w(X) with distinguished point and its quadratic differential equals .φ = (dz)2 . Notice that the quadratic differential .φ does not have singular points, the distinguished point is also a regular point. Such a situation is exceptional and especially easy to treat. The following proposition on the entropy of the mapping holds. Proposition 2.3 .h(ϕ) = h(ϕ0 ) = log λ . ∼ w( ˜ → Proof Choose a small positive number .ε0 so that the projection .C = ˜ X) w(X) is injective on open squares of side length .2λ 0 (in coordinates .x + iy on .C). Take .ε < ε0 . Choose the open cover .Aε of .w(X) that consists of projections to .w(X) of all open squares in .C of side length . with sides parallel to the axes. The minimal cardinality .N (Aε ) of a subcover of .Aε does not exceed the minimal cardinality of a cover of the closure of a fundamental domain on .X˜ by squares of side length .ε. This number does not exceed .c+ ε−2 for a positive number .c+ . On the other hand .N (Aε ) is bounded from below by the maximal number of disjoint squares of side length .ε that can be put in the interior of a fundamental domain on ˜ This number is bounded from below by .c− ε−2 for a positive constant .c− . .X. ˜ The cover .ϕ −1 (Aε ) consists of the projection to .w(X) of all rectangles in .w( ˜ X) 0

with sides parallel to the axes of horizontal side length .λ−1 and vertical side length ˜ intersects an .ε-square in .w( ˜ then by the choice of .λε. If such a rectangle in .w( ˜ X) ˜ X), .ε the projection is injective on the union of the square and the rectangle. Hence, the ˜ cover .Aε ∨ ϕ0−1 (Aε ) consists of the projection to .w(X) of all rectangles in .w( ˜ X) −1 with sides parallel to the axes of horizontal side length at most .λ ε and vertical side length at most .ε. By induction .Aε ∨ ϕ0−1 (Aε ) ∨ . . . ∨ ϕ0−n (Aε ) consists of projections to .w(X) ˜ of horizontal side length at most .λ−n ε and vertical side of all rectangles in .w( ˜ X) length at most .ε. Indeed, if this is proved for a natural number n it is obtained for .n + 1 as follows. The cover .ϕ0−1 (Aε ) ∨ . . . ∨ ϕ0−n−1 (Aε ) is the image under .(ϕ0 )−1 of the cover −n .Aε ∨ . . . ∨ ϕ 0 (Aε ). Hence, it consists of rectangles of horizontal side length at most .λ−n−1 ε and vertical side length at most .λε. The cover .Aε ∨ ϕ0−1 (Aε ) ∨ . . . ∨ ϕ0−n−1 (Aε ) consists of intersections of such rectangles with .ε-squares.

2.3 The Entropy of Pseudo-Anosov Self-homeomorphisms of Surfaces with. . .

73

For each n a lift of each set of a minimal subcover of .Aε can be covered by .λn +1 rectangles of horizontal side length .λ−n ε and vertical side length .ε. On the other hand at least .λn rectangles of horizontal side length at most .λ−n ε and vertical side length at most .ε are needed to cover a lift a single .ε-square of a minimal subcover of .Aε . We obtain λn ≤ N (Aε ∨ ϕ0−1 (Aε ) ∨ . . . ∨ ϕ0−n (Aε )) ≤ c+ ε−2 (λn + 1) .

.

(2.23)

Hence, h(ϕ0 , Aε ) = lim sup

.

n→∞

1 log N (Aε ∨ ϕ0−1 (Aε ) ∨ . . . ∨ ϕ0−n (Aε )) = log λ . n

For any decreasing sequence of small numbers .εn the sequence of coverings .Aεn is refining. Indeed, it is clear that .Aε1 ≺ Aε2 for .ε1 < ε2 . Any open cover .B has a finite subcover. For small enough .ε > 0 each set in .Aε is contained in a set of the finite subcover, hence, .Aε ≺ B. Hence, h(ϕ0 ) = log λ .

.

The proposition is proved.

 

We will now prove Theorem 2.1 in the general case. In the rest of the section X will be a closed connected Riemann surface of genus g with a set .Em of .m ≥ 0 distinguished points, .3g − 3 + m > 0. Let .ϕ0 be an absolutely extremal selfhomeomorphism of X with set of distinguished points .Em . Suppose .ϕ0 is not conformal and has quasiconformal dilatation .K(ϕ0 ) = λ2 > 1. Then there exists a quadratic differential .φ on X with the following properties. .ϕ0 maps the set of critical points of .φ to itself, it maps horizontal leaves of .φ to horizontal leaves and vertical leaves to vertical leaves. Let .ε > 0 be a small number. We prepare the definition of an open cover .Aε of X. We will call an open .φ-rectangle with both side lengths equal to .ε an open .ε-square. An open .φ-rectangle with length of the horizontal sides equal to .ε1 and length of the vertical sides equal to .ε2 will be called an open .(ε1 , ε2 )-rectangle. An .ε-star at a singular point of .φ is defined as follows. Consider a simply connected neighbourhood U of a singular point of .φ and distinguished coordinates 2 a 2 z on it in which .φ has the form .φ(z) = ( a+2 2 ) z (dz) where .a is the order of the singular point. Since there are only finitely many singular points, we may assume that in these coordinates U has the form .{|z| < c} for a constant c that does not i depend on the singular point. The horizontal bisectrices .{r exp( 2πj a+2 ), 0 < r < c}, .j = 0, . . . , a + 1, are segments of horizontal trajectories, the vertical bisectrices 2π(j + 1 )i

{r exp( a+22 ), 0 < r < c}, .j = 0, . . . , a + 1, are segments of vertical trajectories. Consider the “half-sectors” between a horizontal bisectrix and a nearest

.

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2 The Entropy of Surface Homeomorphisms

Fig. 2.3 An .ε-star and a .(2ε, 2ε )-star at a .φ-singular point

vertical bisectrix. There are .2(a + 2) such “half-sectors”. They have the form 2πj i +1)i sj = {r exp(xi) : x ∈ ( 2(a+2) , 2π(j 2(a+2) ), 0 < r < c} , j = 0, . . . , 2(a + 2) − 1.

.

def

Denote by .shj = Int(s2j ∪ s2j +1 ) the sectors between two horizontal bisectrices. a+2

Each half-sector .sj is taken by a branch .gj of the mapping .z → z 2 conformally onto a quarter-disc with center 0 in one of the quarters .{±Imz > 0, ±Rez > 0} √ a+2 of the complex plane. For . 2ε < c 2 we intersect the quarter disc with an open square in the complex plane with center 0 and side length .2ε. We obtain an open square .Qj (ε) of side length .ε. The union of the preimages .gj−1 (Qj (ε)), j = 1, . . . , 2(a + 2) − 1, with the .2(a + 2) open segments of .φ-length .ε emerging from the singular point and contained in a bisectrix, is a punctured neighbourhood of the singular point. Its union with the singular point is the required .ε-star at the singular point (see Fig. 2.3). The .ε-star is bounded by .a + 2 horizontal segments of .φ-length .2ε and .a + 2 vertical segments of .φ-length .2ε. The distance in the .φ-metric from the center of the .ε-star to the horizontal and vertical sides in the boundary of the star is equal to .ε. Hence, the .φ-distance from the center of an .ε-star to its boundary equals .ε. The √ .ε-star is contained in a . 2ε-neighbourhood (in the .φ-metric) of its center. The definition of a star at a distinguished point implies immediately that for any a+2 positive .ε < √1 c 2 any .ε-star punctured at its singular point of order a can be 2 covered by .4(a + 2) .ε-squares contained in the .ε-star. Indeed, the part of the star contained in the half-sectors .sj is covered by .2(a + 2) .ε-squares, and each bisectrix is covered by an open .ε-square. The latter follows from the fact that e.g. each horizontal bisectrix is contained in the sector between two neighbouring vertical a+2 bisectrices, and a branch of the mapping .z → z 2 takes this sector conformally onto a half-disc with center 0 in the complex plane.

2.3 The Entropy of Pseudo-Anosov Self-homeomorphisms of Surfaces with. . .

75

a+2 An .(ε1 , ε2 )-star at a singular point is defined similarly. Let . ε12 + ε22 < c 2 . Consider again the half-sectors .sj at the given singular point and the respective a+2 branches .gj of the mapping .z → z 2 on the half-sectors. Intersect each image .gj (sj ) (which is a quarter disc at 0) with an open rectangle with center 0, horizontal side length .2ε1 , and vertical side length .2ε2 . The intersection is a rectangle .Qj (ε1 , ε2 ) of horizontal side length .ε1 , and vertical side length .ε2 . We take the preimage .gj−1 (Qj (ε1 , ε2 )). Consider the closure of the union of the obtained preimages. Its interior is the .φ-rectangle of horizontal side length .ε1 and vertical side length .ε2 at the given singular point. (See Fig. 2.3.) √ a+2 Lemma 2.2 Let .ε1 and .ε2 be positive numbers such that . 2(ε1 +ε2 ) < c 2 for the number c introduced above. If a .φ-rectangle with both side length’s not exceeding .ε2 intersects an .ε1 -star .Sε1 , then the .(ε1 + ε2 )-star .Sε1 +ε2 at the singular point of .Sε1 exists, and the rectangle is contained in the intersection of .Sε1 +ε2 with a sector between two consecutive horizontal bisectrices (or two consecutive vertical bisectrices). Proof Since the distance√(in the .φ-metric) of each point of the .ε1 -star from the center does not exceed . 2ε1 and the√diameter of the rectangle does not exceed √ . 2ε2 , the rectangle is contained in the . 2(ε1 +ε2 )-neighbourhood (in distinguished coordinates) of the distinguished point. Suppose the rectangle intersects a vertical bisectrix at a point x. Let .shj be the sector between two neighbouring horizontal bisectrices that contains the vertical bisectrix. The rectangle can be obtained as follows. Take the maximal open vertical segment s through x that is contained in it. It has .φ-length at most .ε2 . For each point in s we take the maximal open horizontal segment through this point that is contained in the rectangle. Each has .φ-length at most .ε2 . Notice that the open rectangle does not contain singular points. Take the union of all these segments. The points on these segments are reachable starting from the singular point along a piece of a vertical trajectory of length smaller than .ε1 + ε2 followed by a piece of a horizontal trajectory of length at most .ε2 . Hence, the rectangle is contained in the .(ε1 +ε2 )-star. If the rectangle intersects a horizontal bisectrix or does not intersect any bisectrix, the proof is the same.   Lemma 2.3 There exists a positive number .δ0 such that there exist disjoint .δ0 -stars at the singular points. Moreover, there exists a positive number .ε0 < δ0 such that for each .ε < ε0 and for each point .x ∈ X which is not in an .ε-star at a singular point there exists an .ε-square with center x. Recall that a square in X is required to be contained in the .φ-regular part of X. Proof It is clear, that for some .δ0 > 0 there exist disjoint .δ0 -stars at the singular points. For each point x in the complement .Xδ0 of all . 12 δ0 -stars there is an .ε(x)square with center at this point for some positive number .ε(x) depending on x. Since .Xδ0 is compact it can be covered by a finite number of . 12 ε(xj )-squares, .j = 1, . . . , N . Let .ε0 be the minimum of the numbers . 12 ε(xj ). Then for each .x ∈ Xδ0 there is an .ε0 -square with center at this point.

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2 The Entropy of Surface Homeomorphisms

Let x be in a .δ0 -star at a singular point but not in an .ε-star. Suppose the distance of the point x to a vertical bisectrix is equal to its distance to the union of all bisectrices. Let .shj be the sector of the .δ0 -star, that is bounded by two consecutive horizontal bisectrices and contains the latter vertical bisectrix. Let .gjh be a branch of a+2

the mapping .z → z 2 that maps .shj to the upper or lower half-plane. It defines flat coordinates on .shj . The point .gjh (x) is contained in the intersection of the upper or lower half-plane with a square of side length .2δ0 and center 0, but not in the square of side length .2ε and center 0. Moreover, the distance of .gjh (x) to the imaginary axis does not exceed its distance to the real axis. Hence, the intersection of the half-plane with the square of side length .2δ0 and center 0 contains a square of side length .ε with center .gjh (x). The existence of an .ε-square with center x is proved.   We prove now the following proposition which is one of the estimates of Theorem 2.1. Proposition 2.4 Let X be a (closed connected) Riemann surface of genus g with a set .Em of .m ≥ 0 distinguished points, .3g − 3 + m > 0. Let .ϕ0 ∈ Hom(X; ∅, Em ) be a non-periodic absolutely extremal self-homeomorphism of X with set of distinguished points .Em . Then h(ϕ0 ) ≤

.

1 log K(ϕ0 ) 2 a+2

Proof Let .ε0 be as in Lemma 2.3. Take a positive number .ε < min{ ελ0 , c√2 }. Let 4 2λ .Aε be the set which consists of all .ε-squares and all .ε-stars at distinguished points. Then .Aε is an open cover of X. The cover .ϕ0−1 (Aε ) consists of all .(λ−1 ε, λε)-stars at distinguished points and all −1 ε, λε)-rectangles. Indeed, the image under .ϕ −1 of an .ε-star at a singular point .(λ 0 of .φ is a .(λ−1 ε, λε)-star at some singular point of .φ. Vice versa, take a .(λ−1 ε, λε)star .S. Its image .ϕ0 (S) under .ϕ0 is an .ε-star and .S is the image of the .ε-star .ϕ0 (S) under .ϕ0−1 . The argument for rectangles is the same. Each .(λ−1 ε, ε)-rectangle is the intersection of an .ε-square with a .(λ−1 ε, λε)rectangle. Indeed, if the midpoint of the .(λ−1 ε, ε)-rectangle is outside the .λε-stars, then by Lemma 2.3 there exists a .λε-square with center at this point. This square contains an .ε-square and a .(λ−1 ε, λε)-rectangle centered at this point. The intersection of the latter two sets is the required .(λ−1 ε, ε)-rectangle. Suppose the center x of a .(λ−1 ε, ε)-rectangle R is in a .λε-star .Sλε . Lemma 2.2 √ √ a+2 with .ε1 = λε, .ε2 = ε, and . 2(ε1 + ε2 ) = 2(λ + 1)ε < 12 c 2 implies that the rectangle R is contained in a sector of the .(λ+1)ε-star .S(λ+1)ε between consecutive horizontal bisectrices, or between consecutive vertical bisectrices. We assume that R is contained in a sector .shj ∩ S(λ+1)ε of the .(λ + 1)ε-star between two consecutive horizontal bisectrices. The remaining case is treated similarly and is even slightly simpler.

2.3 The Entropy of Pseudo-Anosov Self-homeomorphisms of Surfaces with. . .

77

By our choice of .ε the .2(λ + 1)ε-star .S2(λ+1)ε at the given singular point exists. The image .gjh (shj ∩ S2(λ+1)ε ) is the intersection of the upper or lower half-plane with a Euclidean square of side length .4(λ + 1)ε. The image .gjh (R) is a Euclidean rectangle of horizontal side length .λ−1 ε and vertical side length .ε contained in the Euclidean rectangle .gjh (shj ∩S2(λ+1)ε ) of horizontal side length .4(λ+1)ε and vertical side length .2(λ + 1)ε. Hence, .gjh (R) can be written as intersection of a Euclidean rectangle of side lengths .λ−1 ε and .λε, and a Euclidean .ε-square, both contained in the Euclidean rectangle .gjh (shj ∩ S2(λ+1)ε ). Hence, R can be written as the desired intersection. We showed that each .(λ−1 ε, ε)-rectangle can be written as .A0 ∩ ϕ0−1 (A1 ) ∈ Aε ∨ ϕ0−1 (Aε ) for .ε-squares .A0 and .A1 . Any .(λ−n ε, ε)-rectangle can be written as .A0 ∩ ϕ0−1 (A1 ) ∩ . . . ∩ ϕ0−n (An ) ∈ Aε ∨ ϕ0−1 (Aε ) ∨ . . . ∨ ϕ0−n (Aε ) for .ε-squares .Aj , .j = 0, 1, . . . , n. The proof goes by induction. Assume the claim is true for a natural number n. We prove that it is true for .n + 1. Suppose first the midpoint of the .(λ−n−1 ε, ε)-rectangle A is outside the .λε-stars. Then by Lemma 2.3 the .λε-square around this point exists. Hence, there is an .ε-square .A and a .(λ−n−1 ε, λε)-rectangle .A with the same center so that the intersection .A ∩ A is equal to A. The image .ϕ0 (A ) is a .(λ−n ε, ε)-rectangle which is an element of .Aε ∨ ϕ0−1 (Aε ) ∨ . . . ∨ ϕ0−n (Aε ) by the claim for the number n. Hence, its preimage .A under .ϕ0 is in .ϕ0−1 (Aε ) ∨ ϕ0−2 (Aε ) ∨ . . . ∨ ϕ0−1−n (Aε ). Since .A = A ∩ A the claim is proved in this case. In the remaining case the midpoint of the .(λ−n−1 ε, ε)-rectangle A is inside a .λεstar. The statement was proved for .n = 1. For .n > 1 we may proceed by induction similarly as above. Each .(λ−n ε, ε)-star can be written as .A0 ∩ ϕ0−1 (A1 ) ∩ . . . ∩ ϕ0−n (An ) ∈ Aε ∨ −1 ϕ0 (Aε ) ∨ . . . ∨ ϕ0−n (Aε ) for .ε-stars .Aj , .j = 0, . . . , n. This is true for .n = 1 since each .(λ−1 ε, ε)-star is the intersection of a .(λ−1 ε, λε)-star with an .ε-star. We suppose that the claim is true for n, and prove it for .n + 1. Each .(λ−n−1 ε, ε)-star is the intersection of an .ε-star with a .(λ−n−1 ε, λε)-star. The latter is a preimage under .ϕ0 of a .(λ−n ε, ε)-star, hence, by the induction hypothesis it can be written as .ϕ0−1 (A1 ) ∩ . . . ∩ ϕ0−n−1 (An+1 ) ∈ ϕ0−1 (Aε ) ∨ . . . ∨ ϕ0−n−1 (Aε ). This proves the claim. Cover X by a finite subcover of .Aε . We may assume that this subcover consists of all .ε-stars and a finite number of .ε-squares that cover the complement of the singular points in X. Let .C1 be the number of squares in the subcover, and .C2 the number of stars. Each square can be covered by no more than .λn + 1 sets that are .(λ−n ε, ε)-rectangles. These rectangles are in .Aε ∨ ϕ0−1 (Aε ) ∨ . . . ∨ ϕ0−n (Aε ). Each singular point can be covered by a .(λ−n ε, ε)-star which is also contained in −1 −n n .Aε ∨ϕ 0 (Aε )∨. . .∨ϕ0 (Aε ). We obtain a subcover of cardinality at most .C1 (λ + n 1) + C2 ≤ C(λ + 1). Hence, for any .ε < ε0 h(ϕ0 , Aε ) ≤ lim sup

.

1 log(C(λn + 1)) = log λ . n

(2.24)

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2 The Entropy of Surface Homeomorphisms

Take a sequence of .ε’s decreasing to 0. We get (2.24) for a refining sequence of coverings. Hence, h(ϕ0 ) ≤ log λ =

.

1 log K(ϕ0 ) . 2  

The proposition is proved. The following proposition states the opposite inequality.

Proposition 2.5 Let X be a (closed connected) Riemann surface of genus g with a set .Em of .m ≥ 0 distinguished points, .3g − 3 + m > 0. Let .ϕ0 ∈ Hom(X; ∅, Em ) be a non-periodic absolutely extremal self-homeomorphism of X with set of distinguished points .Em . Then h(ϕ0 ) ≥

.

a+2

1 log K(ϕ0 ) 2

ε0 Proof Let .ε < min{ 3λ+1 , c√2 }. We consider the cover .Bε which consist of all 4 2λ ε .ε-squares and all . -stars at singular points. We prove first that each non-empty set 2 of the cover .Bε ∨ ϕ0−1 (Bε ) is either a .(λ−1 2ε , 2ε )-star at a singular point of .φ or an −1 ε and .ε ≤ ε. .(ε1 , ε2 )-rectangle with .ε1 ≤ λ 2 We have to consider non-empty intersections .A1 ∩ ϕ0−1 (A2 ) for two sets in .Bε . Suppose both, .A1 and .A2 are .ε-squares, hence, .ϕ0−1 (A2 ) is an .(λ−1 ε, λε)-rectangle. If the center .x0 of .A1 is not in a .2λε-star, then by Lemma 2.3 there exists a .2λεsquare with center at the same point .x0 . .ϕ0−1 (A2 ) is a .(λ−1 ε, λε)-rectangle and intersects .A1 , hence .A1 ∩ ϕ0−1 (A2 ) is contained in the .2λε-square centered .x0 . It is now clear that .A1 ∩ ϕ0−1 (A2 ) is an .(ε1 , ε2 )-rectangle with .ε1 ≤ λ−1 ε and .ε2 ≤ ε. Suppose .A1 and .A2 are .ε-rectangles and the centre of .A1 is in a .2λε-star. Then by Lemma 2.2 .A1 is contained in the .(2λ + 1)ε-star and .ϕ0−1 (A2 ) intersects the .(2λ + 1)ε-star. Hence, by the same Lemma 2.2 .ϕ0−1 (A2 ) is contained in the −1 .(3λ + 1)ε-star. Each of the .A1 and .ϕ 0 (A2 ) is contained in a sector between two consecutive horizontal or between two consecutive vertical bisectrices. Moreover, the two sectors intersect at least along a half-sector. The union of the two sectors, equipped with flat coordinates, is an open subset of a half-plane or a three-quarter plane with Euclidean coordinates. It is now clear that the intersection of .A1 and −1 −1 .ϕ 0 (A2 ) is an .(ε1 , ε2 )-rectangle with .ε1 ≤ λ ε and .ε2 ≤ ε. ε If .A1 is an . 2 -star, further .A2 is an .ε-square, and .A1 ∩ ϕ0−1 (A2 ) is non empty, then by Lemma 2.2 .A1 ∪ ϕ0−1 (A2 ) is contained in the .( 2ε + λε)-star. Moreover, −1 .ϕ 0 (A2 ) is contained in a sector of this star between two consecutive horizontal or two consecutive vertical bisectrices. It is now clear that the intersection of .A1 and −1 −1 .ϕ 0 (A2 ) is an .(ε1 , ε2 )-rectangle with .ε1 ≤ λ ε and .ε2 ≤ ε. ε Let .A1 be an .ε-square and let .A2 be an . 2 -star. Then .ϕ0−1 (A2 ) is a .(λ−1 2ε , λ 2ε )-star. If the intersection .A1 ∩ ϕ0−1 (A2 ) is non-empty, then by Lemma 2.2 .A1 is contained

2.3 The Entropy of Pseudo-Anosov Self-homeomorphisms of Surfaces with. . .

79

in a sector of a .( λ2 + 1)ε-star between two consecutive separatrices of the same kind (both horizontal or both vertical) with the same center. In flat coordinates on this set we obtain the intersection of an .ε-square with an .(ε1 , ε2 )-rectangle with .ε1 ≤ λ−1 2ε and .ε2 ≤ λ 2ε . The claim is obtained also in this case. If both, .A1 and .A2 are stars they have the same center if the intersection is not empty, and the claim is clear. We prove now by induction that each non-empty set of the form A0 ∩ ϕ0−1 (A1 ) ∩ . . . ∩ ϕ0−n (An )

.

(2.25)

(with .Aj , j = 0, . . . n, being sets in .Bε ), is either an .(ε1 , ε2 )-rectangle with .ε1 ≤ λ−n ε and .ε2 ≤ ε or a .(λ−n 2ε , 2ε )-star at a singular point of .φ. Suppose this is true for a natural number n. Prove it for .n+1. We have to consider intersections .A0 ∩ϕ0−1 (B) with .A0 ∈ Bε and .B ∈ Bε ∨ ϕ0−1 (Bε ) ∨ . . . ∨ ϕ0−n (Bε ). Let .A0 be an .ε-square. We have to intersect it either with an .(ε1 , ε2 )-rectangle with .ε1 ≤ λ−n−1 ε and .ε2 ≤ λε, or with a .(λ−n−1 2ε , λ 2ε )-star at singular a point of .φ. In the same way as before we obtain that the intersection is a .(ε1 , ε2 )-rectangle with .ε1 ≤ λ−n−1 ε and .ε2 ≤ ε. Let .A0 be an . 2ε -star at a singular point of .φ. We have to intersect it either with a ε ε −n−1 .(λ 2 , λ 2 )-star at the same singular point, or with an .(ε1 , ε2 )-rectangle with .ε1 ≤ λ−n−1 ε and .ε2 ≤ λε. In the first case we obtain a .(λ−n−1 2ε , 2ε )-star. In the second case the rectangle is contained in a sector between two consecutive bisectrices of the same kind of a larger star and we obtain in flat coordinates the intersection of the .(ε1 , ε2 )-rectangle with an .(ε1 , ε2 )-rectangle, where .ε1 and .ε2 do not exceed .ε. The smaller horizontal side length is .λ−n−1 ε, the smaller vertical side length is .ε. We proved the statement. The proof of the proposition √ is completed as follows. Consider an .ε-square Q with .φ-distance bigger than . 2ε from any singular point. Then for any non-negative integer n no star in the cover .Bε ∨ϕ0−1 (Bε )∨. . .∨ϕ0−n (Bε ) intersects Q. Fix a natural number n. We want to estimate .N (Bε ∨ ϕ0−1 (Bε ) ∨ . . . ∨ ϕ0−n (Bε )) from below. The sets in the cover of X that intersect Q are .(ε1 , ε2 )-rectangles with .ε1 ≤ λ−n ε and n .ε2 ≤ ε. To cover Q at least .λ sets are needed. Hence, h(ϕ0 , B ) ≥ lim sup

.

n→∞

≥ lim sup n→∞

1 log N (Bε ∨ ϕ0−1 (Bε ) ∨ . . . ∨ ϕ0−n (Bε )) n 1 log λn = λ . n

(2.26)

Take a sequence of .ε’s decreasing to 0 we obtain a refining sequence of coverings. We proved that .h(ϕ0 ) ≥ λ.   The first equality in Theorem 2.1 is proved. The second one is Corollary 1.1. Theorem 2.1 is proved.  

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2 The Entropy of Surface Homeomorphisms

2.4 Pseudo-Anosov Self-homeomorphisms of Closed Surfaces Are Entropy Minimizing In this section we will prove Theorem 2.2 along the lines of [25]. Let X be a closed surface of genus at least two and let .x0 be a chosen base point of X. Let f be an arbitrary self-homeomorphism of X. If .f (x0 ) = x0 the mapping f induces a  . Indeed, for each loop .γ with base point .x0 the homomorphism f∗ : π1 (X, x0 )⊃ image .f ◦ γ is a loop with base point .x0 , whose class in .π1 (X, x0 ) depends only on the class of .γ in .π1 (X, x0 ). In the general situation the plan is the following. We will show that the mapping ˜ X) of f induces a conjugacy class of homomorphisms .f# from the group .Deck(X, covering transformations (identified with the fundamental group of X) into itself. We will define a quantity .Γf# that depends only on the conjugacy class .f# and provides a lower bound for the entropy of f . Moreover, the quantity is more geometric than the entropy. In the case when f is a non-periodic absolutely extremal homeomorphism the quantity can be estimated from below by . 12 log K(f ). Before coming to the fundamental group we consider any group G. Let .G = {g1 , . . . , gr } be any subset of G that generates G. For an element .g ∈ G we denote by .LG (g) the minimal length of a word in the .gj ’s and .gj−1 ’s that represents g. (The length of a word in the .gj is the sum of the absolute values of the powers of exponents of the .gj that appear in the word.) For any other subset .G  = {g1 , . . . , gr  } of G that generates G the inequality .LG  (g) ≤ (max LG  (gj ))LG (g) holds. For a group homomorphism .A : G → G we put ΓA = sup lim sup

.

g ∈G n→∞

1 log LG (An g) n

(2.27)

The following lemma is a straightforward consequence of the definition. Lemma 2.4 ΓA = max lim sup

.

gj ∈G

n→∞

1 log LG (An gj ) . n

Hence, .ΓA is finite. For an element .g ∈ G we consider the group homomorphism .gAg −1 : G → G defined by .gAg −1 (x) = gA(x)g −1 . For convenience of the reader we formulate the following technical lemma from [25] (see Lemma 10.6 in [25]). The proof is a straightforward calculation and is left to the reader. ∞ Lemma 2.5 Let .(an )∞ n=1 and .(bn )n=1 be two sequences of positive numbers. Then

 (1) .lim sup n1 log(an + bn ) = max lim sup n1 log(an ) , lim sup n1 log(bn );

 (2) .lim sup n1 log(an ) ≤ lim sup n1 log(a1 + . . . + an ) ≤ max 0, lim sup n1 log(an ) .

2.4 Pseudo-Anosov Self-homeomorphisms of Closed Surfaces Are Entropy. . .

81

The following proposition is proved in [25] (see Proposition 10.5 of [25] and its proof). Proposition 2.6 The equality ΓA = Γg Ag −1

.

holds. Proof By the definition we have .(gAg −1 )(x) = gA(x)g −1 , g ∈ G. Hence, (gAg −1 )2 (x) = gA(gA(x)g −1 )g −1 = gA(g)A2 (x)A(g −1 )g −1 .

.

By induction (gAg −1 )n (x) = gA(g) . . . An−1 (g)An (x)A−n+1 (g) . . . A(g −1 )g −1 .

.

Hence, since .LG (y) = LG (y −1 ) for each .y ∈ G we obtain 1 log(LG (g . . . An−1 (g)An (x)A(g −n+1 ) . . . g −1 )) n x∈G 

  1 ≤sup lim sup log 2 LG (g) + . . . + LG (An−1 (g)) + LG (An (x) . n x∈G (2.28)

Γg Ag −1 =sup lim sup

.

If .An0 (g) is the identity for some .n0 , then it follows from Statement (1) of Lemma 2.5, that the right hand side of (2.28) does not exceed .

lim sup

1 log LG (An (x)). n

In the remaining case .LG (An (g)) ≥ 1 for each n, and by Statements (1) and (2) of Lemma 2.5 .

  1 log LG (gAg −1 )(x) n   1 1 ≤ max lim sup log LG (An (g)), lim sup log LG (An (x)) . n n

lim sup

Taking the maximum over g running along a finite set .gj of elements that generates G, we obtain .Γg Ag −1 ≤ ΓA . The opposite inequality follows by symmetry.   Let X be a closed surface and .x0 ∈ X a base point. Denote by .P : X˜ → X the universal covering of X. Recall that each point of .X˜ that projects to a point .x ∈ X can be identified with an element .ex0 ,x ∈ π1 (X; x0 , x) (see e.g. [27] and Sect. 1.2). Here .π1 (X; x0 , x) denotes the set of homotopy classes of arcs in X with initial point

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2 The Entropy of Surface Homeomorphisms

x0 and terminal point x. We write .{ex0 ,x } if we mean the point of .X˜ associated to the element .ex0 ,x ∈ π1 (X; x0 , x). The projection assigns to each point .{ex0 ,x } the point x. Let .f : X → X be a continuous mapping. For each pair

of points .x1 , .x2 in X the mapping f induces a mapping .f∗ : π1 (X; x1 , x) → π1 X; f (x1 ), f (x) . Namely, .f∗ (ex1 ,x ) is the class represented by .f ◦γx1 ,x for a curve .γx1 ,x that represents .{ex1 ,x }. Let .x0 ∈ X be the base point of X. A lift of f to .X˜ is a continuous mapping .f˜ such that .f ◦ P = P ◦ f˜. A lift of f to .X˜ can be described as follows (see also [27]). Choose and fix a homotopy class of curves .ex00 ,f (x0 ) ∈ π1 (X; x0 , f (x0 )) with fixed endpoints .x0 and .f (x0 ). Put .

f˜({ex0 ,x }) = {ex00 ,f (x0 ) f∗ (ex0 ,x )}, {ex0 ,x } ∈ X˜ .

.

(2.29)

Here .ex00 ,f (x0 ) f (ex0 ,x ) is the homotopy class of arcs represented by first traveling along an arc representing .ex00 ,f (x0 ) and then along an arc representing the image .f∗ (ex0 ,x ) of .ex0 ,x under the induced mapping .f∗ : π1 (X; x0 , x) → π1 (X; f (x0 ), f (x)). All other lifts are obtained by replacing .ex00 ,f (x0 ) by another element of .π1 (X; x0 , f (x0 )). Equivalently, for two lifts .f˜1 and .f˜2 of f there exists a covering transformation .θ ∈ π1 (X, x0 ) such that .f˜1 = θ ◦ f˜2 . Indeed, if .P ◦ f˜1 = P ◦ f˜2 = f ◦ P, then for each .x ˜ ∈ X˜ the points .f˜1 (x) ˜ and .f˜2 (x) ˜ differ by a covering transformation. The covering transformations depend continuously on .x, ˜ hence, there is a single covering transformation .θ such that .f˜1 = θ ◦ f˜2 . The following lemma gives a lower bound for the entropy of a mapping f in terms of metric properties of the lift .f˜ of f . Lemma 2.6 Let X be a closed surface with metric d, .X˜ its universal covering with the lifted metric also denoted by d, .f : X → X a self-homeomorphism of X and .f˜ ˜ Then for any points .x, a lift of f to .X. ˜ y˜ ∈ X˜ .

1 lim sup d(f˜n (x), ˜ f˜n (y)) ˜ ≤ h(f ). n→∞ n

(2.30)

Proof Take any curve .γ˜ in .X˜ joining .x˜ and .y. ˜ Let .γ be the projection of .γ˜ to X. ˜ there is a finite constant l such that each point of Since .γ˜ is a compact subset of .X, .γ is covered at most l times by points in .γ˜ . For small .ε > 0 we denote by .Aε the cover of X by all discs of radius .ε in the metric d. For any decreasing sequence of .ε’s we obtain a refining sequence of covers. Fix any small .ε > 0 and any positive number .δ. If n is large there is a cover .Aε,n of X by at most .exp(n(h(f, Aε ) + δ)) elements of the cover .Aε ∨ f −1 (Aε ) ∨ . . . ∨ f −n (Aε ). Note that .Aε,n is a subcover of .Aε ∨ f −1 (Aε ) ∨ . . . ∨ f −n (Aε ). For each element of the subcover .Aε,n there is an .ε-disc .Aε,n such that the element is contained in .f −n (Aε,n ). Since .f −n is a homeomorphism we see that .f n (γ ) can be covered by .exp(n(h(f, Aε )+δ)) .ε-discs. ˜ Indeed, .f˜ lifts f , i.e. .P ◦ f˜(x) The mapping .(f˜)n is a lift of .f n to .X. ˜ = f ◦ P(x) ˜ ˜ By induction .P◦ f˜n+1 (x) for .x˜ ∈ X. ˜ = P◦ f˜n (f˜(x)) ˜ = f n ◦P(f˜(x)) ˜ = f n+1 ◦P(x) ˜

2.4 Pseudo-Anosov Self-homeomorphisms of Closed Surfaces Are Entropy. . .

83

˜ The curve .f˜n (γ˜ ) is a lift of .f n (γ ) to .X. ˜ Hence, each point of the for .x˜ ∈ X. n curve .f˜ (γ˜ ) is covered by a lift of one of the .ε-discs. Since each point of .f n (γ ) is the projection of at most l points of .f˜n (γ˜ ) the latter curve can be covered by .l exp(n(h(f, Aε ) + δ)) .ε-discs. This implies that the distance between its two endpoints .d(f˜n (x), f˜n (y)), does not exceed .2εl exp(n(h(f, Aε ) + δ)). Hence, .

lim sup n→∞

1 log d(f˜n (x), f˜n (y)) ≤ h(f, Aε ) + δ. n

(2.31)

Since .δ > 0 is arbitrary, Eq. (2.31) holds with .h(f, Aε ) + δ replaced by .h(f, Aε ). For a sequence of .ε’s we obtain a refining sequence of coverings. The statement of the lemma follows.   We will consider now a lift .f˜ of f , and associate to it an action .f˜# on the fundamental group. Recall how to associate to each element .α of the fundamental group .π1 (X, x0 ) a covering transformation. We take a point .x˜0 ∈ X˜ that projects to .x0 . Consider the lift .α˜ of a representative of .α with initial point .x˜0 . Associate to x˜ −1 .α the covering transformation .(Is 0 ) (α) that maps .x˜0 to the terminal point of .α. ˜ The point .x˜0 will be fixed in this section. We denote the covering transformation x˜ −1 .(Is 0 ) (α) associated to .α ∈ π1 (X, x0 ) by .α cov . Recall that .(α1 α2 )cov = α2cov α1cov . The covering transformation .α cov , acts as follows. Write the elements of .X˜ as .x ˜∼ = {ex0 ,x }. Then α cov ({ex0 ,x }) = {α ex0 ,x }, {ex0 ,x } ∈ X˜ .

.

(2.32)

Fix a lift .f˜ of f . Since for each .α ∈ π1 (X, x0 ) the mapping .f˜ ◦ α cov is a lift of f , there is an element .(f˜)# (α) ∈ π1 (X, x0 ) depending on .α such that

cov ◦ f˜ . f˜ ◦ α cov = (f˜)# (α)

.

(2.33)

Since ˜ = f˜ ◦ (α2cov ◦ α1cov )(x) ˜ f˜ ◦ (α1 α2 )cov (x)

cov = (f˜)# (α2 ) ◦ f˜(α1cov (x)) ˜ 

cov

cov  ◦ f˜(x) ˜ ◦ (f˜)# (α1 ) = (f˜)# (α2 )

.

the mapping .α → (f˜)# (α) is a group isomorphism of .π1 (X, x0 ). This isomorphism can be given explicitly as follows. Let .f˜ be given by (2.29). For any element .α ∈ π1 (X, x0 ) we get the equality ˜ = f˜({αex0 ,x }) ={ex00 ,f (x0 ) f∗ (αex0 ,x )} f˜ ◦ α cov (x)

.

={ex00 ,f (x0 ) f∗ (α)f∗ (ex0 ,x )}.

(2.34)

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2 The Entropy of Surface Homeomorphisms

Recall that .f˜(x) ˜ = {ex00 ,f (x0 ) f∗ (ex0 ,x )}. Then

−1 (f˜)# (α) =ex00 ,f (x0 ) f∗ (αex0 ,x ) ex00 ,f (x0 ) f∗ (ex0 ,x )

.

=ex00 ,f (x0 ) f∗ (α)(ex00 ,f (x0 ) )−1 .

(2.35)

Equality (2.33) implies by induction

cov n ˜ = (f˜)n# (α) (f˜ (x)), ˜ x˜ ∈ X˜ . f˜n ◦ α cov (x)

.

(2.36)

Indeed, 

 cov n−1   n cov n

f˜ (x) ˜ = f˜ (f˜)n−1 ˜ = (f˜)# (α) (f˜ (x)) ˜ . f˜ f˜n−1 ◦ α cov (x) # (α)

.

Lemma 2.7 If .f0 and .f1 are two self-homeomorphisms of a closed surface X that are isotopic to each other then .(f˜0 )# = (f˜1 )# . Proof Let .ft be a homotopy of self-homeomorphisms of X joining .f0 and .f1 . Denote by .f : [0, 1] × X → [0, 1] × X the mapping for which .f (t, x) = ft (x), t ∈ [0, 1], x ∈ X. Put .f −1 (t, x) = (ft )−1 (x), t ∈ [0, 1], x ∈ X. Denote by .f˜ a lift of ˜ Fix a covering transformation .α ∈ π1 (X, x0 ). Let .X˜ 0 be a connected f to .[0, 1]× X. compact subset of .X˜ whose interior covers X. The mappings .f˜ and .f˜−1 , .f˜−1 (t, x) = (f˜t )−1 (x), ˜ t ∈ [0, 1], x˜ ∈ X, are uniformly continuous on .[0, 1] × (X˜ 0 ∪ α cov (X˜ 0 )). Hence, for each .ε > 0 there exists .δ > 0 such that the implication (t1 , x˜1 ), (t2 , x˜2 ) ∈ [0, 1]×(X˜ 0 ∪ α cov (X˜ 0 )),

.

|t1 − t2 | < δ, d(x˜1 , x˜2 ) < δ ⇒ d(f˜t1 (x˜1 ), f˜t2 (x˜2 )) < ε

(2.37)

holds, and the respective implication holds for .f˜−1 . ˜ = f˜t1 (x). ˜ We want to show that Put .ξ(x) d

.



 cov

cov (f˜t2 )# (α) (ξ(x)), ˜ (f˜t1 )# (α) (ξ(x) ˜ < 2ε

(2.38)

for .x˜ ∈ X˜ 0 and .|t1 − t 2 | < δ. Equation

(2.38) shows cov cov that the values of the covering transformations . (f˜t1 )# (α) and . (f˜t2 )# (α) are .2ε-close on the set ˜ 0 ). Since .f˜t1 is a lift of a self-homeomorphism of X this set covers X. If .ε is .f˜t1 (X can only hold, if the covering transformations coincide. This small, inequality

(2.38)cov implies that . (f˜t )# (α) is locally constant for .t ∈ [0, 1], hence, it is constant and

cov

cov ˜ ˜ . (f0 )# (α) = (f1 )# (α) . Since .α was an arbitrary covering transformation the lemma will be proved if inequality (2.38) is proved. We prove now inequality (2.38). By the implication (2.37) the inequality cov (x), .d(f˜t1 ◦α ˜ f˜t2 ◦α cov (x)) ˜ < ε holds for .x˜ ∈ X˜ 0 and .t1 , t2 ∈ [0, 1], |t1 −t2 | < δ.

2.4 Pseudo-Anosov Self-homeomorphisms of Closed Surfaces Are Entropy. . .

85

Hence, by (2.33) d

.



 cov

cov (f˜t1 )# (α) (f˜t1 (x)), ˜ (f˜t2 )# (α) (f˜t2 (x)) ˜ 0, such that the .δ-neighbourhood .Vδ (zj ) in the .-metric of any singular point .zj of .φ is simply connected. Any loop that is contained in such a neighbourhood .Vδ (zj ) is contractible to the singular point .zj . Hence, if .αˆ contains a loop contained in .Vδ (zj ), than both, .φ (α) ˆ and . (α) ˆ are equal to zero. In this case the inequalities (2.51) hold. On the other hand, there exist positive numbers .c1 and .c2 depending on .δ such that for any curve .γ  avoiding the . 2δ -neighbourhoods in the .-metric of the singular points the inequalities c1 φ (γ  ) ≤  (γ  ) ≤ c2 φ (γ  )

(2.52)

.

hold. Moreover, there exists a constant .c ≥ 1 such that for each j

  (∂Vδ (zj )) ≤2c d ∂Vδ (zj ), ∂V δ (zj ) , 2

 φ (∂Vδ (zj )) ≤2c dφ ∂Vδ (zj ), ∂V δ (zj ) .

.

2

(2.53)

Suppose the free homotopy class of curves .αˆ cannot be represented by a loop contained in one of the .Vδ (zj ). Take a curve .γ ∈ αˆ which almost minimizes the .-length, . (γ ) ≤  (α) ˆ + ε for a small positive number .ε. For each part of .γ that is contained in .X \ ∪V δ (zj ) (i.e. lies outside the . 2δ -neighbourhoods of the critical 2 points) the estimates (2.52) hold. Suppose there is a j and a connected component .γ˜ of .γ ∩Vδ (zj ) which intersects .V δ (zj ). The connected component .γ˜ is an arc with the two endpoints on .∂Vδ (zj ). 2 Replace .γ˜ by an arc .γ˜  on .∂Vδ (zj ) which is homotopic to .γ˜ with fixed endpoints. By (2.53)  (γ˜  ) ≤  (∂Vδ (zj )) ≤ 2c d (∂Vδ (zj )), ∂V δ (zj )) ≤ c  (γ˜ ) .

.

2

(2.54)

Consider for each singular point .zj all connected components of the intersection γ ∩ Vδ (zj ) which have points in .V δ (zj ). Replace in the same way as above each 2 such component by an arc which avoids the sets .V δ (zj ). We obtain a curve .γ  that is

.

2

homotopic to .γ , avoids the . 2δ -neighbourhoods in the .-metric of the singular points, and satisfies the estimates  (γ  ) ≤ c  (γ ) ≤ c ( (α) ˆ + ) .

.



(2.55)

Hence, by the inequalities (2.52) the inequality .φ (α) ˆ ≤ cc ( (α) ˆ + ε) holds. Since 1 .ε can be taken to be an arbitrary positive number, the first inequality in (2.51) holds.   The second inequality follows by interchanging the role of . and .φ. The following proposition is proved in [25] (see Proposition 5.7. there).

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2 The Entropy of Surface Homeomorphisms

Proposition 2.7 Let X be a closed Riemann surface, and let .φ be a holomorphic quadratic differential on X. Suppose .γ is a simple closed (connected and smooth) curve in X which is contained in the .φ-regular part of X and is transversal to the vertical foliation. Let .γ  be any closed (connected and piecewise smooth) curve in X which is free homotopic to .γ . Then the following estimate for the horizontal .φ-length holds φ,h (γ ) ≤ φ,h (γ  ) .

.

(2.56)

Proof Let .x0 be the base point of .γ . Denote by .α the class of .γ in the fundamental group .π1 (X, x0 ), and let .P : X˜ → X be the universal covering of X. Consider the def cov for the covering transformation .α cov associated to .α and ˜ quotient .Xˆ = Xα the fixed base point .x˜0 ∈ P−1 (x0 ). The quotient is conformally equivalent to an annulus. The universal covering .P : X˜ → X induces a covering .pˆ : Xˆ → X. (For more details see also Sect. 11.2). The holomorphic quadratic differential .φ on X lifts to a holomorphic quadratic differential .φˆ on .Xˆ and the free homotopic loops .γ and   ˆ Notice that . ˆ (γˆ ) = φ,h (γ ) .γ on X lift to free homotopic loops .γˆ and .γˆ on .X. φ,h   and .φ,h ˆ (γˆ ) = φ,h (γ ). Hence, it is sufficient to prove the theorem in the case when X is an annulus with a holomorphic quadratic differential, and both loops are free homotopic to the positive generator of the fundamental group of the annulus. We may assume that .γ  is also simple closed, removing otherwise some parts of .γ  which does not increase its horizontal .φ-length. We will now prove the Theorem under these conditions. Assume first that .γ and .γ  are disjoint. Then the set .γ ∪γ  consists of all boundary points of an annulus .A ⊂ X. Indeed, after a conformal mapping we may assume that X is a subset of the plane. Then the claim follows from the Jordan Curve Theorem applied to each of the curves. For each point .p ∈ γ we consider the vertical trajectory ray .rp with initial point p that enters A at p. We call a trajectory ray .rp exceptional if it stays in A and runs into a critical point contained in A. We claim that each non-exceptional vertical trajectory ray .rp , p ∈ γ , meets the boundary .∂A after p. Indeed, if the non-exceptional ray .rp is critical, it meets .∂A at a point after p by the definition of exceptional rays. Suppose .rp is non-critical. Consider an open .φ-rectangle .R that contains .p ∈ γ . Shrinking .R in the horizontal direction if necessary, we may assume that the connected component .γp , that contains p, of the intersection .γ ∩ R is an arc with endpoints on opposite vertical sides of .R. Denote by .γph the connected component containing p of the intersection of .R with the horizontal trajectory through p. By (the proof of) Theorem 2.4 the divergent vertical trajectory ray .rp intersects .γph after p infinitely often in the same direction. Hence, the ray .rp contains points that are not in A and therefore .rp intersects .∂A. The claim is proved. By Proposition 2.8 for any non-exceptional trajectory ray .rp , p ∈ γ , the first point after p on the boundary .∂A is contained in .γ  . Indeed, suppose it is a point

2.4 Pseudo-Anosov Self-homeomorphisms of Closed Surfaces Are Entropy. . .

93

Fig. 2.4 Curves that are transversal to the vertical foliation are .φ,h -minimizing in their free homotopy class

p on .γ . The union of one of the arcs of .γ with endpoints p and .p and the arc of  .rp between p and .p bounds a disc. But the first arc is transversal to the vertical foliation and the second is an arc of a vertical leaf. This is impossible. We assign to each non-exceptional ray .rp , p ∈ γ , the first point .g(p) > p of the trajectory ray .rp which is on .γ  . Let .rp be the segment on .rp between p and .g(p). There are only finitely many exceptional trajectory rays. Let .pj , j = 1, . . . , N, be the initial points of the exceptional trajectory rays, ordered cyclically with respect to some orientation of .γ . Let .s(pj ) be the singular point in A which is the limit point of .rpj . Then for .pj = pj  the singular points .s(pj ) and .s(pj  ) are different. Indeed, suppose not. Then there are two neighbouring points .pj and .pj +1 with .s(pj ) = s(pj +1 ). (If the two neighbouring points are .pN and .p1 we relabel the .pj .) Then .rpj and .rpj +1 , are segments on neighbouring bisectrices at the singular point .s(pj ) = s(pj  ), and the two segments together with an arc of .γ between .pj and .pj +1 bound a sector which is laminated by vertical arcs with the two endpoints on the arc of .γ between .pj and .pj +1 . This is impossible by Theorem 2.8 (see Fig. 2.4). It is enough to prove the proposition for the case when .γ  does not contain singular points of the quadratic differential and is piecewise linear in flat coordinates. Indeed, one can achieve this by a deformation of .γ  that changes the .φ-length no more than by an a priory given amount. Consider the open arc .γj contained in .γ whose endpoints are two neighbouring points .pj and .pj +1 (.pN and .p1 for .j = N). We consider the set .γj = g(γj ). .

94

2 The Entropy of Surface Homeomorphisms

The obtained subset .γj of .γ  is the union of a finite number of open linear segments contained in .γ  and a finite number of points on .γ  (that are endpoints of some of the just mentioned linear segments). Hence, the set .γj is measurable for the horizontal  .φ-length. Moreover, each maximal open linear piece of .γ has the same horizontal j .φ-length as the corresponding part on .γj which is the preimage of this linear piece under .g. Indeed, each point in the linear piece of .γ  is joined with its preimage under .g by a vertical segment. Hence, .φ,h (γj ) = φ,h (γj ). The union of the .γj equals .γ \ ∪{pj }. The union of the .γj , j = 1, . . . N, may be much smaller than .γ  . The proposition is proved for the case when .γ and .γ  are disjoint. In the general case we may assume again that the curve .γ  does not contain singular points of the quadratic differential and is piecewise linear in flat coordinates. Consider the set .γ  \ γ of points that are in .γ  but not in .γ . It consists of finitely many connected components, each being a piecewise linear open arc. Take any component .γj . There is an arc .γj on .γ so that the union of the closures of .γj and  .γ (each with suitable orientation) bounds a disc in A. The same argument as before j shows that .φ,h (γj ) ≤ φ,h (γj ). Instead of .γ  we consider the closed piecewise smooth curve .γ  obtained by replacing each .γj by .γj oriented so that .γj and .γj are homotopic with fixed endpoints. The horizontal .φ-length .φ,h (γ  ) of the new curve  does not exceed .   .γ φ,h (γ ). All points of .γ lie on .γ . Each point of .γ is covered  by at least one point of .γ , but some parts of .γ may be covered multiply. Hence   .φ,h (γ ) ≤ φ,h (γ ) ≤ φ,h (γ ).   Proof of Theorem 2.2. Let .ϕ0 be a non-periodic absolutely extremal selfhomeomorphism of a closed Riemann surface X of genus .g ≥ 2, and let .ϕ be a self-homeomorphism of X that is isotopic to .ϕ0 . We will prove that the entropy of .ϕ is not smaller than . 12 log K(ϕ0 ). Theorem 2.2 will then be a consequence of Theorem 2.1. Denote by .φ the quadratic differential of .ϕ0 . It is holomorphic on X. Let .γ be a simple closed curve in X with base point .x0 that is contained in the .φ-regular part of X, is transversal to the vertical foliation of .φ, and has positive .φ,h -length. By Theorem 2.5 such a curve exists. By Theorem 2.8 .γ is not contractible. Denote by .α the class of .γ in .π1 (X, x0 ). Let .dhyp be the hyperbolic metric on the universal covering .X˜ ∼ = C+ . We use the same notation for the induced metric on X. By Lemma 2.9 and Proposition 2.7 the inequalities hyp ( α ) ≥ c1 φ ( α ) ≥ c1 φ,h ( α ) = c1 φ,h (γ ) > 0

.

(2.57)

hold for the free homotopy class . α represented by .α. Denote by .x˜0 ∈ X˜ the point corresponding to the identity .Idx0 ,x0 in .π1 (X, x0 , x0 ). Let .α˜ be the lift of .α to the universal covering with initial point .x˜0 . Let as before cov be the covering transformation that maps .x˜ to .α( .α ˜ x˜0 ). Since .dhyp (x˜0 , α cov (x˜0 )) 0  is the infimum of the hyperbolic lengths .hyp (γ˜ ) over all curves .γ˜  in .X˜ with initial point .x˜0 and terminal point .α cov (x˜0 ) and .hyp (γ˜  ) = hyp (γ  ), we obtain by (2.57) dhyp (x˜0 , α cov (x˜0 )) ≥ c1 φ,h (γ ) .

.

(2.58)

2.5 The Entropy of Mapping Classes on Second Kind Riemann Surfaces

95

Since .ϕ0 is a non-periodic absolutely extremal self-homeomorphism of X with quadratic differential .φ, for any natural number n the curve .(ϕ0 )n (γ ) is a simple closed curve in X that avoids the .φ-singular points and is transversal to the vertical foliation. The mapping .(ϕ0 )n expands the horizontal .φ-length of arcs by the factor n n .K 2 . This implies the following estimate of the horizontal .φ-length of .(ϕ0 ) (γ ) n

φ,h ((ϕ0 )n (γ )) ≥ K 2 φ,h (γ ) .

(2.59)

.

The loop .(ϕ0 )n (γ ) represents the free homotopy class of .((ϕ0 )# )n (α) (considered as element of the fundamental group .π1 (X, x0 )). Indeed, .(ϕ0 )n (γ ) lifts to the curve n ˜ with initial point .(ϕ˜0 )n (x˜0 ) and terminal point .(ϕ˜0 )n (α cov (x˜0 )) = .(ϕ˜0 ) (γ˜ ) in .X 

n cov

 (ϕ0 )# (α) (ϕ˜0 )n (x˜0 ) . The last equality follows from equality (2.36). Lemma 2.9 and Proposition 2.7 apply to the loop .(ϕ0 )n (γ ) in the same way as to the loop .γ . Hence, inequality (2.57) with .(ϕ0 )n (γ ) instead of .γ and .((ϕ0 )# )n (α) instead of .α holds. Moreover, inequality (2.58) is true with .α cov replaced by n cov and .γ replaced by .(ϕ )n (γ ). Using inequality (2.59), we obtain .(((ϕ0 )# ) (α)) 0 n

dhyp (x˜0 , (((ϕ0 )# )n (α))cov (x˜0 )) ≥ c1 φ,h ((ϕ0 )n (γ )) ≥ c1 K 2 φ,h (γ ) .

.

(2.60)

Consider the self-homeomorphism .ϕ of X that is isotopic to .ϕ0 . Since by Theorem 2.6, Lemma 2.7, and equality (2.43) h(ϕ) ≥ Γϕ# = Γ(ϕ0 )# ≥ lim sup

.

we obtain .h(ϕ) ≥

1 2

1 log dhyp (x˜0 , (((ϕ˜0 )# )n (α))cov (x˜0 )) , n

log K . The Theorem is proved.

 

2.5 The Entropy of Mapping Classes on Second Kind Riemann Surfaces The following Theorem 2.9 allows us to treat the entropy of mapping classes of Riemann surfaces of second kind. Theorem 2.9 Let X be a connected closed Riemann surface with a set E of distinguished points. Assume that .X\E is hyperbolic (i.e. covered by .C+ ). Let .ϕ be an absolutely extremal non-periodic self-homeomorphism of X that fixes E setwise. Suppose .z0 ∈ E is a fixed point of .ϕ, .ϕ(z0 ) = z0 . Then .ϕ is isotopic through self-homeomorphisms of X that fix E to a homeomorphism .ϕ0 that fixes a topological disc .Δ around .z0 pointwise and has the same entropy .h(ϕ0 ) = h(ϕ) as .ϕ. Moreover, the homeomorphism .ϕ0 can be chosen to represent any a priory given preimage under the mapping .Hz0 : M(X \Δ ; ∂Δ , E \ {z0 }) → M(X; {z0 }, E \ {z0 }) of the class of .ϕ in .M(X; {z0 }, E \ {z0 }). (For the notation see Sect. 1.4.)

96

2 The Entropy of Surface Homeomorphisms

Notice that the complement .X0 in X of an open disc around .z0 is a bordered Riemann surface (the closure of a Riemann surface of second kind), and the restriction .ϕ0 | X0 is a self-homeomorphism of .X0 with entropy .h(ϕ0 | X0 ) = h(ϕ). We will now prepare the proof of Theorem 2.9. In a neighbourhood of the distinguished point .z0 of X we consider distinguished coordinates .ζ with .ζ (z0 ) = 0. In these coordinates the quadratic differential of .ϕ 2 a 2 has the form .φ(ζ )(dζ )2 = ( a+2 2 ) ζ (dζ ) . Take a disc .Δ = {|ζ | < c} in these coordinates. The following lemma says that the restriction .ϕ | X \ {z0 } has an extension to a self-homeomorphism of the “real blowup” of X at .z0 . iθ

ϕ(re ) Lemma 2.10 In distinguished coordinates .ζ = reiθ the quotient . |ϕ(re iθ )| does not depend on .r ∈ (0, c) and defines a homeomorphism def

h(eiθ ) =

.

ϕ(reiθ ) |ϕ(reiθ )|

(2.61)

of the unit circle onto itself. Proof The distinguished coordinates are unique up to multiplication by an .(a +2)nd root of unity. In other words  .

2π i

ζ e a+2 ·l

a  2 2π i dζ e a+2 ·l = ζ a (dζ )2

(2.62)

for any integer l, and these are the only holomorphic changes of coordinates under which the quadratic differential has again the canonical form. The initial quadratic differential of .ϕ coincides with its terminal quadratic differential. Hence in the distinguished coordinates .ζ the mapping .ϕ has the form 2π i

ϕ(ζ ) = Za (ζ ) · e a+2 ·

.

for some integer ,

(2.63)

where  Za (ζ ) =

.

ζ a+2 + 2k |ζ |a+2 + k 2 ζ¯ a+2 1 − k2

1  a+2

.

(2.64)

Here .k = K−1 K+1 , and .K = K(ϕ) is the quasiconformal dilatation of .ϕ. We take the root which is positive on the positive real axis. (See e.g. [11], formula (2.3) and Theorem 6 of [11] or Chap. 1, Eq. (1.27)). i The mapping .Za fixes each horizontal bisectrix .{r exp( 2πj a+2 ), 0 < r < c}, 2π(j + 21 )i a+2 ), 0 < r < c}, i 2π(j +1)i θ ∈ ( 2πj a+2 , a+2 ), 0
1 the .Sjk+1 subdivide the .SjNkk . Define sectors k+1  (SjNkk ) ⊃ SjNkk \ {z0 } as above and so that each .(SjN+1 ) is contained in some .(SjNkk ) .

N

.

Consider for each k the cover .Aεk ,Nk (z0 ) of .Xz0 obtained for the choice of .εk , .Nk , and .(SjNk ) . Then the .Aεk ,Nk (z0 ) provide a refining sequence of open covers of .Xz0 .

This is not difficult to see, taking into account the choice of the .(SjNk ) , the fact that the sequence of coverings .Aεk of X is refining, and the following observation. Fix k and consider the elements of the cover .Aεk ,Nk (z0 ) contained in .Xz0 ∩ Δ. Let .σk be the supremum of the diameters (in coordinates .ζ on .Δ) of such sets. Then the iθ on .Δ ∩ X x0 each .σk tend to zero for .k → ∞, since in polar coordinates .ζ = e z −1  set .(ψ 0 ) (A) with .A ∈ Aεk ,Nk (z0 ) has width comparable to .εk in the .-direction and width equal to the maximal angle of the .(SjNk ) in the .θ direction. An Upper Bound for the Entropy .h(ϕ z0 , Aεk ,Nk (z0 )) We will take a large number k, put .N = Nk , .ε = εk , and give an upper bound of the entropy .h(ϕ z0 , Aε,N (z0 )) of z .ϕ 0 with respect to the cover .Aε,N (z0 ) of .X z0 . For each set A in the above defined cover .Aε of X and one of the N sectors .SjN of .Δ \ {z0 } we define .A ∩∗ SjN to be equal to A if A does not intersect .Δ0 and equal to .A ∩ SjN if A intersects .Δ0 . We prove first the following claim. For each .(λ−n ε, ε)-rectangle and each .(λ−n ε, ε)-star written in the form A0 ∩ . . . ∩ ϕ −n (An ) ∈ Aε ∨ . . . ∨ ϕ −n (Aε )

.

(2.68)

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2 The Entropy of Surface Homeomorphisms

with either all .Aj being .ε-squares or all .Aj being .ε-stars, there are at most .1 + (n+1)N ((n+1)N +1) different non-empty sets of the form 2    −n

 def

... ϕ An ∩∗ SjNn A = A0 ∩∗ SjN0

(2.69)

.

where each .SjNl is one of the N sectors of .Δ. The claim is proved as follows. Suppose first that the set (2.68) is a .(λ−n ε, ε)rectangle, i.e. it is the image of a .φ-rectangle .F : R → A0 ∩ . . . ∩ ϕ −n (An ) for a rectangle R in the complex plane with sides parallel to the axes. If a set .Al intersects .Δ0 then by a similar argument as used in the proof of Lemma 2.2 it is contained in a sector .shjl (or .svjl ) of .Δ between two consecutive horizontal bisectrices (or between 2

two consecutive vertical bisectrices). A branch of the mapping .ζ → ζ a+2 defines flat coordinates on the sector and maps each of the N radii that are contained in the sector to a relatively closed straight line segment in the image. The mapping .ϕ is real linear in flat coordinates. Hence the mapping .ϕ −l takes each of the N radii that are contained in the sector to a relatively closed curve in the image which is a straight line segment in flat coordinates. This implies that each set of the form (2.69) is obtained as follows. Take the rectangle R in the plane which is the preimage .F −1 (A) of a set A of the form (2.68). For each l the collection of preimages .F −1 (A ∗ ϕ −l (Al ∩ SjNl )) is the collection of intersections of R with connected components of the complement of the union of no more than N lines in the complex plane. We have to count nonempty sets each of which is the  intersection over .l = 0, . . . , n, of an element of the collection of preimages .F −1 (A ϕ −l (Al ∩∗ SjNl )). Each such intersection Q is a connected component of the intersection of R with a connected component .Q of the complement of the union of no more than .(n+1)N lines in the complex plane. Indeed, Q is contained in such a component .Q , and since the boundary of Q is the union of segments of some of the lines, it has no boundary points in .Q , hence it coincides with .Q . The estimate of the number of connected components of the complement of the union of N different real lines in the complex plane can be given by induction on the number N of lines. If N equals 1 the number is obviously equal to 2. Suppose an upper bound for the number is found for N lines. The observation for .N + 1 lines is the following. The .(N + 1)-st line intersects a number of lines among the N previous lines, and each of them is intersected once. The intersection points with the N previous lines divide the .(N + 1)-st line into no more than .N + 1 connected components. Each connected component of the last line is contained in exactly one connected component of the complement of the first N lines in .C. This component is divided into two parts by the last line. Hence, adding an .(N + 1)-st line increases the number of connected components of the lines at most by .N + 1. By induction, the number of connected components of the complement of N lines in .C does not exceed .1 + N (N2+1) . We proved the claim for the case when A is a .(λ−n ε, ε)rectangle.

2.5 The Entropy of Mapping Classes on Second Kind Riemann Surfaces

101

If the set (2.68) is a .(λ−n ε, ε)-star and intersects .Δ0 , then all .Aj are stars at .z0 , and for all .Aj that are stars at .z0 we obtain at most N radii in each .Aj . Hence, for the stars in .Aε ∨ ϕ −1 (Aε ) ∨ . . . ∨ ϕ −n (Aε ) there are at most .(n + 1)N different non-empty sets of the form (2.69). The claim is proved. For each small .ε > 0 there is a finite number .C1 depending on .ε, such that the set .X \ {z0 } can be covered by .C1 sets, each either an .ε-square or an .ε-star. We may cover each .ε-square by .[λn + 1] sets that are .(λ−n ε, ε)-rectangles. Consider an .ε-star centered at a singular point of order .a  . Each of the .2a  + 2 sectors of the .ε-star between a horizontal and the nearest vertical bisectrices is an open .ε-square.

 Hence, the punctured star can be covered by no more than .(2a  + 2) · λn + 1 sets that are .(λ−n ε, ε)-rectangles, and the puncture is covered by a .(λ−n ε, ε)-star. By the proof of Proposition 2.4 each .(λ−n ε, ε)-rectangle can be written in the form (2.68) for .ε-squares .Al , and each .(λ−n ε, ε)-star can be written in the form (2.68) for .ε-stars .Al . Hence, there is a finite constant .C2 , such that for each natural number n the set X can be covered by no more than .C2 · (λn + 1) sets of the form (2.68). By the claim, .X \ {z0 } can be covered by no more than (n+1)(N (n+1)+1) · (λn + 1) different non-empty sets of the form (2.69) .C2 1 + 2 N with each .Sl replaced by the slightly bigger .(SlN ) . These sets form a subcover of    −1 −n .A ε,N ∨ ϕ (Aε,N ) ∨ . . . ∨ ϕ (Aε,N ). We obtain a cover of .Xz0 by considering for each element of .Aε,N the respective element of .Aε,N (z0 ). This gives the inequality

 N Aε,N (z0 ) ∨ ϕ −1 (Aε,N (z0 )) ∨ . . . ∨ ϕ −n (Aε,N (z0 ))  (n + 1)N ((n + 1)N + 1)  · (λn + 1) . ≤ C2 1 + 2

.

Hence, h(ϕ z0 )

.



 1 log N Aε,N (z0 ) ∨ ϕ −1 (Aε,N (z0 )) ∨ . . . ∨ ϕ −n (Aε,N (z0 ) n 

 1 N(n + 1)(N (n + 1) + 1)  · (λn + 1) = log λ . ≤ lim sup log C2 1 + n 2 (2.70)

= lim sup

Lemma 2.11 is proved.

 

The restriction of the homeomorphism .ϕ z0 to .∂Xz0 is not equal to the identity mapping on .∂Xz0 . The Lemmas 2.12 and 2.14 below produce a selfhomeomorphism of a slightly larger bordered Riemann surface whose restriction to the boundary is the identity mapping.

102

2 The Entropy of Surface Homeomorphisms

Lemma 2.12 For each integer . there exists a self-homeomorphism r  .ϕ˜ of r r z ≤ |ζ | ≤ of entropy zero which equals .ϕ 0 on . |ζ | = ⊂ Δ ⊂ X 4 2 2 and   2π i r 2(a+2) is equal to multiplication by .e on . |ζ | = 4 for the integer ..

.

2πj i Recall that .r exp( 2(a+2) ), j = 1, . . . , 2(a + 2), 0 < r < c, are the points on  2πj ≤ the vertical and horizontal bisectrices. Consider the arc .γj = 2r eiθ : 2(a+2)  +1) θ ≤ 2π(j 2(a+2) , .j = 0, 1, . . . , 2(a + 2) − 1. We will obtain Lemma 2.12 from the following lemma.

Lemma 2.13 Let .ψj be a self-homeomorphism of .γj which fixes the endpoints of γj pointwise and does not fix any other point. Then there is a self-homeomorphism   ¯ j = ρ eiθ : r ≤ ρ ≤ r , r eiθ ∈ γj which equals the .ψ˜j of the truncated sector .Ω 4 2 2 identity on the boundary part .∂ Ωj \Int γj of .∂ Ωj , equals .ψj on .γj , and has entropy zero. .

Proof We may assume that .j = 0. Put .(t) = 12 t + 14 (1 − t), .t ∈ [0, 1]. Then 1 1 1 1 .(0) = 4 , .(1) = 2 , and . is a linear homeomorphism from .[0, 1] onto .[ 4 , 2 ]. Put def

2π I = [0, 2a+2 ]. The mapping

.

g

[0, 1] × I  (t, θ ) − → (t)eiθ ∈ Ω0

.

is a homeomorphism. The segment .{1} × I is mapped homeomorphically onto .γ0 . Put .Ψ = g −1 ◦ ψ0 ◦ g | {1} × I . The mapping .Ψ is a self-homeomorphism of .{1} × I that is strictly increasing in the variable .θ ∈ I . We need to find a selfhomeomorphism .Ψ˜ of .[0, 1] × I such that .Ψ˜ | {1} × I = Ψ , the restriction of .Ψ˜ to the rest of the boundary of .[0, 1] × I is the identity, and the entropy of .Ψ˜ equals def zero. Then the mapping .ψ˜ 0 = g ◦ Ψ˜ ◦ g −1 satisfies the requirement of the lemma for .j = 0. We put Ψ˜ (t, θ ) = (t, t Ψ (θ ) + (1 − t) θ ),

.

(t, θ ) ∈ [0, 1] × I ,

(2.71)

and .Ψt (θ ) = t Ψ (θ ) + (1 − t) θ . The mapping .Ψ˜ fixes the coordinate t, hence, it fixes each maximal vertical segment in .[0, 1] × I , and maps the horizontal line segment with endpoints .(0, θ ) and .(1, θ ) onto the line segment with endpoints .(0, θ ) and .(1, Ψ (θ )). We will prove now that the homeomorphism .Ψ˜ has entropy zero. We choose open covers of .[0, 1] × I as follows. For a natural number N we consider N vertical line segments in .[0, 1] × I which divide .[0, 1] × I into .N + 1 vertical strips of equal width, and N horizontal line segments in .[0, 1]×I which divide .[0, 1]×I into .N +1 horizontal strips of equal width. The complement of the lines in .[0, 1] × I consists of .(N + 1)2 rectangles .Rl . Let .AN be the cover of .[0, 1] × I by open .(N + 1)2 rectangles each containing the closure of one of the .Rl and having diameter close to

2.5 The Entropy of Mapping Classes on Second Kind Riemann Surfaces

103

that of .Rl . For a suitable sequence of N ’s we get a refining sequence of covers of [0, 1] × I . We prove now that for each N the entropy .h(Ψ˜ , AN ) is equal to zero. The mapping .Ψ has no fixed point on the interior .IntI of I . Hence, either .Ψ (θ ) > θ for each .θ ∈ IntI , or .Ψ (θ ) < θ for each .θ in .IntI . We assume that the first case holds. (In the second case we conjugate the mapping by .(t, θ ) → (t, −θ ).) Then for each .θ ∈ IntI the mapping .t → Ψt (θ ), t ∈ I, is strictly increasing. Since .Ψ is a strictly increasing function on .{1} × I , also each the mapping .θ → Ψt (θ ), θ ∈ I, is strictly increasing for each .t ∈ [0, 1]. Let .Ψtk be the k-th iterate of .Ψt (considered as function of .θ for fixed parameter t). Note that each iterate .Ψtk is strictly increasing in .θ . With .Ψ˜ (t, θ ) = (t, Ψt (θ )), for the k-th iterate .(Ψ˜ )k we have by induction

.





 Ψ˜ k t, θ = Ψ˜ t, Ψtk−1 (θ ) = t, Ψtk (θ ) .

.

We claim that for each .k ≥ 1 the mapping .t → Ψtk (θ )) is strictly increasing for each .θ in the interior of I . This is clear for the linear mapping .t → Ψt (θ ) for .θ ∈ IntI . By induction the claim for .k + 1 is obtained as follows. Suppose we know that for all .0 < t1 < t2 < 1 and .θ ∈ IntI the inequality k k .Ψt (θ ) > Ψt (θ ) holds. We have for all .t ∈ [0, 1] 2 1

 Ψtk+1 (θ ) =t Ψ Ψtk (θ ) + (1 − t) Ψtk (θ ) .

.

(2.72)

Since .Ψ strictly increases and the mapping .t → Ψtk (θ ) strictly increases, we obtain



 t Ψ Ψtk1 (θ ) ) + (1 − t) (Ψt1 )k (θ ) < t Ψ Ψtk2 (θ ) + (1 − t) Ψtk2 (θ )

.

(2.73)

for each .t ∈ [0, 1], .θ ∈ IntI , and .t1 , t2 ∈ [0, 1], .t1 < t2 . Since .Ψ (θ  ) > θ  for each  .θ in the interior of I , we obtain the inequality



 t1 Ψ Ψtk2 (θ ) + (1 − t1 ) Ψtk2 (θ ) < t2 Ψ Ψtk2 (θ ) + (1 − t2 ) Ψtk2 (θ ) .

.

(2.74)

Combining the last inequality (2.74) with inequality (2.73) for .t = t1 and equality (2.72) we obtain the claim. The claim implies that for each natural number k and for each .θ in the interior of I the curve

  def

Ckθ = { t, Ψtk (θ ) , t ∈ [0, 1]} = Ψ˜ k [0, 1] × {θ }

.

(2.75)

intersects each vertical line and each horizontal line in .[0, 1] × I at most once. Since θ Ck+l = Ψ˜ k (Clθ ) the intersection of two curves of form (2.75) consists of at most one point. The intersection behaviour of the union of the set of vertical and horizontal lines and the set curves .Clθ is the same as the intersection behaviour of straight lines.

.

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2 The Entropy of Surface Homeomorphisms

A similar argument as in the proof of Lemma 2.11 proves that the entropy .h(Ψ˜ , AN ) equals zero for each N. Hence, .h(Ψ˜ ) = 0. Lemma 2.13 is proved.   The following lemma states in particular that for a bordered Riemann surface .X one can change a mapping class .m ∈ M(X, ∂X) without increasing entropy by a product of Dehn twists about circles that are homologous to boundary curves.   Lemma 2.14 Let .ϕˆ be a twist on an annulus . 8r ≤ |ζ | ≤ 4r , more precisely, for r some real constant .ν we have .ϕ(ζ ˆ ) = ζ · eiν·(log |ζ |−log 8 ) . Then .h(ϕ) ˆ = 0.  r Proof Denote by A the annulus .A = 8 ≤ |ζ | ≤ 4r . Let N be a natural number.   Divide A by .N − 1 circles . Circj = |ζ | = 8r · (1 + Nj ) , j = 1, . . . , N − 1, and   2π i .N radii .Radj = ρ · e N j : 8r ≤ ρ ≤ 4r , j = 1, . . . , N into a collection .QN of connected open sets .QN l . Locally the circles and the radii are straight line segments in logarithmic coordinates, and the .QN l are open rectangles in these coordinates.  Similarly as in the proof of Lemma 2.11 we take for each l an open rectangle .(QN l ) in local logarithmic coordinates that contains the closure .QN l and has diameter not . We obtain an open cover .AN of the annulus exceeding twice the diameter of .QN l A. For an increasing sequence .Nk of natural numbers with .Nk+1 being an integer k  multiple of .Nk and a suitable choice of the .(QN l ) we obtain a refining sequence of open covers.  For each N the annulus can be covered by the collection of sets .(QN 0 ) ∩ ... ∩ N N −n N  −n N N ϕˆ ((Qn ) ) for which the sets .Q0 ∩ . . . ∩ ϕˆ (Qn ) ( .Ql ∈ Q ) are not empty.  These sets .(Q0 ) N ∩ . . . ∩ ϕˆ −n ((QN n ) ) are elements of a subcover of the cover −n .AN ∨ . . . ∨ ϕ ˆ (AN ). We will estimate the number of these sets for each N and n. Each iterate .ϕˆ − ,  = 1, . . . , n, is real linear in local logarithmic coordinates. It maps each circle .Circj , j = 1, . . . , N − 1, onto itself and each radius .Radj , j = 1, . . . N, onto the curve  2π r r r . .ϕ ˆ − (Radj ) = ex · ei N j + i··ν (x−log 8 ) : log ≤ x ≤ log (2.76) 8 4 N Each curve (2.76) intersects each given set .QN l ∈ Q along at most .c(ν, N) ·  connected components, each a straight line segment in logarithmic coordinates − on .QN l . Hence, as in the proof of Lemma 2.11 the union of the .ϕˆ (Radj ), j = N 1, . . . , N,  = 1, . . . , n, intersects the given set .Ql along at most .c(ν, N )(N + . . . + N n) ≤ c (ν, N )n2 straight line segments in logarithmic coordinates. They  4 divide .QN l into at most .C (ν, N )n connected components. Hence, the number of N N N non-empty sets of the form .Q0 ∩ . . . ϕˆ −n (QN n ) for sets .Ql ∈ Q does not exceed 4   .C(ν, N)n . Here .c(ν, N), c (ν, N ), C (ν, N ), and .C(ν, N ) are positive constants depending only on .ν and N. Hence, .N (AN ∨ . . . ∨ ϕˆ −n (AN )) ≤ C(ν, N)n4 . This implies that the entropy of .ϕ ˆ with respect to the cover .AN does not exceed .limn→∞ n1 log(C(ν, N ) · n4 ) = 0.

2.5 The Entropy of Mapping Classes on Second Kind Riemann Surfaces

105

The estimate is obtained for each element of a refining sequence of open covers of the annulus. Hence .h(ϕ) ˆ = 0.   Proof of Lemma 2.12. Let . be the integer from equality (2.63) and let k be the smallest positive integer for which .k ·  is a multiple of .2(a + 2). Then .ϕ z0 permutes the .γj in cycles of length k and .(ϕ z0 )k fixes each .γj setwise. In particular, for each j the mapping .(ϕ z0 )k | γj satisfies the conditions stated for the function .ψj of ϕ z0

ϕ z0

Lemma 2.13. Let .γ1 −−−−→ γ2 −→ . . . −→ γk −−−−→ γ1 be one of the cycles for .ϕ z0 . For each .j , .j = 1, . . . , k − 1, we take any homeomorphism .ϕ˜j : 2π i Ω¯ j → Ω¯ j +1 which equals .ϕ z0 | γj on .γj and equals multiplication by .e 2(a+2) on the rest of the boundary of .Ω¯ j . Let .ψ1 be the self-homeomorphism of .Ω¯ 1 obtained

by applying Lemma 2.13 to .(ϕ z0 )k | γ1 . We put .ϕ˜k = ψ1 ◦ (ϕ˜k−1 ◦ . . . ◦ ϕ˜1 )−1 . Then on .γk the equality .ϕ˜k = (ϕ z0 )k ◦ (ϕ z0 )−k+1 = ϕ z0 holds. Denote by .ϕ˜cyc the mapping that is defined by .ϕ˜j on .Ω¯ j for .j = 1, . . . , k. The mapping that equals .ψ1 = ϕ˜ k ◦ . . . ◦ ϕ˜2 ◦ ϕ˜1 on .Ω¯ 1 and .ϕ˜j −1 ◦ . . . ◦ ϕ˜1 ◦ ϕ˜k ◦ . . . ◦ ϕ˜j +1 ◦ ϕ˜ j on .Ω¯ j has entropy zero, since by Lemma 2.14 .h(ψ1 ) = 0 and the mapping on .Ω¯ j is conjugate to .ψ1 . Moreover, the mapping is equal to the k-th iterate .ϕ˜cyc . Hence, .h(ϕ˜ cyc ) = 0. Proceed in the same way with all cycles of .γj under iteration by .ϕ z0 . 2π i The obtained mappings .ϕ˜j map .Ω¯ j homeomorphically onto .e 2(a+2) Ω¯ j . They match together to give a well-defined self-homeomorphism .ϕ˜ of the annulus   r  2π i . ≤ |ζ | ≤ 2r which equals .ϕ z0 on . |ζ | = 2r and equals rotation by .e 2(a+2) on 4  r . |ζ | = 4 . The .ψj match together to give a self-homeomorphism .ψ of the annulus ˜ = 0.   such that .ϕ˜ k = ψ. Since .h(ψ) = 0 we have .h(ϕ) Proof of Theorem 2.9. Put .ϕ0 = ϕ z0 on .Xz0 , take .r = c, and put .ϕ0 equal to the mapping .ϕ˜ of Lemma 2.12 on the annulus . 4c ≤ |ζ | ≤ 2c with distinguished coordinates .ζ around .z0 . Let .ϕ0 be equal .ϕ ˆ of Lemma 2.14  cto the homeomorphism  2π  c with .ν = − 2(a+2)·log on the annulus . ≤ |ζ | ≤ and equal to the identity on 2 8 4   c . |ζ | ≤ 8 . Then .ϕ0 is isotopic to .ϕ through self-homeomorphisms of X that fix E,    def .ϕ0 fixes the disc .Δ = |ζ | ≤ 8c around .z0 and has the same entropy as .ϕ. It follows that the isotopy classes of the two mappings .ϕ and .ϕ0 differ by a Dehn twist about a curve that is homologous to .∂Δ . By Lemma 2.14 the homeomorphism .ϕ0 can be chosen to represent any a priory given preimage of the class of .ϕ in   .M(X; {z0 }, E \ {z0 }) under the mapping .Hz0 : M(X \ Δ ; ∂Δ , E \ {z0 }) → M(X; {z0 }, E \ {z0 }). Theorem 2.9 is proved.   The following theorem concerns the slightly more general case when the self homeomorphism .ϕ of a Riemann surface of first kind is changed without increasing entropy on the union of discs around points of a subset of the set of distinguished points (rather than on a single disc around a fixed distinguished point). Theorem 2.10 Let X be a connected closed Riemann surface with a set E of distinguished points. Assume that .X\E is hyperbolic. Let .ϕ be a non-periodic

106

2 The Entropy of Surface Homeomorphisms

absolutely extremal self-homeomorphism of X which fixes E setwise. Suppose there is a .ϕ-invariant subset .E  ⊂ E. Then . ϕ is isotopic through self-homeomorphisms which fix . E setwise to a selfhomeomorphism .ϕ0 of the same entropy .h(ϕ0 ) = h(ϕ) with the following property. For each .z ∈ E  there is a closed round disc .δ z in distinguished coordinates for the quadratic differential of .ϕ, such that z is the center of .δz , and .ϕ0 maps each .δz ,  .z ∈ E , conformally onto another disc of the collection and maps the center of the source disc to the center of the target disc. Moreover, if a .ϕ-cycle of points in .E  has length k, then the iterate .ϕ0k fixes pointwise the disc .δz around each point z of the cycle. Denote by .X the bordered Riemann surface which is the complement of the union of the open discs .∪z∈E  δz . The isotopy class of the restriction .ϕ0 |X of the mapping .ϕ0 of Theorem 2.10 is determined by .ϕ up to a product of powers of Dehn twists around simple closed curves that are homologous to the boundary circles of .X . The mapping .ϕ0 of Theorem 2.10 can be chosen so that the restriction .ϕ0 |X represents the class corresponding to an a priory chosen product of powers of Dehn twists about the boundary circles. The proof of Theorem 2.10 follows along the same lines as the proof of Theorem 2.9 and is left to the reader. We have the following corollaries. We formulate Corollary 2.2 concerning mapping classes of braids separately because it is simple and useful, although it is a particular case of Corollary 2.3. We call a braid b irreducible if the associated mapping class .mb,∞ is irreducible. Corollary 2.2 Let .b ∈ Bn be an irreducible braid and let .mb = Θn (b) ∈ M(D; ∂ D, En ) be its mapping class. Then 





) = h(mb,∞ ) h(mb ) = h(mb · mΔ2k n

.

for each integer k. Recall that .mb,∞ = H∞ (mb ) (see Sect. 1.5), .Δn is the Garside element in .Bn  we denote the conjugacy class of a mapping class .m. and by .m Proof Identify the set of elements of .mb with the set of elements .mb ⊂ mb,∞ which are equal to the identity outside the unit disc .D. The entropy of a class is the infimum  ) ≥ of entropies of mappings in the class. We obtain the inequality .h( mb ) = h(m b  h(m b,∞ ). On the other hand Theorem 2.9 assigns to each absolutely extremal representative   of .m b,∞ a representative of .m b,∞ which has the same entropy and equals the identity   . Hence .h(m  in a neighbourhood of infinity and, thus represents .m b,∞ ) ≥ h(mb ) = b h( mb ). The first equality of the statement of the corollary is Lemma 2.14 and the fact that the mapping class corresponding to .Δ2n is the Dehn twist about a curve that is homologous in .D \ En to .∂D.   Corollary 2.3 Let X be a bordered Riemann surface, and let E be a finite set of distinguished points in .Int X. Let .m be an irreducible relative isotopy class of mappings, .m ∈ M(X; ∂X, E). Then for the mapping .Hζ defined by Eq. (1.9) in

2.5 The Entropy of Mapping Classes on Second Kind Riemann Surfaces

107

Sect. 1.6 the following equalities hold   h( m) = h(m · mD ) = h(H ζ m)

.

where .mD is the mapping class of an arbitrary product of powers of Dehn twists about simple closed curves which are free homotopic to boundary curves of X. The proof follows along the same lines as the proof of Corollary 2.2. It relies on Theorem 2.10.

Chapter 3

Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

In this chapter we define the entropy and the conformal module of conjugacy classes of braids. We formulate our Main Theorem, stating that the entropy of each conjugacy class of braids is inversely proportional with factor . π2 to its conformal module. Here we prove this theorem in the case of irreducible braids. The general case will be proved in the next two chapters. The Main Theorem allows to apply methods and known results concerning the more intensively studied entropy to problems whose solutions are based on the concept of the conformal module. Vice versa, for instance methods of quasiconformal mappings related to the concept of extremal length and conformal module can be applied to give upper and lower bounds for the entropy of 3-braids, the bounds differing by universal multiplicative constants (see Chap. 9). We also define the conformal module of conjugacy classes of elements of the fundamental group of complex manifolds, in particular, of the twice punctured complex plane. A version of the definition for elements of fundamental groups, not merely of their conjugacy classes, appears more effective for applications to quantitative versions of the Gromov-Oka Principle that will be given in later Chapters. In this chapter we compute these quantities explicitly in a number of examples.

3.1 Ahlfors’ Extremal Length and Conformal Module Ahlfors defined the extremal length of a family of curves in the complex plane as follows. Let .Γ be a family each member of which consists of the union of no more than countably many connected locally rectifiable (open, half-open, or closed) arcs or connected closed curves (loops) in the complex plane. (We do not require that this union reparametrizes to a single connected curve.) In this context we will call also the elements of .Γ “curves”. Ahlfors defined the extremal length of the family © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025 B. Jöricke, Braids, Conformal Module, Entropy, and Gromov’s Oka Principle, Lecture Notes in Mathematics 2360, https://doi.org/10.1007/978-3-031-67288-0_3

109

110

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

Γ as follows. Fora non-negative measurable function . in the complex plane he 2 defines .A()  = C  . For an element .γ ∈ Γ and such a function . he puts .Lγ () = γ |dz|, if . is measurable on .γ with respect to arc length and .Lγ () = ∞ otherwise. Put .L() = infγ ∈Γ Lγ (). The extremal length of the family .Γ is the following value .

λ(Γ ) = sup

.



L()2 , A()

where the supremum is taken over all non-negative measurable functions . for which .A() is finite and does not vanish. It is not hard to see from this definition that the extremal length is invariant under conformal mappings (Theorem 3 in [4], Chapter I.D). The conformal module .M(Γ ) of the family .Γ is defined to be M(Γ ) =

.

1 . λ(Γ )

Example 3.1 Let R be an open rectangle in the complex plane .C. Unless said otherwise the considered rectangles will always have sides parallel to the coordinate axes. Denote the length of the horizontal sides of R by .b and the length of the vertical sides by .a. (For instance, we may consider .R = {z = x + iy : 0 < x < b, 0 < y < a }.) The extremal length of the rectangle is defined to be the extremal length of the family of connected open arcs in the rectangle that join the two horizontal sides. An open arc is said to join the two horizontal sides if its limit sets at the two endpoints of the arc are contained in the closure of different horizontal sides. A small computation shows that the extremal length of a .R = {z = x + iy : 0 < x < b, 0 < y < a } equals .λ(R) = b and the conformal b module equals .m(R) = a (see Example 1 in [4], Chapter I.D). For a conformal mapping .ω : R → U of the rectangle R onto a domain .U ⊂ C the image U is called a curvilinear rectangle, if .ω extends to a continuous mapping ¯ and the restriction to each (open) side of R is a homeomorphism on the closure .R, onto its image. The images of the vertical (horizontal, respectively) sides of R are called the vertical (horizontal, respectively) curvilinear sides of the curvilinear rectangle .ω(R). The extremal length of the curvilinear rectangle U is the extremal length of the family of open arcs in U that join the two horizontal curvilinear sides. Thus the extremal length of U equals the extremal length of R. (See [4], Chapter I.D). Example 3.2 Ahlfors [4], Chapter I.D, defined the extremal length of an annulus A = {z ∈ C : r1 < |z| < r2 } in the complex plane as the extremal length of the family of closed curves which are contained in the annulus and represent the conjugacy class of the positively oriented generator of the fundamental group of the annulus. A simple computation (Example 3 in [4], Chapter I.D) shows that the

.

3.1 Ahlfors’ Extremal Length and Conformal Module

111

extremal length of an annulus . A = {z ∈ C : r1 < |z| < r2 } is equal to .λ(A) = r2 2π 1 r2 , and the conformal module of this annulus equals . m(A) = 2π log( r ) . Two 1 log( r ) 1

annuli of finite conformal module are conformally equivalent iff they have equal conformal module. If a manifold .Ω is conformally equivalent to an annulus A in the complex plane, its conformal module is defined to be .m(A). Recall that any domain in the complex plane with fundamental group isomorphic to the group of integer numbers .Z is conformally equivalent to an annulus. Example 2 in [4], Chapter I.D, shows that the extremal length of the family of open arcs .γ in .A = {z ∈ C : r1 < |z| < r2 } that join the two boundary circles is equal r

to .

log r2 1 2π

.

Example 3.3 The following example is a generalization of Example 3.1. Let .Φ be a real .C 1 -function .Φ on the closure of an open interval J , and let .b be a positive def

number. We consider the curvilinear rectangle .RJ,Φ,b = {x + iy ∈ C : y ∈ J, x ∈ (Φ(y), Φ(y) + b)} whose horizontal sides are the horizontal pieces of its boundary. The following lemma holds. Lemma 3.1 Let .Φ be a real .C 1 -function on the closure of an open interval J and let .b be a positive number. Denote by .ΓΦ the set of curves in the curvilinear rectangle .RJ,Φ,b = {x +iy ∈ C : y ∈ J, x ∈ (Φ(y), Φ(y)+b)} which join the two horizontal curvilinear sides. Suppose the absolute value .|Φ  | of the derivative of .Φ is bounded by the constant C. Then λ(ΓΦ ) ≤ (1 + C 2 )λ(Γ0 ),

.

where .Γ0 is the family corresponding to the function .Φ0 which is identically equal to zero. Proof The proof is similar to the arguments used in Example 1 in Chapter 1 of [4], Chapter I.D. For any measurable function . on .C and any .x ∈ (0, b) we have  .

J

 (x + Φ(y) + iy) 1 + Φ  (y)2 dy ≥ LΓΦ ().

Integrate over the interval .(0, b) and apply Fubini’s Theorem and Hölder’s inequality. Using the bound for .|Φ  | we obtain  

  2 · (1 + C 2 )dm2

dm2

.

RΦ,b

RΦ,b

1 2

≥ bLΓΦ ().

Denote by .|J | the length of the interval J . We obtain b|J |(1 + C 2 )A() ≥ b2 LΓΦ ()2 .

.

112

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

Hence, .

|J | LΓΦ ()2 ≤ · (1 + C 2 ) = λ(Γ0 )(1 + C 2 ). A() b

Taking the supremum over all measurable functions . with finite non-vanishing integral we obtain λ(ΓΦ ) ≤ (1 + C 2 )λ(Γ0 ).

.



The lemma is proved.

For later use we formulate three theorems of Ahlfors. For two families .Γ1 and .Γ2 as above the following relation is introduced by Ahlfors: .Γ1 < Γ2 if each “curve” .γ2 ∈ Γ2 contains a “curve” .γ1 ∈ Γ1 . Suppose .Γ1 and .Γ2 are contained in disjoint measurable sets. Ahlfors defines the sum .Γ1 + Γ2 of two such families as follows. Consider a “curve” .γ1 ∈ Γ1 and a “curve” .γ2 ∈ Γ2 . The sum .γ1 + γ2 is the ” curve” consisting of the union of .γ1 and .γ2 . The set .Γ1 + Γ2 is the set consisting of all sums .γ1 + γ2 for .γ1 ∈ Γ1 and .γ2 ∈ Γ2 . The following theorems were proved by Ahlfors. Theorem A ([4], Ch.1.D Theorem 2) If .Γ  < Γ then .λ(Γ  ) < λ(Γ ). Theorem B ([4], Ch.1.D 4) If the families .Γj are contained in disjoint  Theorem  measurable sets then . λ(Γj ) ≤ λ( Γj ). Corollary 3.1 Let R be a rectangle and let .α be an open arc in R, that extends to a closed arc with endpoints on different open vertical sides of R. The arc .α cuts R into two curvilinear rectangles .R1 and .R2 . The following estimate holds: λ(R) ≥ λ(R1 ) + λ(R2 ) .

.

(3.1)

Further, let .A = {z ∈ C : r1 < |z| < r2 } be an annulus in the complex plane and α an open arc in A that extends to a closed arc in .A¯ with endpoints on different boundary circles. The arc cuts A into a curvilinear rectangle R. The following inequality holds

.

λ(A) ≥ λ(R) .

.

(3.2)

Proof To prove inequality (3.1) we let .R1 be the curvilinear rectangle below of .α and .R2 the curvilinear rectangle above of .α. Let .Γj , j = 1, 2, be the family of open arcs in .Rj that join the two open horizontal curvilinear sides of .Rj , and let .Γ be the family of open arcs in R that join the two open horizontal sides of R. Then by Theorem A (see [4], Ch.1 Theorem 2) .λ(Γ ) ≥ λ(Γ1 + Γ2 ). By Theorem B (see [4], Ch.1 Theorem 4) .λ(Γ1 + Γ2 ) ≥ λ(Γ1 ) + λ(Γ2 ). By Example 3.1 of Sect. 3.1 the equalities .λ(Γ ) = λ(R), .λ(Γj ) = λ(Rj ), .j = 1, 2 hold.

3.2 Another Notion of Extremal Length

113

For the proof of inequality (3.2) we let .Γ (A) be the family of curves in A that represent the positively oriented generator of the fundamental group of A. By Example 3.2 of Sect. 3.1 the extremal length .λ(A) is equal to the extremal length .λ(Γ (A)). Further, the set .A \ α is a curvilinear rectangle with horizontal sides being the strands of .α that are reachable from .A \ α by moving clockwise or counterclockwise, respectively. Its extremal length .λ(A \ α) is the extremal length .λ(Γ (A \ α)) in the sense of Ahlfors [4] of the family .Γ (A \ α) of curves in the curvilinear rectangle .A \ α that join the two horizontal sides of the curvilinear rectangle. By Theorem A (see [4], Ch.1 Theorem 2) the inequality λ(Γ (A \ α)) ≤ λ(Γ (A))

.

(3.3)



holds.

3.2 Another Notion of Extremal Length We will use another notion of the extremal length of families of curves. This notion is in the spirit of the definition of the Kobayashi metric and applies also to arbitrary families of curves, that are homotopy classes of curves contained in complex manifolds, maybe, of dimension bigger than one. Definition 3.1 Let .X be a complex manifold, X a domain in .X , and let .E1 and .E2 be relatively closed subsets of X. Let .h =E1 h E2 be a homotopy class of curves in X with initial point in .E1 and terminal point in .E2 . If .E1 = E2 we write .h E1 instead of .h =E h E . A continuous mapping f from an open rectangle into X which admits a 1 2 continuous extension to the closure of the rectangle (denoted again by f ) is said to represent .h if the lower open horizontal side is mapped to .E1 , the upper horizontal side is mapped to .E2 and the restriction of f to the closure of each maximal vertical segment in the rectangle represents .h. The extremal length of homotopy classes of such curves is defined as follows. Definition 3.2 For a complex manifold .X , a domain X in .X , two relatively closed connected subsets .E1 and .E2 of X and a homotopy class .h =E1 hE2 of curves in X with initial point in .E1 and terminal point in .E2 the extremal length .Λ(h) is defined as Λ(h) = inf{λ(R) : R a rectangle which admits a holomorphic mapping to

.

X that represents h} .

(3.4)

The conformal module .M(h) of the class .h is defined as M(h) =

.

1 = sup{m(R) : R a rectangle which admits a holomorphic Λ(h)

mapping to X that represents h} .

(3.5)

114

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

A similar definition can be given for free homotopy classes of curves in a complex manifold, in other words for curves representing a conjugacy class of elements of the fundamental group of the manifold. Definition 3.3 Let X be a complex manifold, and .eˆ a conjugacy class of elements of the fundamental group of X. A continuous mapping from an annulus 1 .A = {z ∈ C : ˆ r < |z| < r} with .1 < r ≤ ∞ into X is said to represent .e, if the restriction to the circle .{|z| = 1} with positive orientation (and, hence, the restriction to each curve that is free homotopic to this circle) represents .e. ˆ The extremal length and the conformal module of free homotopy classes of curves is defined as follows. Definition 3.4 Let X be a complex manifold, and .eˆ a conjugacy class of elements of the fundamental group of X. The extremal length .Λ(e) ˆ of .eˆ is defined as def

Λ(e) ˆ = inf{λ(A) : A an annulus that admits a holomorphic mapping to

.

X that represents e} ˆ .

(3.6)

The conformal module .M(e) ˆ of .eˆ is defined as def

M(e) ˆ =

.

1 = sup{m(A) : A an annulus that admits a holomorphic Λ(e) ˆ

mapping to X that represents e} ˆ .

(3.7)

For a curve .γ we consider the curve .γ −1 obtained from .γ by inverting orientation. For a family of curves .Γ we let .Γ −1 be the family of curves .γ −1 with .γ ∈ Γ . It follows immediately from the definitions that .Λ(h) = Λ(h−1 ) −1 ). Indeed, suppose a mapping .f : R → X represents .h. Put and .Λ(e) ˆ = Λ(e −1 . The .−R = {−z : z ∈ R} and .f− (z) = f (−z). Then .f− : −R → X represents .h argument for .eˆ is similar. We need the following two lemmas. Lemma 3.2 Let R and .R  be rectangles with sides parallel to the axes. Suppose  .S is the vertical strip bounded by the two vertical lines which are prolongations of the vertical sides of the rectangle .R  . Let .f : R → S  be a holomorphic mapping with continuous extension to the closure that takes the two horizontal sides of R into different horizontal sides of .R  . Then λ(R) ≥ λ(R  ) .

.

Equality holds if and only if the mapping is a surjective conformal mapping from R onto .R  . The following lemma concerns holomorphic mappings between annuli.

3.2 Another Notion of Extremal Length

115

Lemma 3.3 Let A and .A be two annuli and let f be a holomorphic mapping from A into .A which induces an isomorphism on fundamental groups. Then λ(A) ≥ λ(A ) .

.

Equality holds if and only if the mapping is a conformal mapping from A onto .A . Proof of Lemma 3.2 Normalize the rectangles and the mapping so that .R = {x + iy : x ∈ (0, 1), y ∈ (0, a)} and .R  = {x + iy : x ∈ (0, 1), y ∈ (0, a )} . Denote the continuous extension of f to the closure of R again by f . We may assume that f maps the upper side of .R to the upper side of .R  and the lower side of .R to the lower side of .R  . Put .u = Re f and .v = Im f . Then a =



1

.

a dx =

0



1



0

=

 a 0

≤

 dy

 a

 a

1

(v(x, a) − v(x, 0))dx =

dx 0

0

1

dx 0

∂ u(x, y) = ∂x

 a

dy

∂ v(x, y) ∂y

(3.8)

dy (u(1, y) − u(0, y))

0

1 dy = a.

0

We used the Cauchy-Riemann equations. To justify, for instance, the third equality we each .x ∈ (0, 1) the limit of the equality .v(x, a − ε) − v(x, ε) =  a−εtake for ∂ dy v(x, y) for .ε → +0 and use that v is continuous on the closure of R. ε ∂y The relation (3.8) implies the inequality .a ≤ a. If .a = a then .u(1, y) − u(0, y) = 1 for each .y ∈ (0, a). Hence, the left side of .R is mapped to the left side of .R  and the right side of .R is mapped to the right side of .R  . Since also the lower side of .R is mapped to the lower side of .R  and the upper side of .R is mapped to the upper side of .R  the image of the positively oriented boundary curve of .R has index 1 with respect to any point of .R  and index ¯ By the argument principle .f (R) = R  and f 0 with respect to each point in .C \ R.  takes each value in .R exactly once. Hence, f is a conformal map of R onto .R  .

Proof of Lemma 3.3 Assume the annuli A and .A have center 0, smaller radius 1 and larger radius r and .r  , respectively. The set .A \ (0, ∞) is conformally equivalent to the rectangle .R = {ξ + iη : ξ ∈ (0, log r), η ∈ (0, 2π )}. The exponential function covers the annulus .A by the strip .S  = {ξ + iη : ξ ∈ (0, log r  ), η ∈ R}. We obtain a holomorphic mapping .g = U + iV from .R to .S  for which either .V (ξ, 2π ) = V (ξ, 0) + 2π or .V (ξ, 2π ) = V (ξ, 0) − 2π for .ξ ∈ (0, log r). Assume without loss of generality that the first option holds. Then  2π log r =

.

0

log r

 (V (ξ, 2π ) − V (ξ, 0))dξ =



log r

dξ 0

2π

dη 0

∂ V (ξ, η) ∂η (3.9)

116

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

Fig. 3.1 The domains .Ga,b and .Ga±,b

 =



log r

2π

dξ 

0

≤

0 2π

∂ dη U (ξ, η) = ∂ξ



2π

(U (log r, η) − U (0, η))dη

0

log r  dη = 2π log r  .

0

Equality .r = r  holds iff f maps the bigger circle of A to the bigger circle of .A and maps the smaller circle of A to the smaller circle of .A . Since the map f induces an isomorphism of fundamental groups an application of the argument principle shows that f is a conformal mapping of A onto .A .

Example 3.4 Let .0 < a < b be two positive numbers. Denote by .Γ a,b the family a ,b of curves in the upper half-plane .C+ that join the two half-circles .C± = {z ∈ a+b b−a C+ : |z ∓ 2 | = 2 } with diameter .(a, b) and .(−b, −a), respectively. Similarly, a,b let .Γ+a,b be the family of curves in the upper half-plane .C+ that join .C+ with the imaginary half-axis.

Let .Ga+,b be the domain .Ga+,b = {z ∈ C : Rez > 0, Imz > 0, |z − a+2 b | > b − a},

a a,b (.Ga,b , respectively) as and .Ga,b = {z ∈ C+ : |z ± a+2 b | > b− + 2 }. We consider .G

a,b a,b curvilinear rectangles with curvilinear horizontal sides equal to .C± (equal to .C+ and .{z ∈ C+ : Rez > 0}, respectively). See Fig. 3.1. Notice, that the elements of a,b .Γ+ are not supposed to be contained in .Ga+,b , and the respective remark concerns a,b and .Γ a,b . .Γ −

3.2 Another Notion of Extremal Length

117

The following lemma holds. Lemma 3.4 a,b .λ(G± )

√ √ 1 ( a + b)2 = log , π b−a

a,b

λ(G

√ √ 2 ( a + b)2 ) = log . π b−a

(3.10)

√ √ ( a + b)2 2 . ) = log π b−a

(3.11)

Moreover, a,b .Λ(Γ± )

√ √ ( a + b)2 1 , = log π b−a

Λ(Γ

a,b

Recall that .Λ(Γ±a,b ) ( .Λ(Γ a,b ), respectively) denotes the extremal length of .Γ±a,b (.Γ a,b , respectively) in the sense of Definition 3.2. Proof To prove equality (3.10) we consider the Möbius transformation .T a,b that maps 0 to 0, .∞ to 1, .a to a positive real number .t < 12 , and .b to .1 − t. .T a,b is a

conformal mapping of the Riemann sphere .P1 onto itself, that takes .Ga+,b to a halfannulus and maps the horizontal curvilinear sides of .Ga+,b to the half-circles. Each a,b Möbius transformation has the form .z → az+b cz+d . The first two conditions for .T a imply that .b = 0 and . c = 1. To determine the Möbius transformation we ignore the normalization by the condition that the matrix with entries .a, b, c, d has determinant a = t, 1 and put .a = c = 1. The remaining two conditions give the equations . a+d b . = 1 − t, hence by eliminating d, b+d a(1 − t)2 = bt 2 .

.

We obtain the quadratic equation (b − a)t 2 + 2at − a = 0

.

in t. Its solutions are a,b t± (= t± )=

.

√ −a ± ab . b−a

(3.12)

The value .t− is negative, so the required solution is .t+ < 1 − t+ . The half-annulus, √ whose boundary circles have center . 12 and radii . 12 − t+ = 12 − −ab+−aab and . 12 , is a curvilinear rectangle which is the conformal image of a true rectangle with 1 . Hence, .λ(Ga+,b ) = horizontal side length .π and vertical side length .log 1−2t + 1 π

1 log 1−2t . Since +

√ √ ( a + b)2 1 1 b−a √ , . = = √ = 1 − 2t+ b−a b + a − 2 ab 1 − 2 −ab+−aab

(3.13)

118

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

the extremal length .λ(Ga+,b ) of the curvilinear rectangle .Ga+,b with horizontal sides a,b a = {z ∈ C+ : |z − a+2 b | = b− being .C+ 2 } and the imaginary half-axis is equal to √

√

2 log ( ab+−ab) . Hence, we obtain equality (3.10) for .λ(Ga+,b ). Equalities (3.10) for a,b a,b ) are obtained by the same reasoning. .λ(G− ) and for .λ(G For the proof of equality (3.11) we consider the conformal mapping .R → Ga,b a,b of a true rectangle R onto .Ga,b that maps the lower horizontal side of R onto .C− a,b and upper side to .C+ . Apply Schwarz Reflection Principle to each horizontal side of R. We obtain a conformal mapping from a true rectangle, denoted by 3R, that has the same horizontal side length as R and three times the vertical side length of R, onto the upper half-plane with two smaller half-discs removed. The removed half-discs are symmetric with respect to the imaginary axis. After repeating n times the application of the reflection principle we obtain a conformal mapping from a rectangle .3n R onto a domain .Ωn which is equal to the left half-plane with two half-discs removed. The half-discs are symmetric with respect to the imaginary axis. The domains .Ωn are increasing. The diameter of the removed half-discs at step n tends to zero for .n → ∞ since the extremal length of n .3 R tends to .∞. We obtain a conformal mapping, denoted by .c, of an infinite strip  .S onto the left half-plane. Let f be any holomorphic mapping from a rectangle to the left half-plane whose a,b a,b and the lower side to .C− . extension to the closure takes the upper side to .C+ −1 a , b Apply Lemma 3.2 to the mapping .c ◦ f . Equality (3.11) for .Λ(Γ ) follows.

The equality for .Λ(Γ±a,b ) is proved in the same way. .

1 π

The Fundamental Group and the Relative Fundamental Group of the Twice Punctured Complex Plane The fundamental group .π1 (C \ {0, 1}, 12 ) of the twice punctured complex plane with base point is a free group in two generators .a1 and .a2 , were .a1 is represented by a curve that surrounds 0 counterclockwise, and .a2 is represented by a curve that surrounds 1 counterclockwise. The fundamental group 1 .π1 (C \ {0, 1}, ) is canonically isomorphic to the relative fundamental group .π1 (C \ 2 {0, 1}, (0, 1)). The elements of the latter group are homotopy classes of arcs in .C \ {0, 1} with initial point and terminal point in the interval .(0, 1). (The initial point may differ from the terminal point.) The isomorphism assigns to the element .e ∈ π1 (C \ {0, 1}, 12 ) represented by a curve .γ with base point . 12 the element in the relative fundamental group represented by .γ . For an element .e ∈ π1 (C \ {0, 1}, 12 ) we denote be .e(0,1) the image of e under the mentioned canonical isomorphism. Example 3.5 The extremal length of .(a1k )(0,1) and .(a2k )(0,1) for .k ∈ Z equals zero, √ and the extremal length of .(a1−1 a2 )(0,1) equals . π2 log(2 + 5). This can be seen as follows. The class .(a1 )(0,1) can be represented by the holomorphic mapping .z → ez from the rectangle .R = {z = x + iy : x ∈ (−∞, 0), y ∈ (0, 2π )}, of extremal length .λ(R) = 0 into .C \ {0, 1}. There is a similar representation for all powers of .(a1 )(0,1) and for all powers of .(a2 )(0,1) , showing that the extremal length of these elements of the relative fundamental group equals zero.

3.2 Another Notion of Extremal Length

119

Fig. 3.2 A lift of .a1−1 a2 to the universal covering

A representing arc for .(a2 )(0,1) lifts under the universal covering map .P : C+ → C \ {0, 1} to an arc in .C+ that joins the half-circle with diameter .(0, 1) with the half-circle with diameter.( 23 , 1) (see Sect. 1.2). A representing arc for .(a1 )(0,1) lifts under the universal covering map .P : C+ → C \ {0, 1} to an arc in .C+ that joins the half-circle with diameter .(0, 1) with the half-circle with diameter −1 .(−1, 0). Hence, a representing arc for .(a 1 a2 )(0,1) lifts to an arc in .C+ that joins the half-circle with diameter .(−1, 0) with the half-circle with diameter .( 23 , 1). (See Fig. 3.2.) Any holomorphic mapping f from a rectangle R into .C \ {0, 1}, that represents .(a1−1 a2 )(0,1) , lifts to a holomorphic mapping from R into .C+ that takes the horizontal sides to the mentioned circles. The Möbius transformation z .z → z+1 maps .C+ onto itself and takes .−1 to .∞ and 0 to 0. Moreover it 2 maps . 3 to . 25 and 1 to . 12 . Compose this mapping with the mapping .z → 10z. The composition .c is a conformal self-mapping of .C+ that takes .−1, 0, 23 , 1 to .∞, 0, 4, 5. Composing the lift of f to .C+ with this conformal self-mapping .c of .C+ , we obtain a holomorphic mapping .c ◦ f from R into .C+ that takes the lower horizontal side to the imaginary half-axis and the upper horizontal side to the circle with diameter .(4, 5). Vice versa, for each holomorphic mapping g from a rectangle into .C+ , that takes the lower horizontal side into the imaginary axis and the upper horizontal side into the circle with diameter .(4, 5), the composition .c−1 ◦g is a holomorphic mapping of a rectangle into the twice punctured complex plane that represents .(a1−1 a2 )(0,1) . Lemma 3.4 implies the equality .Λ((a1−1 a2 )(0,1) ) = √ 2 √ 1 2 λ(G4,5 + ) = π log(2 + 5) = π log(2 + 5). Example 3.6 The extremal length of the free homotopy class of closed curves √  −1 .a a equals . 2 log(3 + 2 2). 1

2

π

120

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

This can be seen as follows. We lift a representative .γ of .a2 ∈ π1 (C \ {0, 1}, 12 ) to a curve in .C+ with initial point q on the half-circle with diameter .(0, 1). The lift joins the half-circle with diameter .(0, 1) with the half-circle with diameter .( 23 , 1). There is a unique covering transformation .Ta2 that maps the initial point of this lift to its terminal point. To obtain an explicit expression for the mapping .Ta2 , we make a change of the base point along a curve .α in .C \ {0, 1}. More precisely, suppose .α : [0, 1] → C \ {0, 1} is an arc with initial point . 12 , such that .α((0, 1]) is contained in .C+ . For .t ∈ [0, 1] we let .αt be the curves obtained by moving along .α from the initial point .α(0) = 12 to the point .α(t). Consider the continuous family .Isαt (γ ), t ∈ [0, 1], of representatives of the free homotopy class .a2 , and their lift to the universal covering with initial point in .D0 if .t ∈ (0, 1]. The terminal point of each such lift is contained in the geodesic triangle .D2 with vertices . 23 , 1, 12 (see Sect. 1.2). The set of covering transformations is discrete, and the family of covering transformations, that map the initial point of the considered lift of .Isαt (γ ) to its terminal point, depends continuously on the parameter. Hence, each transformation of the family equals .Ta2 . Hence, .Ta2 maps .D0 into .D2 and maps the half-circle with diameter .(0, 1) to the half-circle with diameter .( 23 , 1). Consider in the same way the mapping .Ta −1 ,

2

that is associated to the inverse .a2−1 , and its lift with initial point in .D2 . We see that the covering transformation .Ta −1 maps the geodesic triangle .D2 with vertices

.

2

2 1 3 , 1, 2

into .D0 , and maps the half-circle with diameter .( 23 , 1) into the half-circle with diameter .(0, 1). The composition .Ta −1 Ta2 is the identity. Hence, .Ta2 maps .D0 2

conformally onto the geodesic triangle .D2 with vertices . 23 , 1, 12 , and maps the halfcircle with diameter .(0, 1) onto the half-circle with diameter .( 23 , 1). Hence, .Ta2 |D0 coincides with the double reflection. This can be used for explicit computation of .Ta2 . One can also use the fact that .Ta2 is a conformal self-mapping of .C+ , hence extends to a Möbius transformation, that takes the vertex 0 (adjacent to the sides 2 1 . and .C) to . , the vertex .∞ (adjacent to the sides . and r) to . , and the vertex 3 2 z−2 . Its 1 (adjacent to r and .C) to 1. This Möbius transformation equals .Ta2 = 2z−3 −3z+2 −1 inverse equals .Ta2 = Ta −1 = −2z+1 . 2

In the same way we will now associate to .a1 ∈ π1 (C \ {0, 1}, 12 ) a covering transformation .Ta1 . We lift a representative of .a1 to a curve with initial point in the half-circle with diameter .(0, 1). This lift joins the half-circle with diameter .(0, 1) with the half-circle with diameter .(−1, 0). The covering transformation .Ta1 that takes the initial point of the lift to its terminal point, maps .D0 conformally onto the geodesic triangle with vertices .−1, − 12 , 0, and maps the half-circle with diameter .(0, 1) to the half-circle with diameter .(−1, 0). Hence, .Ta1 maps 0 to 0, 1 to .−1, z . The inverse mapping equals and .∞ to .− 12 . A small computation gives .Ta1 = 1−2z z −1 .Ta = Ta −1 = 2z+1 . 1 1

3.2 Another Notion of Extremal Length

121

The covering transformation corresponding to .a1−1 a2 equals .Ta −1 a2 (z) = Ta2 ◦ 1

 √ 32 3z+2 −1 Ta1 (z) = 4z+3 . The matrix .A = has two real eigenvalues .t± = 3 ± 2 2 43 whose product equals the

determinant 1. The matrix A can be conjugated to a  t+ 0 −1 by a matrix B with real entries whose columns diagonal matrix .B AB = 0 t− are eigenvectors. The Möbius transformation corresponding to B maps the real axis to itself. Changing if needed the direction of one of the eigenvectors we may assume that the Möbius transformation maps .C+ to itself. We see that the mapping Ta −1 a2 : C+ ⊃ can be conjugated by a conformal self-mapping of .C+ to the mapping 1 √ 2 √ 2 def t+ z 2 .Tλ , Tλ (z) = t− = t+ z = (3 + 2 2) z with .λ = (3 + 2 2) . The quotient .C+ T −1  is conformally equivalent to the quotient .C+ Tλ . This quotient is an a1 a2 √ √ annulus of extremal length . π1 log(3 + 2 2)2 = π2 log(3 + 2 2). Consider a holomorphic mapping f of an annulus .A = {z ∈ C : r < |z| < R} −1 into .C \ {0, 1} that represents the free homotopy class .a a . Cut the annulus along 1

2

the arc . = (r, R), and lift the restriction .f |A \  to a mapping .f˜ into .C+ . Let .− and .+ be the strands of . that are accessible from a point in .A \  by moving clockwise, or counterclockwise, respectively, and for each point .p ∈  we let .p± ∈ ± be the points corresponding to p. Then the equality .f˜(p+ ) = Ta −1 a2 (f˜(p− )) 1 holds for the continuous extension of .f˜ to .± . This means, that the mapping .f˜ from .(A \ ) ∪ − ∪ + descends to a holomorphic mapping from A into .C+ Tλ −1 that represents .a 3.3 the extremal length of A is not smaller than 1 a2 . By Lemma √ 2 .λ(C+ Tλ ) = log(3 + 2 2), and there is a conformal mapping of an annulus A π √ −1 of extremal length . 2 log(3 + 2 2) onto .C T that represents .a a . π

+

λ

1

2

The Extremal Length of Conjugacy Classes of Braids and the Extremal Length of Braids with Totally Real Horizontal Boundary Values These are conformal invariants of braids and conjugacy classes of braids which will play a key role later. Recall that for a subset A of the complex plane .C we defined the configuration space .Cn (A) = {(z1 , . . . , zn ) ∈ An : zi = zj for .i = j } of n particles moving along A without collision. Each permutation in the symmetric group .Sn acts on .Cn (A) by permuting the coordinates. The quotient .Cn (A)Sn is called the symmetrized configuration space related to A. Recall that the natural projection .Cn (C) → Cn (C)Sn is denoted by .Psym . Choose a base point .En ∈ Cn (R)Sn . Recall that braids on n strands (nbraids for short) with base point .En are homotopy classes of loops with base point .En in the symmetrized configuration space, equivalently, they are elements of the fundamental group .π1 (Cn (C)Sn , En ) of the symmetrized configuration space with base point .En .

122

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

Recall that conjugacy classes of n-braids are free homotopy classes of loops ˆ of a in .Cn (C)Sn ∼ = Pn . According to Definition 3.4 the extremal length .Λ(b) ˆ ˆ conjugacy class of n-braids .b is defined as .Λ(b) = infA∈A λ(A), where .A denotes the set of all annuli which admit a holomorphic mapping into .Cn (C)Sn that ˆ represents .b. To define the extremal length of braids with totally real horizontal boundary def

values we consider the totally real subspace .Etrn = Cn (R)Sn of .Cn (C)Sn . The totally real subspace .Cn (R)Sn of . Cn (C)Sn is connected and simply connected. Indeed, the totally real subspace .Cn (R) of .Cn (C) is the union of the connected components . {(x1 , . . . , xn ) ∈ Rn : xσ (1) < xσ (2) < . . . < xσ (n) } over all permutations .σ ∈ Sn . Thus .Cn (R) is invariant under the action of .Sn and the quotient is homeomorphic to .{(x1 , . . . , xn ) ∈ Rn : x1 < x2 < . . . < xn } . Hence the claim. The fundamental group .π1 ( Cn (C)Sn , En ) is isomorphic to the relative fundamental group .π1 ( Cn (C)Sn , Cn (R)Sn ). The elements of the latter group are homotopy classes of arcs in . Cn (C)Sn with endpoints in . Cn (R)Sn . The isomorphism between the two groups is obtained as follows. Since the fundamental groups with different base point are isomorphic, we may assume that .En is contained in the totally real subspace . Cn (R)Sn . Each element of .π1 ( Cn (C)Sn , En ) is a subset of an element of .π1 ( Cn (C)Sn , Cn (R)Sn ). Vice versa, since . Cn (R)Sn is connected and .En is contained in . Cn (R)Sn , each class in .π1 ( Cn (C)Sn , Cn (R)Sn ) contains a class in .π1 ( Cn (C)Sn , En ). Since . Cn (R)Sn is simply connected, each class in .π1 ( Cn (C)Sn , Cn (R)Sn ) contains no more than one class of .π1 ( Cn (C)Sn , En ). Indeed, if two loops in . Cn (C)Sn with base point .En are homotopic as loops in . Cn (C)Sn with varying base point in . Cn (R)Sn then they are homotopic as loops in . Cn (C)Sn with fixed base point .En . Let .b ∈ Bn be a braid. Denote its image in the relative fundamental group .π1 ( Cn (C)Sn , Cn (R)Sn ) by .btr . We are now ready to define for any braid its extremal length with totally real boundary values (and the conformal module with totally real boundary values, respectively). Definition 3.5 Let .b ∈ Bn be an n-braid. The extremal length .Λtr (b) with totally real horizontal boundary values is defined as Λtr (b) = inf{λ(R) : R a rectangle which admits a holomorphic mapping to

.

Cn (C)Sn that represents btr } . The conformal module .Mtr (b) of b with totally real horizontal boundary values, respectively, is defined as Mtr (b) = sup{m(R) : R a rectangle which admits a holomorphic mapping to

.

Cn (C)Sn that represents btr } .

3.2 Another Notion of Extremal Length

123

We will also use the notation .Λ(btr ) for .Λtr (b) and the notation .M(btr ) for Mtr (b). Note that the two invariants are inverse to each other. Sometimes it is more convenient to work with the extremal length, some other times it is more appropriate to speak about the conformal module. Recall that .Δn denotes the Garside element in the braid group .Bn .

.

2 ) and .Λtr (b) = ˆ = Λ(bΔ Lemma 3.5 For each braid .b ∈ Bn the equalities .Λ(b) n Λtr (bΔn ) = Λtr (Δn b) hold. Proof Let .R = {x + iy : x ∈ (0, 1), y ∈ (0, a)}, and suppose a holomorphic mapping .f = {f1 , f2 , . . . , fn } : R → Cn (Cn )Sn represents .btr . The π π mapping .ζ → e a ζ f (ζ ) = e a ζ {f1 (ζ ), f2 (ζ ), . . . , fn (ζ )} is holomorphic on R and represents .(b Δn )tr . Since .Λtr (b−1 ) = Λtr (b) for each braid b, we obtain ˆ is proved in the same way. .Λtr (Δn b) = Λtr (b). The stated relation for .Λ(b)

Example 3.7 The extremal length with √ totally real horizontal boundary values of 2) σ1−1 σ2 equals .Λ((σ1−1 σ2 )tr ) = log(3+2 . π

.

To prove this fact, we let .γ be an arc in .C3 (C)S3 that represents the homotopy class .(σ1−1 σ2 )tr ∈ π1 (C3 (C)S3 , C3 (R)S3 ). The initial point (and also the terminal point) of .γ is an unordered triple of distinct points, each contained in the real axis. Lift .γ to an arc .γ˜ = (γ˜1 , γ˜2 , γ˜3 ) in .C3 (C), .Psym (γ˜ ) = γ , so that the initial point of the lift .γ˜ is an ordered triple .(x 1 , x2 , x3 ) with .xz22−z 0, Imz > 0, |z − | > }, such that the horizontal sides of R 4 4 correspond to the imaginary half-axis, and to the half-circle .C+ ∩{|z − 34 | = 14 }. The

composition .P ◦ c of .c with the universal covering map .P is a holomorphic mapping from the rectangle into .C \ {0, 1} that represents the curve .Mγ˜ of Fig. 3.3 and takes the horizontal sides of the rectangle into .(−∞, 0) and .(1, ∞), respectively. The mapping .R  z → (0, P ◦ c(z), 1) ∈ C3 (C) projects under .Psym to a mapping into −1 .C3 (C)S3 that represents .(σ 1 σ2 )tr . In the same way we see that the extremal length .Λ((σ1−1 σ22 σ1−1 )tr ) equals √ 2 log(3+2 2) . . For a curve .γ representing .(σ −1 σ 2 σ −1 )tr the curve .M˜ γ˜ joins the two π

half-circles .{|z ± 34 | = 14 } ∩ C+ .

1

2 1



Example 3.8 The extremal length of the conjugacy class of 3-braids .σ1−1 σ2 equals

√

Λ(σ1−1 σ2 ) = π2 log 3+2 5 . This is the smallest non-vanishing extremal length among conjugacy classes of 3-braids.

.

To prove these facts, we will first describe the action on the Teichmüller space T (0, 4) ∼ = C+ of the modular transformations .Tσ1 and .Tσ2 corresponding to .σ1  and .σ2 . We will use the mapping .PA ,n : C3 (C) → (C \ {0, 1} and the covering  .P : T (0, 4) → C \ {0, 1} identified with .P : C+ → C \ {0, 1} as in Example 3.7 T of this section. Recall that the modular transformation .Tb associated to an n-braid b is the modular transformation .ϕb∗ of a self-homeomorphism .ϕb of .P1 that represents the mapping class .mb,∞ corresponding to b. We consider the modular transformation .Tb for the 3-braids .b = σ1 and .b = σ2 . With the same normalization as in Example 3.7 of this section the mapping .ϕb fixes .∞ and maps the set .E3 = {0, 12 , 1} onto itself. Such a homeomorphism .ϕb can be obtained using a parameterizing isotopy .ϕt for a .

3.2 Another Notion of Extremal Length

125

geometric braid .γ representing b, i.e. a continuous family of self-homeomorphisms of .P1 such that .ϕt (E3 ) = γ (t), t ∈ [0, 1]. We may take .ϕb = ϕ 1 and .Tb = (ϕ 1 )∗ . Lift the base point .E3 = {0, 12 , 1} ∈ C3 (C)S3 under .Psym to the point .E˜ 3 = (0, 1 , 1) of .C3 (C). Then .P  (E˜ 3 ) = 1 . Take the preimage . 1+i of . 1 under .P. The A,n

2

2

2

2

image of . 1+i 2 under .Tσ1 can be obtained as follows. Consider a curve .γ1 in .C3 (C)S3 that represents .σ1 with base point .E3 and its  (γ˜ ) (identified with lift .γ˜1 under .Psym with initial point .E˜ 3 . The projection .PA ,n 1 the curve .t → Mγ˜1 (t) ) is a curve in .C \ {0, 1} with initial point . 12 . The curve .γ1 may  (γ˜ ) joins . 1 with a point in the negative real axis, and the be chosen so that .PA 2 ,n 1  (γ˜ ) are contained in upper half-plane .C . interior points of the curve .PA + ,n 1 Consider the lift .M˜ γ˜1 of .Mγ˜1 under .P : C+ → C \ {0, 1} with initial point 1+i 1+i ˜ . 2 . The lift .Mγ˜1 of .Mγ˜1 joins . 2 with a point on the positive imaginary axis. The interior of .M˜ γ˜1 is contained in .D0 . The terminal point of .M˜ γ˜1 is the image of . 1+i 2 under .Tσ1 . Indeed, the following commutative diagram (see also Lemma 1.5 )

.

implies the equality Tσ1 (

.

1+i ) = ϕσ∗1 ([Id]) = [ϕσ1 ] = [ϕ 1 ] = M˜ γ˜1 (1) . 2

We proved that .Tσ1 ( 1+i 2 ) is contained in the imaginary half-axis. 0,1 We will now prove that .Tσ1 maps the half-circle .C+ = {|z − 12 | = 12 } ∩ C+ 0,1 into the positive imaginary half-axis. Take any point in .C+ and join it with . 1+i 2 0,1 by an arc .α(s), s ∈ [0, 1], in .C+ . Let .ws , s ∈ [0, 1], be a continuous family of def

self-homeomorphisms of .P1 , that fix .∞, such that .w0 = Id and .τs = [ws ] = α(s). def def (See also Lemma 1.4.) Notice that for .E˜ 3s = ws (E˜ 3 ) the point .x(s) = ME˜ s is 3 contained in .(0, 1). The equality Tσ1 (τs ) = (ϕσ11 )∗ ([ws ]) = [ws ◦ ϕσ11 ]

.

holds. Since .ϕσ11 acts on .E˜ 3 by permuting the first two coordinates, the homeomorphism .ws ◦ ϕσ11 acts on .ws (E˜ n ) = E˜ ns by permuting the first two coordinates. This

126

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem def

means that .x  (s) = Mws ◦ϕ 1

˜

σ 1 (E 3 )

of .Mws ◦ϕ 1

˜

σ 1 (E3 )

∈ (−∞, 0) ⊂ C\{0, 1}. Hence, the lift .[ws ◦ϕσ11 ]

under .P is contained in the imaginary half-axis.

Each member of the continuous family of curves .{Mws ◦ϕ t

˜

σ 1 (E 3 )

, t ∈ [0, 1]}s∈[0,1]

in .C\{0, 1} has initial point on .(0, 1) and terminal point in .(−∞, 0). The family lifts 0,1 and to a continuous family of curves in .C+ with initial point in the half-circle .C+ terminal point in the imaginary half-axis, and the interior .{Mw0 ◦ϕ t (E˜ 3 ) , t ∈ (0, 1)} σ1

of the curve corresponding to the parameter .s = 0, is contained in the upper half0,1 into the positive imaginary half-axis. plane. Hence, .Tσ1 maps the half-circle .C+ −1 The same argument for the braid .σ1 shows that the modular transformation .Tσ1 maps the half-circle .{z ∈ C+ : |z − 12 | = 1} onto the positive imaginary half-axis. We show now that .Tσ1 maps the geodesic triangle .D1 with vertices .0, 12 , 1 onto 1+i .D0 . Let .α(s), s ∈ [0, 1], be an arc in .C+ with .α(0) = 2 and .α((0, 1]) ⊂ D1 . The 1 projection .P(α((0, 1])) to .C \ {0, 1} is contained in .C− and .P(α(0)) = P( 1+i 2 ) = 2. 1 Let again .ws be a continuous family of self-homeomorphisms of .P for which def

w0 = Id and .τs = [ws ] = αs . Consider again the continuous family of curves .{M ws ◦ϕ t (E˜ 0 ) , s ∈ [0, 1]} in .C \ {0, 1} with initial points .{Mws (E˜ 0 ) , s ∈ [0, 1]} .

σ1

3

in .(0, 1) ∪ C− . The terminal points .{Mws ◦ϕ 1

˜0 σ 1 (E 3 )

3

, s ∈ [0, 1]} are contained in

(−∞, 0) ∪ C+ , since .ϕσ11 acts on .E˜ 30 by permuting the first two coordinates. Indeed, for a point .(z1 , z2 , z3 ) ∈ C3 (C) the inclusion .M(z1 ,z2 ,z3 ) ∈ C− means, that the point .z2 lies on the right of the line through .z1 and .z3 , oriented in the direction from .z1 to .z3 . Then .z1 lies on the left of the line through .z2 and .z3 , oriented in the direction from .z2 to .z3 . This means, .M(z2 ,z1 ,z3 ) ∈ C+ . The lift under .P of the family of curves .{M ws ◦ϕ t (E˜ 0 ) , t ∈ [0, 1]}s∈[0,1] , in .C\{0, 1} provides a continuous family of curves .

σ1

3

0,1 in .C+ with initial points in .D1 ∪ C+ and terminal points in .D0 ∪ {Rez = 0} ∩ C+ . The terminating point of each curve is obtained from its initial point by applying the modular transformation .Tσ1 . We showed that .Tσ1 maps the geodesic triangle with vertices .0, 12 , 1 into .D0 . Similar arguments for the braid .σ1−1 imply that .Tσ1 maps the geodesic triangle with vertices .0, 12 , 1 onto .D0 . Since .Tσ1 maps the half-circle 1 .{z ∈ C+ : |z − | = 1} onto the positive imaginary half-axis, .Tσ1 takes 0 to 0, 1 to 2 1 z z .∞, and . to 1. Hence, .Tσ1 (z) = 2 1−z and .Tσ1−1 (z) = 1+z . The modular transformation .Tσ2 is computed similarly. Represent .σ2 by a closed curve .γ2 : [0, 1] → C3 (C)S3 with initial point .E3x = {0, x, 1}, where .x ∈ (0, 1). We associate to it a curve .M˜ (γ˜2 )t in .C+ with initial point contained in the half-circle 1 1 3 1 .{|z − | = 2 2 }. The terminal point is contained in the half-circle .{|z − 4 | = 4 }. By the same arguments as used for .σ1 we see that .Tσ2 is a Möbius transformation that maps the half-circle .{|z − 12 | = 12 } onto the half-circle .{|z − 34 | = 14 }, and maps .D0 conformally onto the geodesic triangle with vertices .0, 12 , 1. Hence, .Tσ2 maps 0 to 1 1 . , 1 to 1, and .∞ to 0. This implies that the mapping equals .Tσ2 (z) = 2 −z+2 . (See also Fig. 3.3.)

3.3 Invariants of Conjugacy Classes of Braids: Statement of the Main Theorem

127

The modular transformation corresponding to .σ1−1 σ2 equals .Tσ −1 σ2 = Tσ2 ◦ 1

 √ 11 3± 5 z+1 Tσ −1 (z) = z+2 . The eigenvalues of the matrix . are .t± = 2 . The same 1 12

√

arguments as in Example 3.6 of this section show that .Λ(σ1−1 σ2 ) = π2 log 3+2 5 . For each 3-braid b the respective modular transformation .Tb is a finite product of powers of the .Tσj . Hence, each .Tb is an element of .P SL2 (Z). If for the trace .a +d of the matrix corresponding to .Tb the equality .|a+d| = 2 holds, then .Tb is conjugate to a translation by a real number, and the quotient .C+ Tb  is conformally equivalent to the once punctured plane, hence has extremal length 0. If .|a + d| < 2, then a power of the mapping is the identity. Hence, in a neighbourhood of the fixed point 2π the mapping is conjugate to the mapping .z → ei k z. The quotient .{0 < |z| < 2π δ}(z ∼ ei k z) is a punctured disc and has extremal length 0. Hence, .C+ Tb has extremal length 0. If .|a + d| > 2 the extremal length of .C+ Tb equals . π2 log |t+ | √ (a+d)2 −4 for the larger absolute value .|t+ | = |a+d| + among the eigenvalues of 2 2 the matrix. The smallest among these values over all matrices in .SL2 (Z) is obtained √ for .|a + d| = 3. Hence, . π2 log 3+2 5 is the smallest non-vanishing extremal length among conjugacy classes of 3-braids.

Notice that the trace of the matrix corresponding to .Tσj , .j = 1, 2, equals 2, and both eigenvalues are equal to 1. Hence, the matrix is conjugate by a matrix in .SL2 (R) to the sum of the unit matrix and an upper diagonal matrix. Hence, both eigenvalues of all non-trivial powers of .Tσ1 and of all non-trivial powers of .Tσ2 are k ) = Λ(σ k ) = 0 for all integers .k = 0. Moreover, equal to 1. This implies that .Λ(σ 1 2 z−1 for .Tσ1 σ2 (z) = Tσ2 ◦ Tσ1 (z) = 3z−2 the associated matrix has trace 1. The mapping has a fixed point in .C+ . The eigenvalues of the associated matrix are . 12 ± Sect. 1.2). The power .Tσ31 σ2 is equal to the identity and .Λ(σ1 σ2 ) = 0.

√ 3 2 i

(see

3.3 Invariants of Conjugacy Classes of Braids: Statement of the Main Theorem The Entropy of Braids Recall that the entropy .h(b) of a braid .b ∈ Bn is defined as the infimum of the entropies of self-homeomorphims of the closed disc .D that are contained in the mapping class .mb associated to b, h(b) = inf{h(ϕ) : ϕ ∈ mb } .

.

(3.14)

The value .h(ϕ) is invariant under conjugation with self-homeomorphisms of the closed disc .D, hence it does not depend on the position of the set of distinguished ˆ We write points and on the choice of the representative of the conjugacy class .b.

128

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

ˆ = h(b). In Chap. 2 we studied the entropy of irreducible mapping classes and h(b) braids. The entropy of a mapping class is an important dynamical invariant which measures the complexity of its behaviour in terms of iterations. It has received a lot of attention and has been studied intensively. As for braids (and the associated mapping classes), it has been known that the entropy of any irreducible braid is the logarithm of an algebraic number. Further, the lowest non-vanishing entropy .hn among irreducible braids on n strands, .n ≥ 3, has been estimated from below by log 2 −1 . ([70]) and has been computed for small n. The entropy of a few more braids 4 n has been computed explicitly. It has √been known that the smallest non-vanishing entropy among 3-braids equals .log 3+2 5 and is attained on the braid .σ1−1 σ2 . Further, there is an algorithm which detects in principle whether a braid (respectively, a mapping class) is pseudo-Anosov and in this case it gives a computer assisted computation of the entropy ([13]). This approach uses so-called train tracks. Further, fluid mechanics related to stirring devises uses the entropy of the arising closed braids as a measure of complexity. .

The Conformal Module of Conjugacy Classes of Braids On the other hand, the conformal module (or its inverse, the extremal length) is a conformal invariant of conjugacy classes of braids, and is even older than the entropy. The conformal module of conjugacy classes of braids appeared first (without name) in the paper [33] in connection with the interest of the authors in Hilbert’s Thirteen’s Problem. This invariant of conjugacy classes of braids is undeservedly almost forgotten, although it appeared again in Gromov’s seminal paper [35]. Gromov observed that the conformal module of conjugacy classes of braids defines restrictions for the validity of his Oka Principle concerning homotopies of continuous or smooth objects involving braids to the respective holomorphic objects. We recall the definition of the conformal module of conjugacy classes of braids. (See Definition 3.4 for the definition of the conformal module of the conjugacy classes of elements of the fundamental group of a complex manifold.) We say that a continuous mapping f of a round annulus A = {z ∈ C : r < |z| < n if for some R}, 0 ≤ r < R ≤ ∞, into Cn (C)Sn represents an element bˆ ∈ B (and hence for any) circle {|z| = ρ} ⊂ A the loop f : {|z| = ρ} → Cn (C)Sn ˆ represents b. ˆ of bˆ is Let bˆ be a conjugacy class of n-braids, n ≥ 2. The conformal module M(b) ˆ = sup ˆ m(A), where A(b) ˆ denotes the set of all annuli which defined as M(b) A(b) ˆ The extremal admit a holomorphic mapping into Cn (C)Sn which represents b. ˆ of bˆ is defined as Λ(b) ˆ = inf ˆ λ(A). length Λ(b) A(b) The Relation Between Entropy and Conformal Module The interesting point is that the two invariants of conjugacy classes of braids, the entropy, which is a dynamical invariant, and the conformal module, which is a conformal invariant related to complex and algebraic geometry, carry the same information on the braid class. More precisely, the following theorem holds.

3.4 Main Theorem. The Upper Bound for the Conformal Module. The. . .

129

Main Theorem For each conjugacy class of braids bˆ ∈ Bˆn , n ≥ 2, the following equality holds ˆ = M(b)

π 1 . ˆ 2 h(b)

ˆ = Λ(b)

2 ˆ h(b) . π

.

Equivalently, .

The Main Theorem allows to apply methods and results known for the more intensively studied entropy to problems whose solutions are based on the concept of the conformal module. For instance, the Main Theorem together with the lower bounds for the entropy of irreducible n-braids allow to give a new conceptual proof of a slightly improved version of the Theorem of [33], which was the first theorem that used the concept of conformal module (see Chap. 6). The Main Theorem together with Example 3.8 of Sect. 3.2 give another proof√of the fact that the smallest non-vanishing entropy among 3-braids equals log 3+2 5 . Methods of quasi-conformal mappings related to the concept of extremal length and conformal module are applied to give upper and lower bounds, differing by a universal multiplicative constant, for the entropy of 3-braids (see Chap. 9). In the present chapter we will give a proof of the Main Theorem for the case of irreducible braids. In Chap. 4 we will give its proof for arbitrary pure braids, and in Chap. 5 the proof of the general case will be completed. In Chaps. 6, 7, 8, 9, 10, and 11 we will give applications of the concept of the conformal module. Chapters 7, 8, and 11 are devoted to Gromov’s Oka Principle. In these Chapters we apply the concept of the conformal module to describe restrictions for the existence of homotopies of continuous objects involving braids to the respective holomorphic objects.

3.4 Main Theorem. The Upper Bound for the Conformal Module. The Irreducible Case Let .bˆ ∈ Bˆn be a conjugacy class of (maybe, reducible) braids and let .f : A → Cn (C)Sn be a holomorphic mapping of an annulus A into the symmetrized ˆ Our goal is to give an upper bound for the configuration space which represents .b. conformal module .m(A). It will be convenient to identify the annulus with the quotient of the upper halfplane by an automorphism of the upper half-plane as follows. For a number .ρ > 1 we denote by .Λρ the linear map .z → ρz on the upper half-plane .C+ = {z ∈ C : Im z > 0}. The quotient .C+ Λρ is conformally equivalent to an annulus of

130

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

conformal module . logπ ρ . Indeed, the half-open curvilinear rectangle .{reiθ : 1 ≤ r < ρ , 0 < θ < π } is a fundamental polygon for the covering .Λρ : C+ → C+ Λρ . The logarithm maps it to .{x + iy : 0 ≤ x < log ρ , 0 < y < π }. Identifying points on the vertical sides with equal y-coordinate we obtain an annulus of conformal module . logπ ρ . Royden’s Theorem and Translation Length The key ingredient for obtaining the upper bound for the conformal module is Royden’s Theorem 1.5 on equality of the Kobayashi and the Teichmüller metric on the Teichmüller space .T (0, n + 1). Let .dhyp be the hyperbolic metric . |dz| 2y on .C+ . By Royden’s Theorem 1.5 any holomorphic mapping .F : C+ → T (0, n + 1) is a contraction from .(C+ , dhyp ) to .(T (0, n + 1), dT ). In particular, for any positive number .ρ dT (F(i), F(ρi)) ≤ dhyp (i, ρi) =

.

1 log ρ . 2

(3.15)

Let .ϕ be a self-homeomorphism of .P1 with set of distinguished points .En0 ∪ {∞}, and .ϕ ∗ its modular transformation on .T (0, n + 1). Suppose .F has the following invariance property F(ρz) = ϕ ∗ (F(z)), z ∈ C+ .

.

(3.16)

The invariance property allows to associate to .F a holomorphic map of the annulus C+ z ∼ ρz of conformal module . logπ ρ to the quotient .T (0, n + 1)(τ ∼ ϕ ∗ (τ )). By Royden’s Theorem

.

.

1 log ρ ≥ L(ϕ ∗ ) . 2

(3.17)

Indeed, for the term on the left hand side of (3.15) the inequality dT (F(i), F(ρ i)) = dT (F(i), ϕ ∗ (F(i)))

.

≥

inf

τ ∈T (0,n+1)

dT (τ, ϕ ∗ (τ )) = L(ϕ ∗ )

(3.18)

holds. The Mapping to .T (0, n+1) Associated to a Mapping to .Cn (C)Sn The relation between the symmetrized configuration space .Cn (C)Sn and the Teichmüller space of the .(n + 1)-punctured Riemann sphere (see Chap. 1) will allow us to apply Teichmüller theory. Take any holomorphic mapping .f : CΛρ → Cn (C)Sn . The set .CΛρ is identified with an annulus of conformal module . logπ ρ . Lift .f to a .Λρ -equivariant mapping from .C+ to .Cn (C)Sn which we denote by f .

3.4 Main Theorem. The Upper Bound for the Conformal Module. The. . .

131

Fig. 3.4 The holomorphic mapping to Teichmüller space associated to a braid

Take a lift .f˜ : C+ → Cn (C) of f , i.e. a mapping .f˜ for which .Psym f˜ = f . The mapping .PA f˜ is a holomorphic mapping from .C+ to .Cn (C)A. For the holomorphic isomorphism .Isn : Cn (C)A → {0} × { n1 } × Cn−2 (C \ {−1, 1}) we lift the mapping .PA,n f˜ = Isn ◦ PA f˜ with respect to the holomorphic projection 1 .PT : T (0, n + 1) → {0} × { } × Cn−2 (C \ {−1, 1}) and obtain a holomorphic n map .F : C+ → T (0, n + 1), such that .PT F = PA,n f˜. We have the commutative diagram Fig. 3.4. (For the definition of .Psym , .PA,n and .Isn see Chap. 1.) Denote by .En the point .En = f (i) = f (iρ) and by .(z1 , . . . , zn ) ∈ Cn (C) the point .f˜(i). Note that .Psym ((z1 , . . . , zn )) = En . For notational convenience we put  1 .En = En ∪ {∞}. Choose a smooth self-homeomorphism .ψi of .P that is the identity 1 n−1 0 outside a large disc containing .En = {0, n , . . . , n } and .En , and maps .∞ to .∞, and . j −1 n to .zj for .j = 1, . . . , n, and has the property .F(i) = [ψi ]. Recall that each braid b with base point .En0 corresponds to a mapping class .mb of self-homeomorphisms of the unit disc which fix the set .En0 setwise and the boundary ∗ circle pointwise. We denoted by .ϕb,∞ the modular transformation on .T (0, n + 1) of an element of the mapping class .mb,∞ ∈ T (0, n + 1). Recall that a representative of the mapping class .mb,∞ may be obtained by extending an element of .mb to the Riemann sphere putting it equal to the identity outside the disc. For a selfhomeomorphism .ψ of .P1 that fixes .∞ and an element .{ζ1 , . . . , ζn } ∈ Cn (C)Sn we put as before .ψ({ζ1 , . . . , ζn }) = {ψ(ζ1 ), . . . , ψ(ζn )}. The following lemma holds. Lemma 3.6 Let .f˜ : C+ → Cn (C) be a smooth mapping such that .f = Psym (f˜) is .Λρ -invariant. Let .F : C+ → T (0, n + 1) be a smooth mapping such that the diagram Fig. 3.4 commutes. Denote by b the braid with base point .En0 represented by the geometric braid ψi−1 ◦ f | [i, iρ], t ∈ [1, ρ]

.

(3.19)

132

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

where .ψi ∈ F(i) is the self-diffeomorphism of .P1 that was chosen above. Then for all .z ∈ C+ ∗ F(ρ z) = ϕb,∞ (F(z)).

.

(3.20)

Proof Let z be an arbitrary point in .C+ , and let .αz be a smooth curve in .C+ that joins the points i and z. By the proof of Proposition 1.1 (applied to .ψi−1 ◦ f ) there exists a smooth family of self-diffeomorphisms .ψζ of .P1 , .ζ ∈ αz , that are equal to the identity outside a large disc, such that .ψζ (En0 ) = f (ζ ), and for .ζ = i the diffeomorphism of the family coincides with the diffeomorphism .ψi in the statement of the lemma. As a consequence .[ψζ ] = F(ζ ), .ζ ∈ αz , since both mappings, .ζ → [ψζ ] and .ζ → F(ζ ) lift .PA,n (f˜(ζ )) by the commutative diagram of Lemma 1.5. The family ζ →

.

 

t, ψζ−1 (f (ζ t)) , t ∈ [1, ρ] , ζ ∈ αz ,

(3.21)

is an isotopy of geometric braids with base point .En0 that joins the geometric braid (3.19) with the geometric braid  

t, ψz−1 (f (zt)) , t ∈ [1, ρ] .

.

(3.22)

Fix .z ∈ C+ and take a parameterizing isotopy for the geometric braid (3.22). More precisely, for a large positive number R we take a smooth family of homeomorphisms .ϕtz ∈ Hom+ (P1 ; P1 \ RD), t ∈ [1, ρ], such that evEn0 ϕtz = ψz−1 (f (z t)) ,

.

t ∈ [1, ρ] ,

(3.23)

and .ϕ1z is the identity. Then the mapping .ϕρi represents the mapping class .mb,∞ ∈ z z M(P1 ; {∞}, En0 ). Write .ϕρz = ϕb,∞ . Since the .ϕb,∞ are isotopic with base point ∗ 0 .En , their modular transformations do not depend on z. Let .ϕ b,∞ be the modular z transformation induced on .T (0, n + 1) by the .ϕb,∞ . Equation (3.23) can also be written as .evEn0 (ψz ◦ϕtz ) = f (z t), .t ∈ [1, ρ]. Hence, the mapping .t → en (ψz ◦ ϕtz ) is a lift of the mapping .t → f (zt), .t ∈ [0, 1]. Clearly also .t → f˜(zt), t ∈ [0, 1], lifts the mapping .t → f (zt), .t ∈ [0, 1]. Moreover, both mappings, .en (ψz ◦ ϕtz ) and .f˜(z t), take the value .f˜(i) = (z1 , . . . , zn ) for .t = 1. For z z .en (ψz ◦ ϕt ) this follows from the definition of .ψi and the fact that .ϕ = id. Hence, 1 z .en (ψz ◦ ϕt ) = f˜(z t), .t ∈ [1, ρ]. Using also Lemma 1.5 we obtain the commutative diagram Fig. 3.5. The diagrams Figs. 3.4 and 3.5 show that both mappings, .t → F(z t) and .t → [ψz ◦ ϕtz ], lift the mapping .t → PA,n (f˜(z t)) to a mapping from .[1, ρ] to the Teichmüller space .T (0, n + 1). Moreover, since .F(z) = [ψz ] and .ϕ1z = id, we may

3.4 Main Theorem. The Upper Bound for the Conformal Module. The. . .

133

Fig. 3.5 Another lift to the Teichmüller space

write .F(z) = [ψz ◦ ϕ1z ], which is the value of .[ψz ◦ ϕtz ] for .t = 1. Therefore, the two lifts coincide: F(z t) = [ψz ◦ ϕtz ] ,

.

t ∈ [1, ρ] .

(3.24)

In particular, z ∗ F(ρ z) = [ψz ◦ ϕρz ] = [ψz ◦ ϕb,∞ ] = (ϕb,∞ )∗ ([ψz ]) = ϕb,∞ (F(z)) .

.

We obtained (3.20) for the arbitrarily chosen point z. The lemma is proved.

(3.25)



Translation Length and the Upper Bound for the Conformal Module: The General Case The following proposition gives an upper bound for the conformal module of a conjugacy class of braids in terms of the translation length of the associated modular transformation that holds for all (maybe, reducible) n-braids. Proposition 3.1 Let b be an n-braid with base point .En0 and .mb its mapping class (considered as mapping class on a large n-punctured disc). For the associated mapping class .mb,∞ = H∞ (mb ) ∈ M( P1 ; ∞, En0 ) on the Riemann sphere ∗ we consider a representative .ϕb,∞ of .mb,∞ , its modular transformation .ϕb,∞ on ∗ .T (0, n + 1) , and the translation length .L(ϕ ). Then b,∞ ∗ L(ϕb,∞ )≤

.

π 1 . ˆ 2 M(b)

(3.26)

Proof As before we take an arbitrary holomorphic mapping .f that takes the annulus A = C+ Λρ of conformal module .m(A) = logπ ρ to the symmetrized configuration space, and represents the conjugacy class .bˆ of the braid b. Then by Lemma 3.6 and ∗ Royden’s Theorem the inequality (3.17) holds for the modular transformation .ϕb,∞ and the mapping .F associated to .f by the diagram Fig. 3.4. Hence, in terms of the conformal module of A

.

.

1 π ∗ ≥ L(ϕb,∞ ). 2 m(A)

134

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

Taking the supremum over the conformal modules of all annuli admitting a ˆ we obtain (3.26). holomorphic mapping representing .b,

Entropy and the Upper Bound for the Conformal Module: The Irreducible Case Proposition 3.1 implies the upper bound for the conformal module of irreducible braids in terms of the entropy. More precisely, the following proposition holds. Proposition 3.2 For each irreducible conjugacy class of braids .bˆ ∈ Bˆn the inequality ˆ ≤ h(b)

.

π 1 ˆ 2 M(b)

holds. Proof If a braid b with base point .En0 is irreducible, then, .ϕb,∞ is irreducible and ∗ therefore .ϕb,∞ is either elliptic or hyperbolic. By Corollary 1.1 there is an absolutely extremal self-homeomorphism .ϕ˜b,∞ of a Riemann surface .C\E˜ n which is obtained from .ϕb,∞ by isotopy and conjugation. More precisely, there is a homeomorphism 0 ˜ n and a self-homeomorphism .ϕˆ b,∞ of .C\En0 which is isotopic .w : C\En → C\E 0 to .ϕb,∞ on .C\En so that .ϕ˜ b,∞ = w ◦ ϕˆ b,∞ ◦ w −1 : C\E˜ n → C\E˜ n is absolutely ∗ extremal. By Theorems 1.8 and 1.10 the mapping .ϕ˜b,∞ is pseudo-Anosov if .ϕb,∞ is ∗ hyperbolic, and is a periodic conformal mapping if .ϕb,∞ is elliptic. The entropy of a periodic mapping and its quasi-conformal dilatation vanish. Hence, by Theorem 2.1 and Corollary 1.1 for the absolutely extremal mapping .ϕ˜b,∞ .

1 ∗  ) = h(ϕ˜ b,∞ ) = h(m log K(ϕ˜ b,∞ ) = L(ϕb,∞ b,∞ ) . 2

(3.27)

In the pseudo-Anosov case Theorem 2.9 provides an isotopy of .ϕ˜b,∞ through selfhomeomorphisms of .C\E˜ n to a homeomorphism .ϕ˜b which is the identity outside the disc .R D and has the same entropy as .ϕ˜b,∞ : h(ϕ˜ b ) = h(ϕ˜b,∞ ) .

(3.28)

.

∗ Suppose .ϕb,∞ is elliptic. Then there is a periodic conformal map .ϕ˜b,∞ ∈ + Hom (C; ∅, En ) which is obtained from .ϕb,∞ by isotopy and conjugation. Here .En ⊂ C is a set consisting of n points. The mapping .ϕ˜b,∞ is a Möbius transformation that fixes .∞ and, being periodic, has a fixed point on .P1 different from .∞. Conjugate this fixed point to zero. We obtain that .ϕ˜b,∞ is 2π i p conjugate to multiplication by a root of unity .ω = e q for integer numbers .p and .q, .q = 0. Since   .ϕ˜ b,∞ (En ) = En , the set .En is either equal to .

2π i n

2π i(n−1)

2π ip

,...,r e n for some .r > 0, and in this case .ω = e n ; or it is   2π i(n−2) 2π i for some .r > 0, and in this case equal to .En = 0, r, re n−1 , . . . , r e n−1 r, r e

3.5 The Lower Bound for the Conformal Module. The Irreducible Case

135

2π ip

ω = e n−1 . Lemma 2.14 provides a self-homeomorphism .ϕ˜b of .P1 that is isotopic to .ϕ ˜b,∞ by an isotopy that fixes .En , equals .ϕ˜ b,∞ on the disc .R1 D for a large number .R1 , equals the identity outside .RD for some .R > R1 , is a twist on the annulus .RD \ R1 D, and, finally, it has the same entropy as .ϕ ˜b,∞ . In both cases for the obtained homeomorphism .ϕ˜b the mapping .w −1 ◦ ϕ˜b ◦ w is isotopic to .ϕb,∞ through self-homeomorphisms of .C\En0 and is the identity outside the disc .R D. By Theorem 1.3 the mapping class .mb associated to b, and the mapping class of .w −1 ◦ ϕ˜b ◦ w | R D in . M(R D; ∂(R D), En0 ) differ by a power of a Dehn twist about a circle in .R D of large radius. By Corollary 2.2 the entropies of the two mapping classes are equal. We obtain .

ˆ = h(b) = inf{h(ϕ) : ϕ ∈ mb } = inf{h(ϕ) : ϕ ∈ mw−1 ◦ϕ˜ ◦w|RD } h(b) b

.

≤ h(w−1 ◦ ϕ˜b ◦ w | R D) .

(3.29)

Hence we obtain from (3.27), (3.28), and (3.29) ∗ ˆ ≤ h(w−1 ◦ ϕ˜b ◦ w) = h(ϕ˜ b ) = h(ϕ˜b,∞ ) = L(ϕb,∞ h(b) ).

.

ˆ ≤π 1 . By (3.26) we have .h(b) 2 M(b) ˆ The proposition is proved.



3.5 The Lower Bound for the Conformal Module. The Irreducible Case We will prove here the following proposition for the irreducible case. Proposition 3.3 Let .bˆ ∈ Bˆn be an irreducible conjugacy class of braids. Then for any annulus A of conformal module .m(A) = π2 1ˆ the class .bˆ can be represented h(b) by a holomorphic map .f : A → Cn (C)Sn . Proposition 3.3 is stronger than the statement of the Main Theorem in the irreducible case. Indeed, the Proposition 3.3 asserts furthermore that in the irreducible case the supremum in Definition 3.4 is attained. Moreover, Proposition 3.3 says, that in case the supremum is infinite, it is attained on the punctured complex plane, not merely on the punctured disc. Take an irreducible conjugacy class of n-braids .bˆ ∈ Bˆn . Let .b ∈ Bn be a braid ˆ Let .ϕb be a self-homeomorphism of .D\En0 which with base point .En0 representing .b. represents the mapping class .mb = Θn (b) ∈ M(D; ∂ D, En0 ) of b. Let .ϕb,∞ ∈ Hom+ (C; ∅, En0 ) be the extension of .ϕb to the whole plane which is the identity outside the unit disc. Since the class .bˆ is irreducible, the mapping .ϕb,∞ is irreducible. By The∗ orem 1.11 the induced modular transformation .ϕb,∞ on the Teichmüller space .T (0, n + 1) is either elliptic or hyperbolic.

136

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

Proof of Proposition 3.3 to .ϕ˜b,∞ (z) = e The Elliptic Case In this case the mapping .ϕ˜b,∞ is conjugate  on .P1

with set of distinguished points .En either equal to . r, r e

2π i n

,...,r e

2π i pq

2π i(n−1) n

z

2π ip n

for in ; or it is equal to .En =  some2π i.r > 0, 2πand  this case .ω = e 2π ip i(n−2) 0, r, re n−1 , . . . , r e n−1 , and in this case .ω = e n−1 . As above .p and .q are integer numbers. We may assume that the mapping itself has this form. Consider the universal covering C  ζ = ξ + iη → eξ +iη ∈ C∗

.

of the annulus .C∗ = C\{0} = {z ∈ C : 0 < |z| < ∞}. Denote by .F˜ the following holomorphic mapping from .C into the space .A of complex affine mappings, ˜ ) = a(ζ ) ∈ A , where F(ζ

.

p

a(ζ )(z) = e q

·ζ

·z.

(3.30)

Notice that a(0)(z) = z

and

.

a(2π i)(z) = e

2π i pq

z = ϕ˜b,∞ (z) .

(3.31)

The evaluation map C  ζ → evEn a(ζ ) ∈ Cn (C)Sn

.

(3.32)

is holomorphic and is periodic with period .2π i. Hence, the evaluation map induces a holomorphic map f from .C∗ to .Cn (C)Sn . f represents a conjugacy class .bˆ  ∈ Bˆn . By Theorem 1.3 the mapping classes corresponding to the braids .b ∈ bˆ  , and .b ∈ bˆ differ by a power of a Dehn twist. Replacing .p by .p + 2π kq for a suitable integer number k we may achieve that .f : C∗ → Cn (C)Sn represents the conjugacy class ˆ ∈ Bˆn . The proposition is proved for the elliptic case. .b

Note that in the elliptic case .r = ∞, and we may take .A = C∗ . The Hyperbolic Case. Teichmüller Discs The plan of the proof in the hyperbolic case is the following. We will associate to the irreducible braid b a Teichmüller disc, i.e. a holomorphic map .F : C+ → T (0, n + 1) (see Fig. 3.4). Given .F, we need to find a mapping .f˜ : C+ → Cn (C) for which .PA,n f˜ = PT F (which means to find a holomorphic section), such that .Psym f˜ has the equivariance property for which the Diagram Fig. 3.4 is commutative.   Associate to the irreducible braid .b ∈ bˆ with base point .En0 = 0, n1 , . . . , n−1 n a representative .ϕb of the mapping class .mb ∈ M(D; ∂ D, En0 ). Let .ϕb,∞ ∈ Hom+ (C; ∅, En0 ) be the extension of .ϕb to .C which is the identity outside the

3.5 The Lower Bound for the Conformal Module. The Irreducible Case

137

∗ unit disc .D. Consider the modular transformation .ϕb,∞ on .T (0, n + 1) that is associated to b, and let .ϕ˜b,∞ be the absolutely extremal map that is obtained from 0 1 0  .ϕb,∞ by isotopy and conjugation. Put .X0 = C \ En = P \ (En ) . Denote by .w0 : X0 → w0 (X0 ) = X the .ϕ ˜b,∞ -minimal conformal structure on .X0 . Let 0 .X = w0 (X0 ) and .En = w0 (En ). The mapping .ϕ ˜ b,∞ is a self-homeomorphism of X, equivalently, .ϕ˜ b,∞ defines a self-homeomorphism of .P1 with set of distinguished points .En ∪ ∞, and .ϕ˜b,∞ is isotopic to .w0 ◦ ϕb,∞ ◦ w0−1 . mϕb,∞ ) = h( b). Again we identify selfBy Corollary 2.2 .h(ϕ˜ b,∞ ) = h( homeomorphisms of punctured surfaces with self-homeomorphisms of closed surfaces with distinguished points obtained by extension to the punctures. The Beltrami differential of the self-homeomorphism .ϕ˜b,∞ of X has the form φ¯ 1 .k |φ| for a number .k ∈ (0, 1) and a meromorphic quadratic differential .φ on .P which is holomorphic on .C\En and has at worst simple poles at the points of .En ∪ {∞}. The quasiconformal dilatation of .ϕ ˜b,∞ equals .K = 1+k 1−k . Recall that 1 by Theorem 2.1 we have . 2 log K = h(ϕ˜ b,∞ ). ¯

φ For .μz = z |φ| , z ∈ D, we let .W μz be the normalized solution on X of the Beltrami equation for .μz (see Definition 1.1). Consider the Teichmüller disc .D  z → [W μz ] ∈ DX,φ in .T (X) which is associated to the quadratic differential .φ on ∗ X (see Sect. 1.7). The modular transformation .ϕ˜b,∞ maps the disc .DX,φ onto itself. ¯

φ Indeed, .ϕ˜b,∞ has Beltrami differential .k |φ| . For each z in the open unit disc the μ z mapping .W ◦ ϕ˜b,∞ is a Teichmüller mapping with quadratic differential .const · φ ∗ ([W μz ]) has the form .{μ  } for some (see Sect. 1.7). Hence .[W μz ◦ ϕ˜b,∞ ] = ϕ˜b,∞ z    .z ∈ D. If z is real then also .z is real. Each element .{μz }, z ∈ D, is in the image ∗ ∗ under .ϕ˜ b,∞ of the Teichmüller disc . Since .W 0 is the identity mapping, .ϕ˜b,∞ maps the point .{μ0 } to .{μk }. The conformal structure .w0 realizes a canonical isomorphism between the Teichmüller space .T (X) and the canonical Teichmüller space .T (X0 ) = T (0, n+1). Indeed, associate to each conformal structure w on X the conformal structure .w ◦w0 on .X0 . Its class .[w ◦ w0 ] in .T (0, n + 1) depends only on the class .[w] in .T (X). Put ∗ ∗ .w ([w]) = [w◦w0 ], .[w] ∈ T (X). The mapping .w : T (X) → T (0, n+1) gives the 0 0 ∗ canonical isomorphism. .w0 is a holomorphic mapping between Teichmüller spaces. We denote the image .w0∗ (DX,φ ) in .T (0, n + 1) by .Dφ0 . The mapping .w0∗ takes the Teichmüller disc in .T (X) to a Teichmüller disc e

D  z −−−→ w0∗ ({μz }) ∈ Dφ0 ⊂ T (0, n + 1)

.

(3.33)

in .T (0, n + 1). It will be convenient to reparameterize the Teichmüller disc by a mapping from the upper half-plane to Teichmüller space. For this purpose we consider the composition F : C+ → Dφ0 ,

.

with the conformal mapping .c(z) = i

1+z 1−z

F = e ◦ c−1 from .D onto .C+ .

(3.34)

138

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

∗ ∗ The mapping .ϕb,∞ = w0∗ ◦ ϕ˜b,∞ ◦ (w0∗ )−1 takes .Dφ0 onto itself. Conjugate the ∗ restriction of the modular transformation .ϕb,∞ | Dφ0 by .F. We obtain a holomorphic homeomorphism .Λ of the upper half-plane onto itself, ∗ Λ = F −1 ◦ ϕb,∞ ◦ F : C+ → C+ .

(3.35)

.

∗ Since .ϕ˜b,∞ maps each point .{μx } with x real to a point .{μx  } with real .x  , and maps 1+k .{μ0 } to .{μk }, the mapping .Λ fixes the imaginary axis and maps i to .i 1−k = iK. Hence def

Λ(ζ ) = ΛK (ζ ) = Kζ ,

.

ζ ∈ C+ .

(3.36)

By (3.35) and (3.36) ∗ F(Kζ ) = ϕb,∞ (F(ζ )).

(3.37)

.

The annulus .A = C+ ΛK has conformal module ∗ . equivalent to the quotient .Dφ0 ϕb,∞

.

π 1

and is conformally

2 log K 2

The Mapping Representing the Braid Class Using a mapping .F : C+ → Dφ0 that satisfies (3.37), we will now represent the braid class .bˆ by a holomorphic mapping from an annulus to .Cn (C)Sn . Associate to .F the mapping .PT ◦ F from .C+ into .{0} × { n1 } × Cn−2 (C \ {1, n1 }) = Isn (Cn (C)A) (see Sect. 1.7). The diagram Fig. 3.4 and the diagram of Lemma 1.5 suggest to find a suitable holomorphic section for the projection C+ × Cn (C)  (ζ, z) → (ζ, zA) ∈ C+ × Cn (C)A .

.

def

We need a suitable section only for .C+ replaced by a neighbourhood .U (ε) = {ζ ∈ C+ : 1 − ε < |ζ | < (1 + ε)K} of the closure .{ζ ∈ C+ : 1 ≤ |ζ | ≤ K} of a fundamental domain in the universal covering .C+ of the annulus .C+ Λ. Each such holomorphic section has the following form U (ε) × Cn (C)A  (ζ, Z) → (ζ, A(ζ )(Isn (Z))) ∈ U (ε) × Cn (C)

.

for a holomorphic map .A : U (ε) → A that assigns to each point .ζ ∈ U (ε) a complex linear self-mapping .A(ζ ) ∈ A of .C. Recall that for an element .A of .A the point   def   1 1 A (0, , z3 , . . . , zn ) = A(0), A( ), A(z3 ), . . . , A(zn ) n n

.

(3.38)

is the result of the diagonal action of the complex affine mapping .A on the element (0, n1 , z3 , . . . , zn ) ∈ {0} × { n1 } × Cn−2 (C\{0, n1 }).

.

3.5 The Lower Bound for the Conformal Module. The Irreducible Case

139

For .ζ ∈ C+ we write 1

PT (F(ζ )) = 0, , z3 (ζ ), . . . , zn (ζ ) . n

.

(3.39)

For a holomorphic mapping .A : U (ε) → A we define the holomorphic mapping fA : U (ε) → Cn (C) by

.

  def fA (ζ ) = A(ζ )(PT F(ζ )) .

.

The following lemma guarantees the existence of a mapping .A that is suitable for our purposes. Lemma 3.7 There exists a holomorphic map .A : U (ε) → A such that the mapping def

f (ζ ) = Psym fA (ζ ), .ζ ∈ U (ε), has the following properties:

.

f (Kζ ) = f (ζ )

.

for

def

ζ ∈ U  (ε) = {ζ ∈ C : 1 − ε < |ζ | < 1 + ε} ,

(3.40)

and the family .f (it), .t ∈ [1, K], defines a closed path in .Cn (C)Sn in the free  2 b for some . ∈ Z. isotopy class .Δ n Proof of Lemma 3.7 By Lemma 1.4 in a neighbourhood of each point .ζ in .C+ we may choose a smooth family .wζ ∈ QC∞ (0, n + 1), such that .wζ fixes 0 and . n1 , .[wζ ] = F(ζ ), and the equality .PT (F(ζ )) = en (wζ ) holds. The mapping .ϕb,∞ is a self-homeomorphism of .C \ En0 . Its extension to .C acts on .En0 by permutation: j j ϕb,∞ ( ) = Sb ( ), j = 0, 1, . . . , n − 1, n n

.

(3.41)

for the permutation .Sb = τ (b) that is related to the braid b and maps the set {0, n1 , . . . , n−1 n } onto itself. Equation (3.41) is equivalent to the equation .en (ϕb,∞ ) = Sb (en (Id)). Since .wζ represents .F(ζ ), the mapping .wζ ◦ ϕb,∞ represents .F(Kζ ) = ∗ (F(ζ )). Hence, the point .e (w ◦ ϕ ϕb,∞ n ζ b,∞ ) represents the class of .PT (F(Kζ )) in the quotient .Cn (C)A of .Cn (C) by the action of the group .A. The locally defined family .ζ → en (wζ ) does not depend on the local choice of the .ωζ and is holomorphic since it coincides with the holomorphic mapping .ζ →



PT (F(ζ )). For .en (wζ ) = 0, n1 , . . . , zn (ζ ) = z1 (ζ ), . . . , zn (ζ ) we have .en (wζ ◦

ϕb,∞ ) = zS(1) (ζ ), . . . , zS(n) (ζ ) for the permutation .S, S(j ) = nSb ( j −1 n ) + 1 on .{1, . . . , n}, that is induced by .Sb . Put . S(en (wζ )) = .S (z1 (ζ ), . . . , zn (ζ )) = (zS(1) (ζ ), . . . , zS(n) (ζ )). We obtained

.

en (wζ ◦ ϕb,∞ ) = S(en (wζ )) .

.

(3.42)

140

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

Since .wζ represents .F(ζ ), and by Eq. (3.37) .wζ ◦ ϕb,∞ represents .F(Kζ  ) = ∗ ϕb,∞ (F(ζ )), the equality (3.42) means that .PT (F(Kζ )) differs from .S(PT F(ζ )) by the diagonal action of a Möbius transformation depending on .ζ ∈ C+ . This Möbius transformation is determined by its action on the first two coordinates of  .PT (F(Kζ )) and .S(PT F(ζ )) .

The following lemma holds. Lemma 3.8 There exists a holomorphic mapping .A : U (ε) →A such that foreach

 .ζ ∈ U (ε) the first two coordinates of the two n-tuples .A(ζ ) S(PT F(ζ )) and 

 .A(Kζ ) PT F(Kζ ) coincide on .U  (ε). We will prove this lemma after the proof of the present Lemma 3.7 is finished. Continuation of Proof of Lemma 3.7 Let .A be the mapping of Lemma 3.8. We   def

put . fA (ζ ) = A(ζ )(PT F(ζ )) , .fA : U (ε) → Cn (C). Since the first two  



 coordinates of .A Kζ ) PT F(Kζ ) and .A(ζ ) S(PT F(ζ )) coincide on .U (ε), Eq. (3.42) implies  



 A(Kζ ) PT F(Kζ ) = A(ζ ) S PT (F(ζ )) ,

.

ζ ∈ U  (ε) .

(3.43)

ζ ∈ U  (ε) .

(3.44)

Since the action of .A(ζ ) commutes with S, we have  



 A(Kζ ) PT F(Kζ ) = S A(ζ ) PT (F(ζ )) ,

.

Equation (3.44) means that .fA (Kζ ) = S(fA (ζ )) for .ζ ∈ U  (ε), hence, Psym fA (Kζ ) = Psym fA (ζ ) , ζ ∈ U  (ε) .

.

(3.45)

Equality (3.40) is proved. The second statement of Lemma 3.7 is proved as follows. Take the smooth family .t → wit ∈ QC∞ (0, n + 1), t ∈ [1, K], of the beginning of the proof of Lemma 3.7, for which .F(it) = [wit ], and .PT (F(it)) = PT ([wit ]) = en (wit ). We need to prove that .f (t) = Psym (A(it)en (wit )), t ∈ [1, K], represents .bˆ up to a power of a Dehn twist. By Eq. (3.44) the equality



A(Kit) en (wKit ) = S A(it)(en (wit )

.



def holds. The mapping .f˜(it) = A(it) en (w it ) , t ∈ [1, K], it ) = en A(it)(w

is a lift to .Cn (C) of the mapping .f (it) = Psym A(it)en (wit ) = evEn0 A(it)(wit ) . Hence,

3.5 The Lower Bound for the Conformal Module. The Irreducible Case

141

t → A(it)(wit ) is a parameterizing isotopy of the geometric braid .t → f (it). Since [wiK ] = [wi ◦ ϕb,∞ ], also the equality

. .

[A(iK)(wiK )] = [A(i)(wi ◦ ϕb,∞ )]

.

holds. Since

en A(iK) ◦ wiK

.





= A(iK) en (wiK ) = A(iK) PT (F(iK)) ,

en (A(i) ◦ wi ◦ ϕb,∞ ) = A(i)(S(en (wi ))) = S(A(i)(PT (F(i)) , Eq. (3.43) and Lemma 1.3 imply that the Teichmüller equivalent mappings A(iK)(wiK ) and .A(i)(wi ◦ ϕb,∞ ) are isotopic. Hence, the class of .ϕb,∞ is the mapping class in .M(P1 ; ∞, En0 ) of the geometric braid .t → f (it), t ∈ [1, K]. Hence, the conjugacy classes of the braid b and of the braid represented by .t → f (it), t ∈ [1, K], differ by a power of a Dehn twist. Lemma 3.7 is proved.

.

Proof of Lemma 3.8 Write

PT (F(ζ )) = z1 (ζ ), z2 (ζ ), z3 (ζ ), . . . , zn (ζ )

.

with .z1 (ζ ) ≡ 0, .z2 (ζ ) ≡ n1 . For the proof of the lemma we have to find a holomorphic mapping .A : U (ε) → A to the space .A of affine mappings so that the following condition is satisfied   A(Kζ ) 0

 1 A(Kζ ) n

.

≡ A(Kζ )(z1 (Kζ )) = A(ζ )(zS(1) (ζ )) , ≡ A(Kζ )(z2 (Kζ )) = A(ζ )(zS(2) (ζ )),

ζ ∈ U  (ε) .

(3.46)

) We write .A(ζ ) in the form .A(ζ ) (z) = z−b(ζ a(ζ ) , .z ∈ C, for holomorphic functions a and b in .C+ with .a = 0 in .C+ . For a number .K > 1 and any function q on .U (ε) we use the notation .q K (ζ ), .ζ ∈ U  (ε), for the function .q K (ζ ) = q(Kζ ). The Eq. (3.46) can be written on .U  (ε) as

.

−bK zS(1) − b = K , a a zS(2) − b = a

1 n

− bK . aK

Put def

χ =

.

1 . n(zS(2) − zS(1) )

(3.47)

142

3 Conformal Invariants of Homotopy Classes of Curves. The Main Theorem

χ is an analytic function on .U  (ε), .χ = 0 on .U  (ε). The Eq. (3.47) can be rewritten as

.

.

aK =χ, a

bK zS(1) b , − =− aK a a

ζ ∈ U  (ε) .

(3.48)

The first equation leads to a Second Cousin Problem for the coefficient a on the annulus .C+ ΛK (see [31], [40], or Appendix A.1). Indeed, let .ε > 0 be small compared to K. Cover the annulus .C+ ΛK by the open sets U1 = {ζ ∈ C+ : 1 − ε < |ζ | < 1 + 3ε}ΛK ,

.

U2 = {ζ ∈ C+ : 1 + 2ε < |ζ | < 1 + 5ε}ΛK , and U3 = {ζ ∈ C+ : 1 + 4ε < |ζ | < (1 + ε)K}ΛK .

(3.49)

Then . U1 ∩ U2 ∩ U3 = ∅, and .U3 ∩ U1 = U  (ε)ΛK . On .U3,1 = U3 ∩ U1 we take the holomorphic transition function .g3,1 (ζ ΛK ) = χ (ζ ), ζ ∈ U  (ε), on all other intersections of covering sets we take the holomorphic functions that are identically equal to one. We obtained a Cousin II distribution. Since the intersections of two covering sets are simply connected and an annulus is a Stein manifold, the Second Cousin Problem has a solution. This means, there exist nowhere vanishing g holomorphic functions .gj on .Uj such that .gj,k = gjk . Cover .Uε by lifts .U˜j of .Uj , .j = 1, 2, 3. Let .g ˜j be the lift of .gj to .Uj The function a on .U (ε) which is equal to .g ˜j on .U˜j , .j = 1, 2, 3, satisfies the first equation of (3.48). After we found a function a satisfying the first equation, we are looking for a function . ab which satisfies the second equation. This leads to a First Cousin Problem on A which is solvable. The lemma is proved.

End of Proof of Proposition 3.3 To prove the Proposition 3.3 we consider the mapping .a (ζ )(f (ζ )), .ζ ∈ U (ε). Here f is the mapping of Lemma 3.7 and 2π i

a (ζ )(z) = e log K

.

·log

ζ i

·z,

ζ ∈ U (ε) ,

z ∈ C.

(3.50)

Equation (3.50) defines a holomorphic mapping from .A = C+ ΛK to .Cn (C)Sn . When .ζ ranges over .[i, iK] the function .log ζi ranges over .[1, log K], and .a (ζ ), .ζ ∈ [i, iK], defines the .-th power of the full twists about the circle .[i, iK]ΛK . Hence, for a suitable integer number . the path .a (ζ )(f (ζ )), .ζ ∈ [i, iK], represents ˆ The conformal module .m(A) = m(C+ ΛK ) equals the free isotopy class .b. .

π 1 π 1 . = ˆ 2 log K2 2 h(b)

Proposition 3.3 is proved in the hyperbolic case.



Chapter 4

Reducible Pure Braids. Irreducible Nodal Components, Irreducible Braid Components, and the Proof of the Main Theorem

The purpose of this chapter is the proof of the Main Theorem for conjugacy classes of reducible pure braids. The idea is to reduce the proof to the case of irreducible braids and mapping classes. We will describe a decomposition of reducible elements  1 ; {∞} ∪ En ) into irreducible nodal components (See Sect. 4.1). The of .M(P  1 ; {∞} ∪ En ) can be reconstructed from the irreducible nodal elements of .M(P components up to a family of commuting powers of Dehn twists. On the other hand we provide a decomposition of conjugacy classes of reducible pure braids into irreducible braid components. The conjugacy class of a reducible pure braid can be recovered from its irreducible braid components. In Sect. 4.4 we will establish the relation between the irreducible braid components of the conjugacy class of pure braids and the irreducible nodal components of the conjugacy class of mapping classes associated to the pure braid. In Sect. 4.5 we prove that the conformal module of a conjugacy class .bˆ of pure braids is equal to the minimum of the conformal modules of the irreducible braid components, and the entropy of the class .bˆ is equal to the maximum of the entropies ˆ The Main of the irreducible nodal components of the mapping class associated to .b. Theorem for reducible pure braids is proved by applying the version of the Main Theorem for irreducible braids to the irreducible braid components and the related irreducible nodal components.

4.1 Irreducible Nodal Components. Pure Braids Throughout this chapter .b ∈ Bn will be a reducible pure braid and .mb ∈ M(D; ∂D ∪ En ) its mapping class. Here .En ⊂ D is a set consisting of n points. b into irreducible comWe will describe the decomposition of the conjugacy class .m ponents. For this purpose we again represent the mapping class .mb,∞ = H∞ (mb ) ∈ © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025 B. Jöricke, Braids, Conformal Module, Entropy, and Gromov’s Oka Principle, Lecture Notes in Mathematics 2360, https://doi.org/10.1007/978-3-031-67288-0_4

143

144

4 Reducible Pure Braids. Irreducible Nodal Components, Irreducible Braid. . .

M(P1 ; {∞}∪En ) by a self-homeomorphism .ϕb,∞ of .P1 which is the identity outside the unit disc .D. We assume that .ϕb,∞ is completely reduced by an admissible system .ϕb,∞ leaves the union of curves .C = {C1 , . . . , Ck } in .D\En ⊂ C\En . In 1particular,  . C invariant, and also leaves the complement .P \ C invariant. Notice that the C∈C

C∈C

decomposition into irreducible components in general depends on the isotopy class of the admissible system .C. For simplicity of notation we put .En = En ∪ ∞.

The Jordan CurveTheorem induces a partial order on the set of connected components of .P1 \ C. The partial order is described as follows. Let S be a C∈C  C. Its boundary .∂S is the union of some curves connected component of .P1 \ C∈C

that are elements of .C. By the Jordan Curve Theorem each connected component 1 .∂j of .∂S divides .P into two connected components. If S does not contain .∞ then there exists exactly one boundary component of S denoted by .∂E S for which S is contained in the bounded component of .C\∂E S. We call .∂E S the exterior boundary of S. If S is contained in the component of .P1 \ ∂j that contains .∞ the component is called an interior boundary component of S . The union .∂S \ ∂E S of the interior boundary components of S is called the interior boundary  of S and denoted by .∂I S. Denote by .S 1,1 the connected component of .P1 \ C that contains the point C∈C

∞. The exterior boundary of .S 1,1 is empty. We call .S 1,1 the outermost connected  1 component of .P \ C.

.

C∈C

The components of the interior boundary .∂I S 1,1 are called the curves of generation 2. By the Jordan Curve Theorem each connected component of the interior boundary .∂I S 1,1 bounds a disc contained in .D. (Recall that all curves of the system .C are contained in .D.) Denote the discs by .δ 2,j , j= 1, . . . , k2 . For each j there is exactly one connected component .S 2,j of .C\ C which is C∈C

contained in .δ 2,j and has a boundary component in common with .S 1,1 . (For short, 2,j is adjacent to a boundary component of .S 1,1 .) This boundary component is the .S exterior boundary component .∂E S 2,j , .j = 1, . . . , k2 of .S 2,j . The components .S 2,j are called the components of second generation. k  The connected components of .P1 \ Cj of generation . are defined by induction j =1

as follows. Consider the union of all connected components of .P1 \ generation not exceeding . − 1. Take its closure def

Q−1 =





1≤ ≤−1

1≤j ≤k

.



S  ,j ,

k 

Cj of

j =1

(4.1)

which is the closure of a domain .Q−1 ⊂ P1 . .Q−1 is the union of all components of generation not exceeding . − 1 and the exterior boundaries of all components

4.1 Irreducible Nodal Components. Pure Braids

145

of generation between 2 and . − 1. Its boundary .∂Q−1 is the union of the interior boundary components of all components .S −1,j .j = 1, . . . , k−1 , of generation k  1 . − 1. The connected components of .P \ Cj which share a boundary component j =1

with .Q−1 are called the components of generation . and are labeled by .S ,j , .j = 1, . . . , k . Each .S ,j has exactly one boundary component in common with .Q−1 , the exterior boundary .∂E S ,j . The exterior boundaries of the components .S ,j are also called the curves of generation .. We put def

Q =





.



S  ,j .

(4.2)

1≤ ≤ 1≤j ≤k

The process terminates after we obtained components .S N,j of some generation N that all have empty interior boundary. The boundary of .QN is empty, i.e. .QN coincides with .P1 . The mapping .ϕb,∞ fixes each connected component .S ,j setwise. Indeed, it fixes 1,1 .S setwise since it fixes .∞ ∈ S 1,1 . Hence it fixes the interior boundary of .S 1,1 setwise and therefore, it fixes the union of the discs .δ 2,j setwise. Further, since the system of curves .C is admissible, each disc .δ 2,j contains a point of .En . Hence, since b is a pure braid, the mapping .ϕb,∞ fixes the point. Hence, it fixes each disc .δ 2,j setwise, and therefore it fixes setwise the component .S 2,j whose exterior boundary component is the boundary of the disc .δ 2,j . By an induction on the number of generation we see that .ϕb,∞ fixes each connected component .S ,j setwise. Let w : C \ En → Y

.

be a continuous surjection onto a nodal surface Y associated to the isotopy class of the curve system .C in .P1 (see Sect. 1.8). Notice that the nodal surface Y has punctures. The mapping w maps each component . S ,j \ En of . P1 \ (En ∪ Cj ) homeomorphically onto a part .Y ,j of Y . (Recall that a part of a nodal surface Y with set of nodes .N is a connected component of .Y \N .) The correspondence between the components .S ,j \ En and the parts .Y ,j is a bijection. We conjugate the restriction .ϕb,∞ |P1 \(En ∪C) of the self-homeomorphism .ϕb,∞ with the inverse of w. We obtain a self-homeomorphism of .Y \ N that extends to a self-homeomorphism of Y . The isotopy class of this extension is denoted by .mb, and is called the isotopy class of self-mappings of the nodal surface Y determined by  .mb,∞ and the isotopy class of the admissible system .C. Let .m b, be the conjugacy class of .mb, . Let .Y c be the compact nodal surface obtained by filling the punctures of Y . The surjection w extends to a continuous surjection .w : P1 → Y c which by an abuse of notation we denote again by w. The elements of the class .mb, extend across the punctures of Y to self-homeomorphisms of .Y c with set of distinguished points

146

4 Reducible Pure Braids. Irreducible Nodal Components, Irreducible Braid. . .

N ∪w(En ). By an abuse of notation we will also denote the class of these extensions  by .mb, . The conjugacy class .m b, is called the nodal conjugacy class associated  to .m b,∞ and the admissible system of curves .C. More detailed, the mapping .ϕb,∞ fixes each .S ,j \ En setwise, hence each mapping in .mb, fixes each nodal component .Y ,j setwise. For each .(, j ) the

.

def

def

restriction .mb, = mb, |Y ,j is obtained in the following way. Put .E  ,j = ,j

En ∩ S ,j . We conjugate the restriction of .ϕb,∞ to .S ,j \ E  ,j by the inverse of def

the restriction .w ,j = w|S ,j \ E  ,j . The class .mb, is the mapping class of the conjugated mapping which is a self-homeomorphism of .Y ,j . The elements of .mb, |Y ,j extend continuously across the punctures of .Y ,j . The extensions are self-homeomorphisms of the closed Riemann surface .(Y ,j )c ∼ = P1 ,j  ,j c with distinguished points .w(E ) ∪ (N ∩ (Y ) ). The set of extensions is denoted ,j by the same letter .mb, . The conjugate of .ϕb,∞ | S ,j \ E  ,j by .(w ,j )−1 represents ,j

the class .mb, on .Y ,j , and, hence, .ϕb,∞ | S ,j \E  ,j represents the conjugacy class  ,j .m b, .  ,j The conjugacy classes .mb, will be called the irreducible nodal components of  ,j  the class .m b,∞ with respect to the system .C. The conjugacy classes .mb, of the ,j

,j

 restrictions .mb, depend only on the conjugacy class .m b,∞ and the isotopy class of the curve system .C. Each such class is indeed irreducible (see Remark 5.1 for the case of not necessarily pure braids and its proof). Notice, that given the isotopy class  of curve system .C, the irreducible nodal components determine the class .m b,∞ up to products of powers of some Dehn twists. (See Remark 4.1 below.)

4.2 The Irreducible Braid Components. Pure Braids In this section we will describe the decomposition of reducible braids into irreducible braid components. The decomposition depends again on the isotopy class of an admissible system of curves that completely reduces the mapping class associated to the braid. We start with defining a few objects. This will be done here for the case of arbitrary (not necessarily pure) braids. A tube in the “infinite” cylinder .[0, 1] × C is the image of a “finite” cylinder .[0, 1] × D under a diffeomorphism of .[0, 1] × C onto itself which preserves each fiber .{t} × C, t ∈ [0, 1]. Let .Ω n2 = {Ω1 , . . . , Ωn2 } be an unordered .n2 -tuple of pairwise disjoint closed topological discs in the complex plane, and .En1 = {z1 , . . . , zn1 } an unordered .n1 -tuple of points in .C \ ∪Ωj . A tubular geometric .(n1 , n2 )-braid with base point .(En1 , Ω n2 ) consists of the following two objects. The first object is a collection of .n2 mutually disjoint tubes in the cylinder .[0, 1] × C, each of which intersects the fibers .{0} × C and .{1} × C

4.2 The Irreducible Braid Components. Pure Braids

147

along a copy of .Ω n2 . The second object is a geometric .n1 -braid with base point .En1 whose strands do not meet the .n2 tubes. Take a point .En1 ∈ Cn1 (C)Sn1 and a point .E n2 ∈ Cn2 (C)Sn2 such that .En1 ∪ E n2 ∈ Cn1 +n2 (C)Sn1 +n2 . A geometric fat braid (of type .(n1 , n2 )) with base point .En1 ∪ E n2 is a geometric .(n1 + n2 )-braid such that all strands with initial point in .E n2 have also their endpoints in .E n2 and are declared to be fat. The strands with initial point in .En1 will be called ordinary strands. A tubular geometric braid is associated to a fat braid if there is a one-to-one correspondence between the tubes and the fat strands so that each fat strand is contained in the corresponding tube and is a deformation retraction of this tube. A fat braid is an isotopy class of geometric fat braids with fixed base point. Two geometric fat braids are called isotopic if they are isotopic as geometric braids and the isotopy moves the fat strands to fat strands and the ordinary strands to ordinary strands. A geometric tubular fat braid is a tubular geometric braid some of whose strands are declared to be fat. We consider now again pure braids. Take a reducible pure braid .b ∈ Bn with base point .En0 , and describe the decomposition into irreducible braid components. As in Sect. 4.1 we consider a mapping .ϕb,∞ ∈ mb,∞ ∈ M(P1 ; {∞} ∪ En0 ) that is the identity outside the unit disc and is completely reduced by an admissible system .C of curves contained in the unit disc. (Notice that in general the decomposition into irreducible braid components depends on the isotopy class of the admissible system .C.) Replacing .ϕb,∞ by an isotopic mapping which differs from the original mapping only in small neighbourhoods of the admissible curves we may assume that .ϕb,∞ fixes each curve .C pointwise. Consider a path .ϕt ∈ Hom+ (P1 ; P1 \ D), .t ∈ [0, 1], which joins .ϕb,∞ with the identity. More detailed, .ϕt is a continuous family of self-homeomorphisms of .P1 which fix the complement .P1 \ D of the unit disc pointwise, such that .ϕ0 = id and .ϕ1 = ϕb,∞ . The evaluation map   [0, 1] t → evEn ϕt = ϕt (z1 ), . . . , ϕt (zn ) ∈ Cn (C)Sn

.

(4.3)

with .En = {z1 , . . . , zn }, gives a geometric braid that is contained in the cylinder [0, 1] × D (i.e. .evEn ϕt ∈ D for each t), has base point .En = ϕ0 (En ) = ϕ1 (En ) , and represents b. The family .ϕt , t ∈ [0, 1] is a parameterizing isotopy of the geometric braid (4.3). We will write the geometric braid (4.3) as continuous mapping .g : [0, 1] → Cn (C)Sn , .

g(t) = ϕt (En ), t ∈ [0, 1] ,

.

(4.4)

and label the coordinate functions of g so that for .z ∈ En the coordinate function  .gz of g has initial point .gz (0) = gz (1) = z . We will associate to the parameterizing isotopy tubular geometric braids and geometric fat braids. The classes obtained from the fat braids by isotopy and conjugation are the irreducible braid components. This is done as follows.

148

4 Reducible Pure Braids. Irreducible Nodal Components, Irreducible Braid. . .

 The Outermost Braid Component .b(1, 1) of a Pure Braid b Recall that we regard a point in .Cn (C) sometimes as ordered subset of .C, and sometimes as a point in .Cn . We will associate now a fat braid .b(1, 1) to the outermost component .S 1,1 . Put def

E 1,1 = En ∩ S 1,1 . Let .C 2,j , .j = 1, . . . , k2 , be the interior boundary components of .S 1,1 , labeled so that .C 2,j is the exterior boundary of the component .S 2,j of .P1 \ ∪C∈C C, .j = 1, . . . , k2 . Denote by .δ 2,j ⊂ D the topological disc bounded by .C 2,j . The tubular geometric braid associated to the parameterizing isotopy .ϕt and the component .S 1,1 is

.

⎛ {t} × ϕt ⎝E 1,1 ∪

k2 

.

⎞ δ 2,j ⎠ ,

t ∈ [0, 1] .

(4.5)

j =1

For each tube .{t} × ϕt (δ 2,j ), t ∈ [0, 1], we consider a strand contained in it and call it a fat strand. More detailed, for each j we pick a point .z2,j ∈ δ 2,j ∩ En and denote it by the fat letter .z2,j . Such a point exists by the definition of an admissible system of curves and an induction argument. Since the braid is pure we have .ϕb,∞ (z2,j ) = z2,j . The respective fat strand is .ϕt (z2,j ), t ∈ [0, 1]. Put .E 1,1 = {z2,1 , . . . , z2,k2 }. Note that the choice of the points in .E 1,1 may not be unique. We associate to .S 1,1 the set of points .E 1,1 ∪ E 1,1 . The geometric fat braid associated to .S 1,1 and the parameterizing isotopy equals ϕt (E 1,1 ∪ E 1,1 ) ,

.

t ∈ [0, 1] .

(4.6)

The strands with initial point in .E 1,1 are the ordinary strands, and the strands with initial point in .E 1,1 are the fat strands. The geometric fat braid (4.6) is obtained from the geometric tubular braid (4.5) by taking a deformation retract of each tube to a fat strand. Let .n(1, 1) be the number of points of .E 1,1 ∪ E 1,1 . We will write the geometric fat braid (4.6) as mapping g 1,1 : [0, 1] → Cn(1,1) (C)Sn(1,1) ,

.

g 1,1 (0) = g 1,1 (1) = E 1,1 ∪ E 1,1 .

(4.7)

If needed, the “strands” of .g 1,1 will be labeled by points of .E 1,1 ∪ E 1,1 , so that for  1,1 ∪ E 1,1 the strand corresponding to .g 1,1 has initial point .g 1,1 (0) = z . The .z ∈ E z z fat strands are those with initial points in .E 1,1 . We define the fat braid .b(1, 1) with base point .E 1,1 ∪ E 1,1 as the isotopy class of the geometric fat braid (4.6) (equivalently, (4.7)) with given base point. The fat strands are those with initial point in .E 1,1 . Notice that .b(1, 1) is obtained from the braid b by discarding all strands with initial point not in .E 1,1 ∪ E 1,1 and indicating the set of fat strands.

4.2 The Irreducible Braid Components. Pure Braids

149

Fig. 4.1 The irreducible  1) of a braid component .b(1, pure braid b

 Finally we take the conjugacy class .b(1, 1) of the fat braid .b(1, 1). This is the ˆ irreducible braid component of .b associated to b and the component .S 1,1 . The class  .b(1, 1) depends only on the conjugacy class .bˆ and on the isotopy class of the curve system .C. See Fig. 4.1 for an example of the geometric fat braid .g 1,1 and the respective tubular geometric braid. In the figure the set .E 1,1 consists of a single point. The strand  with this initial point is the only ordinary strand. There are two components of .C\ C of generation 2, .S 2,1 and .S 2,2 . The respective distinguished points are C∈C

z2,1 and .z2,2 . These are the points in .E 1,1 . The strands with these initial points are the fat strands of .g 1,1 .

.

 The Irreducible Braid Component .b(, j ) associated to .S ,j with . > 1 Take a component .S ,j of generation .. Recall that .E ,j = E ∩ S ,j . Consider the interior boundary components of .S ,j . Each such  boundary component is the  exterior boundary of a component .S +1,j of .P1 \ Ci of generation . + 1. Let 

C∈C





δ +1,j ⊂ C be the topological disc bounded by .∂E S +1,j , and let .z+1,j be a   point in .E ∩ δ +1,j . Let .E ,j be the collection of .z+1,j obtained in this way (one such point is assigned to each interior boundary component of .S ,j ).

.

150

4 Reducible Pure Braids. Irreducible Nodal Components, Irreducible Braid. . .

We obtain the tubular geometric braid ⎛ ϕt ⎝E ,j ∪



⎞

k+1

.

δ

+1,j 

⎠,

t ∈ [0, 1] ,

(4.8)

j =1

and the geometric fat braid

ϕt E ,j ∪ E ,j ,

.

t ∈ [0, 1] .

(4.9)

The geometric fat braid

is obtained by taking a deformation retraction of each of the  cylinders .ϕt δ +1,j , t ∈ [0, 1] , to the fat strand of b contained in this cylinder. Let .n(, j ) be the number of points of .E ,j ∪E ,j . We will write the geometric fat braid (4.9) as a mapping .g ,j : [0, 1] → Cn(,j ) (C)Sn(,j ) from the unit interval into .Cn(,j ) (C)Sn(,j ) . Denote the isotopy class of the geometric fat braid by .b(, j ). Intuitively, the fat braid .b(, j ) is obtained from the braid b by discarding all strands with initial point not in .E ,j ∪ E ,j and indicating the set of fat strands.  The conjugacy class .b(, j ) of the fat braid .b(, j ) is the irreducible component of the braid b associated to .S ,j . It depends only on the conjugacy class .bˆ of the braid and the isotopy class of the curve system .C. The decomposition of the conjugacy class of a braid into irreducible braid components is described.

4.3 The Relation Between Irreducible Nodal Components and Irreducible Braid Components  ,j Recall that for each .(, j ) we denoted by .mb, the irreducible nodal component that is associated to .mb,∞ and the isotopy class of the admissible system of curves  .C. For the irreducible braid component .b(, j ), that is a conjugacy class of fat braids which was associated to b and the isotopy class of .C, we now consider the  j ) by declaring all strands to be respective conjugacy class of ordinary braids .b(, ordinary. Let .mb(,j ),∞ be the conjugacy class of mapping classes corresponding to the braid class .b(, j ). The following lemma holds. 



Lemma 4.1 For each .(, j ) the equality  ,j mb, = mb(,j ),∞



.

holds.

(4.10)

4.3 The Relation Between Irreducible Nodal Components and Irreducible. . .

151

In the statement of the lemma either both conjugacy classes are considered as classes of self-homeomorphisms of a punctured Riemann surface or both are considered as classes of self-homeomorphisms of a closed Riemann surface with distinguished points. Proof of Lemma 4.1 Recall that .En = En ∪ {∞}, and the mapping .ϕb,∞ |S ,j \  ,j  ,j ). We will En represents the class .mb, considered as element of .M(Y

write again .E  ,j = En ∩ S ,j . As before we will identify the con ,j ) on the punctured surface with the conjugacy class jugacy class .M(Y

,j c  (Y ) ; (N ∩ (Y ,j )c ) ∪ w ,j (E  ,j ) on the closed surface. .M  j ) is represented by the geometric The irreducible braid component .b(, braid (4.9) after forgetting that some strands are declared to be fat. The parameterizing family .ϕt that was chosen for the geometric braid (4.4) representing the braid b serves also as parameterizing family for the latter geometric braid. The mapping class .mb(,j ),∞ ∈ M(P1 ; E  ,j ∪ E ,j ) that is associated to the braid .b(, j ) (see (4.9)) is the class of .ϕ1 = ϕb,∞ in .M(P1 ; E  ,j ∪ E ,j ) identified with .M(P1 \ (E  ,j ∪ E ,j )). This follows from the fact that .ϕt is a parameterizing isotopy also for the geometric braid (4.9) that represents .b(, j ). The mapping .ϕb,∞ | (P1 \ (E  ,j ∪ E ,j )) is obtained from the mapping ,j \ E  ) by isotopy and conjugation. Indeed, consider all holes of .S ,j . .ϕb,∞ | (S n  These are all discs .δ +1,i that are adjacent to an interior boundary component of  ,i . Each disc .δ +1,i contains a point .z+1,i  ∈ E ,j and each point in .E ,j is .S contained in one of the discs. If .(, j ) = (1, 1) then there is one more hole of .S ,j , namely the complement of .δ ,j . This disc contains .∞ and no other point of .E  ,j . ,j We denote it by .δ∞ .   ,j ,j For each hole .δ +1,i (and .δ∞ , if . = 1) we take an annulus .A+1,i (and .A∞ ,j respectively, if . = 1), that is contained in .S and shares a boundary component with the boundary of the hole. The annuli are chosen to have disjoint closure. For    ,j  each disc .δ +1,i ( .δ∞ , respectively) we consider the disc .δ  +1,i = δ +1,i ∪A+1,i (and .δ  ,j ∞ = δ∞ ∪ A∞ , respectively). There is a homeomorphism .w˜ ,j from .S ,j \E  ,j onto .P1 \(E  ,j ∪E ,j ) with the following properties. .w˜ ,j is equal to the identical injection on the complement of all  ,j annuli .A+1,i (and .A∞ respectively, if . = 1) in .S ,j . Then .w˜ ,j maps each annulus  ,j +1,i .A (.A∞ , respectively) homeomorphically onto the respective punctured disc  +1,i   .δ \ {z+1,i } (.δ  ,j ∞ \ {∞}, respectively) and its continuous extension to the   ,j boundary component .∂δ  +1,i of .A+1,i (to the boundary component .∂δ  ,j ∞ of .A∞ , respectively) equals the identity on the boundary component. The mapping .(w˜ ,j )−1 conjugates .ϕb,∞ | S ,j \ E  ,j to a self-homeomorphism  1  ,j ∪ E ,j ) that differs from .ϕ 1  ,j ∪ E ,j ) only on the .ϕ b,∞ | P \ (E b,∞ of .P \ (E ,j

,j





union of the punctured discs .δ  +1,i \ {z+1,i } (.δ  ,j ∞ \ {∞}, respectively) and maps −1  each punctured disc onto itself. Moreover, .ϕb,∞ ◦ ϕb,∞ is equal to the identity on

152

4 Reducible Pure Braids. Irreducible Nodal Components, Irreducible Braid. . .

the boundary of each disc. Since two self-homeomorphisms of a punctured disc that fix the boundary circle pointwise are isotopic to the identity through selfhomeomorphisms of the punctured disc that fix the boundary circle pointwise, the conjugated homeomorphism is isotopic to .ϕb,∞ | P1 \ E  ,j . The equality (4.10) is proved.   Lemma 4.1 (together with Remark 5.1) shows in particular that the irreducible braid components are indeed irreducible conjugacy classes of braids.

4.4 Recovery of the Conjugacy Class of a Pure Braid from the Irreducible Braid Components We will consider now again the isotopy class of an admissible system of curves that completely reduces the class .mb,∞ of the braid and consider the connected components .S ,j of the complement of the curves. We will show how to recover the conjugacy class of the pure braid b from knowing the irreducible braid components  .b(, j ) associated to the .S ,j . For this purpose we will take arbitrary geometric fat braids, that are contained in .[0, 1] × D and represent the irreducible braid  components .b(, j ). By induction on the generation . we will “insert the geometric fat braids representing the irreducible braid components of generation . into tubes around fat strands of the geometric fat braids representing the irreducible braid components of generation . − 1”. We have to show that indeed for arbitrary choices  of geometric fat braids in .[0, 1] × D representing the .b(, j ) the geometric braid ˆ For this purpose obtained by the inductive construction represents the class .b. we will compare an arbitrary inductively constructed geometric braid with the ˆ geometric braid (4.3) that represents .b. We will use the following operation on points of symmetrized configuration spaces. Consider a point .E ∪ E ∈ Cn (C)Sn (which is also considered as a subset of .C). The points in the set .E are declared to be fat points. Let .z ∈ E be one of the fat points. Take a subset .E  of .C that contains .n points (also considered as point in   .Cn (C)Sn . ) .E may contain fat points or not. Suppose that .E does not intersect   .E ∪ (E \ {z }). Remove the point .z from the set .E ∪ E and add the points of the set  .E . The resulting set is denoted by def

(E ∪ E) z E  = (E ∪ E  \ {z }) ∪ E  .

.

(4.11)

The set (4.11) may be considered as subset of .C consisting of .n + n − 1 points, or as point in .Cn+n −1 (C)Sn+n −1 . The respective operation on mappings into symmetrized configuration spaces is defined pointwise. For instance, let .g (t), t ∈ [0, 1], be a geometric fat braid, let .g 1 (t), t ∈ [0, 1], be a fat strand of .g , and let .T = ∪t∈[0,1] T(t) be a tube that contains the fat strand .g 1 (i.e. .g 1 (t) ∈ T(t), t ∈ [0, 1],) and does not intersect any

4.4 Recovery of the Conjugacy Class of a Pure Braid from the Irreducible. . .

153

other strand (i.e. .T(t) does not intersect the set .g (t) \ g 1 (t), .t ∈ [0, 1]). Suppose g  (t), t ∈ [0, 1], is another geometric fat braid, such that .g  is contained in the tube  .T, i.e. .g (t) ∈ T(t), t ∈ [0, 1]. By removing the fat strand .g 1 and inserting the geometric fat braid .g  which is contained in the tube .T, we obtain a new geometric fat braid .g (t) g 1 (t) g  (t), t ∈ [0, 1], that is denoted by .g g 1 g  . We will need also the following operation on points in .Cn (C)Sn and on mappings from an interval to .Cn (C)Sn . For an element .E ∈ Cn (C)Sn and a number .ε we denote by .εE the element .E ∈ Cn (C)Sn which is obtained by multiplying each point in E (considered as unordered set of points in .C) by .ε. For a point .z ∈ C we let .

z  εE

.

(4.12)

be the set that is obtained from .εE by adding z to each point in .εE. Similarly, for a set .Ω and a mapping .f1 : Ω → C, a number .ε, and a mapping .f 2 : Ω → Cn (C)Sn we denote by f1  εf 2

.

(4.13)

the mapping from .Ω to symmetrized configuration space defined for each .t ∈ Ω by f1 (t)  εf 2 (t). The following two lemmas on isotopies of geometric fat braids will be needed for the recovery.

.

Lemma 4.2 Let .g 0 and .g 1 be two geometric fat braids contained in a closed tube 0 1 .T (i.e. for each .t ∈ [0, 1] the points .g (t) and .g (t) are contained in .T ∩ ({t} × C)). 0 1 If .g and .g are isotopic then there exists an isotopy of geometric fat braids which is contained in .T and joins the two braids. Proof Let .g s , s ∈ [0, 1], be the smooth family of geometric fat braids defined by the isotopy. Let .R > 0 be a number such that all geometric braids .g s , s ∈ [0, 1], together with the tube .T, are contained in .[0, 1] × RD. Let .T0 ⊂ T be a tube which contains both geometric braids .g 0 and .g 1 , such that the fiber .T0 ∩ ({t} × C) is relatively compact in the fiber .T ∩ ({t} × C) for each .t ∈ [0, 1]. Let .χ be a self-homeomorphism of .[0, 1] × C which preserves each fiber .{t} × C, is equal to the identity on .T0 and outside .[0, 1] × R1 D for a large number .R1 > R, and maps the set .[0, 1] × RD onto .T. Then the equalities .χ ◦ g 0 = g 0 and .χ ◦ g 1 = g 1 hold, and .χ ◦ g s , s ∈ [0, 1], is a smooth family of geometric fat braids contained in .T that joins .g 0 with .g 1 . The   lemma is proved. Lemma 4.3 Let .f 1 , .f 2 , .g 1 , and .g 2 be pure geometric fat braids, such that .f 1 j is free isotopic to .f 2 , and .g 1 is free isotopic to .g 2 . For .j = 0, 1 we let .f 1 be fat strands of .f j that correspond to each other under the isotopy. Further, for .j = 1, 2

154

4 Reducible Pure Braids. Irreducible Nodal Components, Irreducible Braid. . . j

j

we let .T(f 1 ) be a tube around .f 1 which does not meet the other strands of .f j and contains .g j . Then .f 1 f 1 g 1 and .f 2 f 2 g 2 are free isotopic geometric fat braids. 1

1

Proof Note first that for any sufficiently small number .ε the braids defined by the mappings .g 1 and .f 11  εg 1 are free isotopic. Further, for small .ε the fat braid .f 11  εg 1 is contained in the tube .T(f 11 ) around the fat strand .f 11 of .f 1 . Since, .g 1 is also contained in .T(f 11 ), Lemma 4.2 (which applies to fat braids as well) provides an isotopy that is contained in .T(f 11 ) and joins .g 1 and .f 11  εg 1 . As a consequence, there is a free isotopy joining .f 1 f 1 g 1 and .f 1 f 1 (f 11 εg 1 ). In the same way for 1

1

small .ε the braids .f 2 f 2 g 2 and .f 2 f 2 (f 21 εg 2 ) are free isotopic. Hence, for any 1 1 a priory given small enough positive number .ε we may assume from the beginning that .g 1 = f 11  εg˜ 1 , and .g 2 = f 21  εg˜ 2 , where .g˜ 1 , and .g˜ 2 are geometric fat braids that are contained in the cylinder .[0, 1] × D. Let .f s , s ∈ [1, 2], be an isotopy of braids joining .f 1 with .f 2 , and let .f s1 be the family of strands joining .f 11 with .f 21 . Also, let .g˜ s , s ∈ [1, 2], be an isotopy of braids contained in the cylinder .[0, 1] × D, that join .g˜ 1 and .g˜ 2 . Choose .ε > 0 so that for each .s ∈ [1, 2] the .ε-tubes around the strand of .f s are disjoint. Then s s .f f s (f  ε g ˜ s ), s ∈ [1, 2], is an isotopy of braids that joins .f 1 f 1 (f 11  εg˜ 1 ) 1 1 with .f 2 f 2 (f 21  εg˜ 2 ). The Lemma is proved. 1

1

 

We describe now in detail the recovery procedure for the conjugacy class .bˆ knowing the isotopy class of the admissible set of curves .C ⊂ D, and, hence, up  to isotopy, the sets .S ,j , and knowing for all . and j the sets of the fat braids .b(, j) ,j associated to the .S .  For this purpose we will take arbitrary representatives of the .b(, j ) contained in the cylinder .[0, 1] × D and describe an inductive procedure to put them together after isotopy and normalization. To show that any geometric braid obtained by this ˆ we will use Lemmas 4.2 and 4.3 to procedure represents the conjugacy class .b, compare the geometric fat braid obtained at each inductive step with a geometric fat braid related to .t → ϕt (E), t ∈ [0, 1], where .E ⊂ D and .ϕt is a continuous family of self-homeomorphisms of .P1 such that .ϕ0 = id, and .ϕ1 represents .mb,∞ and fixes the admissible set of curves .C ⊂ D. Take a connected component .S ,j of .P1 \C. Recall that the conjugacy class of fat  braids .b(, j ) was defined in terms of the set E, and the family .ϕt , as the class of the geometric fat braid .g ,j : [0, 1] → Cn(,j ) (C)Sn(,j ) with g ,j (t) = ϕt (E ,j ∪ E ,j )

.

(4.14)

for the set .E ,j = E ∩ S ,j and the set .E ,j . The set .E ,j = {z,1 , . . .} was constructed so that for each interior boundary component .∂  of .S ,j it contains exactly one point of E that is contained in the bounded connected component of the complement of .∂  in .C. Here .n(, j ) is the total number of points in .E ,j ∪ E ,j .

4.4 Recovery of the Conjugacy Class of a Pure Braid from the Irreducible. . .

155

 We represent each .b(, j ) by an arbitrary geometric fat braid f ,j : [0, 1] → Cn(,j ) (C)Sn(,j )

.

˜ ,j for some which is contained in the tube .[0, 1] × D and has base point .E˜ ,j ∪ E ˜ ,j . The set of initial points of the fat strands equals .E ˜ ,j and .n(, j ) sets .E˜ ,j and .E is the total number of strands (fat strands and ordinary strands together) of .f ,j . ˜ 1,1 = {˜z2,1 , . . . , z˜ 2,k2 } and .E 1,1 = {z2,1 , . . . , z2,k2 }. Denote by .f 2,j the Let .E strand of .f 1,1 with initial point .f 2,j (0) = z˜ 2,j . We may choose the label so that for each j the strand .ϕt (z2,j ), t ∈ [0, 1], of .g 1,1 corresponds to .f 2,j under the isotopy of .g 1,1 and .f 1,1 . Take a small positive number .ε2 such that the .ε2 -neighbourhoods of the strands of .f 1,1 are pairwise disjoint. We consider the pure geometric fat braid f 1,1 f 2,1 (f 2,1  ε2 f 2,1 ) : [0, 1] → Cn(1,1)+n(2,1)−1 (C)Sn(1,1)+n(2,1)−1 , (4.15)

.

that is obtained by removing the strand .f 2,1 from .f 1,1 and inserting into a tube around .f 2,1 a geometric fat braid that is obtained from .f 2,1 by the operations . and .. The base point of the obtained geometric braid (4.15) is the point ˜ (E˜ 1,1 ∪ E

.

1,1

˜ ) z˜ 2,1 (˜z2,1  ε2 (E˜ 2,1 ∪ E

2,1

)) .

(4.16)

 Since the geometric fat braids .f 1,1 and .g 1,1 (see (4.14)) both represent .b(1, 1), and 2,1 2,1  the geometric fat braids .f and .g (see (4.14)) both represent .b(2, 1), Lemma 4.3 shows that for small .ε2 the geometric fat braids (4.15) and .g 1,1 g 2,1 g 2,1 are free isotopic. The geometric braid .g 1,1 g 2,1 g 2,1 equals ϕt ((E 1,1 ∪ E 1,1 )

.

z2,1

(E 2,1 ∪ E 2,1 ))

= ϕt ((E 1,1 ∪ E 1,1 ) \ z2,1 )∪(E 2,1 ∪ E 2,1 )), t ∈ [0, 1] .

(4.17)

We proceed in the same way by induction on .j = 2, . . . , k2 , . We arrive at a geometric fat braid f 1,1 f 2,1 (f 2,1  ε2 f 2,1 )  . . . f 2,k2 (f 2,k2  ε2 f 2,k2 ) .

.

(4.18)

Since .E 1,1 = {z2,1 , . . . , z2,k2 }, the equality (E 1,1 ∪ E 1,1 ) z2,1 (E 2,1 ∪ E 2,1 )  . . . z2,k2 (E 2,k2 ∪ E 2,k2 ) =

.

E 1,1 ∪ (E 2,1 ∪ E 2,1 ) ∪ . . . ∪ (E 2,k2 ∪ E 2,k2 ) = (E ∩ Q2 ) ∪ (E 2,1 ∪ . . . ∪ E 2,k2 )

156

4 Reducible Pure Braids. Irreducible Nodal Components, Irreducible Braid. . .

˜ 1,1 , .E˜ 2,j , and .E ˜ 2,j . Hence, repeated holds. The respective equality holds for .E˜ 1,1 , .E application of Lemma 4.3 shows that (4.18) is free isotopic to the geometric fat braid   ϕt (E ∩ Q2 ) ∪ (E 2,1 ∪ . . . ∪ E 2,k2 ) , t ∈ [0, 1] ,

.

(4.19)

all whose fat strands have initial points in discs .δ 3,i bounded by curves of third generation. We make an induction on the number of generation . = 2, . . . , N. Suppose we found by induction small positive numbers .εl and obtained a geometric fat braid def

2,1 f Q = f 1,1 )  . . . f 2,k2 (f 2,k2  ε2 f 2,k2 ) 1 f 2,1 (f 2,1  ε2 f

.

 . . . f ,k (f ,k  ε f ,k )

(4.20)

that is free homotopic to the geometric fat braid   ϕt (E ∩ Q ) ∪ (E ,1 ∪ . . . ∪ E ,k ) , t ∈ [0, 1] ,

.

(4.21)

Denote the fat strands of (4.20) by .f +1,j . Choose a positive number .ε+1 such that the .ε+1 -neighbourhoods of the strands of the geometric fat braid (4.20) are pairwise disjoint. Add strands to (4.20) by the operation f Q f +1,1 (f +1,1  ε+1 f +1,1 )  . . . f +1,k+1 (f +1,k+1  ε+1 f +1,k+1 ) . (4.22)

.

We obtain a geometric fat braid of the same form as (4.20) with . replaced by .+ 1. Repeated application of Lemma 4.3 shows that (4.22) is isotopic to the geometric fat braid (4.21) with . replaced by . + 1. Since the boundary of .QN is empty, the geometric fat braid .f QN is an ordinary braid that is isotopic to ϕt (E), t ∈ [0, 1] .

.

(4.23)

We described a procedure that gives for arbitrary geometric fat braids .f i,j in  [0, 1] × D, that represent .b(i, j ), the ordinary geometric braid .fQN (see (4.20) with ˆ The recovery . replaced by N ), that is free isotopic to (4.22), hence, represents .b. procedure is described. .

4.5 Pure Braids, the Reducible Case. Proof of the Main Theorem In this section we will prove the Main Theorem for reducible pure braids. The theorem will follow from the Propositions 4.1 and 4.2 below which are of independent interest.

4.5 Pure Braids, the Reducible Case. Proof of the Main Theorem

157

We will represent a conjugacy class of pure braids .bˆ by a holomorphic map from an annulus A to the symmetrized configuration space .Cn (C)Sn (or by ¯ The set a holomorphic map on A that extends continuously to the closure .A). ¯ respectively) is the union of n connected .{(z, f (z)); z ∈ A} (or .{(z, f (z)); z ∈ A}, components which we also call strands. A map with some strands declared to be fat will be called a fat map and denoted by .f . Proposition 4.1 The conformal module of a conjugacy class .bˆ of pure n-braids is equal to the smallest conformal module among the irreducible braid components of ˆ In other words, the equality .b.

 ˆ = min M b(, M(b) j)

.

,j

(4.24)

holds. ˆ Then there exists Proof Let A be any annulus of conformal module .m(A) < M(b). a holomorphic mapping f : A → Cn (C)Sn

.

(4.25)

 ˆ Recall that for all .(, j ) the class .b(, j ) is obtained from .bˆ by forgetrepresenting .b. ting some strands. Forgetting the respective strands of the mapping (4.25) provides  j ). us a holomorphic map .f ,j : A → Cn(,j ) (C)Sn(,j ) that represents .b(,  ˆ Hence, for each annulus A with .m(A) < M(b) the inequality .M(b(, j )) > m(A) holds for all .(, j ). Hence,

 ˆ ≤ min M b(, M(b) j) .

.

,j

To prove the opposite inequality, let A be an annulus of conformal module

 m(A) < min,j M b(, j ) . Then for each .(, j ) there is a continuous fat map ,j : A ¯ → Cn(,j ) (D)Sn(,j ) that is holomorphic on A and represents the .f  conjugacy class of fat braids .b(, j ). We will denote by .f +1,j  the strands of ,j  that correspond to fat strands of .b(, j ). As in the recovery the mappings .f procedure of the conjugacy class of braids from the irreducible braid components we successively choose small numbers .ε ,  ≤ N, and consider the mapping .

f 1,1 f 2,1 (f 2,1  ε2 f 2,1 )  . . . f 2,k2 (f 2,k2  ε2 f 2,k2 )

.

 . . . f ,k (f ,k  ε f ,k )

(4.26)

that is defined pointwise for .z ∈ A¯ in the same way as the respective objects were defined in Sect. 4.4 for geometric fat braids. The .ε are chosen successively

158

4 Reducible Pure Braids. Irreducible Nodal Components, Irreducible Braid. . .

so that the .ε -neighbourhoods of the strands of the fat map defined by (4.26) for  replaced by . − 1 are pairwise disjoint. For . = N the mapping (4.26) is an ˆ Since all mappings .f ,j and ordinary mapping and represents the conjugacy class .b. .f  ,j  are holomorphic on A, the mapping (4.26) is holomorphic by construction.

 ˆ ≥ m(A) for any A with .m(A) < min,j M b(, j ) . We obtained Hence, .M(b) .

the inequality

 ˆ ≥ min M b(, M(b) j) .

.

,j

 

The proposition is proved.

Proposition 4.2 For the mapping class .mb,∞ associated to a pure n-braid b and  ,j its irreducible nodal components .mb, the equality  ,j  h(m b,∞ ) = max h(mb, )

.

,j

(4.27)

holds. In the lemma all conjugacy classes are considered as classes of selfhomeomorphisms of compact Riemann surfaces with distinguished points. Proof We prove first the inequality  ,j  h(m b,∞ ) ≥ max h(mb, ).

.

,j

(4.28)

We take again a self-homeomorphism .ϕb,∞ of .P1 with distinguished points .En = En ∪ {∞} that fixes pointwise the complement of the unit disc and represents the mapping class .mb,∞ ∈ M(P1 ; En ) of the braid b. We require that .ϕb,∞ is completely reduced by the admissible system of curves .C that is contained in the unit disc .D. As in Sect. 4.2 we choose the set .E ,j ⊂ En as follows. For each hole of .S ,j bounded by an interior boundary component of .S ,j we chose a point of E contained in it. These points constitute the set .E ,j . Then .mb(,j ),∞ is the mapping class of .ϕb,∞ in .M(P1 ; E  ,j ∪ E ,j ). We obtain h(mb(,j ),∞ ) = inf{h(ϕ) : ϕ is Hom+ (P1 ; E 

.

,j

∪ E ,j ) − isotopic to ϕb,∞ } . (4.29)

On the other hand, h(mb,∞ ) = inf{h(ϕ) : ϕ is Hom+ (P1 ; En ) − isotopic to ϕb,∞ } .

.

(4.30)

4.5 Pure Braids, the Reducible Case. Proof of the Main Theorem

159

Since the space which appears in (4.30) is contained in the space which appears in (4.29), the infimum in (4.30) is not smaller than the infimum in (4.29): h(mb,∞ ) ≥ h(mb(,j ),∞ )

.

for each , j.

(4.31)

 ,j Since by Lemma 4.1 the equality .mb, = mb(,j ),∞ holds, the inequality (4.28) is proved. The proof of the opposite inequality is based on Theorem 2.10. Recall that the  ,j  ,j ) for a punctured Riemann sphere (considered as element of .M(Y class .m 

b,

Y ,j ) is represented by the mapping .ϕb,∞ |(S ,j \ En ) on a Riemann surface of second kind. Take an absolutely extremal representative of this class. This is a self-homeomorphism of a punctured Riemann surface .Y˜ ,j . It extends across the ,j punctures to a self-homeomorphism .ϕ˜ ,j of .P1 with distinguished points .E˜  ∪ ˜ ,j corresponds to the holes ˜ ,j . (The set .E˜  ,j corresponds to .S ,j ∩ En , the set .E E of .S ,j .) .ϕ˜ ,j is entropy minimizing in the class of self-homeomorphisms of .P1 with distinguished points that is obtained from .ϕb,∞ |(S ,j \ En ) by isotopy, conjugation and extension across punctures, i.e.

.

 ,j h(ϕ˜ ,j ) = inf{h(ϕ) : ϕ ∈ mb, } .

.

By Theorem 2.10 for each .(, j ) there is a closed topological disc around each ˜ ,j and a self-homeomorphism .ϕ˜ ,j of .P1 with distinguished distinguished point in .E 0 ,j ˜ ,j that is isotopic (with distinguished points .E˜  ,j ∪ E ˜ ,j ) to .ϕ˜ ,j , points .E˜  ∪ E ,j

has the same entropy .h(ϕ˜0 ) = h(ϕ˜ ,j ), and equals the identity in a neighbourhood of each of the discs.  ,j Since .ϕ˜ ,j ∈ mb, there exists a homeomorphism .w˜ ,j : S ,j \ E  ,j → Y˜ ,j ˜ ,j ) to a self-homeomorphism of .S ,j \ E  ,j that conjugates .ϕ˜ ,j | P1 \ (E  ,j ∪ E

that is isotopic on this set to .ϕb,∞ | S ,j \E  ,j . The conjugate .(w˜ ,j )−1 ◦ ϕ˜0 ◦ w˜ ,j ,j of the isotopic homeomorphism .ϕ˜0 equals the identity in a neighbourhood of all boundary components of .S ,j . Hence, it extends to the boundary of .S ,j as a self,j homeomorphism .ϕ0 of .S ,j \ En that fixes the boundary pointwise. The isotopy ,j class in .M(S ,j \ En ; ∂S ,j ) of the extension .ϕ0 differs from that of the restriction  ,j that .ϕb,∞ | S ,j \ En by products of powers of Dehn twists about curves in .S ,j are homologous to one of the boundary components of .S . By Lemma 2.14 for ,j each .(, j ) the entropy minimizing homeomorphism .ϕ˜0 may be chosen from the ,j beginning so that .ϕ0 and .ϕb,∞ | S ,j \ En are .M(S ,j \ En ; ∂S ,j )-isotopic. Consider the self-homeomorphism .ϕ0 of .P1 \ En whose restriction to .S ,j \ En ,j equals .ϕ0 for all . and j . (The self-homeomorphism is well defined since each ,j ,j .ϕ 0 is equal to the identity near the boundary .∂S .) The mapping .ϕ0 is isotopic ,j

160

4 Reducible Pure Braids. Irreducible Nodal Components, Irreducible Braid. . .

to .ϕb,∞ in .M(P1 \ En ), hence it represents .mb,∞ (considered as class of self,j homeomorphisms of a punctured surface). For the extensions .ϕˆ 0 and .ϕˆ0 of the ,j ,j mappings .ϕ0 and .ϕ0 across the punctures the inequality .h(ϕˆ0 ) ≤ max,j h(ϕˆ 0 ) holds. Hence, the opposite inequality  ,j  h(m b,∞ ) ≤ max h(mb, )

.

,j

holds and the proposition is proved.

(4.32)  

The Proof of the Main Theorem for Reducible Pure Braids is an immediate consequence of Propositions 4.1 and 4.2, Lemma 4.1, and the Main Theorem for irreducible braids.    Remark 4.1 The mapping class .m b,∞ can be recovered from the irreducible nodal components of the braid up to a product of Dehn twists about simple closed curves that are homologous to the admissible curves. Indeed, this statement was obtained by the proof of inequality (4.32).

Chapter 5

The General Case. Irreducible Nodal Components, Irreducible Braid Components, and the Proof of the Main Theorem

In this chapter we prove the Main Theorem for reducible non-pure braids. While the strategy of the proof in this case is similar to that in the case of pure braids, there are some difficulties that need additional considerations. For instance, for each conjugacy class of pure braids the irreducible braid components can be obtained by forgetting strands. This approach does not work for non-pure braids. We define irreducible braid components and irreducible nodal components in the non-pure case, and describe the respective decompositions into irreducible components and the recovery procedures.

5.1 Irreducible Nodal Components. The General Case In this Chapter we consider a reducible, not necessarily pure, braid .b ∈ Bn and its mapping class .mb ∈ M(D; ∂D, En ), .En ⊂ D . We represent the mapping class 1 .mb,∞ = H∞ (mb ) ∈ M(P ; ∞, En ) (see equality (1.8)) by a self-homeomorphism 1 .ϕb,∞ of .P which is the identity outside the unit disc .D. We may choose .ϕb,∞ so that it is completely reduced by an admissible system of curves .C = {C1 , . . . , Ck } 1   in .D\E n ⊂ P \En . As before .En = En ∪ {∞}. In particular,  .ϕb,∞ leaves the union . C invariant, and also leaves the complement .P1 \ C invariant. The C∈C

C∈C

nodal components and the irreducible braid components will be associated to the isotopy class of the chosen admissible system of curves.  As in Chap. 4 the connected component of .P1 \ C∈C C that contains .∞ is denoted by .S 1,1 and is said to be of generation 1. By induction for . ≥ 2 the interior of the union of the closure of all components  of generation not exceeding 1 . − 1 is denoted by .Q−1 . The components of .P \ C∈C C that share a boundary component with .Q−1 are called the components of generation .. However, since b is not required to be a pure braid, the connected components of .C \ C of generation © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025 B. Jöricke, Braids, Conformal Module, Entropy, and Gromov’s Oka Principle, Lecture Notes in Mathematics 2360, https://doi.org/10.1007/978-3-031-67288-0_5

161

162

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

 are not necessarily fixed by .ϕb,∞ . But the components of each generation . are permuted along cycles. Indeed, since .ϕ fixes .∞ ∈ Q−1 it fixes .Q−1 setwise and, hence, permutes its boundary components, and therefore permutes the components  of .P1 \ C that share a boundary component with .Q−1 . C∈C  We label the connected components of .P1 \ C of generation . ≥ 2 as follows. C∈C  C of generation . a set of the cycle Choose for each cycle of components of .P1 \

.

C∈C

and denote it by .S1,i . Let .k(, i) be the length of the cycle, i.e. the smallest positive k(,i) integer for which .ϕb,∞ maps .S1,i onto itself. Put def

Sj,i+1 = ϕb,∞ (Sj,i ) , j = 1, . . . , k(, i) − 1.

(5.1)

,i ϕb (Sk(,i) ) = S1,i ,

(5.2)

.

For all ., and i .

 The .ϕb,∞ -cycles of connected components of generation . of .P1 \ C are denoted C∈C  ,i ,i  by .cyc,i 0 = S1 , . . . , Sk(,i) , i = 1, . . . , k , ϕb,∞

ϕb,∞

ϕb,∞

ϕb,∞

,i S1,i −→ S2,i −→ . . . −→ Sk(,i) −→ S1,i .

.

Note that the number of cycles .k does not exceed the number of components .k of generation .. ,i  The set of distinguished points .E  ,i j = Sj ∩ En is permuted in the same way: ,i ϕb,∞

,i ϕb,∞

ϕb,∞

ϕb,∞

E  1 −→ E  2 −→ . . . −→ E  k(,i) −→ E  1 .

.

,i

,i

    The cycles of connected components of .P1 \ ( C) En of generation . are C∈C   ,i \ En , i = 1, . . . , k . denoted by .cyc,i = S1,i \ En , . . . , Sk(,i) Consider the .ϕb,∞ -cycle of curves .∂E Sj,i , .j = 1, . . . , k(, i), of generation . and the cycle of discs .δj,i in .C that are bounded by .∂δj,i = ∂E Sj,i . The homeomorphism ,i ϕb,∞ permutes the discs along the cycle .(δ1,i , δ2,i , . . . , δk(,i) ),

.

ϕb,∞

ϕb,∞

ϕb,∞

ϕb,∞

,i δ1,i −→ δ2,i −→ . . . −→ δk(,i) −→ δ1,i .

.

As in the case of pure braids (see also Sect. 4.1 and also Sect. 1.8) we associate to b a nodal surface Y and an isotopy class .mb, of self-homeomorphisms of Y , using

5.1 Irreducible Nodal Components. The General Case

163

a continuous surjection .w : P1 \ En → Y whose preimage of each node is a curve .C ∈ C. Recall w maps.P1 \  that the set of nodes of Y is denoted by .N . The surjection  1 (En ∪ C) homeomorphically onto .Y \ N . Conjugate .ϕb,∞ | P \ (En ∪ C) C∈C C∈ C  by the inverse of .w | P1 \ (En ∪ C) and denote the conjugate by .ϕb, . Its C∈C

isotopy class of self-homeomorphisms of .Y \ N is denoted by .mb, . The conjugacy  class .m b, is called the nodal conjugacy class associated to .mb,∞ and the system of admissible curves .C. Notice that the mapping w extends across the punctures to a continuous surjection from .P1 onto the compact nodal surface .Y c obtained by filling the punctures of Y . The extended mapping is also denoted by w. The elements of the class .mb,

extend across the punctures of the nodal surface Y to self-homeomorphisms of the compact nodal surface .Y c . By an abuse of notation we denote the class of extended  mappings also by .mb, , and denote the respective conjugacy class also by .m b, . Let .Yj,i be the component .w(Sj,i \En ) of .Y \N . The mapping .ϕb, permutes the

components .Yj,i of .Y \N along cycles corresponding to the cycles of the .Sj,i \E  ,i j .

We denote the cycles of components .Yj,i by .cyc,i b, .  ,i ,i The conjugacy classes .m,i b, of the restrictions .mb, of .mb, to the cycles .cycb, ,  are called the irreducible nodal components of the class .m b,∞ . The irreducible nodal  components determine the class .m up to products of powers of some Dehn b,∞ twists. This can be seen in the same way as in the case of pure braids. (See later the Remark 5.2.) Instead of the cycles of components .Yj,i , .j = 1, . . . , we may, equivalently, consider the respective cycles of compact Riemann surfaces .(Yj,i )c , .j = 1, . . . , and extend the homeomorphisms between the .Yj,i , .j = 1, . . . , to homeomorphisms between the compact surfaces. The obtained conjugacy classes will also be denoted  by .m,i b, .  If the length of the cycle is bigger than one, the class .m,i b, is the conjugacy class of a mapping class on a not connected Riemann surface. Lemma 5.1 below  states that each irreducible nodal component .m,i b, is determined by a conjugacy class of mapping classes of self-homeomorphisms of a connected Riemann surface. For instance, for the length .k(, i) of the cycle .cyc,i b, , we take the .k(, i)-th power of .ϕb, , and restrict it to a component of the cycle, say to .Y1,i . The lemma states 

that the conjugacy class .mbk(,i) , | Y1,i of the restricted mapping determines the  k(,i) irreducible nodal component .m,i | Yj,i , j = b, . Notice that the mappings .ϕb,



1, . . . , k(, i) , are conjugate, hence, the class .mbk(,i) , | Y1,i is defined by the cycle

164

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

cyc,i b, and is independent on the choice of the set in the cycle to which the mapping

.

k(,i)

ϕb, is restricted. The key ingredient of the proof of the following Lemma 5.1 is a simple Lemma on Conjugation, which was used by Bers [11]. For convenience of the reader we include the Lemma on Conjugation together with a proof in the Appendix A.2.

.

Lemma 5.1 Suppose a self-homeomorphism .ψ of . Y1,i represents the class



mbk(,i) , | Y1,i for the . k(, i)-th power of the braid . b . Then any self,i , that moves the sets .Yj,i homeomorphism . ϕ of the union .Y1,i ∪ . . . ∪ Yk(,i) along the cycle

.

ϕ

ϕ

ϕ

ϕ

,i Y1,i −→ Y2,i −→ . . . −→ Yk(,i) −→ Y1,i ,

.

(5.3)

and satisfies the condition k(,i)

ϕ

.

| Y1,i = ψ ,

(5.4)



represents .mb, | cyc,i

. A homeomorphisms .ϕ with these properties always exists. ,i that represents Vice versa, for any self-homeomorphism .ϕ of .Y1,i ∪ . . . ∪ Yk(,i) 



k(,i)

mb, | cyc,i

the restriction .ϕ

.

| Y1,i represents . mbk(,i) , | Y1,i . 

k(,i)

Proof It is clear that for each .ϕ ∈ mb, | cyc,i

the inclusion .ϕ



mbk(,i) , | Y1,i holds.

| Y1,i ∈



Prove the first part of the lemma. Let .ψ in .mbk(,i) , | Y1,i . Define a self

homeomorphisms .ϕ ∈ mb, | cyc,i

as follows. For each .j = 1, . . . , k(, i) − 1, we take any homeomorphism .ϕ ,j from .Yj,i onto .Yj,i +1 . For .j = k(, i) we take the

,i onto .Y1,i for which homeomorphism .ϕ ,k(,i) from .Yk(,i)

ϕ ,k(,i) ◦ . . . ◦ ϕ ,1 = ψ

.

k(,i) 

Consider the self-homeomorphism .ϕ of . k(,i)

ϕ

.

j =1

on

Y1,i .

(5.5)

Yj,i which equals .ϕ ,j on .Yj,i . Then

| Y1,i = ψ .

We have to prove that any mapping .ϕ obtained in this way is in the class



from .P1 \ En onto Y that mb, | cyc,i

. Let w be as before the continuous surjection  1  C) onto .Y \ N . Let .w ,i be the restricts to a homeomorphism from .P \ (En ∪

.

C∈C

5.2 Irreducible Braid Components. The General Case k(,i) 

restriction of w to .

j =1

165

Sj,i \En . The mapping .w ,i takes .Sj,i \En homeomorphically

onto .Yj,i for each j . Conjugate .ϕb,∞ |

k(,i) 

Sj,i \En by the inverse of .w ,i and

j =1 k(,i) 

denote the obtained self-homeomorphism of .

j =1



Yj,i by .ϕ˜ . Then .ϕ˜ represents

,i the class .mb, | cyc,i

. The mappings .ϕ and .ϕ˜ move the .Yj along the same k(,i)

cycle, and both, .ϕ˜



k(,i)

| Y1,i and .ϕ

| Y1,i , represent the same conjugacy class



mbk(,i) , | Y1,i . Since .ϕ˜ represents .mb, | cyc,i

, by the Lemma on Conjugation (see Appendix A.2) also the mapping .ϕ represents this class. The lemma is proved.



.

Remark 5.1 The restriction .ϕb,∞ | S1,i \ En , equivalently, the mapping .ϕb,

k(,i)

Y1,i ,

k(,i)

|

is irreducible.

Suppose the contrary. Then there exists an .M(P1 \En )-isotopy that joins .ϕb,∞ with k(,i)

k(,i)

a self-homeomorphism .ψ of .P1 and changes the values of .ϕb,∞ only on .S1,i , such that .ψ has the following property. It fixes setwise an admissible system .γ1 ∪ . . . ∪ γk  of simple closed curves in .S1,i \ En . Then the mapping .ϕb,∞ , that equals .ϕb,∞ on

,i P1 \ S1,i (in particular, on .S2,i ∪ . . . Sk(,i) ) and equals .(ϕb,∞ )−k(,i)+1 ◦ ψ on .S1,i , fixes the set of loops

.

γ1 ∪ . . . ∪ γk

.



 ϕb,∞ (γ1 ∪ . . . ∪ γk )



...

  (ϕb,∞ )k(,i)−1 (γ1 ∪ . . . ∪ γk )

setwise and is .M(P1 \ En )-isotopic to .ϕb,∞ by the Lemma on conjugation (see the Appendix A.2). This contradicts the fact that the system of curves .C was maximal.

5.2 Irreducible Braid Components. The General Case We continue to consider non-pure reducible braids b with base point .En ⊂ D + 1 1 together with the associated homeomorphism .ϕb,∞ ∈ Hom (P ; P \ D , En ), that fixes setwise an admissible system of curves . C contained in .D and is completely C∈C

reduced by this system. Notice that a cycle of discs .δj,i of length .k(, i) corresponding to a cycle of components of .P1 \ C may not contain a cycle of distinguished points of this C∈P1

length. However, the following lemma holds.

166

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

Lemma 5.2 Let . > 1. Take any .ϕb -cycle .cyc,i of sets .Sj,i , j = 1, . . . k(, i) of length .k(, i), and the associated cycles of discs .δj,i and of curves .∂E Sj,i . Then there is an .M(P1 ; P1 \ D, En )-isotopy of .ϕb,∞ , that changes .ϕb,∞ only in  a small neighbourhood of .∂E S1,i , and provides a homeomorphism .ϕb,∞ which has ,i ⊂ δj,i of length .k(, i), a cycle of perhaps non-distinguished points .z,i j ∈ Sj

,i ,i ,i   ϕb,∞ (z,i j ) = zj +1 , j = 1, . . . , k(, i) − 1, ϕb,∞ (zk(,i) ) = z1 . Moreover, the  has the following property. new homeomorphism .ϕb,∞

.

For 1 ≤ j ≤ k(, i), ϕ  b,∞ fixes a neighbourhood of ∂E Sj,i ∪ {z,i j } pointwise. (5.6) k(,i)

.

We will apply Lemma 5.2 to all cycles of discs .δj,i . Recall that the collection of the δj,i , i = 1, . . . , k , j = 1, . . . k(, i), is the set of all holes of .Q−1 . Hence, there

.

is a one-to-one correspondence between the set of holes of .Q−1 and the points .z,i j contained in the holes.  Proof of Lemma 5.2 With the requirement that .ϕb,∞ = ϕb,∞ in a neighbourhood

of .∂E Sj,i , j = 2, . . . , k(, i), the equation  ϕb,∞ = ϕb,∞

.

1−k(,i)

(5.7)

in a neighbourhood of .∂E S1,i implies .ϕ  b,∞ = Id in a neighbourhood of .∂E S1,i . k(,i)

Then this equation also holds in a neighbourhood of .∂E Sj,i , .j = 2, . . . , k(, i), j −2

,i since the restriction of .ϕ  k(,i) equals .ϕb,∞ ◦ b,∞ to a neighbourhood of .∂E Sj k(,i)−j +1

 ◦ ϕb,∞ ϕb,∞

. It is clear that there is an .M(P1 ; P1 \ D, En )-isotopy of

ϕb,∞ that changes .ϕb,∞ only in a small neighbourhood of .∂E S1,i and provides a  that satisfies (5.7) in a neighbourhood of .∂E S1,i . homeomorphism .ϕb,∞

.

,i It remains to choose a point .z,i 1 ∈ δ1 that is contained in this neighbourhood. ,i  The homeomorphism .ϕb,∞ maps .z1 ∈ δ1,i along the required cycle and the .k(, i) fixes a neighbourhood of .z,i th iterate of .ϕb,∞ 1 . The lemma is proved.



In the following the homeomorphism .ϕb,∞ will satisfy all conditions that are  obtained for the homeomorphism .ϕb,∞ of Lemma 5.2. + 1 1 Let .ϕt ∈ Hom (P ; P \ D), .t ∈ [0, 1], be a continuous family of selfhomeomorphisms of .C such that .ϕ0 = id and .ϕ1 = ϕb,∞ . The geometric braid .{(t, ϕt (En )), t ∈ [0, 1]} is contained in the cylinder .[0, 1] × D and represents b. Put .g(t) = ϕt (En ), t ∈ [0, 1], .g : [0, 1] → Cn (C)Sn . The Outermost Braid The tubular braid associated to the outermost component S 1,1 can be defined similarly as in Chap. 4. Consider the set of all holes of the outermost component. We give each hole the label of the unique .Sj2,i whose exterior

.

5.2 Irreducible Braid Components. The General Case

167

boundary is the boundary of the hole, and denote the respective hole by .δj2,i . Put 1,1 = E ∩ S 1,1 . .E Associate the tubular geometric braid  k2 k(2,i)   2,i  t, ϕt (E 1,1 ∪ . δj ) ,

t ∈ [0, 1]

(5.8)

i=1 j =1

to b and .S 1,1 . The set correctly defines a tubular braid, since .ϕ0 is the identity and k2 k(2,i) 2,i 1,1 ∪ .ϕ1 = ϕb,∞ maps the set .E i=1 j =1 δj onto itself. We define a geometric fat braid that is a deformation retract of the tubular braid. 2,i 2,i For each cycle .cyc2,i 0 of sets .Sj of length .k(2, i) we take the cycle of points .zj ∈ Sj2,i chosen in Lemma 5.2 (see also relation (5.6)) as initial points of the fat strands. 2,i ⊂ δj2,i , .i = In other words, let .E 1,1 be the collection of all points .z2,i j ∈ Sj 1, . . . , k2 , .j = 1, . . . , k(2, i) , assigned to the set of holes of .S 1,1 in Lemma 5.2,

E 1,1 =

k2 

.

2,i {z2,i 1 , . . . , zk(2,i) } .

(5.9)

i=1

Associate to .S 1,1 the geometric fat braid   t, ϕt (E 1,1 ∪ E 1,1 ) ,

.

t ∈ [0, 1] .

(5.10)

 1) be Let .B(1, 1) be the isotopy class of the geometric fat braid (5.10), and let .B(1,  1) the outermost irreducible braid component of its conjugacy class. We call .B(1, the braid b. See Fig. 5.1, where .E 1,1 = ∅. In this figure the connected components of  .C\ C of generation 2 are .S12,1 , S22,1 , S32,1 , which are moved along a 3-cycle by C∈C

2,1 2,1 ϕb,∞ . The set .E 1,1 equals .E 1,1 = {z2,1 1 , z2 , z3 }. Denote by .n(1, 1) the number of points in .E 1,1 ∪ E 1,1 . We write the mapping defining the geometric fat braid (5.10) as a fat map .g 1,1 ,

.

g 1,1 : [0, 1] → Cn(1,1) (C)Sn(1,1) , g 1,1 (t) = ϕt (E 1,1 ∪ E 1,1 ) .

.

(5.11)

The following equality holds Lemma 5.3 

mB(1,1),∞ = m1,1 b, .



.

(5.12)

168

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

Proof of Lemma 5.3 The situation differs from that of Lemma 4.1 by the fact that b is not necessarily a pure braid and, hence, the interior boundary components of 1,1 may be permuted by the mapping .ϕ 1,1 \ E  1,1 .S b,∞ . Again, the mapping .ϕb,∞ | S

1,1 ). The mapping .ϕb,∞ | P1 \ (E 1,1 ∪ E  1,1 ) represents the class .m1,1 ∈ M(Y 

b,

1 \ (E 1,1 ∪ E  1,1 )). (Recall that .E  1,1 = represents the class .mB(1,1),∞ ∈ M(P 1,1 E ∪ {∞}.) Similarly as in the proof of Lemma 4.1 we take for each first labeled set .δ12,i of a cycle whose exterior boundary is an interior boundary component of .S 1,1 1,1 \ E  1,1 and shares a boundary an open annulus .A2,i 1 that is contained in .S component with the boundary of .δ11,1 . Moreover we require that .A2,i 1 is contained k(2,i) 2,i in the neighbourhood of .∂δ1 which is fixed by the iterate .ϕb,∞ . We put .A2,i j = 

j −1

ϕb,∞ (A2,i 1 ), j = 2, . . . , k(2, 1). The annuli are taken small enough so that all obtained annuli for .i = 1, . . . , k2 , .j = 1, . . . , k(2, i) are pairwise disjoint. There is a homeomorphism .w˜ 1,1 from .S 1,1 \ E  1,1 onto .P1 \ (E  1,1 ∪ E 1,1 ) with the following properties. .w˜ 1,1 is equal to the identical injection on the set .S 1,1 \  ˜ 1,1 takes each annulus .A2,i (E  1,1 ∪ A2,i j ). Moreover, for each j the mapping .w j def

2,i  2,i = A2,i ∪ δ 2,i . Conjugate .ϕ onto the punctured disc .δ  2,i b,∞ | j \ {zj }, where .δ j j j def

1,1 1,1 S 1,1 \ E  1,1 with the inverse of .w˜ 1,1 . The conjugate .ϕ˜b,∞ is related to .ϕb,∞ =

ϕb,∞ | P1 \ (E 1,1 ∪ E  1,1 ) by isotopy and conjugation. Indeed, the two mappings, 1,1 \ E  1,1 and .ϕ˜ 1,1 , differ only on the punctured discs .δ  2,i \ z2,i around .ϕb,∞ | S j b,∞ j 1,1 and map the punctured discs along the same cycles. They the points .z2,i j of .E

are equal on the boundary of each .δ  2,i j . Moreover, for each i the .k(2, i)-th iterate

of both mappings is the identity on each .∂δ  2,i j , .j = 1, . . . , k(i, j ), since by (5.6)

k(2,i) ϕb,∞ is the identity on each .A2,i 1 . Apply for each i the Lemma on conjugation (see  1,1 1,1 Appendix A.2) to the restriction of the two mappings .ϕb,∞ and .ϕ˜b,∞ to . k(2,i) δ  2,i j , j and take into account the following fact. If a self-homeomorphism of a closed disc punctured at a point fixes the boundary circle pointwise, then it is isotopic to the identity through self-homeomorphisms that fix the boundary circle pointwise. We 1,1 1,1 and .ϕ˜b,∞ are related by isotopy and conjugation.

proved that .ϕb,∞

.

i) Associated to the .ϕb,∞ -Cycles The Irreducible Braid Components .B(, ,i .cyc , . > 1 Recall that, for instance, the ordinary part of the geometric fat braid representing .B(1, 1) was found by forgetting all strands of the original braid with initial points not in .E 1,1 = En ∩ S 1,1 . It was important that the set .E 1,1 is invariant under .ϕb,∞ . i) In order to apply this recipe for getting the irreducible braid component .B(, for . > 1 we wish to start with a .ϕb,∞ -invariant set of points. The intersection of ,i ,i has this property. The problem .En with the union of the sets .S j of the cycle .cyc is that the geometric braid obtained by forgetting all strands with initial point not

5.2 Irreducible Braid Components. The General Case

169

   contained in this set carries also information about the tubular braid . t, ϕt ( j δj,i ) , which is part of the information carried by the geometric fat braids of generation less k(,i) than .. However, the power .ϕb,∞ leaves each .Sj,i , .j = 1, . . . , k(, i), invariant k(,i) by forgetting the strands, and, for instance, the geometric braid related to .ϕb,∞

whose base points are not contained in .S1,i ∩ En , does not contain information about the geometric fat braids of generation less than .. We will therefore relate  i) to the power .bk(,i) of the braid b similarly the irreducible braid component .B(, as we related the irreducible nodal component to the power .bk(,i) . Recall that, the  ,i irreducible nodal component .m,i b, can be identified with the class .mbk(,i) , | Y1



that is related to the power .bk(,i) of the braid b and a set .Y1,i of the cycle .cyc,i . In detail, we extend the family .ϕt , t ∈ [0, 1], to a family defined for all .t ∈ R def

by putting .ϕt = ϕt−k ◦ (ϕb,∞ )k , t ∈ [k, k + 1]. Since .ϕ0 = Id and .ϕ1 = ϕb,∞ , the family is correctly defined. The family .ϕt , t ∈ [0, k(, i)], is a parameterizing isotopy for .bk(,i) (normalized on the interval .[0, k(, i)] rather than on the interval .[0, 1]). In other words, the geometric braid  .

 t, ϕt (En ) ,

t ∈ [0, k(, i)] ,

(5.13)

k(,i)

represents .bk(,i) . Indeed, .ϕ0 = Id, and .ϕk(,i) = ϕb,∞ . Define a tubular braid in .[0, k(, i)] × C as follows. Let as before .Sj,i , j = def

,i 1, . . . , k(, i), be the sets of the cycle .cyc,i = E ∩ S1,i . Further, 0 . Put .E1 consider all interior boundary components of .S1,i and the closed topological discs of generation .+1 in .C that are bounded by them (in other words, consider all “interior 



holes” of .S1,i ). They are of the form .δj+1,i , where .δj+1,i is the bounded disc in .C   

,j

that is bounded by the exterior boundary component of .Sj+1,i . Let .H (S1 ) be the  ,j

set of “interior” holes of .S1 . Consider the tubular braid ⎧ ⎪ ⎨  t, ϕt E1,i ∪ . ⎪ ⎩ in the tube .[0, k(, i)] × C.

⎫ ⎪ ⎬   +1,i   , t ∈ [0, k(, i)] δj  ⎪ ⎭ ,j

(5.14)

H (S1 )



,j

,j



∈ H (S1 ) of .S1 we consider the point .z+1,i ∈ For each “interior hole” .δj+1,i  j 





Sj+1,i ⊂ δj+1,i associated to .δj+1,i by Lemma 5.2. Let .E ,i    1 be the set of all such 

. The geometric fat braid which is a deformation retract of (5.14) is points .z+1,i j defined as   t, ϕt (E1,i ∪ E ,i . ) , t ∈ [0, k(, i)] (5.15) 1

170

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

 1) of a non-pure braid b Fig. 5.1 The irreducible braid component .B(2,

3,1 3,2 2,1 (See Fig. 5.1 where .E 2,1 1 = {z1 , z2 }, .E1 = ∅.) Denote by .n(, i) the number of points in .E1,i ∪ E ,i 1 . We write the fat mapping defining the geometric fat braid (5.15) as k(,i) ,i : [0, k(, i)] → Cn(,i) (C)Sn(,i) , g

.

k(,i) ,i (t) = ϕt (E1,i ∪ E ,i g 1 ).

(5.16)

k(,i) ,i Reparameterize the geometric fat braid .g to obtain a mapping defined on the unit interval, in other words, put k(,i) ,i g ,i (t) = g (k(, i) t) , g ,i : [0, 1] → Cn(,i) (C)Sn(,i) .

.

(5.17)

5.3 The Building Block of the Recovery. The General Case

171

The isotopy class of the geometric fat braid .g ,i is denoted by .B(, i). Its conjugacy i) is the irreducible braid component of the braid b corresponding to the class .B(, i) can be identified with the free isotopy class cycle .cyc,i . The conjugacy class .B(, ,i of .g . It does not depend on the choice of the set .S1,i of the cycle .cyc,i . The equality  

mbk(,i) , | Y1,i = mB(,i),∞

(5.18)

.

k(,i)

follows by applying the arguments proving Eq. (5.12) to the mapping .ϕb,∞ | (S1,i \ 

k(,i)

E1,i ) representing .mbk(,i) , | Y1,i and the mapping .ϕb,∞ representing .mB(,i),∞ . 

,i | P1 \ (E  ,i 1 ∪ E1 )

5.3 The Building Block of the Recovery. The General Case We want to describe a procedure that recovers the conjugacy class of a non-pure braid given an admissible system of curves that completely reduces a homeomorphism that represents the mapping class of the braid and knowing the irreducible braid components related to this system. In the case of non-pure braids it is more transparent to work with the associated closed braids rather than with the braids themselves. We obtain a closed geometric fat braid from a geometric fat braid  (t, g (t)), t ∈ [0, 1]

.

(5.19)

in the cylinder .[0, 1]×C by replacing the interval .[0, 1] by the quotient .[0, 1]0 ∼ 1 which is diffeomorphic to the unit circle .∂D. In other words, the closed geometric fat braid is obtained from the geometric braid by gluing together the top and the bottom fiber of the cylinder .[0, 1] × C using the identity mapping of .C. We put .G(e2π it ) = g (t), t ∈ [0, 1], which is well defined since .g (0) = g (1). The associated closed geometric fat braid, denoted by .G, is defined as the subset of the solid torus .∂D × C     e2π it , g (t) , t ∈ [0, 1] = e2π it , G(e2π it ) , e2π it ∈ ∂D .

.

(5.20)

The connected components of the closed geometric fat braid (5.20) are simple closed curves in the solid torus .∂D × C. For each such loop .Γ there is a natural number k such that the loop intersects each fiber .{e2π it } × C along k points. We call k the covering multiplicity of .Γ . p˜ k D × C  (ζ, z) → (ζ k , z) ∈ ∂D × C of .∂D × C. Consider the k-fold covering .∂ k D × C is isomorphic to .∂D × C. We may identify .∂D × C with Notice that .∂ k

172

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

    t k D × C with . e2π i k , z , t ∈ [0, k], z ∈ C . e2π it , z , t ∈ [0, 1], z ∈ C , and .∂

.

t t k Then .p˜ k ((e2π i k , z)) = (e2π it , z) for .(e2π i k , z) ∈ ∂ D × C. A loop of covering multiplicity k can be written as

Γ =

.

  e2π it , γ (t) , t ∈ [0, k]

(5.21)

for a function .γ with .γ (0) = γ (k). If .Γ has covering multiplicity k, the lift .Γk of .Γ to the k-fold covering, Γk =

.

  t e2π i k , γ (t) , t ∈ [0, k], z ∈ C

(5.22)

D × C. has covering multiplicity 1 as subset of .∂ A solid torus T in .∂D × C that intersects each fiber .{e2π it } × C along a set of k disjoint closed discs is called a solid torus of covering multiplicity k. A solid torus T of covering multiplicity k can be written as .∪t∈[0,k] {e2π it } × Ut , where for each t the sets .Ut , . . . , Ut+k−1 are disjoint topological discs in .C. Take the lift k = ∪t∈[0,k] {e2π i kt } × Ut under .p˜ k of the solid torus T of covering multiplicity .T k. The torus .Tk has covering multiplicity 1 as subset of the k-fold covering space k  D ×C∼ .∂ = ∂D × C. The following procedure is the basis for the recovery of conjugacy classes of not necessarily pure reducible braids. → Cn (C)S n be a closed geometric  Let .G : ∂D  2π it , g (t) , t ∈ [0, 1] with .G(e2π it ) = g (t), t ∈ fat braid in .∂D×C, written as . e [0, 1] (see Eq. (5.20)). Let .Γ be a fat loop of covering multiplicity k which is a part of the closed geometric fat braid (5.20), k

Γ =

.

  e2π it , γ (t) , t ∈ [0, k]

(5.23)

k can be written as for a fat function .γ with .γ (0) = γ (k). The lift .Γ   t k = (e2π i k , γ (t)) : t ∈ [0, k] . Γ

.

(5.24)

k. k is the graph of a function depending on .e2π i kt ∈ ∂D Γ Take a solid torus T in .∂D × C of covering multiplicity k, that surrounds the fat loop .Γ of covering multiplicity k (more precisely, .Γ is contained in T and is a deformation retract of T ), such that T does not intersect any other loop of the closed geometric fat braid .G. Replacing .Γ by T we obtain a closed geometric tubular fat braid. (A closed geometric tubular fat braid is defined in a similar way as a geometric tubular fat braid.)

.

5.3 The Building Block of the Recovery. The General Case

173

k  k be a closed geometric fat braid in .∂ D × C that is contained in the lift Let .F k . We write .T

  t k (t)) : t ∈ [0, k]  k = (e2π i k , f F

.

(5.25)

 (t) = F  : [0, k] → Cn (C)Sn for some  (e2π it ), t ∈ [0, k] , .f with .f natural number n. We define the following operation that replaces the fat loop .Γ k ) = of the closed fat braid .G by “a copy of the closed fat braid .F = p˜ k (F  2π it k   (t)) : t ∈ [0, k] inserted into .T ”: (e ,f def

k

k

 k def G Γ F =

.

k

  k (t) , t ∈ [0, k] . e2π it , g (t) γ (t) f

(5.26)

 k k def  The operation .g (t) γ (t) f˜ (t) = g (t) \ {γ (t)} ∪ f˜ (t) for every .t ∈ [0, k] is defined as in Chap. 4 Eq. (4.11), namely, for each t we replace the point .γ (t) of the k (t). unordered tuple of points .g (t) by the unordered tuple of points .f k k Since the closed geometric fat braid defined by  .F is contained in .T , the closed  k  (t) , t ∈ [0, k] is contained in T . Hence, (5.26) geometric fat braid . e2π it , f defines a closed geometric fat braid. We need the following analog of Lemma 4.3. Lemma 5.4 Let .G0 and .G1 be (not necessarily pure) closed geometric fat braids in .∂D × C, such that .G0 is free isotopic to .G1 . Let .Γ 0 be a fat loop of .G0 of covering multiplicity k, and let .Γ 1 be the fat loop of .G1 of covering multiplicity k that corresponds to .Γ 0 under the isotopy joining .G0 and .G1 . For .j = 0, 1 we let .T (Γ j ) be a tube around .Γ j which does not meet the other loops of .Gj and has .Γ j as a deformation retract. Suppose for .j = 0, 1 there are k k k  j ) in .∂  D × C that are contained in the lift .T (Γ ) closed geometric fat braids .(F j

of .T (Γ j ). k k k   0 ) and . (F 1 ) are free isotopic in .∂  D × C, then . G0 Γ 0 If in addition . (F k k   0 ) and . G1 1 ) are free isotopic closed geometric fat braids. (F (F Γ1

Proof Let .G , s ∈ [0, 1], be an isotopy of closed geometric fat braids joining .G1 with .G2 , and let .Γ s be the family of loops in .Gs that is obtained by this isotopy and joins .Γ 0 with .Γ 1 . Choose .ε so small thatfor each .s ∈ [0, 1] the .ε-neighbourhood in  2π it s 2π it  e .∂D × C of the closed geometric fat braid . , G (e ) , t ∈ [0, 1] is a closed geometric tubular braid.  k  k t   j j) = For .j = 0, 1 the closed geometric fat braid . (F e2π i k , (f ) (t) , t ∈ k k D ×C is contained in .T (Γj ) . By the proof of Lemma 4.2 for .j = 0, 1 [0, k], in .∂ s

k  j there exists an isotopy of closed geometric fat braids .(F s ) , s ∈ [0, 1], such that

174

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

k k  j (Γ j ) , and for each j the family joins the for .j = 0, 1, each .(F s ) is contained in .T k k  j  j ) with a closed geometric fat braid .(F closed geometric fat braid .(F ) contained 1

j k ) of the lift .Γ j k of .Γ j to the k-fold covering of in the .ε-neighbourhood .Tε (Γ k j as in formula (5.24). .∂D × C. Write .Γ k  j Since .Gj Γ j (F s ) , s ∈ [0, 1], is a free isotopy of closed geometric fat braids j k is contained in the .εfor .j = 0, 1, we may assume from the beginning, that .F j k def j k , j = 0, 1, has the form .Γ ˚ j k ) around .Γ j k . Then .F j k  εF tube .Tε (Γ = k k  2π i t    , that ˚ (t)), t ∈ [0, k] for a closed geometric fat braid .F ˚ (e k , γj (t)  εf j j k  2π i t   k is defined by . (e k , f˚j (t)), t ∈ [0, k] and contained in .∂ D × D. Recall that k   k (t) (considered f (t) is obtained from .˚ for each positive number .ε the set .ε˚ f j

j

as subset of .C) by multiplying each point of the latter set by .ε, and for each .t ∈  k (t) is obtained by adding .γ (t) to each point of the set ˚ [0, k] the set .γ (t)  εf j

0 k 1 k  k (t). Since the closed geometric fat braids .Γ ˚ ˚ 0 k  εF 1 k  εF .ε ˚ f and .Γ are j k k   ˚ and .F ˚ are free isotopic. Let free isotopic, also the closed geometric fat braids .F 0

1

k s k ˚ D ×D , s ∈ [0, 1], be an isotopy of closed geometric fat braids contained in .∂ F k k 0 1 ˚ ˚ and joining .F and .F . s k ˚ s k  εF By the choice of .ε the family .Gs Γ s Γ is a free isotopy of closed k 0 1 k ˚ ˚ 0 k  εF 1 k  εF and .G1 Γ 1 Γ . We geometric fat braids joining .G0 Γ 0 Γ k  0 assumed that the latter two closed geometric fat braids are equal to .G (F 0 )

.

and .G Γ 1 1

k  1 ) , respectively. The Lemma is proved. (F

Γ0



Notice that for any closed geometric fat braid given by a mapping .F and any diffeomorphism .σ : ∂D → ∂D the two closed fat braids     e2π it , F (e2π it ) , t ∈ [0, 1] and e2π it , F (e2π iσ (t) ) , t ∈ [0, 1]

.

are free isotopic.

5.4 Recovery of Conjugacy Classes of Braids from the Irreducible Braid Components. The General Case i) of the reducible Suppose that we know the irreducible braid components .B(, non-pure braid b with respect to the admissible system of curves .C that completely  reduces a homeomorphism that represents the class .m b,∞ of the braid b. Represent

5.4 Recovery of Conjugacy Classes of Braids from the Irreducible Braid. . .

175

i) by an arbitrary closed geometric fat braid each conjugacy class of fat braids .B(, k(,i)  ,i  k(,i) × D which is the .k(, i)-fold covering .F in the bounded solid torus .∂D i) is space of .∂D × D. Here .k(, i) is the length of the cycle of .Sj,i to which .B(,  k(,i)  t k(,i) ,i ,i as . e2π i k(,i) , f (t) , t ∈ [0, k(, i)] for a associated. We write .F

k(,i) ,i continuous mapping .f from .[0, k(, i)] into the symmetrized configuration space of suitable dimension. By induction on the generation . we will (after multiplication by a small positive number and applying the operation .) “insert k(,i) ,i into tubes around respective fat loops of the closed geometric fat braids .F

the closed geometric fat braid obtained by induction at step . − 1”. To show that indeed the geometric fat braids, that are obtained by the inductive construction with i), represent the class .b, ˆ we will compare any choice of the representatives of .B(, at each step the inductively constructed closed geometric fat braids with a closed geometric fat braid related to (5.20). We will use Lemma 5.4. 1 1,1 Start with the closed geometric fat braid .F 1,1 = F written as  1   1,1 2π it e . ,f (t) , t ∈ [0, 1] . Let .ε2 > 0 be so small that the .ε2 -neighbourhood in .∂D × C of .F 1,1 is the union of disjoint solid tori that retract to the loops of the closed geometric fat braid. Let .F 2,i , i = 1, . . . , k2 , be the fat loops of .F 1,1 . The covering multiplicity of .F 2,i is equal to .k(2, i). For each .i = 1, . . . , k2 , we will “insert a suitable closed geometric braid i) into the .ε2 -neighbourhood of the loop .F 2,i ”. For .i = 1 this representing .B(2, means that we consider the closed geometric fat braid given by 1 k(2,1)  1,1 2,1 ). F F 2,1 (F 2,1  ε2 F

.

(5.27)

More detailed, equality (5.27) means the following. Write .F 2,1 as  2π it  (e , f 2,1 (t)), t ∈ [0, k(2, 1)] .

.

k(2,1) 2,1 Then .F 2,1  ε2 F is the closed geometric fat braid defined by

 k(2,1)  2,1 e2π it , f 2,1 (t)  ε2 f (t) , t ∈ [0, k(2, 1)] .

.

The closed geometric fat braid (5.27) is defined by  1 k(2,1)   1,1 2,1 e2π it , f (t) f 2,1 (t) f 2,1 (t)ε2 f (t) , t ∈ [0, k(2, 1)] .

.

(5.28)

We define successively the closed geometic braids 1 k(2,1) k(2,i)  1,1 2,1 2,i ) . . . F 2,i (F 2,i  ε2 F ) F F 2,1 (F 2,1  ε2 F

.

(5.29)

176

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

by “inserting the closed geometric fat braid .F 2,i+1  ε2 F 2,i+1 representing .B(2, i + 1) into the .ε2 -neighbourhood of the fat loop .F 2,i+1 of the closed geometric fat braid (5.29)” to obtain the closed geometric fat braid (5.29) with i replaced .i + 1.  1) is the free isotopy class of the geometric On the other hand we recall that .B(1, fat braid   t, ϕt (E 1,1 ∪ E 1,1 ) , t ∈ [0, 1] .



(see (5.10)). Here .E 1,1 = E ∩ S 1,1 , and .E 1,1 is the collection of all points .z2,i j ∈ Sj2,i ⊂ δj2,i , .i = 1, . . . , k2 , .j = 1, . . . , k(2, i) , assigned to the set of holes of .S 1,1 2 2,i in Lemma 5.2, .E 1,1 = ki=1 {z2,i 1 , . . . , zk(2,i) } . With .n(1, 1) being the number of points in .E 1,1 ∪ E 1,1 , we write the mapping defining the geometric fat braid (5.10) as a fat map .g 1,1 , g 1,1 : [0, 1] → Cn(1,1) (C)Sn(1,1) ,

.

1 1  1,1 1,1 = g 1,1 we consider the closed geometric fat braid .G (see (5.11)). With .g  1 × C, defined by in .∂D

  1 1,1 (t) , t ∈ [0, 1] . e2π it , g

.

1  1,1 The connected components .Γ 2,i of the closed fat braid .G that are fat loops   2π it def 2,i 2,i  correspond to the cycles .cyc0 , .Γ 2,i = e , ϕt (E 1 ) , t ∈ [0, k(2, i)] ⊂   ∂D × C. We also write this fat loop as .Γ 2,i = e2π it , γ 2,i (t) , t ∈ [0, k(2, i)] . Since by the definition of the extension of .ϕt to .R the equalities j k(,i) 2,1 2,1 2,1 2,1 2,1 2,1 .ϕk(,i) (z 1 ) = ϕb,∞ (z1 ) = z1 , and .ϕj (z1 ) = ϕb,∞ (z1 ) = zj +1 hold for .j = 1, . . . , k(, i) − 1, the fat loop .Γ 2,i intersects the fiber .{1} × C at 2,i 2π it } × C along .k(2, i) the points .(1, z2,i 1 ), . . . , (1, zk(,i) ). It intersects each fiber .{e points. For each i the fat loop .Γ 2,i is contained in the tube

⎫ ⎧ k(2,i) ⎬ ⎨  2,i  e2π it , ϕt (E 1,1 ∪ . δj ) , t ∈ [0, 1] . ⎭ ⎩ j =1

of the closed geometric tubular braid ⎧ ⎨ .

⎩

e2π it , ϕt (E 1,1 ∪

k2 k(2,i)   i=1 j =1



⎫ ⎬

δj2,i ) , t ∈ [0, 1] . ⎭

(5.30)

5.4 Recovery of Conjugacy Classes of Braids from the Irreducible Braid. . .

177

Further, we recall that for each cycle .cyc2,i of generation 2 the irreducible braid i) associated to this cycle is, after reparametrizing, equal to the component .B(2, free isotopy class of the geometric braid, that is defined by   t, ϕt (E12,i ∪ E 2,i 1 ) , t ∈ [0, k(2, i)]

.

(see (5.15)) and is denoted by k(2,i) 2,i : [0, k(2, i)] → Cn(2,i) (C)Sn(2,i) g

.

k(2,i) 2,i (t) = ϕt (E12,i ∪ E 2,i g 1 )

(see (5.16)). Here .E12,i = E ∩ S12,i , and the set .E 2,i 1 contains for each interior hole of 2,i .S 1 the point associated to it by Lemma 5.2. We consider for each .(2, i) the closed k(2,i)  2,i  k(2,i) × C, that is defined as in .∂D geometric fat braid . G   t k(2,i) 2,i e2π i k(2,i) , g (t) , t ∈ [0, k(2, i)]

(5.31)

.

k(2,i) 2,i where .g is defined by (5.15) and (5.16) with . = 2. The closed geometric fat k(2,i)  2,i is contained in the tube (5.30). braid . G

1 1  1,1 1,1 are free isotopic, and the fat The closed geometric fat braids . G and . F loops . Γ 2,1 and . F 2,1 correspond to each other under the isotopy. By the choice k(2,1) k(2,1)  2,1 2,1 of . ε the closed geometric fat braids . G and . F  ε F , 2

2,1

2

respectively, are contained in suitable tubes around .Γ 2,1 and .F 2,1 , respectively. By Lemma 5.4 the closed geometric fat braid (5.27) is free isotopic to the closed 1 k(2,1)   1,1 2,1

. By induction the closed geometric fat G geometric fat braid .G Γ 2,1

braid 1 k(2,1)    1,1 2,1 2,k2

Γ 2,1 G

. . . Γ 2,k2 G G

k(2,k2 )

(5.32)

.

is free isotopic to (5.29) with .i = k2 . We claim that the obtained closed geometric fat braid 1 k(2,1)    1,1 2,1 2,k2

Γ 2,1 G

. . . Γ 2,k2 G G

.

k(2,k2 )

(5.33)

 2 k(2,i)  2,i 2,i  is associated to the domain .Q2 = S 1,1 ∪ ki=1 in the sense that j =1 Sj ∪ Cj the set of its ordinary strands intersects the fiber .{1} × C of .∂D × C along the set .{1} × (E ∩ Q2 ), and the set of intersection points of the fat strands with .{1} × C contains exactly one point in .{1} × δj3,i for each hole .δj3,i of .Q2 , and contains no other point.

178

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

1 k(2,1)   1,1 2,1 Indeed, the closed geometric fat braid .G

Γ 2,1 G can be written as

 .

2,1 where .g

 1 k(2,1)  1,1 (t) 2,1 e2π it , g (t) , t ∈ [0, k(2, 1)] , γ 2,1 (t) g

(5.34)

k(2,1)

is defined by (5.15), or equivalently by (5.16), with . = 2, i = 1. 1  1,1 Since the closed geometric fat braid .G can be written by (5.11), the closed geometric fat braid (5.34) is equal to  .

 e2π it , ϕt (E 1,1 ∪ E 1,1 ) ϕ (z2,1 ) ϕt (E12,1 ∪ E 2,1 ) , t ∈ [0, k(2, 1)] . 1 t

(5.35)

1

The following equalities hold: j ϕt+j = ϕt ◦ ϕb,∞ for .t ∈ [0, 1],

.

j

ϕb,∞ (E 1,1 ) = E 1,1 ,

.

j

k(2,1)

2,1 2,1 2,1 ϕb,∞ (z2,1 1 ) = zj +1 for .j = 1, . . . , k(2, 1) − 1, and .ϕb,∞ (z1 ) = z1 ,

.

j

2,1 2,1 ϕb,∞ (E12,1 ∪ E 2,1 1 ) = Ej +1 ∪ E j +1 for .j = 1, . . . , k(2, 1) − 1, and

.

k(2,1)

2,1 2,1 ϕb,∞ (E12,1 ∪ E 2,1 1 ) = E1 ∪ E 1 . Hence, the closed geometric fat braid (5.35) is equal to

.

⎫ ⎧ k(2,1) k(2,1) ⎬ ⎨   2,1    e2π it , ϕt E 1,1 ∪ (E 1,1 \ . {zj }) ∪ (Ej2,1 ∪ E 2,1 ) t ∈ [0, 1] . j ⎭ ⎩ j =1

Notice

that .E 1,1

=

k2 k(2,i)   i=1 j =1

j =1

(5.36) {z2,i j }.

Induction over the closed geometric fat braids

1 k(2,1) k(2,i)    1,1 2,1 2,i

Γ 2,1 G

. . . Γ 2,i G G

.

for .i = 1, . . . , k2 , shows that the closed geometric fat braid (5.33) equals ⎫ ⎧⎛ ⎞ k2 k(2,i) ⎬ ⎨     2,i ⎠ , t ∈ [0, 1] . ⎝e2π it , ϕt E 1,1 ∪ . (Ej ∪ E 2,i ) j ⎭ ⎩ i=1 j =1

Hence, the closed geometric fat braid (5.33) is associated to the domain .Q2 .

(5.37)

5.5 Proof of the Main Theorem for Reducible Braids. The General Case

179

The interior boundary of the domain .QN is empty. Induction over the number of the generation . = 1, . . . , N shows that the closed geometric fat braid 1 k(2,1)    1,1 2,1 2,k2 G

Γ 2,1 G

. . . Γ 2,k2 G

k(2,k2 )

.

 N,1 . . . Γ N,1 G

k(N,1)

 N,kN

. . . Γ N,kN G

k(N,kN )

(5.38)

is an ordinary braid which is equal to the closed geometric braid  closed geometric  2π it , ϕ(t) , t ∈ [0, 1] . G given by . e By induction over the number of the generation . = 1, . . . , N, we find small +1,i 

+1,i of enough numbers .ε+1 and “insert the representatives .F +1,i   ε+1 F  .B( + 1, i ) into the .ε -neighbourhoods of the fat loops .F +1,i  that correspond to the cycles of holes of .Q ”. For . = N we arrive at the closed geometric fat braid 



1 k(2,1)   1,1 2,1 2,k2 ) . . . F 2,k2 (F 2,k2  ε2 F F F 2,1 (F 2,1  ε2 F

k(2,k2 )

)

.

 N,1

. . . F N,1 (F N,1  εN F

k(N,1)

 N,1 ) . . . F N,kN (F N,kN  εN F

k(N,kN )

). (5.39)

An inductive application of Lemma 5.4 shows, that the closed geometric fat braid (5.39) is an ordinary closed braid and is free isotopic to the closed braid (5.38) which is equal to G. We provided a procedure which gives for each choice of closed geometric fat braids, that represent the irreducible braid components .B(, i), the ˆ We recovered .bˆ knowing the irreducible braid closed braid (5.39) that represents .b. components. 

5.5 Proof of the Main Theorem for Reducible Braids. The General Case 

The following lemma describes the entropy of the nodal conjugacy class .mb, , that is associated to .mb,∞ and an admissible system of curves .C ⊂ D that completely reduces .mb,∞ , in terms of the entropy of irreducible nodal components. Recall that we identify mapping classes on (possibly not connected) punctured Riemann surfaces (or on punctured nodal surfaces) with mapping classes on (possibly not connected) closed Riemann surfaces (or on closed nodal surfaces) with distinguished points. If for notational convenience we write .h(m) for a mapping class .m on a punctured surface, we mean the entropy of the respective mapping class on the closed surface with distinguished points. Similarly, we call a selfhomeomorphism of a punctured Riemann surface non-periodic absolutely extremal

180

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

if its extension to the closed Riemann surface with the respective set of distinguished points is so. Lemma 5.5     h mb, = max h mb, | cyc,i = max





.

cyc,i

cyc,i

  1 h mbk(,i) , | Y1,i . k(, i) 

(5.40) Proof The first equality is an easy consequence of Theorem 4 of [1]. To obtain the second equality it is enough to prove that the equation   h mb, | cyc,i =



.

  1 h mbk(,i) , | Y1,i k(, i) 

(5.41)

holds for each nodal cycle .cyc,i

. For each mapping .ϕ representing the class



k(,i)

mb, | cyc,i

the equality .h(ϕ

,i .Y j and for each j the mapping .

j −1

k(,i) ϕ

| Yj,i = (ϕ

.

k(,i)

is a conjugate of .ϕ

k(,i)

) = k(, i)h(ϕ ) holds. Since .ϕ

j −1

k(,i) | Y1,i ) ◦ (ϕ

| Y1,i ) ◦ (ϕ

fixes each

| Y1,i )−1

| Y1,i , the equality k(,i)

h(ϕ

.

| Y1,i ) = k(, i)h(ϕ )

(5.42)

  holds. The entropy .h mb, | cyc,i is equal to the infimum



     h mb, | cyc,i = inf h ϕ : ϕ ∈ mb, | cyc,i







.

= inf



  1 k(,i) {h ϕ

| Y1,i : ϕ ∈ mb, | cyc,i

. k(, i) (5.43) 

On the other hand,  h(mbk(,i) , | Y1,i ) = inf h(ψ) : ψ ∈ mbk(,i) , | Y1,i . 



.

(5.44)



k(,i)

By Lemma 5.1 the inclusion .ψ ∈ mbk(,i) , | Y1,i holds if and only if .ψ = ϕ



Y1,i for a mapping .ϕ ∈ mb, | cyc,i

. The lemma is proved.

|



5.5 Proof of the Main Theorem for Reducible Braids. The General Case

181

By Lemma 5.5 for the proof of the following theorem it is enough to relate the entropy of a mapping class on a surface to the entropy of its nodal mapping class. Theorem 5.1 For an arbitrary braid the following equality holds    ,i · .h (mb ) = h(mb, ) = max h mbk(,i) , | Y 1 



cyc,i

1 k(, i)

 .

(5.45)

Moreover, there exists a mapping with this entropy that represents .mb . We split the proof of Theorem 5.1 into two parts. The first part is the lower bound for the entropy of the braid. Lemma 5.6    ,i · .h (mb ) ≥ max h mbk(,i) , | Y 1 

cyc,i

1 k(, i)

 .

Proof Let N be the smallest natural number for which .bN is a pure braid. Notice that h(mbN ) ≤ N · h(mb ) .

.

(5.46)

Indeed, for a chosen homeomorphism .ϕb ∈ mb  h(mbN ) = inf h(ϕ) : ϕ is Hom+ (D; ∂ D, E)-isotopic to ϕbN ,

.

(5.47)

and  N · h(mb ) = inf h(ψ N ) = N · h(ψ) : ψ is Hom+ (D; ∂ D, E)-isotopic to ϕb .

.

(5.48) Since the infimum in (5.48) is taken over a smaller class than in (5.47), we obtain (5.46).  ,i ,i  Suppose .cyc,i b, = Y1 , . . . , Yk(,i) is a nodal cycle on the nodal surface Y 

such that the mapping class .mbk(,i) , | Y1,i is pseudo-Anosov, i.e. there exists a self-homeomorphism of .Y1,i that represents .mbk(,i) , | Y1,i and is conjugate to a non-periodic absolutely extremal mapping .ϕ˜ ,i . Any power of .ϕ˜ ,i is again pseudo

Anosov. Hence the class .mbN , | Y1,i is pseudo-Anosov, since N is divisible by .k(, i), i.e. N = k(, i) · m(, i)

.

(5.49)

182

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

for an natural number .m(, i) Moreover,     h mbN , | Y1,i = h (ϕ˜ ,i )m(,i) = m(, i) h(ϕ˜ ,i ) 

(5.50)

.

since .(ϕ˜ ,i )m(,i) represents .mbN , | Y1,i . We obtained 



h mbN , |

.

Y1,i









= m(, i) h mbk(,i) , |

Y1,i

(5.51)

.

 

By Proposition 4.2 for the pure braid .bN the inequality .h(mbN ) ≥ h(mbN , | Y1,i ) holds. By (5.46) and (5.51) we obtain the inequality       N · h(mb ) ≥ h mbN ≥ h mbN , | Y1,i = m(, i) h mbk(,i) , | Y1,i 





.

(5.52) for all . and i for which .mbk(,i) , | Y1,i is pseudo-Anosov. Hence, by (5.49)   1 h mbk(,i) , | Y1,i k(, i) 

h(mb ) ≥

.

(5.53)

if .mbk(,i) , | Y1,i is pseudo-Anosov. If .mbk(,i) , | Y1,i is not pseudo-Anosov, it is elliptic, since it is irreducible. In this case the entropy equals zero and inequality (5.53) is trivially satisfied. Lemma 5.6 is proved.

The remaining inequality that is needed for the proof of Theorem 5.1 is stated in the following lemma. Lemma 5.7 The following inequality holds    h (mb ) ≤ max h mbk(,i) , | Y1,i · 

.

(,i)

1 k(, i)

 .

Proof of Lemma 5.7 The goal is to represent .mb by an element .ϕ ∈ Hom+ (P1 ; ∞, En ) that maps the curve system .C ⊂ D onto itself and for each . and 

i the restriction .ϕ k(,i) |S1,i has entropy equal to .h(mbk(,i) , | Y ,i ), equivalently, 

1 ,i the restriction .ϕ|cyc,i 0 has entropy . k(,i) h(mbk(,i) , | Y ). We will use Theorem 2.10. Suppose .ϕb,∞ ∈ Hom+ (P1 ; P1 \ D, En ) represents .mb,∞ and maps the admissible set of curves .C onto itself. After a .Hom+ (P1 ; P1 \ D, En )-isotopy of .ϕb,∞ that changes the values only in a small neighbourhood of .C (see Lemma 5.2)

5.5 Proof of the Main Theorem for Reducible Braids. The General Case

183 k(,i)

we may assume that for each . > 1 and .i = 1, . . . k the iterate .ϕb,∞ equals the identity on a small neighbourhood of the exterior boundary .∂E S1,i of the first labeled set of each cycle .cyc,i 0 . Here .k(, i) is the length of the cycle. For uniformity of notation we write .S11,1 for .S 1,1 . k(,i) ,i 1. A Representative . ψ of the Isotopy Class of . ϕb,∞ | S1,i \ En 

By Corollary 1.1 for each .(, i) the class .mbk(,i) , |Y ,i can be represented by an ,i absolutely extremal self-homeomorphism .ψ˜

of a punctured Riemann surface ,i ,i ˜ ,i . The mapping .ψ˜

.Y has the following properties. .ψ˜

minimizes the entropy 

,i in the class .mbk(,i) , |Y ,i . Moreover, the continuous extension of .ψ˜

to the closed ,i ,i c ˜ ˜ Riemann surface .(Y ) (also denoted by .ψ ) fixes the node corresponding to the exterior boundary component of .S1,i (in the case of .S11,1 it fixes .∞), and it moves the other nodes in .N ∩ (Y˜1,i )c of .(Y˜1,i )c along cycles corresponding to the .ϕbk(,i) , ∞ cycles of the connected components of the interior boundary .∂I S1,i . ,i is pseudo-Anosov, then If the continuous extension of the homeomorphism .ψ˜

,i by Theorem 2.10 there exists a self-homeomorphism .ψ˜  of .(Y˜1,i )c that is isotopic ,i by an isotopy that changes the mapping only in a punctured neighbourhood to .ψ˜

,i , and has the following properties. of the nodes, that has the same entropy as .ψ˜

,i ˜ .ψ fixes pointwise a topological disc around the node corresponding to the ,i -cycle exterior boundary of .(Y˜1,i )c (if .(, i) = (1, 1) it fixes .P1 \ D). For each .ψ˜

,i of nodes there is a .ψ˜  -cycle of closed topological discs around the nodes of the ,i k ) fixes cycle, and, for the length of the cycle being equal to .k ≥ 1, the iterate .(ψ˜  each disc pointwise. ,i If the continuous extension of .ψ˜

is a periodic self-mapping of the Riemann ,i ,i k c ) = Id for a natural number k, sphere .(Y˜1 ) with distinguished points, i.e. .(ψ˜

,i itself maps a collection then for each cycle of nodes of length k the extension of .ψ˜

of topological discs around the distinguished points along a cycle of length k, and ,i k the restriction of the extension of .(ψ˜

) to each of the discs is the identity. The ,i ,i rotates a disc around the node. mapping .ψ˜ may fix a node. In this case .ψ˜

,i ˜ Lemma 2.14 allows to change .ψ without changing it outside the disc and without increasing the entropy, so that the new mapping is the identity on a smaller disc around the node. ˚,i denote the complement in .Y˜ ,i of the collection of all respective Let .Y 1 1 ˚,i is a bordered surface with punctures. There is a open discs. Notice that .Y 1 ˚,i → S ,i \ E  , for which the conjugate .ψ ,i def = homeomorphism .w ,i : Y 1

1

1

n



,i ˚,i ) ◦ (w ,i )−1 is a self-homeomorphism of .S ,i \ E  ,i , whose ◦ (ψ˜

| Y 1 1 1 1 k(,i) ,i ,i ,i ,i  ,i  ,i restriction to .S1 \E  ,i 1 is isotopic to .ϕb,∞ |S1 \E 1 . Moreover, .ψ | S1 \E 1

(w1,i )

184

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . . 

also minimizes the entropy among mappings representing .mbk(,i) , | Y1,i . In other words, 

,i h(ψ ) = h(mbk(,i) , | Y1,i ) .

(5.54)

.



For all .(, i) we found a homeomorphism

,i .ψ

that represents .mbk(,i) , | Y1,i ,

satisfies (5.54), and is equal to the identity at all points of .S1,i that are close to the exterior boundary .∂E S1,i (on .P1 \ D in case of .S11,1 ). Moreover, it moves the interior ,i ,i boundary components along .ψ -cycles. If the length of a .ψ -cycle equals k then ,i k .(ψ ) is the identity near each boundary component of the cycle. 1,1 2. The Homeomorphism .ψ and an Isotopy of .ϕb,∞ Recall that for .l = 1, . . . , N we denoted by .Ql the following closed subset of .P1

Ql =







.

l  =1,...,l i=1,...,kl  j =1,...,k(l  ,i)



Sjl ,i .

Ql is the closure of a domain .Ql in .P1 . Notice that .Q1 = S11,1 . 1,1 1,1 |S1 represent Let first .l = 1, i.e. .Q1 = S11,1 . Both mappings, .ϕb,∞ |S11,1 and .ψ

.



1,1 mb, | Y11,1 . Both mappings are equal to the identity on .P1 \ D. Take any .ψ -cycle of interior boundary components of .S11,1 . Let k be the length of the cycle. Then the 1,1 k mapping .(ψ ) equals the identity on each boundary component of the cycle (and

.



1,1 ) = h(mb, | Y11,1 ). on a small neighbourhood of it in .S11,1 ). Moreover, .h(ψ Consider a .Hom+ (P1 ; (P1 \D), En )-isotopy of .ϕb,∞ , which changes the mapping only in a small annulus around each interior boundary component of .S11,1 , so that the new self-homeomorphism .ϕb,∞,1 of .P1 with distinguished points .En coincides 1,1 with .ψ on .P1 \ D and on all boundary components of .S11,1 . Then the mapping 1,1 1,1 −1 1 .ψ ◦ ϕ b,∞,1 is the identity on .P \ D and on each boundary component of .S1 . 

1,1 1,1 |S1 , represent .mb, | Y 1,1 , the isotopy Since both mappings, .ϕb,∞ |S11,1 and .ψ

classes of their continuous extensions to .S11,1 with distinguished points differ by a product of commuting Dehn twists around curves that are homologous to the boundary circles of .S11,1 or to .∂D. Lemma 2.14 allows to change the isotopy 1,1 by any power of a Dehn twist around the circle .∂D by changing class of .ψ 1,1 .ψ in a small neighbourhood of this circle without increasing entropy. By a further isotopy of .ϕb,∞,1 on .P1 that does not change the mapping outside small neighbourhoods of the interior boundary of .S11,1 we may change the isotopy class of the restriction .ϕb,∞,1 | S11,1 by any product of Dehn twists around curves that 1,1 are homologous to the interior boundary circles of .S11,1 . After changing .ψ and

5.5 Proof of the Main Theorem for Reducible Braids. The General Case

185

ϕb,∞,1 in the described way and keeping previous notation, we may assume that

.

1,1 ψ ◦ (ϕb,∞,1 |S11,1 )−1 is .Hom+ (S11,1 ; (S11,1 ∩ En 1,1 ) ∪ (P1 \ D) ∪ ∂S11,1 )-isotopic to the identity.

.

3. The Inductive Construction of the Entropy Minimizing Element of .mb By induction on l we will find a self-homeomorphism .ϕQl of .Ql with distinguished points .En ∩ Ql and a self-homeomorphism .ϕb,∞,l of .P1 for which the following requirements hold. The entropy of the self-homeomorphism .ϕQl equals   h(mbk(l  ,i) , | Y1l ,i ) · 

h(ϕQl ) =

.

max

(l  ,i):l  ≤l,i≤kl 

1  . k(l  , i)

Moreover, .ϕQl equals the identity on .P1 \ D and moves the boundary components k of .Ql along cycles so that for each .ϕQl -cycle of length k the homeomorphism .ϕQ l is the identity near each component of the boundary belonging to the cycle. The self-homeomorphism .ϕb,∞,l of .P1 is .Hom+ (P1 ; ∞, En )-isotopic to .ϕb,∞ by an isotopy that changes .ϕb,∞ only in a small neighbourhood of the interior boundary of the .Ql  with .l  ≤ l and coincides with .ϕQl on .∂Ql . Moreover, the self-homeomorphism .ϕQl of .Ql is isotopic to .ϕb,∞,l | Ql by an isotopy that does not change the mappings in a small neighbourhood of  1 −1 .∂Ql ∪ (En ∩ Ql ) and on .P \ D (in other words, the mapping .ϕQl ◦ (ϕb,∞,l | Ql )    1 is .Hom Ql ; (En ∩ Ql ) ∪ ∂Ql ∪ (P \ D) -isotopic to the identity on .Ql ). We already found .ϕQ1 and .ϕb,∞,1 . Suppose for some .l ≤ N − 1 we found a selfhomeomorphisms .ϕQl with distinguished points .En ∩Ql and a self-homeomorphism 1  .ϕb,∞,l of .P with distinguished points .En that satisfy the requirements. We have to find these objects for the number .l + 1. The boundary components of .Ql are exactly the curves in .C that are of generation .l + 1. Each such curve is the exterior boundary of a connected component of l+1,i = (S1l+1,i , . . . , Sk(l+1,i) ) generation .l + 1 of .P1 \ C. Take any .ϕb,∞,l -cycle .cycl+1,i 0 of generation .l + 1 of length .k(l + 1, i), .1 ≤ i ≤ kl+1 . We first consider the def

l+1,i instead of .Ql+1 , and will associate set .Ql,i = Ql ∪ S1l+1,i ∪ . . . ∪ Sk(l+1,i) to .Ql,i a self-homeomorphism .ϕQl,i of .Ql,i and a homeomorphism .ϕb,∞,l,i ∈ Hom+ (P1 ; P1 \ D, C ∪ En ). We put

ϕQl,i

.

⎧ ⎪ ⎪ ⎨ϕQl = ϕb,∞,l ⎪ ⎪ ⎩ l+1,i (ϕb,∞,l )−k(l+1,i)+1 ◦ ψ

on Ql , on Sjl+1,i with j = 2, . . . , k(l + 1, i) , on S1l+1,i . (5.55)

186

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

l+1,i Since the homeomorphism .ψ is equal to the identity at all points of .S1l+1,i that are close to the exterior boundary .∂E S1l+1,i and also .(ϕb,∞,l )k(l+1,i) is the identity there, the equality .ϕQl,i = ϕb,∞,l holds on .∂E S1l+1,i . Hence, by the induction hypothesis .ϕQl,i = ϕQl on .∂E Sjl+1,i , j = 1, . . . , k(l + 1, i), and .ϕQl,i is a welldefined self-homeomorphism of .Ql,i . We consider a self-homeomorphism .ϕb,∞,l,i of .P1 that is obtained from .ϕb,∞,l by a .Hom+ (P1 ; ∞, En )-isotopy, that changes the mappings only in a small neighbourhood of the interior boundary components of .S1l+1,i and satisfies the condition .ϕb,∞,l,i = ϕQl,i on all interior boundary components of .S1l+1,i . Hence,

ϕb,∞,l,i = ϕQl,i on .∂Ql and on all boundary components of all .Sjl+1,i . By the induction hypothesis the restrictions of .ϕQl,i (see (5.55)) and of .ϕb,∞,l,i to .Ql are isotopic by an isotopy that does not change the mappings in a small neighbourhood of .∂Ql ∪ (En ∩ Ql ) and on .P1 \ D. k(l+1,i) l+1,i Since the self-homeomorphism .ψ | S1l+1,i \ En is isotopic to .ϕb,∞,l |

.

S1l+1,i \ En , and the continuous extensions to .S1l+1,i \ En of the two mappings coincide on all boundary components of .S1l+1,i , the isotopy classes of their extensions differ by a product of Dehn twists around curves that are homologous to the boundary components of .S1l+1,i . l+1,i If necessary, we may change .ψ without increasing entropy by a power of a Dehn twist about a circle in .S1l+1,i that is homologous to the exterior boundary l+1,i ∂E S1l+1,i of .S1l+1,i , and build the mapping .ϕQl,i from the new .ψ (see Eq. (5.55)), using the previous notation for the changed objects. On the other hand, we may change the .Hom+ (P1 ; ∞, En )-isotopy joining .ϕb,∞,l and .ϕb,∞,l,i , so that the

.

k(l+1,i) | S1l+1,i \ En differs from that of the previous one isotopy class of the new .ϕb,∞,l.i by an a priori given product of Dehn twists around curves that are homologous to def  l+1,i  . the interior boundary components of .S1l+1,i . Put .Ω l+1,i = S1l+1,i ∪ . . . ∪ Sk(l+1,i)

Since for .j = 2, . . . , n the equality .ϕQl,i = ϕb,∞,l,i = ϕb,∞,l holds on .Sjl+1,i \ En , we can choose the Dehn twists so that the self-homeomorphism .ϕQl,i | Ω l+1,i of .Ω l+1,i (see (5.55)) and the mapping .ϕb,∞,l,i | Ω l+1,i are isotopic on .Ω l+1,i with set of distinguished points .En ∩Ω l+1,i by an isotopy that does not change the mappings on the boundary of the set. Hence, the self-homeomorphism .ϕQl,i of .Ql,i and the mapping .ϕb,∞,l,i | Ql,i are isotopic by an isotopy that fixes the boundary of .Ql,i and the set .E ∩ Ql,i . = For the obtained self-homeomorphism . ϕQl,i of . Ql ∪ Ω l+1,i  l+1,i l+1,i  . Ql ∪ S ∪ . . . ∪ Sk(l+1,i) the following equality on entropies holds 1   .h ϕQl,i | Ω l+1,i =



1 h(mbk(l+1,i) , | Y1l+1,i ) . k(l + 1, i)

(5.56)

5.5 Proof of the Main Theorem for Reducible Braids. The General Case

187

Moreover, the self-homeomorphism .ϕb,∞,l,i of .P1 with set of distinguished points  .En , is isotopic to .ϕb,∞ by an isotopy that changes the mapping only in a small neighbourhood of the curves of the system .C of generation at most l and in a neighbourhood of the interior boundary components of the set .S1l+1,i . The homeomorphisms .ϕQl,i and .ϕb,∞,l,i | Ql,i are isotopic by an isotopy that does not change the values on .En ∩ Ql,i , on .P1 \ C, and on the boundary of .Ql,i . 

, j  = 1, . . . , k(l + 1, i  ), correWe add successively to .Ql,i the sets .Sjl+1,i  

of components of .P1 \ C of generation sponding to all other .ϕb,∞,l -cycles .cycl+1,i 0 .l + 1. In the same way as .ϕQl,i was constructed (see (5.55)) we extend .ϕQl   to each cycle . j  =1,...,k(l+1,i  ) Sjl+1,i and, similarly as before, we make further  isotopies of .ϕb,∞,l . We obtain a self-homeomorphism .ϕQ,l+1 of .Ql+1 and a selfhomeomorphism .ϕb,∞,l+1 of .P1 that is .Hom+ (P1 ; ∞, En )-isotopic to .ϕb,∞ by an isotopy that changes .ϕb,∞ only in a small neighbourhood of the interior boundary of the .Ql  with .l  ≤ l + 1. Moreover, the self-homeomorphism .ϕQl+1 of .Ql+1 is isotopic to .ϕb,∞,l+1 | Ql+1 by an isotopy that does not change the mapping on  1 .(En ∩ Ql+1 ) ∪ P \ D and in a small neighbourhood of .∂Ql+1 . Using the induction hypothesis, and also the equality (5.56) for all cycles  l+1,i  . , as well as Theorem 4 of [1] we obtain the estimate for the j  =1,...,k(l+1,i  ) Sj  entropy   h(mbk(l  ,i) , | Y1l ,i ) · 

h(ϕQl+1 ) ≤

.

max

(l  ,i):l  ≤l+1,i≤kl 

1  . k(l  , i)

(5.57)

We will prove now that .ϕQl+1 moves each boundary component of .Ql+1 along a .ϕQl+1 -cycle of curves so that if the length of the cycle equals k then the k homeomorphism .ϕQ is the identity near each component of the cycle. l+1 Each boundary component of .Ql+1 is an interior boundary component of an . element .Sjl+1,i of some cycle .cycl+1,i 0 of the first labeled element Consider first an interior boundary component .∂jl+1,i  S1l+1,i of an arbitrary cycle .cycl+1,i of sets of generation .l + 1 . The mapping 0 l+1,i l+1,i .ψ moves the boundary component along a .ψ -cycle of interior boundary k(l+1,i) l+1,i l+1,i = ϕQl+1 components of .S1 . Denote the length of this cycle by k. Since .ψ .

l+1,i on .S1l+1,i , the .ψ -cycle of length k is part of a .ϕQl+1 -cycle of length .k ·k(l +1, i), l+1,i . Since by the consisting of interior boundary components of .S1l+1,i ∪ . . . ∪ Sk(l+1,i) k(l+1,i)·k l+1,i l+1,i k the iterate .(ψ ) is the identity on .∂jl+1,i , the iterate .ϕQ choice of .ψ  l+1

is the identity on .∂jl+1,i . 

188

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

An arbitrary boundary component .∂jl+1  of .Ql+1 is an interior boundary component of a component .Sjl+1,i of .P1 \ C. Since .ϕQl+1 moves .Sjl+1,i along the cycle 0 0 l+1,i Sjl+1,i → Sjl+1,i → . . . → Sk(l+1,i) → S1l+1,i → . . . Sjl+1,i 0 0 0 +1

.

of length .k(l+1, i), an iterate of .ϕQl+1 takes .∂jl+1  to an interior boundary component of .S1l+1,i . Hence, .ϕQl+1 | ∂jl+1  is conjugate to the restriction of .ϕQl+1 to an interior k(l+1,i)·k

boundary component of .S1l+1,i . Hence, the iterate .ϕQl+1 is the identity on .∂jl+1  . For .l ≤ N − 1 we found the mappings .ϕQl+1 and .ϕb,∞,l+1 with the required properties, provided we have these mappings for the number l. If .l = N − 1 with N being equal to the number of the generations, the sets .SjN,i of the next generation have no interior boundaries. The process stops and we obtain a mapping .ϕQN that represents .mb and has entropy equal to the right hand side of the equality in the statement of the lemma. The lemma is proved.

Remark 5.2 The construction of the proof of Lemma 5.7 allows to recover, up to a product of commuting Dehn twists, the conjugacy class of a mapping class from its irreducible nodal components. Indeed, when we only know the irreducible nodal components of the mapping, we make the construction of the proof of Lemma 5.7 except the corrections related to the Dehn twists about simple closed loops that are homologous to the boundary components of the .S1,i . The Proof of Theorem 5.1 is a consequence of Lemmas 5.6 and 5.7.



The counterpart of Theorem 5.1 for the irreducible braid components is the following theorem. Theorem 5.2      ˆ = min M B(, j ) · k(, j ) . M(b)

.

,j

(5.58)

ˆ The following lemma is the lower bound for the conformal module of .b. Lemma 5.8      ˆ ≥ min M B(, j ) · k(, j ) . M(b)

.

,j

(5.59)

Proof of Lemma 5.8 Let     def  j ) · k(, j ) M0 = min M B(,

.

,j

(5.60)

5.5 Proof of the Main Theorem for Reducible Braids. The General Case

189

be the right hand side of the inequality (5.59) in the lemma. Let .r0 be a positive number such that .r0 < e2π M0 . Consider the annulus

√ 1 A0 = z ∈ C : √ < |z| < r0 , r0

.

whose conformal module equals m(A0 ) =

.

1 log r0 < M0 , 2π

and an annulus .A ⊃ A0 , whose conformal module is also less than .M0 . i) corresponding to the Consider for each . and i the conjugacy class .B(, cycle .cyc,i . Recall that .k(, i) is the length of the cycle. The conjugacy class is  k(,i) × C of the .k(, i)represented by closed geometric fat braids in the product .∂D fold covering of the unit circle with the complex plane, equivalently, by continuous  k(,i) to a symmetrized configuration space. Recall, that we may maps from .∂D consider the points of a symmetrized configuration space as subsets of the complex plane .C. We will define fat holomorphic maps similarly as we defined fat geometric braids. A holomorphic map .F from an annulus .A into .Cm (C)Sm is called a holomorphic fat map if some of the connected components of the relatively closed complex curve .{(z, F (z)), z ∈ A} in .A × C are declared to be fat. i) by a holomorphic fat map We will represent now the conjugacy class .B(, k(,1)  ,i k(,1) of the annulus A into the respective .F of the .k(, i)-fold covering .A symmetrized configuration space .Cn(,i) (D)Sn(,i) . We choose the representatives k(,1) ×D, not merely in .A k(,1) ×C. so that we obtain complex curves contained in .A 1 1 1 i)) ≥ Since .M(B(, k(,i) · M0 > k(,i) m(A) and . k(,i) m(A) is the conformal k(,1) ,i module of the .k(, i)-fold covering of the annulus A, such a mapping .F exists. We will put together these mappings proceeding similarly as in the process of recovery of conjugacy classes of braids from their irreducible braid components. 1 1,1 on A. Choose .ε > 0 so that Start with the holomorphic fat map .F 2

1 1,1 (z)), z ∈ A } {(z, F 0

(5.61)

.

is a deformation retract of its .ε2 -neighbourhood in .A0 ×C. Consider all fat connected components of (5.61). They correspond to the cycles .cyc2,i and are defined by the fat mappings .F 2,i . Define the holomorphic fat map 1 k(2,1)   1,1 2,1 2,k2 F ) . . . F 2,k2 (F 2,k2  ε2 F F 2,1 (F 2,1  ε2 F

.

(k(2,k2 )

)

(5.62)

190

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

in a neighbourhood of .A0 in the same way as we defined the maps that determine the respective geometric fat braids. By induction on the number . ≤ N − 1 of the generation we find a small positive number .ε+1 so that the set defined by 1 k(2,1)   1,1 2,1 2,k2 ) . . . F 2,k2 (F 2,k2  ε2 F F F 2,1 (F 2,1  ε2 F

(k(2,k2 )

)

.

,1

. . . F ,1 (F ,1  ε F

k(,1)

 ,k ) . . . F ,k (F ,k  ε F

k(,k )

)

(5.63)

is a deformation retract of its .ε+1 -neighbourhood in .A0 × C. This allows to define the set (5.63) with . replaced by . + 1. By the same arguments as used for the recovery procedure of conjugacy classes of braids we obtain for . = N a holomorphic map from .A0 into the symmetrized configuration space of suitable ˆ Recall dimension that extends continuously to the closure .A0 and represents .b. 2π M 0 that for an arbitrary number .r0 < e we have chosen an annulus .A0 of 1 conformal module . 2π log r0 and represented b by a holomorphic mapping of .A0 into ˆ > M0 , and Lemma 5.8 is proved. symmetrized configuration space. Hence, .M(b)

The proof of the opposite inequality in the case of pure braids was straightforward, since in that case each mapping that represents a conjugacy class .bˆ provides also a mapping that represents a given irreducible braid component by forgetting suitable strands. This does not work in the case of non-pure braids. We will first prove the Main Theorem and then prove the remaining inequality in Theorem 5.2. For the upper bound of the conformal module of the braid in the Main Theorem we will use the estimate of the conformal module by the translation length of the modular transformation which is provided by Proposition 3.1 in the general case. Further we will use the relation between the translation length of the modular transformation and the quasiconformal dilatation of the absolutely extremal mapping on the nodal Riemann surface, and its relation to the quasiconformal dilatation of the mappings of cycles of parts of the nodal Riemann surface. This will imply the Main Theorem. The proof of the remaining inequality of Theorem 5.2 will follow from relations obtained in the proof of the Main Theorem. One of the inequalities of the Main Theorem is stated in the following corollary of Lemma 5.8. Corollary 5.1 1 π π 1 M( b) ≥ . = 2 h(mb ) 2 h( b)

.

Proof of Lemma 5.7 By the equality (5.18), and the Main Theorem in the irreducible case we obtain for each . and i

5.5 Proof of the Main Theorem for Reducible Braids. The General Case

  π  k(, i) M B(, i) = 2

.

k(, i)  . ,i h mbk(,i) , | Y1 

191

(5.64)

Lemma 5.8 and Theorem 5.1 imply the inequality 1 π 1 π = M( b) ≥ 2 h( mb ) 2 h( b)

.



that is stated in the corollary.

To prove the Main Theorem in the general case it remains to prove the opposite inequality that is stated in Lemma 5.9 below. Lemma 5.9 π 1 M( b) ≤ . 2 h( b)

.

Proof of Lemma 5.7 By Proposition 3.1 1 π M( b) ≤ ∗ ). 2 L(ϕb,∞

.

Here again .ϕb,∞ is a self-homeomorphism of .P1 with the following properties. It is equal to the identity outside .D and on .D it is equal to a homeomorphism ∗ .ϕb representing the mapping class .mb . As before .ϕ b,∞ denotes the modular transformation of .ϕb,∞ . We have ∗ L(ϕb,∞ )=

.

1 log I (ϕb,∞ ) 2

(see (1.46) and (1.47)). As in Chap. 1, .I (ϕb,∞ ) is the infimum of quasiconformal dilations in the class obtained from .ϕb,∞ by isotopy and conjugation. We may assume that .ϕb,∞ is completely reduced by an admissible system of curves .C that are contained in the disc .D. Let Y be a nodal surface associated to the system .C. Let .w˜ : Y → w(Y ˜ ) = Y˜ be the conformal structure of part (1) of Theorem 1.13 and let .ϕ˜ be the absolutely extremal self-homeomorphism of .Y˜ which appears in (1), Theorem 1.13. We have .

1 1 log K(ϕ) ˜ = log I (ϕb,∞ ) . 2 2

,i ˜ we have Then for the .ϕ-cycles ˜ .cyc of .Y

  1 1 ,i log K(ϕ) ˜ = max log K(ϕ˜ | cyc ) . . ,i 2 2

(5.65)

192

5 The General Case. Irreducible Nodal Components, Irreducible Braid. . .

By Lemma 1.8 .

1 1 1 log K(ϕ˜ | cyc,i log K(ϕ˜ k(,i) | Y˜1,i ) .

) = 2 2 k(, i)

(5.66)

By the Theorem of Fathi-Shub for irreducible self-homeomorphisms of connected Riemann surfaces of first kind (see Theorem 2.1) we obtain .

1 1 1 log K(ϕ˜ k(,i) | Y˜1,i ) = h(ϕ˜ k(,i) | Y˜1,i ). 2 k(, i) k(, i)

(5.67)

Notice that   h(ϕ˜ k(,i) | Y˜1,i ) = h mbk(,i) , | Y˜1,i , 

.

(5.68)

since .ϕ˜ k(,i) is an absolutely extremal element of the class .mbk(,i) , induced by .ϕb,∞ on .Y˜ . By Theorem 5.1     1 ,i ˜ ˆ .  h mbk(,i) , | Y1 = h m .max b, = h(mb ) = h(b) ,i k(, i) 

(5.69)

Hence, by (5.65), (5.66), (5.67), (5.68) and (5.69) we have .

  1 1  log I (ϕb,∞ ) = log K(ϕ) ˜ =h m b, . 2 2

Hence ˆ ≤ M(b)

.

1 1 π 1 π π = = . 2 h( 2 h(mb ) 2 h( mb, ) b)

(5.70)

Lemma 5.9, and, hence, the general case of the Main Theorem is proved.



The following corollary of Lemma 5.9 states the remaining inequality that is needed for the proof of Theorem 5.2. Corollary 5.2      i) · k(, i) . M( b) ≤ min M B(,

.

,i

Proof of Lemma 5.7 The corollary follows from inequality (5.70) and Eqs. (5.69) and (5.64).

Theorem 5.1 together with the Main Theorem imply the following corollary.

5.5 Proof of the Main Theorem for Reducible Braids. The General Case

193

Corollary 5.3 For each .bˆ ∈ Bˆn .(n ≥ 2) and each nonzero integer . M(b  ) =

.

1 ˆ . M(b) ||

(5.71)

Proof of Corollary 5.3 By the Main Theorem Corollary 5.3 is equivalent to the equality   ! h m b = || h (mb )

.

for each braid and each non-zero integer .. The equality is easy for irreducible

b is elliptic, the class .m ! braids. Indeed, if the class .m b is so and both sides are zero.

b is pseudo-Anosov and .ϕ˜ is a pseudo-Anosov representative then .ϕ˜  If the class .m ! is a pseudo-Anosov representative of .m b . The equality follows from h(ϕ˜  ) = || h(ϕ) ˜ .

.

(See [1], Theorem 2 or Sect. 2.1). If b is reducible one has to use Theorem 5.2 and equality (5.71) for each irreducible braid component.

Problem 5.1 Define irreducible nodal components of any mapping class on a Riemann surface of genus g with m punctures, and prove the analogue of Theorem 5.1.

Chapter 6

The Conformal Module and Holomorphic Families of Polynomials

In this Chapter we discuss the first result [33] which uses the concept of the conformal module of conjugacy classes of braids. The paper [33] was motivated by the interest of the authors in Hilbert’s Thirteen’s Problem and treats global reducibility of holomorphic families of polynomials. In Sect. 6.1 we give a historical account, in Sect. 6.2 we give a conceptual alternative proof of a slightly improved version of the result of [33].

6.1 Historical Remarks The conformal module of braids and first applications of this concept to families of polynomials, depending holomorphically on a parameter, appeared in a line of research that was initiated by the 13.th Hilbert problem, posed by Hilbert in his famous talk on the International Congress of Mathematicians 1900. Hilbert asked whether each holomorphic function in three variables can be written as finite composition of continuous functions of two variables. The question was answered affirmatively by Kolmogorov and Arnold. In the Proceedings of the Symposium [17] devoted to “Mathematical developments arising from Hilbert Problems”, the problem is commented as follows. “Hilbert posed this question especially in connection with the solution of a general algebraic equation of degree 7. It is reasonable to presume that he formulated it in terms of continuous functions partly because he had an interest in nomography and partly because he expected a negative answer. Now that it is settled affirmatively, one can ask an equally fundamental, and perhaps more interesting, question with algebraic functions instead of continuous functions.”

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025 B. Jöricke, Braids, Conformal Module, Entropy, and Gromov’s Oka Principle, Lecture Notes in Mathematics 2360, https://doi.org/10.1007/978-3-031-67288-0_6

195

196

6 The Conformal Module and Holomorphic Families of Polynomials

This latter question goes back to mathematics of the seventeenth century. Consider an algebraic equation of degree n. zn + an−1 zn−1 + . . . + a0 = 0 .

.

(6.1)

Put .a = (a0 , . . . , an−1 ) ∈ Cn and consider P (z, a) = zn + an−1 zn−1 + . . . + a0

.

as a monic polynomial of degree n whose coefficients are polynomials in a. P is also called an algebraic function of degree n on .Cn . Recall that .Pn is the space of monic polynomials of degree n (maybe, with multiple zeros). The adjective monic refers to the property that the coefficient by the highest order of the variable equals 1. Parameterizing the space .Pn by unordered n-tuples of zeros of polynomials, we get a map which assigns to each .a ∈ Cn the n-tuple .{z1 (a), . . . , zn (a)} of solutions of the equation .P (z, a) = 0, i.e. what is called an “n-valued algebraic function”. Tschirnhaus suggested to “substitute” the unknown variable z in (6.1) by an algebraic function of a new variable w. In other words, put w = zk + bk−1 zk−1 + . . . + b0 ,

.

(6.2)

and let .b = (b0 , . . . , bk−1 ) ∈ Ck . To eliminate z from the two Eqs. (6.1) and (6.2) def

consider P and Q, .Q(z, w) = zk + bk−1 zk−1 + . . . + b0 − w, as polynomials in z with coefficients being polynomials in w. They are relatively prime. The resultant R is a polynomial in w (and in the coefficients a and b). It can be written as R = pP + qQ

.

where p and q are polynomials in z and w (and in the coefficients .a ∈ Cn , .b ∈ Ck ). If z satisfies (6.1) and (6.2) then .R(w) = 0. Vice versa, if .Q(z, w) = 0 and .R(z) = 0, then .P (z) = 0. This allows to think about solutions of (6.1) as part of zeros of a “composition of algebraic functions” determined by the equations .Q(z, w) = 0, respectively by .R(w) = 0. The method consists now in choosing the coefficients b depending on a so that the “algebraic functions” depend on a smaller number of variables. This method brings the general equation of degree 7 to the equation z7 + a z3 + b z2 + c z + 1 = 0 .

.

(6.3)

The 7-tuple of solutions of (6.3) is an algebraic function of degree 7 depending on three complex variables. Hilbert was interested in the “complexity” of the general 7-valued algebraic function, in other words, he wanted to know how far one can go in this process to get formulas beyond radicals. There are two difficulties. First, the “function-composition” may have more zeros than P , and, secondly, the “algebraic functions” obtained by choosing

6.1 Historical Remarks

197

the coefficients b are actually determined by polynomials in one variable with coefficients being rational functions in several complex variables, that may have indeterminacy sets. The composition problem was rigorously formulated for entire algebraic functions (i.e. for polynomials in one variable with coefficients being entire functions of complex variables—the restricted composition problem) and was considered first for the case when the “function-composition” coincides with the original function (the faithful composition problem). Arnold’s interest in the topological invariants of the space .Pn of monic polynomials of degree n without multiple zeros was motivated by finding cohomological obstructions for the restricted faithful composition problem. The restricted composition problem is now answered negatively in a series of papers by several authors. The last step was done by Lin (see [61]). But the original question allowing polynomials in one variable with coefficients being rational functions is widely open. Compare also with the more classical account [60]. See also the papers [82, 83]. On the other hand, .Pn is a complex manifold, in fact, the complement of the algebraic hypersurface .{Dn = 0} in complex Euclidean space .Cn . Here .Dn denotes the discriminant of polynomials in .Pn . Recall that the function .Dn is a polynomial in the coefficients of elements of .Pn . While, in connection with his interest in the Thirteen’s Hilbert Problem, Arnold studied the topological invariants of the space .Pn , the conformal invariants of .Pn (i.e. invariants that are preserved under biholomorphic mappings) are of interest in connection with Gromov’s Oka Principle. The collection of conformal modules of conjugacy classes of elements of the fundamental group of the space of monic polynomials of degree n without multiple zeros is a collection of conformal invariants of .Pn ∼ = Cn \ Dn . Notice, that for any complex manifold the conformal module of conjugacy classes of its fundamental group can be defined. The collection of conformal modules of all conjugacy classes is a biholomorphic invariant of the manifold. We will focus here on the respective invariants of the space .Pn , in other words, on the conformal module of conjugacy classes of braids. In whatever sense one understands “algebraic functions”, for each algebraic function there is a Zariski open set in .Cn such that the restriction of the algebraic function to this set is a separable algebroid function. The restriction of the algebraic function to each loop in this set defines a conjugacy class of braids. Moreover, this conjugacy class is represented by a holomorphic mapping of an annulus into the space of polynomials—any annulus which is mapped holomorphically into the Zariski open set and represents the homotopy class of the loop may serve. We do not know at the moment whether further progress related to the concept of the conformal module of braids may have some impact on the open problems related to the 13.th Hilbert problem or stimulated by it. I am grateful to M. Zaidenberg who asked me this question.

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6 The Conformal Module and Holomorphic Families of Polynomials

6.2 Families of Polynomials, Solvability and Reducibility The concept of the conformal module of conjugacy classes of braids appeared (without name) in the paper [33]. The paper was motivated by the interest of the authors in Hilbert’s Thirteen’s Problem and treats global reducibility of holomorphic families of polynomials. The following objects related to .Pn have been considered in this connection. A continuous mapping from a topological space X into the set of monic polynomials of fixed degree .Pn (maybe, with multiple zeros) is a quasipolynomial. It can be written as a function in two variables .x ∈ X, ζ ∈ C : . f (x, ζ ) = a0 (x) + a1 (x)ζ + ... + an−1 (x)ζ n−1 + ζ n , for continuous functions .aj , j = 1, ..., n , on X. If X is a complex manifold and the mapping is holomorphic it is called an algebroid function. If the image of the map is contained in the space .Pn of monic polynomials of degree n without multiple zeros, it is called separable. A separable quasipolynomial is called solvable if it can be globally written as a product of quasipolynomials of degree 1. It is called irreducible if it can not be written as product of two quasipolynomials of positive degree, and reducible otherwise. We also call a solvable quasipolynomial a globally solvable family of polynomials, and an irreducible quasipolynomial a globally irreducible family of polynomials. Two separable quasipolynomials are isotopic if there is a continuous family of separable quasipolynomials joining them. An algebroid function on .Cn whose coefficients are polynomials is called an algebraic function. In this terminology the result of the paper [33] concerns reducibility of holomorphic quasipolynomials. The space .Pn of monic polynomials of degree n without multiple zeros coincides with the symmetrized configuration space .Cn (C)Sn . The space .Pn can be parameterized either by the coefficients or by the unordered tuple of zeros of polynomials. A separable quasipolynomial of degree n on a connected closed Riemann surface or a connected bordered Riemann surface X is a continuous mapping from X into the symmetrized configuration space .Cn (C)Sn . The restriction of the quasipolynomial to a loop in X with a base point .x0 is a geometric braid. Hence, the monodromy of the mapping to .Cn (C)Sn along each element of the fundamental group .π1 (X, q0 ) of X is a braid. Theorem 1.1 in this situation can be stated as follows (see also [36, 76]) The (free) isotopy classes of separable quasipolynomials of degree n on X are in one-to-one correspondence to conjugacy classes of homomorphisms from .π1 (X, q0 ) into .Bn . We start with a simple Lemma on global solvability of holomorphic families of polynomials. Lemma 6.1 Let X be a closed Riemann surface of positive genus with a closed smoothly bounded topological disc removed. Suppose f is an irreducible separable algebroid function of degree 3 on X. Suppose X contains a domain A, one of whose boundary components coincides with the boundary circle of X, such that

6.2 Families of Polynomials, Solvability and Reducibility

199

A is conformally equivalent to an annulus of conformal module strictly larger than √ 3+ 5 −1 π (log( )) . Then f is solvable over A. 2 2

.

We postpone the proof of Lemma 6.1, and state some simple lemmas which will be needed in the sequel. Let A be an annulus equipped with an orientation (called positive) of simple dividing closed curves. (A simple closed curve is dividing if its complement is not connected.) Let .γ be a positively oriented simple closed dividing curve in A. For a continuous mapping f from an annulus A to the symmetrized configuration space .Cn (C)Sn the restriction .f | γ represents a conjugacy class of braids which is denoted by . bf,A . (It does not depend on the choice of the positively oriented simple closed dividing curve.) Lemma 6.2 Suppose the separable quasipolynomial f of degree n on an annulus A is irreducible and n is prime. Then the induced conjugacy class of braids . bf,A is irreducible. Notice that, on the other hand, conjugacy classes of irreducible pure braids define solvable, hence reducible, quasipolynomials on the circle. Proof Recall that .τn : Bn → Sn is the natural projection from the braid group to the symmetric group. .τn maps conjugacy classes of braids to conjugacy classes of permutations. If f is irreducible the conjugacy class of braids . bf,A projects to a conjugacy class .τn ( bf,A ) of n-cycles. The lemma is now a consequence of the following known lemma (see e.g. [24]).   Lemma 6.3 If n is prime then any braid .b ∈ Bn , for which .τn (b) is an n-cycle, is irreducible. For convenience of the reader we give the short argument. Proof of Lemma 6.3 If b was reducible then a homeomorphism .ϕ which represents the mapping class corresponding to b would fix setwise an admissible system of curves .C. Let .C1 be one of the curves in .C and let .δ1 be the topological disc contained in .D, bounded by .C1 . .δ1 contains at least two distinguished points .z1 and .z2 . Since k .ϕ permutes the distinguished points along an n-cycle there is a power .ϕ of .ϕ which k maps .z1 to .z2 . Since n is prime, .ϕ also permutes the distinguished points along an n-cycle. Hence it maps some distinguished point in .δ1 to the complement of .δ1 . We obtained that .ϕ k (δ1 ) intersects both, .δ1 and its complement. Hence .ϕ k (C1 ) = C1 but .ϕ k (C1 ) intersects .C1 . This contradicts the fact that the system of curves .C was admissible and invariant under .ϕ.   Recall that .Δ3 = σ1 σ2 σ1 = σ2 σ1 σ2 and the group . Δ2 generated by .Δ2 coincides with the center .Z3 of the braid group .B3 . We need the following lemma. Lemma 6.4 A mapping class in .M(D; ∂D, E3 ) is reducible if and only if the braid corresponding to it is conjugate to .σ1k Δ2 3 for integers k and .. A mapping class in .M(P1 ; {∞}, E3 ) is reducible if and only if it corresponds to a conjugate of .σ1k Z3 .

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6 The Conformal Module and Holomorphic Families of Polynomials

Proof Any admissible system of curves in .D with three distinguished points consists of one curve that divides .D into two connected components. One of these components contains .∂D and one of the distinguished points, the other one contains two distinguished points. For a reducible mapping class .m in .M(D; ∂D, E3 ) we take an admissible curve .γ that reduces .m and a representing homeomorphism .ϕ ∈ m that fixes .γ pointwise. Suppose the points .ζ1 , ζ2 , and .ζ3 in .E3 are labelled by using a homomorphism from .π1 (C3 (C)S3 , E3 ) to .π1 (C3 (C)S3 , E30 ) (which is defined up to conjugation) so that one of the connected components .C1 of the complement of .γ contains .ζ1 and .ζ2 , the other connected component .C2 contains .ζ3 and .∂D. Since .ϕ fixes .∂D and .γ pointwise, it also fixes the annulus .C 2 setwise and fixes the point .ζ3 . A self-homeomorphism of an annulus that fixes the boundary pointwise is a twist. After an isotopy of .ϕ on .D that fixes .γ setwise and fixes each distinguished point we may assume that .ϕ fixes pointwise a simple arc that joins .ζ3 with .γ . Then the restriction .ϕ | C 2 represents a power of a Dehn twist in .D about a curve in .C2 that is homotopic in .C 2 \ {ζ3 } to .∂D. Such a Dehn twist corresponds to the braid .Δ23 . The restriction .ϕ | C 1 represents the class in .M(C 1 ; ∂C1 , {ζ1 , ζ2 }), corresponding to .σ k for some integer k. Hence the class represented by .ϕ in .M(D; ∂D, E3 ) corresponds to a braid that is conjugate to .σ1k Δ2 3 for some integer .. Vice versa, it is clear that the mapping class .mσ 2k Δ2 is reducible. 1

3

The statement concerning the reducible elements of .M(P1 ; {∞}, E3 ) is proved   in the same way. We want to point out that the identity in the braid group .Bn with .n ≥ 3 is reducible, the identity in .B2 is irreducible. Recall that by Example 3.8 in Sect. 3.2 the  √ −1 value .η = π2 log 3+2 5 is the largest finite conformal module among irreducible √

3-braids, and hence, by the Main Theorem the value .log 3+2 5 is the smallest nonvanishing entropy among 3-braids. For the latter fact see also [74]. 3 the conformal module Lemma 6.5 Suppose for a conjugacy class of braids . b∈B M( b) satisfies the inequality .M( b) > η. Then the following holds.

.

(a) .M( b) = ∞. (b) If . b is irreducible then . b is the conjugacy class of a periodic braid, i.e. either of   .(σ1 σ2 ) for an integer . not divisible by 3, or of .(σ1 σ2 σ1 ) for an integer . not divisible by 2. (c) If . b is reducible then . b is the conjugacy class of .σ1k Δ2 3 for integers k and .. 2 2 Here .Δ3 = (σ1 σ2 σ1 ) = (σ1 σ2 )3 . (d) If . b is the conjugacy class of a commutator, then it is represented by a pure braid b.  √ −1 Proof Suppose . is the largest finite conforb is irreducible. Since . π2 log 3+2 5 mal module among irreducible 3-braids the equality .M( b) = ∞ holds. (Hence, b) = 0.) This proves (a) in the irreducible case. also .h(

6.2 Families of Polynomials, Solvability and Reducibility

201

Let .b ∈ B3 represent the irreducible class . b, let .mb ∈ M(D; ∂D, E3 ) be the mapping class corresponding to b, and let .H∞ (mb ) ∈ M(P1 ; ∞, E3 ) be the b) = ∞, corresponding mapping class on .P1 with distinguished points. Since .M( the class .H∞ (mb ) is represented by a periodic mapping of the complex plane .C with three distinguished points (see Proposition 3.1, Theorem 1.10, and Theorem 1.11). By a conjugation we may assume that the periodic mapping fixes zero and hence is a rotation by a root of unity. If zero is not a distinguished point, the three distinguished points are equidistributed on a circle with center zero and the mapping is rotation by



a power of .e

2π i 3

. The mapping representing .H∞ (mb ) corresponds to .(σ1 σ2 ) / Δ23



and the mapping representing .mb corresponds to .(σ1 σ2 ) for an integer number .. If zero is a distinguished point, the other two distinguished points are equidistributed on a circle with center zero. The mapping representing .H∞ (mb )is a rotation, precisely, a multiplication by a power of .−1, and corresponds to



.

(σ1 σ2 σ1 ) / Δ23 for an integer number .. (Note that powers of .Δ23 are reducible and the mapping class in .M(P1 ; {∞}, E3 ) corresponding to .Δ23 is the identity.) The 

mapping representing .mb corresponds to .(σ1 σ2 σ1 ) . We proved (b). Suppose . b. Then by b is reducible, i.e. the mapping class .mb is reducible for .b ∈  Lemma 6.4 the class . for integers .k = 0 and .. We b is the conjugacy class of .σ1k Δ2 3 obtained (c). The braid .b = σ1k Δ2 3 has infinite conformal module. This gives (a) in the reducible case. Suppose . b is the conjugacy class of a commutator .b ∈ B3 . Then b has exponent sum zero (more detailed, the sum of exponents of the generators in a representing word equals zero). If .M( b) = ∞ then this is possible only if b is conjugate to k 2 with .k + 6 = 0. Hence, k is even and the braid b is pure. This proves (d). .σ Δ 3 1 The lemma is proved.   Proof of Lemma 6.1 Let .Sf = {(z, ζ ) ∈ X × C : f (z, ζ ) = 0} be the zero set of the algebroid function f . Since f is irreducible, this set is connected. We obtain an unramified covering .Sf → X of degree 3. Hence, for the Euler characteristics .χ (X) and .χ (Sf ) the relation .χ (Sf ) = 3χ (X) holds. Let .m(X) and .m(Sf ) be the number of boundary components of X and .Sf . Then .2 − 2g(Sf ) − m(Sf ) = 3(2 − 2g(X) − m(X)). Hence, since .m(X) = 1, .m(Sf ) equals either 1 or 3. This means that .f | A is either irreducible or solvable. If .f | A is irreducible, it represents an irreducible conjugacy class of braids . bf,A . By Lemma 6.5 .bˆf,A must be the conjugacy class of a periodic braid which corresponds to a 3-cycle. This is impossible for conjugacy classes of products of braid commutators.   We will discuss in more detail the result of Gorin, Lin and Zjuzin ([33, 84]) and give an alternative proof of a slight improvement. For simplicity we will restrict ourselves to the case when the degree of the quasipolynomial is a prime number. (The general case has been considered by Lin and later by Zjuzin. An alternative

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6 The Conformal Module and Holomorphic Families of Polynomials

proof of the general case can be given as well.) In [33] Gorin and Lin proved that for each prime number n there exists a number .rn such that any separable algebroid function of degree n on an annulus A of conformal module strictly larger than .rn is reducible provided the index of its discriminant .z → Dn (fz ), .z ∈ A, is divisible by n. Recall that the value at a point .z ∈ A of a separable algebroid function of degree n on A is a polynomial .fz ∈ Pn . .Dn (fz ) denotes its discriminant. Recall that .Dn is a function on the space of all polynomials .Pn of degree n which vanishes exactly on the set of polynomials with multiple zeros. Explicitly, for a polynomial .p ∈ Pn , n  .p(ζ ) = (ζ − ζj ), its discriminant equals j =1

Dn (p) =



.

(ζi − ζj )2 .

(6.4)

i log 2 n, and f is a separable algebroid function on A of degree n such that the index of its discriminant is divisible by n, then f is reducible.

The present proof of the theorem uses the Main Theorem (see also Theorem 1 of [43]) and the following result of Penner [70] on the smallest non-vanishing entropy among irreducible n-braids. Theorem 6.2 (Penner) Denote by .hm g the smallest non-vanishing entropy among irreducible self-homeomorphisms of Riemann surfaces of genus .g with .m distinguished points .(2g − 2 + m > 0). Then hm g ≥

.

log 2 . 12g − 12 + 4m

6.2 Families of Polynomials, Solvability and Reducibility

203

Penner’s theorem implies that the smallest non-vanishing entropy among irreducible n-braids, .n ≥ 3, is bounded from below by .

log 2 log 2 1 log 2 = ≥ · , 4(n + 1) − 12 4n − 8 4 n

n ≥ 3.

By the Main Theorem the largest finite conformal module among irreducible conjugacy classes of n-braid does not exceed . π2 log4 2 · n for .n ≥ 3. Proof of Theorem 6.1 Suppose f is an irreducible separable algebroid function on the annulus A and m(A) >

.

2π π 4 ·n= · n. 2 log 2 log 2

3 induced by f is irreducible. By the By Lemma 6.2 the conjugacy class . bf,A ∈ B n Main Theorem the entropy .h( bf,A ) of the conjugacy class of braids . bf,A ∈ B log 2 1  induced by f is strictly smaller than . 4 · n . By Penner’s Theorem .h(bf,A ) = 0, bf,A ) = ∞. i.e. .M( This implies that . bf,A is the conjugacy class of a periodic braid corresponding to an n-cycle, hence it is the conjugacy class of .(σ1 σ2 . . . σn )k for an integer k which is not divisible by n. The isotopy class of the algebroid function .f˜(z, ζ ) = ζ n − zk , 2π it , .t ∈ [0, 1], the .z ∈ A, induces this conjugacy class of braids. Indeed, for .z = e set of solutions of the equation .f˜(z, ζ ) = 0 is  2π ik 2π ik(n−1) 2π ik 2π ik 2π ik def .En (t) = e n t , e n + n t , . . . , e n + n t , t ∈ [0, 1] . The path .t → En (t), t ∈ [0, 1], in .Pn defines a geometric braid in the conjugacy class of .(σ1 · σ2 · . . . · σn )k . Compute the discriminant .Dn (f˜z ): Dn (f˜z ) =





e

.

2π ik 2π ik n ·m+ n ·t

−e

2π ik 2π ik n ·+ n ·t

2

0≤m< 3. Hence, .Pn is not a Gromov-Oka manifold for .n ≥ 3. On the other hand, any continuous map from an oriented connected open smooth surface X into .P2 has the Gromov-Oka property, moreover, any such map is homotopic to a holomorphic map for any conformal structure on X. This statement can be proved in the same way as Statement (2) of Theorem 8.2 below. By the definition of the conformal module of a conjugacy class of n-braids (see Definition 7.1) a continuous mapping f from an annulus A to the space .Pn is homotopic to a holomorphic mapping if .m(A) < M( bf,A ) and is not

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7 Gromov’s Oka Principle and Conformal Module

homotopic to a holomorphic mapping if .m(A) > M( bf,A ). In other words, the relation of the conformal module .M( bf,A ) to the conformal invariant .m(A) of A “almost” determines whether f is homotopic to a holomorphic mapping. Recall that .M( bf,A ) is inversely proportional to the entropy of . bf,A . It is reasonable to interpret the entropy as “complexity”: The complexity grows with increasing entropy, while the chance for the mapping to be homotopic to a holomorphic one decreases. For .n = 3 there are effective upper and lower bounds for the conformal module (equivalently, for the entropy) of conjugacy classes of 3-braids that differ by multiplicative constants. We will address this topic in Chap. 9 following [44, 45]. Notice that each element of the fundamental group .π1 (C \ {−1, 1}, 0) can be interpreted as a pure 3-braid and we can associate to its conjugacy class a mapping class. A multiple of the inverse of the conformal modules of a conjugacy class of elements of the fundamental group .π1 (C \ {−1, 1}, 0) (equivalently a multiple of its extremal length) is equal to the entropy of the associated mapping class. The problem of understanding obstructions to Gromov’s Oka Principle is interesting for other target spaces Y as well. In this respect the following problem occurs. Problem 7.2 Study the Gromov-Oka Principle in case the target is the complement of an arbitrary algebraic hypersurface of .Cn or the quotient of the n-dimensional round complex ball by a subgroup of its automorphism group which acts freely and properly discontinuously. Study the conformal module of conjugacy classes of elements of the fundamental group of such spaces. Let Y be a complex manifold. The collection of conformal modules of all conjugacy classes of elements of its fundamental group .π1 (Y, y0 ) is a biholomorphic invariant of Y , which is expected to play a special role for understanding the Gromov-Oka Principle. In the case of the space .Pn this may be opposed to the fact, that in connection with his interest in the Thirteen’s Hilbert Problem Arnold [7] considered the topological (cohomological) invariants of .Pn . The facts known for mappings from annuli to .Pn motivate the following two problems related to the restricted validity of Gromov’s Oka Principle in general. Problem 7.3 Fix a connected open (i.e. non-compact) complex manifold X and a complex manifold Y . Obtain information about the set of continuous or smooth mappings .X → Y that are homotopic to holomorphic mappings. For a smooth manifold X an orientation preserving homeomorphism .ω : X → ω(X) from X to a complex manifold .ω(X) is called a complex structure on X. If X is a smooth surface and .ω(X) is a Riemann surface, we will also call .ω a conformal structure on X. A mapping f from X to a complex manifold Y is said to be holomorphic for the complex structure .ω if the mapping .f ◦ ω−1 is holomorphic on .ω(X). The following statement is known as “soft Oka Principle” (see [30]).

7.1 Gromov’s Oka Principle and the Conformal Module of Conjugacy Classes. . .

209

Soft Oka Principle Any continuous mapping from a Stein manifold X to a complex manifold Y gets holomorphic after changing both, the mapping and the complex structure of X, by a homotopy. (Forstneric, Slapar) If X is a finite open Riemann surface and .Y = C \ {−1, 1}, the soft Oka principle is just Runge’s approximation theorem on compact subsets of Riemann surfaces. Indeed, take any continuous function .f : X → C \ {−1, 1}. Let Q be a standard bouquet of circles for X. The continuous function .f | Q can be approximated uniformly on Q by holomorphic functions on X. See [55] for Mergelyan type approximation (approximation of continuous functions on Q by holomorphic functions in a neighbourhood of Q) and [72] for Runge approximation by meromorphic functions on closed Riemann surfaces (here, approximation of holomorphic functions in a neighbourhood of Q by holomorphic functions on X). We obtain approximating holomorphic functions defined on X which do not necessarily omit .−1 and 1. But their restrictions to Q are close to .f | Q , hence their restrictions to Q have their image in .C \ {−1, 1} and are homotopic to .f | Q as mappings into .C \ {−1, 1}. Hence an approximating holomorphic function F maps a neighbourhood V of Q into .C \ {−1, 1} and .F |V is homotopic to .f | V through mappings into .C \ {−1, 1}. We may take V to be smoothly bounded and such that there exists a continuous family of mappings .ϕt : X → X, t ∈ [0, 1], that map X homeomorphically onto a domain .Xt in X such that .ϕ0 is the identity and .X1 = V . The homeomorphisms .ϕt : X → Xt are considered as new complex structures on X. The mapping .F ◦ ϕ1 is holomorphic on X for the complex structure .ϕ1 . Definition 7.2 Let X be an oriented finite open smooth surface and Y a complex manifold. A continuous mapping .f : X → Y is said to be homotopic to a holomorphic mapping for a conformal structure .ω : X → ω(X) on X, if .f ◦ w −1 is homotopic to a holomorphic mapping from .ω(X) to Y . The “soft Oka Principle” motivates the following problem. Problem 7.4 Consider an oriented connected smooth open manifold X, a complex manifold Y , and a continuous mapping .f : X → Y . Obtain information on the set of orientation preserving complex structures .ω : X → ω(X) on X such that f is homotopic to a holomorphic mapping for .ω. We restrict ourselves to surfaces X as source manifolds. Recall the terminology introduced in Chap. 1. A surface is called finite if its fundamental group is finitely generated. Each finite open Riemann surface X is conformally equivalent to a domain (denoted again by X) on a closed Riemann surface .Xc such that each connected component of the complement .Xc \ X is either a point or a closed topological disc with smooth boundary [77]. The connected components of the complement will be called holes. A Riemann surface is called of first kind, if it is closed, or it is obtained from a closed Riemann surface by removing finitely many points (called punctures). Otherwise the connected Riemann surface is called of second kind. If all holes of a finite open Riemann surface are closed topological discs, the Riemann surface is said to have only thick ends.

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7 Gromov’s Oka Principle and Conformal Module

Definition 7.3 Let X be a connected oriented finite open smooth surface, and let Y be a complex manifold. We will say that a continuous mapping .f : X → Y has the Gromov-Oka property if it is homotopic to a holomorphic mapping for any orientation preserving conformal structure .ω : X → ω(X) with only thick ends (i.e. with .ω(X) having only thick ends). For short we will call mappings with the Gromov-Oka property Gromov-Oka mappings. We look first at the Gromov-Oka property for mappings from annuli into the spaces .Pn . Recall that if .n = 2 then for an annulus A each continuous mapping .f : A → P2 has the Gromov-Oka property, moreover, it is homotopic to a holomorphic mapping for any conformal structure on A. For the case .n = 3 the following lemma holds. Lemma 7.1 A continuous mapping f from an annulus .A = {r1 < |z| < r2 } to .P3 is homotopic to a holomorphic mapping for each orientation preserving conformal structure of second kind on A if and only if it has the Gromov-Oka property. This √ happens if and only if the inequality .M( bf,A ) > π2 (log 3+2 5 )−1 holds. √

Recall that .log 3+2 5 is the smallest non-vanishing entropy among 3-braids. Recall also that an annulus is of second kind if and only if it has at least one thick end, and a mapping .f : A → Y is a Gromov-Oka mapping if it is homotopic to a holomorphic one for any conformal structure on A with only thick ends. Recall that .Pn is the space of monic polynomials of degree n without multiple zeros. It can be identified with .Cn (Cn Sn ) (the zero sets of such polynomials). A mapping into .Pn can be considered as a quasipolynomial or as mapping with values in .Cn (Cn Sn ). Proof of Lemma 7.1 If f is homotopic to a holomorphic mapping for any orientation preserving conformal structure of second kind on A or for any conformal structure with only thick ends, then .M( bf,A ) = ∞. √ bf,A ) > π2 (log( 3+2 5 ))−1 . Then Lemma Suppose on the other hand that .M( 6.5 implies that .M( bf,A ) = ∞ and, therefore, f is homotopic to a holomorphic mapping for any orientation preserving conformal structure on A of finite conformal module. There are two orientation preserving conformal structures of infinite conformal module, one conformally equivalent to the punctured disc .D \ {0}, it is of second kind, and the other conformally equivalent to the punctured complex plane, it is of first kind. By Lemma 6.5 a conjugacy class of braids of infinite conformal module is either the class of a periodic braid, or it is the class of .σ1k Δ2 3 . We claim  that in the case .bf,A is a periodic conjugacy class of braids, there is an integer number k, such that f is isotopic on the punctured plane to the quasipolynomial k 3 k 2 .f1,k (z, ζ ) = z − ζ , z ∈ C \ {0}, ζ ∈ C, or to .f2,k (z, ζ ) = ζ (z − ζ ), z ∈ k  C \ {0}, ζ ∈ C, respectively. If .bf,A is the conjugacy class of .σ1 , we take any constant .R > 1. We claim that the mapping f is isotopic on the punctured disc to k 2 .f3,k (z, ζ ) = (z − ζ )(ζ − R), z ∈ D \ {0}, ζ ∈ C for some integer number k.

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Indeed, for .z = 12 e2π it , .t ∈ [0, 1], the set of solutions of the equation .f1,k (z, ζ ) = 0 is   2π ik 4π ik def − k 2π ik t k 1, e 3 , e 3 , t ∈ [0, 1] . .E1 (t) = 2 3 e 3 The path .t → E1k (t), t ∈ [0, 1], in .P3 defines a geometric braid in the conjugacy class of .(σ1 · σ2 )k . The set of solutions of the equation .f2,k (z, ζ ) = 0, .z = 12 e2π it , .t ∈ [0, 1], is def

k

E2k (t) = 2− 2 e

.

2π ik 2 t

{−1, 0, 1} ,

t ∈ [0, 1] .

The respective path represents .Δk3 . Finally, the set of solutions of the equation .f3,k (z, ζ ) = 0, .z = 12 e2π it , .t ∈ [0, 1], is  k 2π ik  k 2π ik def k .E3 (t) = 2− 2 e 2 t , −2− 2 e 2 t , R , t ∈ [0, 1] . The respective path represents .σ1k . The quasipolynomial .f3 is not isotopic to a holomorphic one on the punctured plane. This follows from Lemma 7.2 below. 

Lemma 7.2 Let .bˆ be a reducible conjugacy class of n-braids of infinite conformal module which is not periodic. Then there is no holomorphic mapping from .C∗ = C\{0} into .Pn representing ˆ .b. Proof We have to prove that there is no holomorphic mapping from .C∗ into .Pn ˆ Assume the contrary. We may assume that .bˆ is represented which represents .b. by a pure braid b. Indeed, if there is a holomorphic mapping from .C∗ into .Pn ˆ then there is also such a holomorphic mapping representing .bk for representing .b, any .k ∈ Z\{0}. The number k can be chosen so that .bk is pure. By our assumption k 2 k .b is not a power of .Δn . Since .b is pure, there is a lift of the mapping that represents ∗  .b k to a mapping .g : C → Cn (C). Associate to each point .z = (z1 , z2 , . . . , zn ) ∈ 1 Cn (C) the complex affine mapping .az on .C, .az (ζ ) = zζ2−z −z1 ., that maps .z1 to 0 ∗ and .z2 to 1. Consider the holomorphic mapping .C ξ → ag (ξ ) (g(ξ )) that   maps

a point .ξ ∈ C∗ to the point . ag (ξ ) (g1 (ξ )), ag (ξ ) (g2 (ξ )), . . . , ag (ξ ) (gn (ξ )) , where   .g(ξ ) = g1 (ξ ), g2 (ξ ), . . . , gn (ξ ) ∈ Cn (C). Then for the first two coordinates of .ag (ξ ) (g(ξ )) the equalities .(ag (ξ ) (g(ξ )))1 = ag (ξ ) (g1 (ξ )) ≡ 0 and .(ag (ξ ) (g(ξ )))2 = ag (ξ ) (g2 (ξ )) ≡ 1 hold. For .j ≥ 3 the coordinate function .(ag (ξ ) (g(ξ )))j is a ˜ ∗ maps mapping from .C∗ to .C \ {0, 1}. Its lift to the universal covering .C ∼ = C the complex plane into .C+ . Hence by Liouville’s Theorem the mapping .ξ → ag (ξ ) (g(ξ )) is constant on .C∗ . Therefore, .bk can be represented by a mapping of

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the form .ξ → Psym ((ag (ξ ) )−1 (c1 , . . . , cn ))) for constants .c1 , . . . , cn . This means that .bk is a power of .Δ2n in contrary to the assumption. 

We obtained the following facts for conjugacy classes .bˆ of n-braids with infinite conformal module. Each annulus of finite conformal module admits a holomorphic ˆ The conjugacy class of each periodic mapping into .Pn , representing such a class .b. braid (irreducible or not) can be represented by a holomorphic mapping on .C∗ (the proof given in Lemma 7.1 for .n = 3 works for arbitrary n). The conjugacy class of non-periodic reducible mappings cannot be represented by a holomorphic mapping on .C∗ , but in some cases there exists a holomorphic representing mapping from ∗ .D = D\{0}. bf ) are homotopic to a We saw that only very few mappings f with infinite .M( holomorphic mapping on an annulus of first kind (i.e. on the punctured plane). This fact together with Lemma 7.1 motivates the use of conformal structures with only thick ends in the definition of the Gromov-Oka property. The following question arises, which is part of Problem 7.4. Problem 7.4a Given a connected oriented finite open smooth surface X and a complex manifold Y with a set of generators whose conjugacy classes have infinite conformal module, which mappings .f : X → Y have the Gromov-Oka property? In particular, which mappings .f : X → Pn have the Gromov-Oka property? Which mappings from X into the n-punctured complex plane have the Gromov-Oka property? We will use the notion of reducible elements of the fundamental group of the npunctured complex plane. The notion is an analog of the notion of reducible braids, or reducible mapping classes, respectively. Definition 7.4 Let .E  be a finite subset of the Riemann sphere .P1 which consists of .n + 1 ≥ 3 points. Let X be a connected finite open Riemann surface with non-trivial fundamental group. A non-contractible continuous map .f : X → P1 \ E  is called reducible if it is free homotopic (as a mapping to .P1 \E  ) to a mapping whose image is contained in .D \ E  for an open topological disc .D ⊂ P1 with .E  \ D containing at least two points of .E  . Otherwise the mapping is called irreducible. Let .E = {z1 , z2 , . . . , zn } be a subset of the complex plane that contains exactly n points and let .E  = E ∪ {∞}. We assign to each continuous mapping .f : [0, 1] → C \ E with .f (0) = f (1) the pure geometric .(n + 1)-braid .F (t) = {f (t), z1 , z2 , . . . , zn }, t ∈ [0, 1]. Approximating we may assume that the mapping f is smooth. Consider a parameterizing isotopy .ϕt , t ∈ [0, 1], for the geometric braid F that is contained in the cylinder .[0, 1]×RD for a large number R. The equality .ϕt ({f (0), z1 , z2 , . . . , zn }) = {f (t), z1 , z2 , . . . , zn }, t ∈ [0, 1] holds. Let .ef be the element of the fundamental group .π1 (C \ E, f (0)) with base point .f (0), that is represented by f . We associate to .ef the braid .bf , that is represented by F , and the mapping class .mf,∞ ∈ M(P1 ; E ∪ {f (0), ∞}) that is represented by the self-homeomorphism .ϕ1 of .P1 , belonging to the family .ϕt . We may assign to

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the homotopy class .ef of f an entropy, namely the entropy of the mapping class mf , which is a measure of the complexity of the mapping f . The following lemma holds.

.

Lemma 7.3 For a subset E of the complex plane that contains exactly .n ≥ 2 points a continuous non-contractible mapping .f : [0, 1] → C \ E is reducible if and only if the associated braid .bf is reducible, equivalently, if the associated mapping class .mf,∞ is reducible. Proof We may assume that the mapping f is smooth and all related objects are smooth. Suppose that the mapping f is reducible. Then there exists a free homotopy .fs , s ∈ [0, 1], of mappings from .[0, 1] into .C \ E with .fs (0) = fs (1) and .f0 = f , such that for all .t ∈ [0, 1] the inclusion .f1 (t) ⊂ D holds for a disc D with .(E ∪ {∞}) \ D containing at least two points. The pure geometric braids .Fs , s ∈ [0, 1], with varying base point, that are defined by .fs , are free isotopic, hence, represent the same conjugacy class of braids . bf . Let .E be the set .{z1 , . . . , zn }, where the label is chosen so that .z1 , . . . , zk are the points of .E that are contained in D. Let .ϕt , t ∈ [0, 1] be a parameterizing isotopy for the geometric braid .{f1 (t), z1 , . . . , zk }, .ϕt ({f1 (0), z1 , . . . , zk }) = {f1 (t), z1 , . . . , zk }, t ∈ [0, 1]. Since .f1 (t) ∈ D for all t, the parameterizing isotopy can be chosen so that all .ϕt , .t ∈ [0, 1], are equal to the identity on .C \ D. In particular .ϕ1 maps .∂D onto itself. Since .(E ∪ {∞}) \ D contains at least two points and .E ∩ D contains at least one point (since .f1 is not contractible), the mapping .ϕ1 of .P1 with set of distinguished points .{f (0), z1 , . . . , zn , ∞} is reducible, and hence the associated mapping class .mf1 ,∞ is reducible. Hence . bf , and thus .bf , is reducible. Vice versa, suppose .mf,∞ is reducible. Since .bf is a pure braid, there exists a mapping .ϕ ∈ mf,∞ that fixes each of the points .f (0), z1 , . . . , zn , and fixes a closed curve C pointwise. Each of the components of .P1 \ C contains at least two points among the .f (0), z1 , . . . , zn , ∞. After applying a Möbius transformation and perhaps relabeling the points, we may assume that .f (0), z1 , . . . , zk are contained in the bounded connected component D of .C \ C, and .zk+1 , . . . , zn are contained ¯ Take a continuous family .ϕt , t ∈ [0, 1], of self-homeomorphisms of .P1 in .C \ D. that fix .C pointwise and join the identity with .ϕ. Then the pure geometric braid   def .F(t) = ϕt {f (0), z1 , . . . , zn } , t ∈ [0, 1], is isotopic (through geometric braids with fixed base point) to .{f (t), z1 , . . . , zn }, t ∈ [0, 1], and the strands of .F with initial points .f (0), z1 , . . . , zk , are contained in the domain D.   We prove now, that the geometric braid . F(t) = ϕt {f (0), z1 , . . . , zn )} , .t ∈ [0, 1], is isotopic (through geometric braids with fixed base point) to a geometric braid of the form .G(t) = {g(t), z1 ,. . . , zn }, .t ∈ [0, 1], with .g(t) ∈ D for .t ∈ [0, 1]. We look at the geometric braid . ϕt {z1 , . . . , zn } , t ∈ [0, 1], that is obtained from .F(t), t ∈ [0, 1], by forgetting the strand with initial point .f (0). This geometric   braid .ϕt {z1 , . . . , zn } , t ∈ [0, 1], is isotopic with fixed base point to the constant geometric braid .{z1 , . . . zk , zk+1 , . . . , zn }, t ∈ [0, 1], by an isotopy that fixes .C pointwise.

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By Remark 1.1 there exists a smooth family of .P1 that  .ψt,s of diffeomorphisms  are equal to the identity on .C, such that .ψt,s {z1 , .. . , zn } realizes an isotopy of  geometric braids with fixed base point that joins .ϕt {z1 , . . . , zn } , t ∈ [0, 1], with the constant geometric braid, and has the following property. For a small positive number .ε we have .ψt,s = ϕt for .0 ≤ s ≤ ε, .ψt,1 = Id, and .ψ0,s = Id. Then the equality .ψt,s (z1 , . . . , zn ) = (z1 , . . . , zn ) holds for .(t, s) in the boundary of the square .[0, 1] × [0, 1] except for .{s = 1, t ∈ [0, 1]}.  We look now at .ψt,s {f (0), z1 , . . . , zn )} , t, s ∈ [0, 1]. Notice that the mapping .[ε, 1] s → ψ1,s (f (0)) defines a curve with base point .f (0) in .D \ {z1 , . . . , zk }. Let h be an orientation preserving self-homeomorphism of .[0, 1] × [0, 1] that equals the identity on .[0, 1]×{0} and on .{0}×[0, 1], maps .{1}×[0, 1] onto .{1}×[0, ε] by a homeomorphism that fixes the point .(1, 0), and takes .[0, 1]×{1} homeomorphically onto the remaining part of the   boundary. The mapping .ψh(t,s) (f (0), z1 , . . . , zk ) equals .ϕt (f (0), z1 , . . . , zk ) for .t ∈ [0, 1] and .s = 0, it equals .(f (0), z1 , . . . , zk ) for .t = 0 and .t = 1, and is of the form .t → (g(t), z1 , . . . , zk ), for .s = 1. Notice that the mapping g is homotopic to the inverse of the mapping .[ε, 1] s → ψ1,s (f (0)). We found an isotopy (through geometric braids with fixed base point) joining . F(t) = ϕt (f (0), z1 , . . . , zn ) , .t ∈ [0, 1], with a geometric braid of the form .G(t) = (g(t), z1 , . . . , zn ), .t ∈ [0, 1], with .g(t) ∈ D for .t ∈ [0, 1]. It remains to prove the following statement. If two pure geometric braids .(f (t), z1 , . . . , zn ), .t ∈ [0, 1], and .(g(t), z1 , . . . , zn ), .t ∈ [0, 1], with .z1 , . . . , zn being distinct complex numbers, are isotopic through geometric braids with fixed base point .(f (0), z1 , . . . , zn ) = (g(0), z1 , . . . , zn ), then g and f are homotopic mappings (with fixed base point) into .C \ {z1 , . . . , zn } = P1 \ {z1 , . . . , zn , ∞}. This statement is again a consequence of Remark 1.1. Let .ft,s be the isotopy of pure geometric braids with fixed base point (contained in the cylinder .[0, 1] × RD for a large number R). We let .f˜t,s be the isotopy of geometric braids obtained from .ft,s by forgetting the first strand. We  take the family  .ϕt,s , (t, s) ∈ [0, 1] × [0, 1] × RD of Remark 1.1, for which .ϕt,s (z1 , . . . , zn ) is equal to .f˜t,s , and .ϕ0,s = ϕt,0 = −1 ϕt,1 = Id. The family .ϕt,s (ft,s ) defines an isotopy of the form .(fs (t), z1 , . . . , zn ) that joins .(f (t), z1 , . . . , zn ) and .(g(t), z1 , . . . , zn ). The family .fs (t) is the required 

homotopy. The lemma is proved. We will mostly consider the case when .E = {−1, 1, ∞}. We will often refer to .P1 \ {−1, 1, ∞} as the thrice punctured Riemann sphere or the twice punctured complex plane .C\{−1, 1}. By Definition 7.4 a continuous mapping from a Riemann surface to the twice punctured complex plane is reducible, iff it is homotopic to a mapping with image in a once punctured disc contained in .P1 \ E. (The puncture may be equal to .∞.) To describe the reducible mappings into .P1 \ {−1, 1, ∞}, we recall, that we denoted by .a1 (.a2 , respectively) the generator of .π1 (C \ {−1, 1}, 0) that is represented by a curve with base point 0 that surrounds .−1 (1, respectively) positively. We called .a1 and .a2 the standard generators of .π1 (C \ {−1, 1}, 0). By Definition 7.4 a mapping from a finite open Riemann surface into .P1 \ {−1, 1, ∞} is reducible if and only if its monodromies are contained in a subgroup generated

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215

by a single element which is a conjugate of a power of an element of the form a1 , a2 , or (a1 a2 )−1 .

.

(7.1)

Indeed, the images under the mapping of all loops on the Riemann surface represent elements of such a subgroup of .P1 \ {−1, 1, ∞} iff the mapping is homotopic to a mapping with image in a punctured neighbourhood of .−1 (.+1, or .∞, respectively). Lemma 7.4 Let X be a connected finite open Riemann surface with only thick ends equipped with base point .q0 . For each homomorphism .h : π1 (X, q0 ) → π1 (C \ {−1, 1}, 0), whose image is contained in a subgroup of .π1 (C \ {−1, 1}, 0) generated by a power of one of the elements of the form (7.1), there exists a holomorphic mapping .f : X → C \ {−1, 1}, whose monodromy homomorphism .f∗ : π1 (X, q0 ) → π1 (C \ {−1, 1}, 0) is conjugate to h. Proof of Lemma 7.4 Let X be a connected finite open Riemann surface with only thick ends, and let .h : π1 (X, q0 ) → Γ be a homomorphism into the group .Γ generated by an element of .π1 (C \ {−1, 1}, 0) of the form (7.1). The Riemann surface X is conformally equivalent to a domain on a closed Riemann surface .Xc such that all connected components of its complement are closed topological discs. Hence, there exists an open Riemann surface .X1 ⊂ Xc such that X is relatively compact in .X1 . Let .ω : X → X1 be a homeomorphism from X onto .X1 that is homotopic on X to the inclusion .X → X1 for which .ω(q0 ) = q0 . Identify the fundamental groups .π1 (X, q0 ) and .π1 (X1 , q0 ) by the mapping induced by .ω. By Lemma 1.1 .X1 is diffeomorphic to a standard neighbourhood of a standard bouquet B of circles for .X1 . Hence, .X1 can be written as union .X1 = D ∪ j Vj of an open disc D and half-open bands .Vj attached to D (see Sect. 1.2). Let .cj be the circle of the bouquet that corresponds to the generator .ej of the fundamental group .π1 (X1 , q0 ). We consider an open cover of .X1 as follows. Put .U0 = D. Then .U0 is an open topological disc in .X1 that contains the base point .ω(q0 ). Cover each .Vj by two simply connected open subsets .Uj + and .Uj − of .X1 with connected and simply connected intersection, so that the .Uj ± are disjoint from the .Uk ± for .j = k, and for each .Uj ± its intersection with .U0 is connected and simply connected. We may also assume that the following holds. The intersection of three different sets among the .U0 , .Uj + and .Uj − is empty. The intersections of each .cj with .Uj + and .Uj − are connected, and each .cj is disjoint from .Uk + ∪ Uk − for .k = j . Label the .Uj ± so that walking on .cj \ U0 (which is contained in .Vj ) in the direction of the orientation of .cj we first meet .Uj − . For each j the set .U0 ∪ Uj − ∪ Uj + is an annulus that contains .cj . Define a Cousin I distribution as follows. For each j we put .F0,j − = 0 on .U0 ∩ Uj − , .Fj − ,j + = 0 on .Uj − ∩ Uj + , and .Fj + ,0 = kj on .Uj + ∩ U0 . Since open Riemann surfaces are Stein manifolds (see e.g. [27]), the respective Cousin Problem has a solution (see [31, 40], or Appendix A.1). This means that there are holomorphic

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functions .F0 on .U0 , .Fj + on .Uj + and .Fj − on .Uj − such that F0 − Fj − = 0 on U0 ∩ Uj − ,

.

Fj − − Fj + = 0 on Uj − ∩ Uj + , Fj + − F0 = kj on Uj + ∩ U0 .

(7.2)

The function ⎧ 2π iF0 ⎪ ⎪ ⎨e .F = e2π iFj − ⎪ ⎪ ⎩e2π iFj +

on U0 , on Uj −

(7.3)

on Uj +

is well-defined and holomorphic in .X1 . Let .e be the generator of the fundamental group of .C \ {0} with base point .F (q0 ). The restriction of F to each .cj represents k .e j . The restriction of F to .X  X1 is bounded. Hence, for a positive number C the mapping .−1 + C1 F maps X into .C \ {−1, 1} so that the monodromy along each .ej   k equals .a1j (after identifying .π1 C \ {−1, 1}, −1 + C1 F (q0 ) with .π1 (C \ {−1, 1}, 0) by a suitable isomorphism). The case when the monodromies are powers of other elements of form (7.1) is treated by composing F with a suitable conformal selfmapping of the Riemann sphere that permutes the points .−1, 1, and .∞. 

In Sect. 7.2 we will address Problem 7.4a for the case when the target manifold is the twice punctured complex plane, and in Sect. 7.3 for the case when .Y = P3 and X is a smooth surface of genus one with a hole. In Chap. 11 we address Problem 7.3. For instance, we give an upper bound for the number of homotopy classes of irreducible mappings from a finite open Riemann surface to the twice punctured complex plane that contain a holomorphic mapping. The estimates can be regarded as a quantitative information concerning the restricted validity of Gromov’s Oka Principle for some cases when the target manifold is not a Gromov-Oka manifold. In Sects. 8.3–8.9 we will consider the Gromov-Oka Principle for fiber bundles over a smooth torus with a hole.

7.2 Description of Gromov-Oka Mappings from Open Riemann Surfaces to C \ {−1, 1} Let .X be a topological space and let .eˆ be a conjugacy class of elements of the fundamental group .π1 (X , x0 ) of .X . Choose a loop .γ in .X that represents .e. ˆ Consider a complex manifold .Y and a continuous mapping .f : X → Y. The conjugacy class of elements of the fundamental group of .Y represented by the

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217

restriction .f ◦ γ depends only on f and on .eˆ and is denoted by .fˆe . Recall that M(fˆe ) = ∞ iff the restriction of f to an annulus representing .eˆ has the GromovOka property.

.

Lemma 7.5 Let X be a connected oriented smooth finite open surface and let e ∈ π1 (X, q0 ) be an element whose conjugacy class can be represented by a simple closed curve in X. Then for any .a > 0 there exists an orientation preserving conformal structure .ω : X → ω(X) on X such that the Riemann surface .ω(X) contains a holomorphic annulus A that represents .eˆ and has conformal module .m(A) > a. .

Notice that the conjugacy class of any element of a standard basis of .π1 (X, q0 ) and the commutator of two elements of a standard basis of .π1 (X, q0 ) corresponding to a handle can be represented by a simple closed curve. Proof of Lemma 7.5 Take any conformal structure .ω : X → ω (X) on X. Let .γ be a smooth simple closed curve that represents the conjugacy class .eˆ of e. Cut .ω (X) along .ω (γ ). Take a neighbourhood of .ω (γ ) that is cut by .ω (γ ) into connected components .X+ and .X− , each being conformally equivalent to an annulus. Take an annulus A of conformal module .m(A) > a, and glue the .X± conformally to annuli .A± in A that are disjoint and adjacent to different boundary components of A. We obtain a Riemann surface and a homeomorphism .ω from X onto this Riemann 

surface. The mapping .ω : X → ω(X) gives the required complex structure. We will consider now the target space .Y = C \ {−1, 1}. Let A be an annulus in the complex plane. By Lemmas 1.2 and 7.1 a mapping .f : A → C \ {−1, 1} has the Gromov-Oka property if and only if for a generator e of the fundamental group of the annulus the conjugacy class .f ∗ (e) has infinite conformal module. By Lemmas 1.2, 6.5, and 7.1 a conjugacy class of elements of .π1 (C \ {−1, 1}, 0) has infinite conformal module iff it is represented by an integer power of one the elements (7.1). By Lemma 7.3 this happens exactly if the mapping f is reducible. Notice that a continuous mapping f from an annulus .A = {r −1 < |z| < r} into .C \ {−1, 1} with the Gromov-Oka property is also homotopic to a holomorphic mapping for any orientation reversing conformal structure with only thick ends on the annulus. Indeed, the mappings .z → f (z) and .z → f ( 1z¯ ) are homotopic on A since they coincide on .{|z| = 1}. The following theorem concerns Problem 7.4a for smooth mappings from oriented finite open smooth surfaces X of positive genus to the twice punctured complex plane and confirms the special role of the conformal module of conjugacy classes of elements of the fundamental group of the target with respect to this problem. It states the existence of finitely many annuli contained in X, such that a smooth orientation preserving mapping .X → C \ {−1, 1} has the GromovOka property if and only if its restriction to each of the mentioned annuli has this property. Theorem 7.1 Let X be a connected smooth oriented surface of positive genus g with .m ≥ 1 holes. There exists a subset .E  of the fundamental group .π1 (X, q0 ) of X

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consisting of at most .(2g + m − 1)3 elements such that the conjugacy class .eˆ of each element of .E  can be represented by a simple closed curve and the following holds. A continuous mapping .f : X → C \ {−1, 1} has the Gromov-Oka property if and only if for each .e ∈ E  the restriction of the mapping to an annulus in X that represents .eˆ has the Gromov-Oka property. The following theorem describes all continuous maps .X → C \ {−1, 1} with the Gromov-Oka property. In this theorem we allow the surface X to have any genus. Theorem 7.2 A continuous mapping .f : X → C \ {−1, 1} from a connected smooth oriented surface X of genus .g ≥ 0 with .m ≥ 1 holes, .2g + m ≥ 3, into the twice punctured complex plane has the Gromov-Oka property, if and only if f is either reducible, or X is a smooth oriented two-sphere .S 2 with at least three holes and the mapping f is homotopic to a mapping that extends to an orientation preserving diffeomorphism .F : S 2 → P1 that maps three points of .S 2 contained in pairwise different holes to the three points .−1, 1, and .∞, respectively. Let .E be a standard system of generators of the fundamental group .π1 (X, q0 ) of the smooth surface X of genus g with m holes with base point .q0 ∈ X. Before proving the Theorems 7.1 and 7.2 we will describe a set .E  of elements of .π1 (X, q0 ) such that the conjugacy class of each element of .E  can be represented by a simple closed curve. Later we will compare the Gromov-Oka property of a mapping .X → C \ {−1, 1} with the Gromov-Oka property of its restriction to each annulus in X that represents .eˆ for an element .e ∈ E  . Let First .g > 0. For the j -th handle we choose three elements .e2j −1 , .e2j , and [e2j −1 , e2j ]. The conjugacy class of each of them can be represented by a simple closed curve. For the .-th hole, . = 1, . . . , m − 1, we take the element .e2j + . There is a simple closed curve that represents its conjugacy class. The obtained .3g + m − 1 elements of the fundamental group will be contained in .E  . We convert each unordered pair of different elements of .E, that is different from a pair .{e2j −1 , .e2j } corresponding to a handle, into an ordered pair .(e , e ). There is a simple closed curve representing .e e = e e . The products .e e constitute a set of . 12 (2g + m − 1)(2g + m − 2) − g elements of the fundamental group. They will be contained in .E  . For each pair .e2j −1 , e2j of elements of .E that corresponds to a handle, and each other element .e among the chosen generators (if there is any) we consider the 3  2 2   3  elements .e2j −1 e2j e , .e2j −1 e2j e , .e2j −1 e2j e , and .e2j −1 e2j e . For the conjugacy class of each such element there is a simple closed curve in X representing it. The obtained .4g (2g + m − 3) elements of the fundamental group will be contained in  .E . Suppose .m > 2. Let .(e1 , e2 ) be the pair of elements of .E that corresponds to the first handle. We convert each unordered pair of different elements of .E corresponding to holes to an ordered pair .(e , e ), so that the conjugacy class of the element .e e1 e e2 e can be represented by a simple closed curve in X. (See Fig. 7.1.) .





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Fig. 7.1 A simple closed curve that represents the free homotopy class of .e e1 e e2 e

We obtain no more than . 12 (m − 1)(m − 2) elements of .π1 (X, q0 ). They will be contained in .E  . Suppose now .g(X) = 0. If .m = 1 the fundamental group is trivial. If .m = 2 it has one generator. The set .E  will be equal to the set .E consisting of the generator. def  −1 is represented by a simple closed Let .m ≥ 3. The element .em = ( jm−1 =1 ej ) curve that surrounds .Cm positively (i.e. walking along the curve the set .Cm is on the left). All elements .e ∈ E ∪ {em } will be contained in .E  . Further, each unordered pair of different elements of .E ∪ {em } is converted into an ordered pair .(e , e ) and the product .e e will be contained in .E  . Finally, we convert each unordered triple of different elements of .E ∪ {em } into an ordered triple .(e , e , e ) so that .e e e can be represented by a simple closed curve. The product .e e e will be contained in .E  . We described all elements of .E  . Recall that the conjugacy class of each of them can be represented by a simple closed curve. We estimate now the total number N of elements of .E  . Let first .g > 0. With .x = 2g + m − 1 ≥ 2 the number N does not exceed



 .

x + g + 12 x(x − 1) − g + 4g(x − 2), x+g +

1 2 x(x

− 1) − g + 4g(x − 2) +

0 < m ≤ 2, 1 2 (m − 1)(m − 2),

m > 2.

Since .2g ≤ x, for .m ≤ 2 the sum does not exceed .x+x(x−1)+2x(x−2). Hence, 2 N ≤ 3x < x 3 . For  − 4x  .m > 2 we use the inequality .m − 1 = x − 2g ≤ x − 2, 1 hence . 4g + 2 (m − 2) (x − 2) ≤ 2x(x − 2). Again .N ≤ x 3 . Let .g = 0. If .m = 2 the set .E  contains a single element, and .N = 1 = (m−1)3 = 3 x . If .x = m − 1 ≥ 2 the number N is not bigger than .x + 1 + 12 (x + 1)x + 16 (x + 1)x(x − 1) = 12 (x + 1)(x + 2) + 16 x(x 2 − 1) = 16 x 3 + 12 x 2 + ( 32 − 16 )x + 1 and since .x ≥ 2, also in this case .N ≤ x 3 . We estimated the number of elements of .E  from above by .x 3 = (2g + m − 1)3 .

.

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For the proof of Theorem 7.1 we need the following lemma. Its proof will given after the proof of the theorems. Lemma 7.6 Let X be a connected oriented smooth open surface of genus g with m holes, .2g + m ≥ 0, and .E  the set of elements of the fundamental group .π1 (X, q0 ) chosen above. Assume the restriction of a continuous mapping .f : X → C\{−1, 1} to each annulus in X representing the conjugacy class .eˆ of an element .e ∈ E  has the Gromov-Oka property. Then one of the following statements holds. 1. The image of the monodromy homomorphism .f∗ : π1 (X, q0 ) → .π1 (C \ {−1, 1}, f (q0 )) ∼ = π1 (C \ {−1, 1}, 0) is contained in the group generated by a conjugate of a power of one of the elements of the form (7.1). In this case the mapping is reducible. 2. X is equal to the smooth oriented 2-sphere denoted by .S 2 with at least three holes removed and the mapping f is homotopic to a mapping that extends to a diffeomorphism .F : S 2 → P1 that maps three points of .S 2 contained in pairwise different holes to the three points .−1, 1, and .∞, respectively. If F is orientation preserving, than F is homotopic to a holomorphic map for any orientation preserving conformal structure on X, including structures of first kind. Moreover, the restriction .F | X is irreducible. If F is orientation reversing, then for any orientation preserving conformal structure on X the mapping F is homotopic to an antiholomorphic mapping, but F is not a Gromov-Oka mapping. Proof of Theorem 7.1 We let .E  be the set of elements of .π1 (X, q0 ) chosen above. If a mapping .f : X → C \ {−1, 1} has the Gromov-Oka property then by Lemma 7.5 the restriction of f to each annulus in X representing the conjugacy class of an element of .E  has the Gromov-Oka property. Vice versa, suppose X has positive genus and the restriction of a continuous mapping .f : X → C \ {−1, 1} to each annulus in X representing the conjugacy class .eˆ of an element .e ∈ E  has the Gromov-Oka property. Then by Lemma 7.6 the monodromy homomorphism .f∗ is contained in the group generated by a conjugate of a power of an element of the form (7.1), equivalently f is reducible. By Lemma 7.4 each reducible map .X → C \ {−1, 1} is homotopic to a holomorphic mapping for each conformal structure on X with only thick ends. Hence, the mapping f has the Gromov-Oka property. 

Proof of Theorem 7.2 As before, .E  is the set of elements of .π1 (X, q0 ) chosen above. Suppose first that .g > 0. If a continuous mapping .f : X → C \ {−1, 1} has the Gromov-Oka property, then by Lemma 7.5 the restriction of f to each annulus represented by a conjugacy class .eˆ for an element .e ∈ E  has the GromovOka property. By Lemma 7.6 the monodromy homomorphism .f∗ is contained in the group generated by a conjugate of a power of an element of the form (7.1) (equivalently f is reducible).

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221

Vice versa, let X be an oriented smooth finite surface of any genus. By Lemma 7.4 each reducible map .f : X → C \ {−1, 1} is homotopic to a holomorphic mapping for each conformal structure on X with only thick ends, i.e. reducible maps have the Gromov-Oka property. Let .g = 0. Then X is a smooth oriented 2-sphere denoted by S with at least 3 holes removed. If an irreducible continuous mapping .f : X → C \ {−1, 1} has the Gromov-Oka property, then by Lemmas 7.5 and 7.6 the mapping f is homotopic to a mapping that extends to a diffeomorphism .F : S → P1 that maps three points contained in different holes to the points .−1, 1, and .∞. If F is orientation preserving then by Lemma 7.6 for any conformal structure on S it is homotopic to a conformal mapping from .P1 onto itself that takes an ordered tuple of three distinct points in .P1 to the tuple .(−1, 1, ∞). If the mapping F is not orientation preserving, F is not homotopic to a holomorphic mapping by Lemma 7.6. 

The theorem shows that for oriented finite open surfaces of positive genus only the reducible mappings .X → C \ {−1, 1} have the Gromov-Oka property. In case X is the oriented two-sphere with at least three holes there are also irreducible homotopy classes of mappings with the Gromov-Oka property. Each consists of mappings with the following property. For each orientation preserving homeomorphism .ω : X → ω(X) onto a Riemann surface .ω(X) (maybe, of first kind) .f ◦ ω−1 is homotopic to a holomorphic mapping that extends to a conformal mapping .P1 → P1 that maps three points in different holes to .−1, 1 and .∞, respectively (in some order depending on the class). These homotopy classes are the only irreducible homotopy classes with the Gromov-Oka property, and they are the only homotopy classes of mappings .X → C \ {−1, 1} that contain a holomorphic mapping for any conformal structure on X including conformal structures of first kind. Indeed, if the image of a non-trivial monodromy homomorphism .f∗ is generated by a conjugate of a power of one of the elements of the form (7.1) the mapping f cannot be homotopic to a holomorphic mapping for a conformal structure of first kind (see the proof of Corollary 11.1). Proof of Lemma 7.6 7.6.1 Identifying the fundamental groups .π1 (C \ {−1, 1}, 0) and .π1 (C \ {−1, 1}, f (q0 )) by a fixed isomorphism, we obtain for each .e ∈ E  the equation −1 .f∗ (e) = we be we for an element .we ∈ π1 (C \ {−1, 1}, 0) and an element .be that is a power of an element of the form (7.1). The images .f∗ (e2j −1 ) and .f∗ (e2j ) of two elements corresponding to a handle commute. This is a particular case of the situation treated in the proof of Theorem 7.4 below, but can be seen easily directly. Indeed, the sum of exponents of the terms of a word representing a commutator is equal to zero. Since the commutator .[f∗ (e2j −1 ), f∗ (e2j )] = f ([e2j −1 , e2j −1 ]) must be a power of a conjugate of an element of the form (7.1) it must be equal to the identity. Recall that a word in the generators of a free group is called reduced, if neighbouring terms are powers of different generators. We will identify elements of

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a free group (in particular of .π1 (C \ {−1, 1}, 0)) with reduced words in generators of the group. A reduced word is called cyclically reduced, if either the word consists of a single term, or it has at least two terms and the first and the last term of the word are powers of different generators. 7.6.2 Suppose for two generators .e(1) , e(2) ∈ E ⊂ E  we have .f∗ (e() ) = k we−1 () be() we() with .be() = aj , . = 1, 2, where each .aj is one of the generators .a1 , .a2 of .π1 (C \ {−1, 1}, 0). We prove now that there exists an element .we(1) ,e(2) of .π1 (C \ {−1, 1}, 0) which conjugates both .f∗ (e() ) to .ajk , . = 1, 2,. Put

(1) e(2) ∈ E  or .e(2) e(1) ∈ E  , the monodromy w = we(2) we−1 (1) . Since either .e (1) e(2) ) = f (e(1) )f (e(2) ) has infinite conformal module and is conjugate .f∗ (e ∗ ∗ k1 −1 k2 to .aj1 w aj2 w. The element .ajk11 w −1 ajk22 w is conjugate to .ajk11 w  −1 ajk22 w  for an .

−k 

−k 

element .w  = aj2 2 waj1 1 ∈ π1 (C \ {−1, 1}, 0) that is either equal to the identity or it can be written as a reduced word that starts with a power of .aj1 and ends with a power of .aj2 . (In the case .aj1 = aj2 we do not exclude so far that .w  is equal to a power of the other generator of .π1 (C \ {−1, 1}, 0).) If .w  is not the identity, then .ajk11 w  −1 ajk22 w  can be written as cyclically reduced word that contains powers of different sign of generators of .π1 (C \ {−1, 1}, 0). Any cyclically reduced word that represents a conjugate of a power of an element of the form (7.1) contains k k

only powers of equal sign of the generators. Hence .w  = Id, and .w = aj22 aj11 , equivalently, −k 

k

aj2 2 we(2) = aj11 we(1) .

.

−k 

(7.4)

k

Put .we(1) ,e(2) = aj2 2 we(2) = aj11 we(1) . Then .we(1) ,e(2) f∗ (e(1) )(we(1) e(2) )−1 = ajk11 and we(1) ,e(2) f∗ (e(2) )(we(1) ,e(2) )−1 = ajk22 .

.

7.6.3 As a corollary we see that .f∗ (e(1) e(2) ) is conjugate to .ajk11 ajk22 . Hence, either at least one of the monodromies .f∗ (e() ), . = 1, 2, is the identity, or .j1 = j2 , or .k1 = k2 = ±1. Further, if the monodromies .f∗ (e() ), . = 1, . . . , k, along a collection of elements of .E are conjugate to powers of a common single generator .aj , there is a single element of .π1 (C \ {−1, 1}, 0) that conjugates the monodromy along each element of this collection to a power of .aj . Indeed, suppose .we() conjugates the monodromy .f∗ (e() ) to a power of .aj . Then equality (7.4) implies to each pair .e(1) , e() ,  = 2, . . . , k, of elements of this collection. By this equation k

() ) to a power of .a . we() = aj  we(1) and, hence .we−1 j (1) conjugates each .f∗ (e (1) (k) Moreover, if each monodromy of a collection .f∗ (e ), . . . , f∗ (e ), .e() ∈ E,  = 1, . . . , k, is conjugate to a power of a common element of form (7.1) and the monodromy .f∗ (e(k+1) ), .e(k+1) ∈ E, is conjugate to a different element of form (7.1), then there exists a single element .w ∈ π1 (C \ {−1, 1}, 0), that conjugates all these monodromies to powers of elements of form (7.1). Indeed, such a w exists for

.

7.2 Description of Gromov-Oka Mappings from Open Riemann Surfaces to. . .

223

the pair .f∗ (e(1) ) and .f∗ (e(k+1) ), and by the preceding arguments w also conjugates (2) ), . . . , f (e(k) ) to powers of elements of form (7.1). .f∗ (e ∗ 7.6.4 The same arguments apply if for two elements .e(1) , e(2) ∈ E the monodromy (1) ) is conjugate to a power of .a and the monodromy .f (e(2) ) is conjugate to a .f∗ (e j ∗ power of .a1 a2 . We replace the generators .a1 , .a2 of the free group .π1 (C \ {−1, 1}, 0) by the generators .A1 = aj and .A2 = (a1 a2 )−1 . Notice that .A2 can be considered as element of the fundamental group of the thrice punctured plane that is represented by loops surrounding .∞ positively. The arguments above imply the following. If the monodromy .f∗ (e) ˜ = w −1 v k w along an element .e˜ ∈ E is a conjugate to the k-th power of an element v among .a1 , .a2 , .a1 a2 , with .|k| > 1, then for any .e ∈ E,   .e = e, ˜ the monodromy .f∗ (e ) equals .w −1 v k w for an integer .k  . 7.6.5 We consider now the case when there is no element in .E the monodromy along which is conjugate to the k-th power of an element of form (7.1) with .|k| > 1. We claim that if .g > 0 and the monodromy along an element .e2j ∈ E, j ≤ g, (i.e. .e2j belongs to a pair corresponding to a handle) is not trivial and, hence, is equal to a conjugate .w −1 v ±1 w for an element v of form (7.1), then all monodromies are powers of the same element .w −1 vw or .w −1 v −1 w, respectively. Indeed, by our assumption all non-trivial monodromies along elements e of .E are conjugate to .ve±1 for an element .ve of the form (7.1). Consider the element .e2j −1 for which the pair .e2j −1 , e2j ∈ E corresponds to a handle. Conjugating all monodromies by a single element, we suppose for instance that .f∗ (e2j ) = a1 . Since the monodromies corresponding to a pair of handles commute (see part 8.6.1 of the proof), the monodromy .f∗ (e2j −1 ) is a power of .a1 , hence, by our assumption .f∗ (e2j −1 ) is equal to .a1±1 , or it is the identity. Since for an element .e ∈ E different from .e2j −1 , e2j 3  2 e and .e  the products .e2j −1 e2j 2j −1 e2j e are contained in .E , the monodromies 3  2  .f∗ (e2j −1 e 2j e ) and .f∗ (e2j −1 e2j e ) must be conjugate to a power of an element of the form (7.1). If .f∗ (e2j −1 ) equals .a1 or the identity, then the monodromy 2  .f∗ (e2j −1 e 2j e ) can only be conjugate to a power of an element of the form (7.1), if 3 e ) can only f∗ (e ) is a power of .a1 . If .f∗ (e2j −1 ) is equal to .a1−1 , then .f∗ (e2j −1 e2j be conjugate to a power of an element of the form (7.1), if .f∗ (e ) is a power of .a1 . The case when .f∗ (e2j ) equals .a1−1 , .a2±1 , or .(a1 a2 )±1 , or when the role of .e2j −1 and .e2j is interchanged, can be treated similarly. The claim is proved.

.

7.6.6 Suppose .g(X) > 0 but all monodromies along elements of .E corresponding to handles are trivial. We claim that still all monodromies are powers of the same conjugate of an element of form (7.1). If the monodromies along all but possibly one element of .E are trivial, there is nothing to prove. Hence, we may assume that .m > 2. By our assumption all non-trivial monodromies along elements of .E are conjugate to an element of the form (7.1) or are inverse to a conjugate of an element of the form (7.1). Let .e1 and .e2 be the elements of .E corresponding to the first labeled handle. Suppose there is an unordered pair .{e , e } of different elements of .E so that the monodromy along each element of the pair is non-trivial. Note that none of the

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elements equals .e1 or .e2 . Convert the unordered pair into an ordered pair .(e , e ), so that the conjugacy class .e e1 e e2 e can be represented by a simple closed curve. Assume that .f∗ (e ) = a1 . If .f∗ (e ) is not a power of .a1 then .f∗ (e e1 e e2 e ) = f∗ (e )2 f∗ (e ) (see Fig. 7.1) cannot be a power of an element of the form (7.1). The remaining cases in which .f∗ (e ) is of the form (7.1) or is inverse to an element of the form (7.1) are treated in the same way. The claim is proved. 

7.6.7 Let .g(X) = 0, i.e. X equals the smooth oriented 2-sphere .S 2 with holes. Suppose .f : X → C \ {−1, 1} is a continuous mapping whose restriction to each annulus representing an element .e ∈ E  has the Gromov-Oka property. Assume that the monodromies are not all conjugate to powers of a single common element of the form (7.1). Then by our assumption no monodromy is conjugate to the k-th power of an element of the form (7.1) with .|k| > 1, and there are at least three holes, and at least two elements .ej  and .ej  in .E with monodromies .f∗ (ej  ) and .f∗ (ej  ) being non-trivial powers of absolute value 1 of conjugates of different elements of the form (7.1).  −1 ∈ π (X, q ) is represented by a loop Recall that the element .em = ( jm−1 1 0 =1 ej ) with base point .q0 that surrounds the last hole .Cm counterclockwise. The product m . j =1 f∗ (ej ) is equal to the identity. The product of two non-trivial powers of different elements of the form (7.1) cannot be equal to the identity. Hence, by part 7.6.3. of the proof there must be an integer number .j  ∈ [1, m] different from .j  and .j  , for which .f∗ (ej  ) = Id. The monodromies along two different elements from .E ∪ {em } cannot be nontrivial powers of the same element of the form (7.1). Otherwise by part 7.6.3. of the proof there would be an ordered couple or an ordered triple of different elements of  .E ∪ {em } whose product is in .E but the monodromy along the product cannot be a conjugate of a power of an element of the form (7.1). Indeed, assume for instance that two monodromies .f∗ (e ) and .f∗ (e ) are conjugate to powers of .a1 and a third momodromy .f∗ (e ) is conjugate to a power of .a2 . By part 7.6.3. of the proof we    may assume that .f∗ (e ) = a1k , .f∗ (e ) = a1k , and .f∗ (e ) = a2k with .|k  | = |k  | = |k  | = 1. If .k  and .k  have different sign, then one of the monodromies .f∗ (e e ) or     .f∗ (e e ) cannot be conjugate to a power of an element of form (7.1). If .k and .k    have equal sign, the monodromy along the product of the three elements .(e , e , e ) in any order is not a power of an element of form (7.1). But the product of the three elements .(e , e , e ) for some order is an element of .E  . For other possible combinations of powers of elements of form (7.1) the arguments are the same. 7.6.8 Part 7.6.7 of the proof shows that the monodromy along at most three elements of .E ∪ {em } is nontrivial. Hence, there are exactly three elements .ej  , .ej  , and .ej  among the .ek , k = 1, . . . , m, with non-trivial monodromy. After conjugating all monodromies by a single element of .π1 (C \ {−1, 1}) the monodromies along two of them are equal to either .a1 and .a2 , respectively, or to −1 −1 −1 −1 .a 1 and .a2 , respectively. (The combinations .a1 , .a2 or .a1 , .a2 are impossible.) Order the three elements by .(e , e , e ) so that the product is in .E  . After a cyclic permutation which does not change the conjugacy class of the product, we may

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225

assume that the monodromy along .e is not equal to a power of an .aj . Then the ordered triple of monodromies along .(e , e , e ) is either .(a1 , a2 , (a1 a2 )−1 ), or −1 ), or .(a −1 , a −1 , a a ), or .(a −1 , a −1 , a a ). The elements .a , a , .(a2 , a1 , (a2 a1 ) 2 1 1 2 1 2 1 2 2 1 −1 and .(a1 a2 ) , respectively, of the fundamental group of the twice punctured complex plane are represented by curves that surround positively the points .−1, 1, and .∞, respectively, the elements .a1−1 , a2−1 , a2 a1 are represented by curves that surround .−1, 1, ∞ negatively. The case .(a1 , a2 , (a1 a2 )−1 ) for the monodromies along .(e , e , e ) corresponds to the homotopy class that contains the following mapping. Let .C  , C  , C  be the holes of X corresponding to .e , e , e , respectively. Take points .p ∈ C  , p ∈ C  , and .p ∈ C  , respectively. Denote by F an orientation preserving diffeomorphism from .S 2 onto .P1 , that maps .p to .−1, .p to 1, and .p to .∞. It is straightforward to check that F has the required monodromies along all generators. Hence, f is homotopic to .F | X. The case .(a2 , a1 , (a2 a1 )−1 ) for the monodromies along    .(e , e , e ) is similar. In these two cases for any orientation preserving conformal structure .ω : X → ω(X), including conformal structures of first kind, the mapping −1 is homotopic to a mapping hat extends to .P1 = ω(X)c as a conformal .(F |X) ◦ ω self-diffeomorphism of .P1 that maps X into .C \ {−1, 1}. If f has monodromies .(a1−1 , a2−1 , a2 a1 ) along .(e , e , e ), then f is homotopic to .F | X for an orientation reversing diffeomorphism F from .S 2 onto .P1 , that maps .p to .−1, .p to 1, and .p to .∞. The case .(a2−1 , a1−1 , a1 a2 ) is similar. In these cases for any orientation preserving conformal structure .ω : X → ω(X), including conformal structures of first kind, the mapping .(F |X) ◦ ω−1 is homotopic to a mapping hat extends to .P1 = ω(X)c as an anti-conformal self-homeomorphism of .P1 that maps X into .C \ {−1, 1}. But the mapping f does not have the GromovOka property. The latter fact can be seen as follows. Assume the contrary. Let .ωn : X → ωn (X) def

be a sequence of conformal structures of second kind on X such that .Xn = ωn (X) can be identified with an increasing sequence of domains in .C \ {−1, 1} whose union equals .C \ {−1, 1}. We may choose the .ωn uniformly converging on compact subsets of X. We identify the fundamental groups of .Xn and of .C \ {−1, 1} by the isomorphism induced by inclusion. Consider the case when the monodromies of f along .(e , e , e ) are equal to −1 −1 −1 .(a 1 , a2 , a2 a1 ). By our assumption for each n the mapping .f ◦ ωn is homotopic to a holomorphic mapping .fn : Xn → C \ {−1, 1} with monodromies .a1−1 along .e and .a2−1 along .e . By Montel’s Theorem there is a subsequence .fnk that converges locally uniformly on .C \ {−1, 1}. The limit function F cannot be a constant (including the values of the constant .−1, 1, or .∞), since for curves .γ  and .γ  representing .e and .e , respectively, the set .fnj (γ  ) ∪ fnj (γ  ) separates .−1, 1 and .∞. The limit function F is a holomorphic mapping from .C \ {−1, 1} to itself. Hence it extends to a meromorphic function on .P1 , and defines therefore a branched covering of .P1 . This is impossible, since the curves .γ  and .γ  are mapped to curves that surround .−1 and 1, respectively, negatively. 

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7.3 Gromov-Oka Mappings from Tori with a Hole to P3 A Riemann surface of genus 1 will be called a torus and a Riemann surface of genus 1 with a hole will be called a torus with a hole. A smooth oriented surface of genus 1 will be called a smooth torus. We consider now a connected smooth oriented surface X of genus one with a hole (i.e. a connected smooth oriented closed surface Xof genus one with a point or a closed disc removed) and address the question which mappings .X → Pn have the Gromov-Oka property. Recall that such a mapping can be considered as a separable quasipolynomial of degree n. Recall that any continuous map from an oriented connected open smooth surface X into .P2 has the Gromov-Oka property. We consider Problem 7.4 and Problem 7.4a for the case when the target manifold equals .P3 . We will say that a separable quasipolynomial on an open Riemann surface X is isotopic to a holomorphic quasipolynomial, if the corresponding mapping to .Pn is homotopic to a holomorphic one. We will say that a separable quasipolynomial f on a finite open oriented smooth surface X is holomorphic for the conformal structure −1 is a holomorphic .ω : X → ω(X) (with .ω(X) being a Riemann surface) if .f ◦ ω quasipolynomial on .ω(X). The soft Oka Principle states that for each separable quasipolynomial f of degree n on a finite open oriented smooth surface X there exists an orientation preserving conformal structure on X for which the quasipolynomial is isotopic to a holomorphic quasipolynomial. In the following theorem we consider quasipolynomials of degree 3 on a torus X with a hole. The theorem shows that the obstructions for a separable quasipolynomial to be isotopic to a holomorphic quasipolynomial are discrete. Theorem 7.3 Let X be a torus with a hole, and let .E = {e1 , e2 } be a standard system of generators of the fundamental group of X with base point .q0 . Denote by −1 −2 −1 −1  .E the set .{e1 , e2 , e2 e 1 , e2 e1 , e1 e2 e1 e2 }. Let f be a separable quasipolynomial of degree 3 on X such that for each .e ∈ E  the quasipolynomial f is isotopic to an algebroid function for an orientation preserving conformal structure .we on X with √ a holomorphic annulus representing e of conformal module larger than . π2 (log 3+2 5 )−1 . Suppose X is a Riemann surface of second kind. Then the quasipolynomial is isotopic to an algebroid function on X. The following statement is formally slightly stronger. It follows from the proof of Theorem 7.3. Let X, .E and .E  be as in the theorem. A continuous mapping .f : X → P3 has the Gromov-Oka property if and only if for each .e ∈ E  the restriction of the mapping to an annulus in X that represents .eˆ has the Gromov-Oka property. The crucial step for the proof of Theorem 7.3 is Theorem 7.4 below. The rest of this chapter is devoted to the proof of Theorem 7.4. Theorem 7.3 and results related

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to bundles will be proved in the next chapter. We first provide two lemmas that are needed for the proof of Theorem 7.4. The proof of the following lemma on permutations can be extracted for instance from [81]. For convenience of the reader we give the short argument. Lemma 7.7 Let n be a prime number. Any Abelian subgroup of the symmetric group .Sn which acts transitively on a set consisting of n points is generated by an n-cycle. Proof Let .Sn be the group of permutations of elements of the set .{1, . . . , n}. Suppose the elements .sj ∈ Sn , .j = 1, . . . , m, commute and the subgroup .s1 , . . . , sm  of .Sn generated by the .sj , .j = 1, . . . , m, acts transitively on the set .{1, . . . , n}. Let .As1 ⊂ {1, . . . , n} be a minimal .s1 -invariant subset. Then .s1 | As1 is a cycle of length .k1 = |As1 |. (The order .|A| of a set A is the number of elements of this set.) For any integer . the set .s2 (As1 ) is minimal .s1 -invariant. Hence, two such sets are either disjoint or equal. Take the minimal union .As1 s2 of sets of the form .s2 (A1 ) for some integers ., which contains .As1 and is invariant under .s2 . Then .s2 moves the .s2 (As1 ) along a cycle of length .k2 such that .|As1 s2 | = k1 · k2 . .As1 s2 is a minimal subset of .{1, . . . , n} which is invariant for both, .s1 and .s2 . Continue in this way. We obtain a minimal set .A = As1 ...sm ⊂ {1, . . . , n} which is invariant under all .sj . By the transitivity condition .A = {1, . . . , n}. The order .|A| equals .k1 · . . . · km . Since n is prime, exactly one of the factors, .kj0 equals n, the other factors equal 1. Then for some number .j0 the element .sj0 is a cycle of length n and, if .j0 = 1, then .|As1 ...sj0 −1 | = 1. Since each .sj commutes with .sj0 , each .sj is a power of .sj0 . Indeed, consider a bijection of .{1, . . . , n} onto the set of n-th roots of unity so that .sj0 corresponds to rotation by the angle . 2π n . In 2π i

other words, put .ζ = e n . The permutation .sj0 acts on .{1, ζ, ζ 2 , . . . , ζ n−1 } by multiplication by .ζ . Consider an arbitrary .sj . Then for some integer .j , .sj (ζ ) = ζ j , i.e. .sj (ζ ) = ζ j −1 · ζ = (sj0 )j −1 (ζ ). Then for any other n-th root of unity .ζ m sj (ζ m ) =sj ((sj0 ) m−1 (ζ ))

.

=(sj0 ) m−1 (sj (ζ )) = ζ m−1 · ζ j = ζ j −1 · ζ m = (sj0 )j −1 (ζ m ) . Hence .sj = (sj0 )j −1 .



 .E

Lemma 7.8 Let .e1 , e2 be generators of a free group .F2 . Let ⊂ F2 be the finite subset .E  = {e1 , e2 , e2 e1−1 , e2 e1−2 } of primitive elements of the group .F2 . Put def

E −1 = {e−1 : e ∈ E  }. Suppose .Ψ : F2 → S3 is a homomorphism from .F2 into the symmetric group .S3 whose image is an Abelian subgroup of .S3 which acts transitively on the set of three elements. Then there are elements .e1 , e2 ∈ E  ∪ E −1 that generate .F2 such that .Ψ (e1 ) is a 3-cycle, .Ψ (e2 ) = id and the commutator .[e1 , e2 ] is conjugate to .[e1 , e2 ]. .

Proof By Lemma 7.7 the image of one of the original generators of .F2 is a 3-cycle. −q If . Ψ (e1 ) is a 3-cycle, then .Ψ (e2 e1 ) is the identity for q being either 0 or 1 or 2.

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(Recall that .Ψ (e2 ) is a power of .Ψ (e1 ) and .Ψ (e1 )3 = id.) Also, −q

−q

[e1 , e2 e1 ] = e1 e2 e1 e1−1 e1 e2−1 = [e1 , e2 ] .

.

q

−q

We may take the pair .e1 = e1 , .e2 = e2 e1 . If .Ψ (e1 ) is the identity then .Ψ (e2 ) is a 3-cycle. The pair . e1 = e2 , .e2 = e1−1 generates .F2 and the commutator . [e1 , e2 ] = e2 e1−1 e2−1 e1 is conjugate to the commutator . [e1 , e2 ] . 

Let X be a torus with a disc removed. Its fundamental group .π1 (X, q0 ) with base point .q0 is isomorphic to the free group .F2 with two generators. Theorem 7.4 Suppose X is a torus with a hole. Let .E = {e1 , e2 } be a standard system of generators of the fundamental group of X with base point .q0 , and −1 −2 −1 −1  .E = {e1 , e2 , e2 e 1 , e2 e1 , e1 e2 e1 e2 }. Let f be a separable quasipolynomial of degree 3 on X such that for each .e ∈ E  the quasipolynomial f is isotopic to an algebroid function for an orientation preserving conformal structure .we on X with a√holomorphic annulus representing .eˆ of conformal module larger than 3+ 5 −1 π . (log 2 2 ) . Then the isotopy class of f corresponds to the conjugacy class of a homomorphism Φ : π1 (X, q0 ) → Γ ⊂ B3

.

for a subgroup .Γ of .B3 which is generated either by .σ1 σ2 , or by .σ1 σ2 σ1 , or by .σ1 and .Δ23 . In particular, in all cases the image .Φ([e1 , e2 ]) of the commutator of the generators of .Γ is the identity. For the proof of the theorem we need two lemmas which we state and prove before proving the theorem. Lemma 7.9 Let .b1 = (σ1 σ2 )±1 and .b2 = w −1 σ12k w for an integer k and a braid 2k ∗ 2∗ ∗ ∗ .w ∈ B3 . If the commutator .[b1 , b2 ] is conjugate to .σ 1 Δ3 for integers .k and . then .b2 is the identity and hence also the commutator is the identity. Proof of Lemma 7.9 Let .b ∈ Bn be a pure braid with base point .En ∈ Cn (C)Sn . Fix a point .(x1 , . . . , xn ) ∈ Cn (C) that projects to .En . Label each strand of b by the number j of its initial point .xj . Let .{i, j } be an unordered pair of distinct integer numbers between 1 and n. The linking number .{i,j } of the i-th and j -th strand is defined as follows. Discard all strands except the i-th and the j -th strand. We obtain a pure braid .σ 2m . We call the integral number m the linking number of the two strands and denote it by .{i,j } . ∗ ∗ ∗ Note that the linking numbers .∗{i,j } of the braid .σ12k Δ2 3 are equal to .{12} = ∗ ∗ ∗ ∗ ∗ ∗ k + , .{23} =  , .{13} =  . Since the braid is conjugate to a commutator, the sum of exponents of generators in a word representing it must be zero. This means that 2k ∗ 2∗ ∗ ∗ .2k + 6 = 0. Hence, the (unordered) collection of linking numbers of .σ 1 Δ3

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is .{−2∗ , ∗ , ∗ }. Since conjugation only permutes linking numbers between pairs of strands of a pure braid, the unordered collection of linking numbers of pairs of strands of .[b1 , b2 ] equals .{−2∗ , ∗ , ∗ } for the integer .∗ . For the pure braid .σ12k the linking number between the first and the second strand equals k, the linking numbers of the remaining pairs of strands equal zero. Consider a not necessarily pure braid .w ∈ B3 with base point .E3 and fix a lift .(x1 , x2 , x3 ) of .E3 . Let .Sw be the permutation .Sw = τ3 (w). The permutation .Sw defines a permutation .sw acting on the set .(1, 2, 3), so that .Sw ((x1 , x2 , x3 )) = (xsw (1) , xsw (2) , xsw (3) ). Order the linking numbers of pairs of strands of a pure braid .b ∈ B3 as .({2,3} , {1,3} , {1,2} ). Notice that the complement of the set .{2, 3} (of the set .{1, 3}, or .{1, 2}, respectively) in .{1, 2, 3} is .{1} ( .{2}, or .{3}, respectively). The ordered tuple of linking numbers of pairs of strands of .w −1 b w equals Sw (({2,3} , {1,3} , {1,2} )) = ({sw (2)sw (3)} , {sw (1)sw (3)} , {sw (1)sw (2)} ).

.

Hence, the ordered tuple of linking numbers between pairs of strands of .b2 = w −1 σ12k w equals .Sw (0, 0, k). Similarly, the ordered tuple of linking numbers between pairs of strands of .b−1 2 is equal to .Sw (0, 0, −k). −1 Consider the commutator .b1 b2 b−1 1 b2 . The ordered tuple of linking numbers of pairs of strands of the pure braid −1  −1 ◦ S (0, 0, k) for the permutation .S  = τ (b ) (see Fig. 7.2). .b1 b2 b w 3 1 1 equals .(S ) Hence the ordered tuple of linking numbers of pairs of strands of the commutator −1 −1  −1 ◦ S (0, 0, k) + S (0, 0, −k). Since .S  is a 3-cycle, .b1 b2 b w w 1 ◦ b2 equals .(S )  the mapping .S (x1 , x2 , x3 ) = (xs  (1) , xs  (2) , xs  (3) ) does not fix (setwise) any pair of points among the .x1 , x2 , x3 . Hence the unordered 3-tuple of linking numbers of −1 −1 .b1 b2 b 1 b2 is .{k, −k, 0}. It can coincide with an unordered 3-tuple of the form ∗ ∗ ∗ ∗ .{−2 ,  ,  } only if .k =  = 0. Hence .b2 is the identity and the commutator is the identity. 

Lemma 7.10 The centralizer of .σ1k in .B3 , .k = 0 an integral number, equals k  2   .{σ 1 Δ3 : k ,  ∈ Z}. Proof Consider for each .b ∈ B3 the modular transformation .Tb on the Teichmüller space .T (0, 4) ∼ = C+ that is associated to a mapping .ϕb,∞ ∈ mb,∞ . Recall that Fig. 7.2 The braid −1 = (σ1 σ2 )b2 (σ1 σ2 )−1

.b1 b2 b1

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7 Gromov’s Oka Principle and Conformal Module

the mapping .B3 S3 bS3 → Tb is a bijection that satisfies equality (1.42) (see Sect. 1.7). The group of modular transformations on . C+ is isomorphic to 1 0 z .SL2 (Z) ± Id. Recall that .Tσ1 (z) = 1−z corresponds to . −1 1  ± Id (see Sect. 3.2, Example 3.8). Suppose the braid b commutes with .σ1k for some non-zero   v v integer k and corresponds to .V  ± I where .V = 11 12 . Then v21 v22  .

1 0 −k 1

     v v 1 0 v11 v12 = 11 12 v21 v22 v21 v22 −k 1

i.e.  .

Since  .k 1 ± −m proved.

v12 v11 −k v11 + v21 −k v12 + v22





v − k v12 v12 = 11 v21 − k v22 v22

 .

= 0 we have .v12 = 0 and .v11 = v22 . Since .det V = 1 we obtain .V = 0 for an integer m. Thus .b = σ1m for some integer m. The lemma is 1 

Proof of Theorem 7.4 Let .Φ : π1 (X, x0 ) → B3 be a homomorphism whose conjugacy class corresponds to the isotopy class of f . Denote by .Ψ = τ3 ◦ Φ : π1 (X, x0 ) → S3 the related homomorphism into the symmetric group. Since the quasipolynomial F is isotopic to an algebroid function for the conformal √ structure .ω[e1 ,e2 ] with an annulus of conformal module larger than . π2 (log 3+2 5 )−1 representing the commutator .[e1 , e2 ], by Lemma 6.5 the conformal module of the commutator .[b1 , b2 ] equals infinity. Hence, by Lemma 6.5 . Ψ ([e1 , e2 ]) = id . Consider first the case when the subgroup .Ψ (π1 (X, x0 )) of .S3 acts transitively on the set of three points. Apply Lemma 7.8 to the free group .π1 (X, x0 ) with generators  −1 .e1 and .e2 and to the homomorphism .Ψ . We obtain new generators .e1 , e2 ∈ E ∪E  −1 of .π1 (X, x0 ) (with .E and .E being the sets of the lemma) such that .Ψ (e1 ) is a 3cycle and .Ψ (e2 ) = id. Put .s1 = Ψ (e1 ), .s2 = Ψ (e2 ), .b1 = Φ(e1 ), .b2 = Φ(e2 ). Then .b2 and .[b1 , b2 ] are pure braids (the latter holds since .[e1 , e2 ] is conjugate to .[e1 , e2 ]). Since f is isotopic to an algebroid function for the conformal structure .we1 , Lemma 6.5 implies .M(b1 ) = ∞. Since .τ3 (b1 ) is a 3-cycle, by the same lemma the braid .b1 must be conjugate to an integral power of .σ1 σ2 . After conjugating .Φ we may assume that .b1 = (σ1 σ2 )3±1 = (σ1 σ2 )±1 · Δ2 3 for an integer .. By Lemma 6.5 we also have .M(b2 ) = ∞ for the pure braid .b2 . Hence, −1 σ 2k Δ2 w for integers k and . and a conjugating braid .w ∈ B . Since .b2 = w 3 3 1 2k ∗ Δ2∗ for integers .k ∗ .M([b1 , b2 ]) = ∞ the commutator .[b1 , b2 ] is conjugate to .σ 1 3 and .∗ .

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Take into account that .Δ2 3 commutes with each 3-braid and apply Lemma 7.9 to ∗ ±1 .b1 = (σ1 σ2 ) and .b2 = w −1 σ12k w. This gives the statement of Theorem 7.4 for the case when the subgroup .Ψ (π1 (X, x0 )) acts transitively on the set of three points. Consider now the case when the subgroup .Ψ (π1 (X, x0 )) of .S3 does not act transitively on the set of three points. Then .Ψ (π1 (X, x0 )) is generated either by a transposition (we may assume the transposition to be .(12) by choosing the mapping .Φ in its conjugacy class) or is equal to the identity in .S3 . There are generators .e1 , e2 ∈ E ∪ E −1 such that .[e1 , e2 ] is conjugate to .[e1 , e2 ] and .Ψ (e2 ) = id. Indeed, this is clear if .Ψ (π1 (X, x0 )) is the identity. Suppose .Ψ (π1 (X; x0 )) is generated by the transposition .(12) and neither .Ψ (e1 ) nor .Ψ (e2 ) is the identity. Then .Ψ (e1 ) = Ψ (e2 ) = (12). Hence, .Ψ (e2 e1−1 ) = id, and for the def

def

generators .e1 = e1 and .e2 = e2 e1−1 the claimed statements hold. Use the notation .b1 = Φ(e1 ), .b2 = Φ(e2 ). By Lemma 6.5 we may assume, that k1 21 k1 k1 .b1 is conjugate to either .σ 1 Δ3 or to .(σ1 σ2 σ1 ) = Δ3 for an odd integer .k1 and 2 an integer .1 , and .b2 is conjugate to .σ12k2 Δ2 3 for integers .k2 and .2 . Conjugating 21 k1 2 .Φ, we may assume that .b1 = B1 Δ 3 with either .B1 = σ1 or .B1 = σ2 σ1 = 2k2 2 σ1−1 (σ1 σ2 σ1 )σ1 . Respectively, .b2 is conjugate to .B2 Δ2 3 with .B2 = σ1 . Since .[b1 , b2 ] is a pure braid with infinite conformal module, Lemma 6.5 implies that 2k ∗ 2∗ .[b1 , b2 ] is conjugate to .σ 1 Δ3 . Since the commutator has degree zero the equality ∗ ∗ .2k + 6 = 0 holds and the unordered tuple of linking numbers of .[b1 , b2 ] equals ∗ ∗ ∗ .{ ,  , −2 }. This implies that the commutator .[b1 , b2 ] and the braid .B2 are equal to the 1 identity. Indeed, suppose .b2 = w −1 σ12k2 w Δ2 for a braid .w ∈ B3 . The ordered 3 tuple of linking numbers of pairs of strands of .B2 = w −1 σ12k2 w equals .Sw (0, 0, k2 ) and the respective ordered triple for .B2−1 is .Sw (0, 0, −k2 ). Here .Sw = τ3 (w). The ordered tuple of linking numbers of pairs of strands of .B1 B2 B1−1 equals  −1 ◦ S (0, 0, k ), where .S  = τ (B ). Since .Δ2 is in the center of .B , the .(S ) w 2 3 1 3 3 unordered tuple of linking numbers of pairs of strands of .[b1 , b2 ] is equal to that of .[B1 , B2 ], i.e. it equals either .{k2 , −k2 , 0} or .{0, 0, 0} (in dependence on .Sw ). Either of these unordered tuples can be equal to .{∗ , ∗ , −2∗ } for some integer .∗ only if .∗ = 0. We obtained that .[b1 , b2 ] = id also in this case. By Lemma 7.10 the group .Φ(π1 (X, x0 )) is generated either by (a power of) .σ1 and .Δ23 , or by (a power of) .σ2 σ12 and .Δ23 . The theorem is proved. 

We do not know the answer to the following problem. Problem 7.5 Can two braids in .B3 of zero entropy have a non-trivial commutator of zero entropy? If the answer is negative, then Theorem 7.4 holds with .E  = E. The proof of Theorem 7.4 gives the following corollary which is weaker than a negative answer to Problem 7.5. Corollary 7.1 Let .b1 and .b2 be braids in .B3 with .h(b1 ) = h(b2 ) = h([b1 , b2 ]) = 0. If one of the braids is pure or

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Fig. 7.3 Two 4-braids of vanishing entropy with non-trivial commutator

h(b2 b1−1 ) = h(b2 b1−2 ) = 0

.

then .[b1 , b2 ] = id. On the other hand D.Calegari and A.Walker suggested the following example of two elements of the braid group .B3 with non-trivial commutator of zero entropy. Example 7.1 The commutator of the non-commuting braids .b1 = (σ2 )−1 σ1 and −1 has entropy zero. .b2 = σ2 (σ1 ) It is enough to show that the commutator of the two braids equals .[b1 , b2 ] = (σ2 )−6 Δ23 . Since b1−1 b2−1 = (σ1 )−1 σ2 σ1 (σ2 )−1

.

= (σ1 )−1 σ2 σ1 σ2 (σ2 )−2

= (σ1 )−1 σ1 σ2 σ1 (σ2 )−2 = σ2 σ1 (σ2 )−2 , b1 b2 = (σ2 )−1 σ1 σ2 (σ1 )−1

= (σ2 )−2 σ2 σ1 σ2 (σ1 )−1

= (σ2 )−2 σ1 σ2 σ1 (σ1 )−1 = (σ2 )−2 σ1 σ2 , we obtain [b1 , b2 ] = b1 b2 b1−1 b2−1 = (σ2 )−3 σ2 σ1 σ2 · σ2 σ1 σ2 (σ2 )−3

.

= (σ2 )−3 Δ23 (σ2 )−3 = (σ2 )−6 Δ23 . It is clear now that the commutator .[b1 , b2 ] has entropy zero. Problem 7.5 has a positive answer for braid groups on more than 3 strands. Consider the braids .b1 = σ1−2 and .b2 = (σ2 )−1 (σ1 )−1 (σ3 )−1 (σ2 )−1 in .B4 of zero entropy. Their commutator equals .[b1 , b2 ] = σ1−2 · σ32 = id and is a non-trivial braid of zero entropy (see Fig. 7.3). Notice the following fact. Let X be a torus with a disc removed. Consider the separable quasipolynomial of degree four on X whose isotopy class corresponds to the conjugacy class of the homomorphism .Φ : π1 (X, x0 ) → B4 with .Φ(e1 ) = b1

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and .Φ(e2 ) = b2 for the just defined 4-braids .b2 and .b2 . By a similar argument as in the proof of Theorem 7.3 the quasipolynomial is isotopic to an algebroid function for each conformal structure of second kind on X. Problem 7.6 Consider separable quasipolynomials of degree .n ≥ 4 on a torus with a hole. Are the obstructions for them to be isotopic to holomorphic quasipolynomials discrete like in the case .n = 3? Is there an analogue of Corollary 7.1 for braid groups on more than three strands?

Chapter 8

Gromov’s Oka Principle for (g, m)-Fiber Bundles .

In this chapter we prove theorems related to the failure and the restricted validity of the Gromov-Oka Principle for bundles whose fibers are Riemann surfaces of type .(1, 1) (in other words, the fibers are once punctured tori). The problem can be reduced to the respective problem for .(0, 4)-bundles (and, hence, for mappings into .P3 ) by considering double branched coverings.

8.1 (g, m)-Fiber Bundles. Statement of the Problem We will consider bundles whose fibers are connected closed oriented surfaces of genus .g ≥ 0 with .m ≥ 0 distinguished points. Recall that a connected smooth closed oriented surface of genus .g with .m distinguished points is called a surface of type .(g, m). In case the surface is equipped with a complex structure we call it a Riemann surface of type .(g, m). Definition 8.1 (Smooth Oriented .(g, m) Fiber Bundles) Let X be a smooth oriented manifold of dimension k, let .X be a smooth (oriented) manifold of dimension .k + 2 and .P : X → X an orientation preserving smooth proper submersion such that for each point .x ∈ X the fiber .P −1 (x) is a smooth closed oriented surface of genus .g. Let .E be a smooth submanifold of .X that intersects each fiber .P −1 (x) along a set .Ex of .m distinguished points. Then the tuple .Fg,m = (X , P, E, X) is called a smooth (oriented) fiber bundle over X with fibers being smooth closed oriented surfaces of genus .g with .m distinguished points (for short, a smooth oriented .(g, m)-bundle). If .m = 0 the set .E is the empty set and we will often denote the bundle by (X , P, X). If .m > 0 the mapping .x → Ex locally defines .m smooth sections of the .(g, 0)-bundle .(X , P, X) that is obtained by forgetting all distinguished points. A section of the bundle .(X , P, X) is a continuous mapping .s : X → X with .P ◦ s

.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025 B. Jöricke, Braids, Conformal Module, Entropy, and Gromov’s Oka Principle, Lecture Notes in Mathematics 2360, https://doi.org/10.1007/978-3-031-67288-0_8

235

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

being the identity on X. .(g, 0)-bundles will also be called genus .g fiber bundles. For g = 1 and .m = 0 the bundle is also called an elliptic fiber bundle. We will consider mostly the case when .2g − 2 + m > 0. Let S be a smooth reference surface of genus .g and .E ⊂ S a set of .m distinguished points. For an open subset U of Xwe consider the trivial bundle (also called product bundle) . U × S, pr1 , U × E, U with set .{x} × E of distinguished points in the fiber .{x} × S over x. Here .pr1 : U × S → U is the projection onto the first factor. By Ehresmann’s Fibration Theorem each smooth .(g, m)-bundle

.

def

Fg,m = (X , P, E, X) with set of distinguished points .Ex = E ∩ P −1 (x) in the fiber over x is locally smoothly trivial, i.e. each point in X has a neighbourhood −1 (U ) → U × S the .U ⊂ X such that for a surjective diffeomorphism .ϕU : P diagram .

.

is commutative and .ϕU maps .E ∩ P −1 (U ) onto .U × E, equivalently, for all .x ∈ U the diffeomorphism .ϕU maps the set of distinguished points .Ex = E ∩ P −1 (x) in the fiber .P −1 (x) to the set of distinguished points .{x} × E in the fiber .{x} × S. The idea of the proof of Ehresmann’s Theorem is the following. Choose smooth coordinates on U by a mapping from a rectangular box in .Rn to U . Consider smooth vector fields .vj on U , which form a basis of the tangent space of U at each point of U . Take smooth vector fields .Vj on .P −1 (U ) that are tangent to .E at points of this set and are mapped to .vj by the differential of .P. Such vector fields can easily be obtained locally. To obtain the globally defined vector fields .Vj on .P −1 (U ) one uses partitions of unity. The required diffeomorphism .ϕU is obtained by composing the flows of the vector fields .Vj (in any fixed order). In this way a trivialization of the bundle can be obtained over any simply connected smooth manifold. In the case when the base manifold is a Riemann surface, a holomorphic .(g,.m) fiber bundle over X is defined as follows. Definition 8.2 Let X be a Riemann surface, let .X be a complex surface, and .P a holomorphic proper submersion from .X onto X, such that each fiber .P −1 (x) is a closed Riemann surface of genus .g. Suppose .E is a complex one-dimensional submanifold of .X that intersects each fiber .P −1 (x) along a set .Ex of .m distinguished points. Then the tuple .Fg,m = (X , P, E, X) is called a holomorphic .(g,.m) fiber bundle over X. Notice that the mapping .x → Ex locally defines .m holomorphic sections of the (g, 0)-bundle .(X , P, X) that is obtained by forgetting all distinguished points. We will call two smooth oriented (holomorphic, respectively) .(g, m) fiber bundles, .F0 = (X 0 , P 0 , E 0 , X0 ) and .F1 = (X 1 , P 1 , E 1 , X1 ), smoothly isomor-

.

8.1 (g, m)-Fiber Bundles. Statement of the Problem

237

phic (holomorphically isomorphic, respectively) if there are smooth (holomorphic, respectively) homeomorphisms .Φ : X 0 → X 1 and .φ : X0 → X1 such that for each .x ∈ X0 the mapping .Φ takes the fiber .(P 0 )−1 (x) onto the fiber .(P 1 )−1 (φ(x)) and the set of distinguished points in .(P 0 )−1 (x) to the set of distinguished points in 1 −1 (φ(x)). Holomorphic bundles that are holomorphically isomorphic will be .(P ) considered the same holomorphic bundles. For a smooth .(g, m)-bundle .F = (X , P, E, X) over X and a homeomorphism .ω : X → ω(X) onto a surface .ω(X) we denote by .Fω the bundle Fω = (X , ω ◦ P, E, ω(X)).

.

(8.1)

The bundles .F and .Fω are isomorphic. Indeed, we may consider the homeomorphisms .ω : X → ω(X) and .Id : X → X . We are interested in smooth deformations of a smooth bundle to a holomorphic one. Two smooth (oriented) .(g, m) fiber bundles over the same oriented smooth base manifold X, .F0 = (X 0 , P 0 , E 0 , X), and .F1 = (X 1 , P 1 , E 1 , X), are called (free) isotopic if for an open interval I containing .[0, 1] there is a smooth .(g, m) fiber bundle .(Y, P, E, I × X) over the base .I × X (called an isotopy) with the following t −1 property. For each .t ∈ [0, 1]we put .Y t = P −1 ({t}×X)  and .E = E∩P 1({t}×X). 0 0 0 0 The bundle .F is equal to . Y , P | Y , E , {0} × X , and the bundle . F is equal   to . Y 1 , P | Y 1 , E 1 , {1} × X .   Notice that for each .t ∈ I the tuple . Y t , P | Y t , E t , {t} × X is automatically a smooth .(g, m)-fiber bundle. Lemma 8.1 Isotopic bundles are isomorphic. Proof Let .I ⊃ [0, 1] be an open interval, and let .(Y, P, E, I × X) be a smooth (g, m)-bundle over .I × X such that the restrictions to .{0} × X and to .{1} × X are equal to given smooth .(g, m)-bundles over X. Here X is a smooth finite surface. Consider the vector field v on .I × X that equals the unit vector in positive direction of I at each point of .I × X, and let V be a smooth vector field on .Y that projects to v under .P and is tangent to .E at points of this set. For each .t ∈ I we let .ϕt be the time t map of the flow of V , more precisely,

.

.

∂ ϕt (y) = V (ϕt (y)), ϕ0 (y) = y, ∂t

t ∈ I, y ∈ P −1 ({0} × X) .

Then .ϕ1 is a diffeomorphism from .P −1 ({0} × X) onto .P −1 ({1} × X), that maps the fiber over .(0, x) onto the fiber over .(1, x) and maps distinguished points to distinguished points.

In Sect. 8.2 we will prove that vice versa, isomorphic bundles are isotopic. The following problem on isotopies of smooth objects to the respective holomorphic objects concerns another version of the restricted validity of Gromov’s Oka Principle.

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

Problem 8.1 Let X be a finite open Riemann surface. Can a given smooth .(g,m)fiber bundle over X be smoothly deformed to a holomorphic fiber bundle over X? More precisely„ does there exist an isotopy .(Y, P, E, X ×I ) over the base .X ×I with .[0, 1] ⊂ I , for which the restriction to .{0} × X is equal to the given bundle, and the restriction to .{1} × X can be equipped with the structure of a holomorphic bundle? The answer is in general negative. There is a similar notion that describes obstructions for the existence of such isotopies for .(g,m)-bundles as for the existence of isotopies of smooth separable quasipolynomials to holomorphic ones. It is called the conformal module of isotopy classes of .(g,m)-fiber bundles over the circle, and is defined as follows. Consider a smooth oriented .(g, m)-fiber bundle .F = F(g,m) = (X , P, E, ∂D) over the circle. Denote by . F the isotopy class of fiber bundles over .∂D that contains .F. Let .Ar,R = {z ∈ C : r < |z| < R}, .r < 1 < R, be an annulus containing the unit circle. A smooth .(g, m)-fiber bundle on .Ar,R is said to represent . F if its restriction to the unit circle .∂D is an element of . F. Definition 8.3 (The Conformal Module of Isotopy Classes of .(g, m)-Fiber Bundles) Let . F = Fg,m be the isotopy class of an oriented .(g, m)-fiber bundle over the circle .∂D. Its conformal module is defined as M( F) = sup{m(Ar,R ) : there exists a holomorphic fiber bundle

.

F}. on Ar,R that represents 

(8.2)

8.2 The Monodromy of (g, m)-Bundles. Isotopy and Isomorphism of Bundles Monodromy and Maping Torus Consider a smooth .(g, m)-bundle .Fg,m = (X , P, E, ∂D) over the unit circle .∂D. Denote as before the set of distinguished points .E ∩ P −1 (x) in the fiber over x by .Ex . Let v be the unit tangent vector field to .∂D. The argument used for the proof of Ehresmann’s Theorem provides a smooth vector field V on .X which is tangent to .E at points of this set and projects to v, p i.e. .(d P)(V ) = v. Cover .∂D by its universal covering .R → ∂D, using the mapping 2π it , P,  E,  R) .p(t) = e , .t ∈ R. Lift the bundle .Fg,m over .∂D to a bundle . Fg,m = (X over .R. −1 (t) with set of distinguished points E˜ t equals P −1 (e2π it ) with set The fiber P −1 (t +2π k) of distinguished points Ee2π it , hence for k ∈ Z and each t ∈ R the sets P −1  ˜ ˜ and P (t), are equal, and Et+2π k is equal to Et . Lift the vector field V to a vector  with the following property. The equality V  on X (z1 ) = V (z2 ) holds, if field V p˜

 are mapped to the same point z ∈ X under the projection X  −−→ X . z1 , z 2 ∈ X

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, we let   be the solution of the differential equation For t ∈ R, ζ ∈ X ϕt (ζ ) ∈ X .

∂ ( ϕt (ζ )),  ϕ0 (ζ ) = ζ.  ϕt (ζ ) = V ∂t

(8.3)

def −1 (0) and E def Put S = P −1 (1) ∼ = E ∩ P −1 (1) ⊂ S. Let ζ ∈ S. Then = P −1   defines a ϕt (ζ ), ζ ∈ S, of the vector field V  ϕt (ζ ) ∈ P (t). The time t map  −1 −1 ∼   homeomorphism from the fiber S = P (0) onto the fiber P (t), that maps the −1 (t). t ⊂ P set of distinguished points E ⊂ S to the set of distinguished points E −1  In the same way  ϕt defines a homeomorphism from the fiber P (t1 ) onto the fiber −1 (t1 + t) that maps the distinguished points E˜ t1 to the distinguished points E˜ t1 +t . P The mappings  ϕt form a group:

, t1 , t2 ∈ R.  ϕt1 +t2 (ζ ) =  ϕt1 ( ϕt2 (ζ )), ζ ∈ X

.

(8.4)

  Let R × S, pr1 , R × E, R be the trivial bundle. Here pr1 : R × S → R is the projection onto the first factor. The mapping Φ,  R × S (t, ζ ) → Φ(t, ζ ) =  ϕt (ζ ) ∈ X def

.

(8.5)

  provides an isomorphism from the trivial bundle R × S, pr1 , R × E, R to the lifted bundle  Fg,m . Define by ϕt the projection p˜

−1 (t)  X˜ ⊃ P ϕt (ζ ) −−→ ϕt (ζ ) = p( ˜ ϕt (ζ )) ∈ P −1 (e2π it ) ∈ X .

.

(8.6)

−1 (t) = P −1 (e2π it ) and E˜ t = Ee2π it , we obtain a smooth family of Since P homeomorphisms ϕt : P −1 (1) → P −1 (e2π it ) that map distinguished points to distinguished points. We call the family ϕt , t ∈ R, a trivializing family of homeomorphisms. Consider the time-1 map ϕ1 : P −1 (1) → P −1 (e2π i ) = P −1 (1)

.

(8.7)

which is a self-homeomorphism of the fiber over 1 that maps the set of distinguished points E ⊂ S ∼ = P −1 (1) to itself. For k ∈ Z we have ϕt+k = ϕt ◦ ϕ1k . For each n ∈ Z the projection (8.6) maps the points Φ(t, ζ ) =  ϕt (ζ ) and Φ(t +n, ϕ1−n (ζ )) =  ϕt+n (ϕ1−n (ζ )) to the same point ϕt (ζ ) ∈ P −1 (e2π it ). The group Z of integer  by numbers acts on Φ(R × S) = X Φ(R × S) Φ(t, ζ ) → Φ(t + n, ϕ1−n (ζ )) ,

.

n ∈ Z.

(8.8)

The quotient   def X1 = R × S (t, ζ ) ∼ (t + 1, ϕ1−1 (ζ ))

.

(8.9)

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

is a smooth manifold that is diffeomorphic to X . Indeed, the mapping p˜ ◦ Φ : R × S → X , p˜ ◦ Φ(t, ζ ) = ϕt (ζ ) ∈ P −1 (e2π it ), satisfies the condition p˜ ◦ Φ t + 1, ϕ1−1 (ζ ) = p˜ ◦ Φ(t, ζ ). Hence, this mapping descends to a mapping from X1 to X , that is a local diffeomorphism and is bijective by the construction. Hence, it is a diffeomorphism from X1 onto X . Put   def E 1 = R × E (t, ζ ) ∼ (t + 1, ϕ1−1 (ζ )) ,

.

(8.10)

and recall that p(Φ(t, ˜ E)) = p( ˜ ϕt (E)) = E ∩ P −1 (e2π it ), since  ϕt maps the distinguished points E ⊂ S = P −1 (1) to the distinguished points E ∩ P −1 (e2π it ) −1 (t) = P −1 (e2π it ). Define the projection P1 : X1 → ∂D, so in the fiber P that P1 takes the value e2π it on the class containing (t, ζ ). We obtain a bundle (X1 , P1 , E 1 , ∂D) that is smoothly isomorphic to (X , P, E, ∂D). We will also say for short that Fg,m is smoothly isomorphic to the mapping torus .

  ([0, 1] × S)  (0, ζ ) ∼ (1, ϕ(ζ ))

(8.11)

where ϕ = ϕ1−1 is a smooth orientation preserving self-homeomorphism of the fiber S = P −1 (1) with set of distinguished points E ⊂ S. The mapping ϕ depends on the trivializing vector field. However, the mapping class of ϕ is independent on this vector field, it is merely determined by the bundle. We denote it by mF (with F = Fg,m ). The mapping class mF ∈ M(S; ∅, E) is called the monodromy mapping class of the bundle F over the circle, or the monodromy, for short. For diffeomorphic closed surfaces S1 and S2 and sets of distinguished points E1 and E2 of the same cardinality the groups M(S1 ; ∅, E1 ) and M(S2 ; ∅, E2 ) are isomorphic. Any diffeomorphism ϕ : S1 → S2 with ϕ(E1 ) = E2 induces an isomorphism Isϕ : M(S1 ; ∅, E1 ) → M(S2 ; ∅, E2 ). Two such isomorphisms M(S1 ; ∅, E1 ) → M(S2 ; ∅, E2 ) differ by conjugation with an element of M(S2 ; ∅, E2 ). Hence, there is a canonical one-to-one correspondence between the sets of conjugacy classes M(S1 ; ∅, E1 ) and M(S2 ; ∅, E2 ), and we will identify conjugacy classes according to this correspondence. Similarly, let h : Fk → M(S1 ; ∅, E1 ) be a homomorphism from a free group Fk of rank k. Any diffeomorphism ϕ : S1 → S2 with ϕ(E1 ) = E2 induces another homomorphism hϕ : Fk → M(S2 ; ∅, E2 ). Replacing ϕ by another diffeomorphism we arrive at a conjugate homomorphism. We obtain a canonical bijection between conjugacy classes of homomorphisms Fk → M(Sj ; ∅, Ej ), j = 1, 2. 



Problem 8.2 Prove that for each isotopy class  Fg,m of (g, m)-bundles over the Fg,m of elements of the mapping class circle and the associated conjugacy class m of the fiber over the point 1 the equality 1 π M( Fg,m ) = 2 h( mFg,m )

.

holds.

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In the sequel we will use the mapping class group M(S; ∅, E) on a reference (Riemann) surface of type (g, m) and the set of its conjugacy classes M(S; ∅, E), or the set of conjugacy classes of mappings from a free group to M(S; ∅, E), respectively. Recall that the group M(S; ∅, E) is isomorphic to the modular group Mod(g, m). The following theorem holds. (See, e.g. [24] for the case of closed fibers of genus g ≥ 2 and paracompact Hausdorff spaces X.)



Theorem 8.1 Let X be a connected smooth finite open oriented surface with base point q0 . The set of isomorphism classes of smooth oriented (g, m)-bundles on X is in one-to-one correspondence to the set of conjugacy classes of homomorphisms from the fundamental group π1 (X, q0 ) into the modular group Mod(g, m). Proof Let first F0 = (X0 , P0 , E 0 , ∂D) and F1 = (X1 , P1 , E 1 , ∂D) be isomorphic bundles over the circle ∂D. Suppose the isomorphism is given by a diffeomorphism φ : ∂D⊃ and a diffeomorphism Φ : X0 → X1 . We may assume that φ preserves the base point 1, hence, Φ maps each fiber P0−1 (e2π it ) of the first bundle onto  . the fiber P1−1 (e2π iφ1 (t) ) of the second bundle for a homeomorphism φ1 : [0, 1]⊃ Let ϕt : S → P0−1 (e2π it ) be a trivialising family of homeomorphisms for the def

bundle F0 . Then ψt = Φ|P0−1 (e2π it ) ◦ ϕt ◦ (Φ|P0−1 (1))−1 is a trivialising family of homeomorphisms of the second bundle. We obtain ψ1 = Φ|P0−1 (1) ◦ ϕ1 ◦ (Φ|P0−1 (1))−1 . This means that ψ1 and φ1 represent the same element of



M(P0−1 (1)). We saw that the monodromies of isomorphic (g, m)-bundles over the circle belong to the same conjugacy class Mod(g, m). Vice versa, suppose the monodromies of two (g, m)-bundles F0 and F1 over the circle with base point 1 represent the same element of Mod(g, m). We will prove that the bundles are isomorphic. The assumption means that the monodromies mj of Fj , j = 0, 1 are represented by mappings ϕj ∈ mj , such that ϕ1 = ϕ ◦ ϕ0 ◦ ϕ −1 for a homeomorphism ϕ : P0−1 (1) → P1−1 (1) that maps the set of distinguished points of P0−1 (1) onto the set of distinguished points of P1−1 (1). The bundle F0 is isomorphic to the mapping torus of ϕ0 . This   mapping torus has total space X0 = R × S  (t, ζ ) ∼ (t + 1, ϕ (ζ )) and set of distinguished points 0  E0 = R × E  (t, ζ ) ∼ (t + 1, ϕ0 (ζ )) . The mapping torus F0 with total space 



    X0 = R × ϕ(S)  (t, ζ ) ∼ (t + 1, ϕ ◦ ϕ0 ◦ ϕ −1 (ζ ))

.

and set of distinguished points     E0 = R × ϕ(E)  (t, ζ ) ∼ (t + 1, ϕ ◦ ϕ0 ◦ ϕ −1 (ζ ))

.

is isomorphic to F0 and has monodromy represented by ϕ ◦ ϕ0 ◦ ϕ −1 . Indeed, the mapping R × S (t, ζ ) → (t, ϕ(ζ )) ∈ R × S descends to a diffeomorphism X0 → X0 that takes fibers to fibers.

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

The mapping torus F0 with total space X0 is isomorphic to F1 . Indeed, the monodromy mappings of F0 and F1 coincide, and the mapping tori of isotopic mappings are isotopic, and, hence, isomorphic. Therefore, the mapping torus F0 is isomorphic to the mapping torus of F1 . Hence, F0 and F1 are isomorphic. Let now X be a connected smooth oriented open surface of genus g with m holes with base point q0 . Suppose F = (X , P, E, X) is a smooth (g, m)-bundle over X. Denote the fiber P −1 (q0 ) over q0 by S and the set of distinguished points E ∩ P −1 (q0 ) by E. We assign to F a homomorphism π1 (X, q0 ) → M(S, ∅, E) as follows. Take smooth closed curves in X parameterized by γj : [0, 1] → X that represent the elements ej of a standard system of generators of the fundamental group π1 (X, q0 ), j = 1, . . . , 2g +m−1. Associate to each generator ej of π1 (X, q0 ) the monodromy mapping class of the restricted bundle F|γj . This is a mapping class on the fiber S = P −1 (q0 ) over q0 with distinguished points E = E ∩ P −1 (q0 ), that depends only on the ej and on the bundle. We obtain a well-defined mapping that associates to each generator of the fundamental group π1 (X, q0 ) a mapping class. This mapping extends to a homomorphism from the fundamental group π1 (X, q0 ) to the mapping class group M(S, ∅, E). In the same way as in the case of bundles over the circle we may prove that the respective homomorphisms corresponding to isomorphic bundles represent the same conjugacy class of homomorphisms from π1 (X, q0 ) to Mod(g, m). Vice versa, take a homomorphism h from the fundamental group π1 (X, q0 ) to the modular group Mod(g, m). A bundle over X whose monodromy homomorphism coincides with h can be obtained as follows. By Lemma 1.1 the surface X is diffeomorphic to a standard neighbourhood of a standard bouquet B of circles for X. Hence, it can be written as union X = D ∪ V of an open disc D and half-open bands Vj attached to D. The disc D contains j j the base point q0 . The boundary ∂D of D in X is smooth. The set B \ D is the union of disjoint closed arcs sj with endpoints on ∂D. Each circle cj of the bouquet contains exactly one of the sj . For each j the set Vj is a neighbourhood in X \ D of sj . P

Consider the universal covering X˜ −−→ X. Take a point q˜0 ∈ X˜ with P(q˜0 ) = q0 . For each j we take the lifts  cj of cj to X˜ with initial point q˜0 , and the ˜ that contains q˜0 , and let   lift Vj of Vj that intersects  cj . Let D0 be the lift of D to X, j , j = 1, . . . , 2g + m − 1, be the lift of D, that contains the endpoint x˜j def =  cj (1) D of  cj . Let U˜ be the domain in X˜ that is the union of D˜ j , j = 0, 1, . . . , 2g + m − 1, and V˜j , j = 1, . . . , 2g + m − 1. Choose for each j ≥ 1 a mapping ϕj in the mapping class h(ej ), and define a bundle over X by taking the trivial bundle over U˜ and making the following j identifications. For each x  ∈ D and each j we glue the fiber over the point x˜j ∈ D       for which p(x˜j ) = x to the fiber over the point x˜0 ∈ D0 for which p(x˜0 ) = x using the mapping ϕj . We obtained a bundle with the given monodromy homomorphism. As in the case of bundles over the circle one can see that each bundle is isomorphic to a bundle that is constructed in the just described way, and bundles constructed in this way with equal conjugacy class of monodromy homomorphisms are isomorphic. We obtained a bijective correspondence of isomorphism classes

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of smooth (g, m)-bundles over a connected finite open smooth oriented surface X of genus g with m > 0 holes and conjugacy classes of homomorphisms from the fundamental group of X to Mod(g, m).

Isotopy Classes and Isomorphism Classes of Bundles We will consider now the relation between isotopy classes and isomorphism classes of (g, m)-bundles, and their relation to the monodromies of the bundles. Lemma 8.1 says that isotopic smooth (g, m)-bundles over the circle are smoothly isomorphic. The following lemma states in particular, that, vice versa, isomorphic bundles are isotopic. Lemma 8.2 Two smooth (g, m)-bundles over a connected smooth finite open oriented surface X are isomorphic if and only of they are isotopic. Proof By Lemma 8.1 and Theorem 8.1 it remains to prove that two smooth (g, m)bundles over X are isotopic if their monodromy homomorphisms represent the same conjugacy class of homomorphisms. We first consider two smooth (g, m)-bundles F0 and F1 over the circle with fiber and distinguished points over the base point 1 being equal. Suppose the bundles (g, m)-bundles F0 and F1 have the same monodromy. We will prove that they are isotopic by an isotopy that fixes the fiber over the base point and the set of distinguished points in this fiber (by a based isotopy, for short). Denote the fiber over 1 by S and the set of distinguished points in S by E. Lift each bundle Fj , j = 0, 1, to a bundle over the vertical line {j } × R ⊂ R2 by the covering {j } × R (j, t) → e2π it . Let ε be a small positive number. Restrict the lift of the bundle Fj to the open interval {j }×(−ε, 1+ε) ⊃ {j }×[0, 1], ε > 0, and extend for each j the restricted bundle to a bundle over the rectangle (j −ε, j +ε)×(−ε, 1+ε), so that the restriction of the extended bundle Fj to each horizontal segment is a product bundle. Consider the product bundle with fiber S and set of distinguished points E over the horizontal line segments (−ε, 1 + ε) × {k}, k = 0, 1. Take an extension of the product bundle over each of the horizontal line segments (−ε, 1 + ε) × {k} to a smooth (g, m)-bundle over the open rectangle (−ε, 1 + ε) × (k − ε, k + ε) around the segment, so that for y ∈ (−ε, ε) and x ∈ (−ε, 1 + ε) the fibers and the distinguished points over (x, y) and (x, y + 1) are the same, and the bundles over the four rectangles (−ε, 1 + ε) × (k − ε, k + ε), k = 0, 1 and (j − ε, j + ˜ ε) × (−ε,  1 + ε), j = 0, 1, match to a smooth bundle F0 over the neighbourhood def

V = (−ε, 1 + ε) × (−ε, 1 + ε) \ (ε, 1 − ε) × (ε, 1 − ε) of the boundary of the square [0, 1] × [0, 1]. Let (0, 0) be the base point of V . The monodromy of the obtained bundle along the generator of the fundamental group π1 (V , 0) of V is the identity. Hence, the bundle is smoothly isomorphic to the trivial bundle. Let ϕ be a diffeomorphism of the total space of the bundle F˜ 0 to the total space of the trivial bundle over V , that maps the fiber of the bundle F˜ 0 over each point in V to the fiber of the trivial bundle over this point, and maps distinguished points to distinguished points. The trivial bundle over V extends to the trivial bundle over the neighbourhood (−ε, 1 + ε) × (−ε, 1 + ε) of the square. Glue the trivial bundle over

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

the square ( 2ε , 1 − 2ε ) × ( 2ε , 1 − 2ε ), that is relatively compact in the open unit square,   to the bundle F˜ 0 over V along V ∩ ( 2ε , 1− 2ε )×( 2ε , 1− 2ε ) using the mapping ϕ. We obtain a smooth bundle over the neighbourhood (−ε, 1 + ε) × (−ε, 1 + ε) of the unit square, that coincides with F˜ 0 over V . For (x, y) ∈ (−ε, 1 + ε) × (−ε, ε) we glue the fiber over (x, y) to the fiber over (x, y + 1) using the identity mapping. We get a bundle over the product of the interval (−ε, 1 + ε) with the circle, which provides an isotopy with fixed fiber over the base point and fixed set of distinguished points (a based isotopy for short). If two bundles over the circle with the same fiber over the point 1 and the same set of distinguished points over the base point have conjugate monodromy, we consider instead of the product bundle on each of the two horizontal line segments (−ε, 1 + ε) × {k}, k = 0, 1, the bundle obtained from the product bundle by the mapping Φ, Φ(t, ζ ) = ϕt (ζ ), t ∈ (−ε, 1 + ε), ζ ∈ S, between total spaces, where ϕt is a smooth family of diffeomorphisms with ϕ0 = Id and ϕ1 representing the conjugating mapping class. Along the same lines as above one can prove that the bundles are free isotopic. If two bundles over the circle have different fibers S1 and S2 over the point 1 with sets of distinguished points E1 and E2 , respectively, and (after an isomorphism) their monodromy maps are conjugate, then the bundles are free isotopic. Indeed, the condition means that the monodromies of the two bundles differ by “conjugation” by a homeomorphism ψ : S1 → S2 that maps E1 onto E2 . Instead of the product bundle on the horizontal segments (−ε, 1+ε)×{k}, k = 0, 1, we use in this case the bundle obtained from the product bundle by the mapping Φ  , Φ  (t, ζ ) = ϕt (ζ ), t ∈ (−ε, 1 + ε), ζ ∈ S, between total spaces, where ϕt : S → St is a smooth family of diffeomorphisms with ϕ0 = Id and ϕ1 = ψ. The proof for bundles over a smooth finite open oriented surface X follows along the same lines using the fact that such a surface X is diffeomorphic to a standard neighbourhood of a standard bouquet of circles. We obtain a bijective correspondence between smooth isomorphism classes and free isotopy classes of (g, m)-bundles over connected oriented smooth finite open surfaces.

A smooth (g, m)-bundle (X , P, E, X) over a connected smooth finite open oriented surface X admits a smooth section, i.e. a smooth mapping X x → s(x) ∈ X \ E for which P ◦ s(x) = x for all x ∈ X. Indeed, let N(B) be a neighbourhood of a standard bouquet of circles for X as in Sect. 1.2, and let D, Vj and sj be as in Sect. 1.2. There is a smooth section over D ∪j sj , where sj are the arcs of ∂D to which the horizontal sides of the Vj are glued. This follows from the fact, that there is a simply connected neighbourhood of this set, and the bundle is smoothly trivial over any simply connected domain. Extend the smooth section to a smooth section over a neighbourhood of D ∪j (sj ∪ sj ). Since a neighbourhood of each Vj is simply connected and there is a smooth deformation retraction of a neighbourhood of Vj

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to a neighbourhood of sj ∪ sj , we obtain a smooth section on the whole N(B). By Lemma 1.1 we obtain a smooth section on the whole surface X. Reducible Bundles and Irreducible Components Thurston’s notion of reducible mapping classes takes over to families of mapping classes on a surface of type (g, m), and therefore to (g, m)-bundles. Namely, an admissible system of curves on a connected oriented closed surface S of genus g with set of m distinguished points E is said to reduce a family of mapping classes mj ∈ M(S; ∅, E), j = 1, . . . , k, if it reduces each mj . If the family consists of a single mapping class we arrive at the notion of a reduced mapping class (see Sect. 1.8). Similarly, a (g, m)-bundle over a smooth finite open connected oriented surface X with fiber S over the base point q0 and set of distinguished points E ⊂ S is called reducible if there is an admissible system of curves in the fiber over the base point that reduces all monodromy mapping classes simultaneously. Otherwise the bundle is called irreducible. Consider any isotopy class of reducible (g, m)-bundles with fiber S and set of distinguished points E ⊂ S over the base point q0 . Take an admissible system C of simple closed curves on S \ E that reduces each monodromy mapping class and is maximal in the sense that there is no strictly larger admissible system with this property. For each monodromy mapping class mj we choose a representing homeomorphism ϕj that fixes C setwise. Then all homeomorphisms ϕj fix S \ C. Decompose S \ C into disjoint open sets Sk , each of which being a minimal union of connected components of S \ C that is invariant under each ϕj . Each Sk is a union of surfaces with holes and possibly with distinguished points and is homeomorphic to ◦

a union S k of closed surfaces with a set Ek of finitely many punctures and, possibly, a set Ek of finitely many distinguished points. The restrictions of the ϕj to each Sk ◦

(possibly, with distinguished points) are conjugate to self-homeomorphisms of S k (possibly with set of distinguished points Ek ). For each k we denote by Skc the (possibly non-connected) closed surface ◦

which is obtained from S k by filling the punctures. The surface Skc is equipped with distinguished points Ek ∪ Ek . For each k we obtain a conjugacy class of homomorphisms from the fundamental group π1 (X, x0 ) to the modular group of the possibly non-connected Riemann surface Skc with set of distinguished points Ek ∪ Ek . By Theorem 8.1 (more precisely, by its analog for possibly non-connected fibers) we obtain for each k an isotopy class of bundles over X with fiber a union of closed surfaces with distinguished points. The obtained isotopy classes of bundles are irreducible and are called irreducible bundle components of the (g, m)bundle with fiber S. (There may be different decompositions into irreducible bundle components.) The reducible bundle can be recovered from the irreducible bundle components up to commuting Dehn twists in the fiber over the base point. (See the proof of Theorem 8.2 below for the case of (0, 4)-bundles.) Locally Holomorphically Trivial and Isotrivial Bundles Let X be a Riemann surface and 2g − 2 + m > 0. A holomorphic (g, m)-bundle over X is called

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

locally holomorphically trivial if it is locally holomorphically isomorphic to the trivial (g, m)-bundle. All fibers of a locally holomorphically trivial (g, m)-bundle F = (X , P, E, X) are conformally equivalent to each other. For a locally trivial holomorphic (g, m)-bundle there exists a finite unramified ˆ E, ˆ of F to Xˆ such that Fˆ is ˆ X) covering Pˆ : Xˆ → X and a lift Fˆ = (Xˆ , P, holomorphically isomorphic to the trivial bundle. This can be seen as follows. All fibers of the bundle F are conformally equivalent and for each fiber there are only finitely many different conformal self-homeomorphisms. Consider the lift F˜ of the ˜ E, ˜ X), ˜ where bundle F to the universal covering P : X˜ → X of X, i.e. F˜ = (X˜ , P, −1 −1 ˜ ˜ ˜ the fiber P (x) ˜ with distinguished points E ∩ P (x) ˜ is conformally equivalent to the fiber P −1 (x) with distinguished points E ∩ P −1 (x) with x = P(x). ˜ Let P˜ : X˜ → X be the respective fiber preserving projection. The bundle F˜ is locally holomorphically trivial. Since X˜ is simply connected, F˜ is holomorphically trivial ˜ i.e., there is a biholomorphic mapping Φ : X˜ → X˜ × S that maps P˜ −1 (x) on X, ˜ to ˜ to X×E. ˜ {x}×S ˜ for each x˜ ∈ X˜ and maps E Here S is the fiber P˜ −1 (q˜0 ) ∼ = P −1 (q0 ) ˜ ∩ P −1 (x). ˜ over a chosen point q˜0 ∈ X˜ over the base point q0 ∈ X and E = E −1 The mapping Φ provides a holomorphic family of conformal mappings ϕ˜x˜ : ˜ Each mapping of the family takes the set of S = P˜ −1 (q˜0 ) → P˜ −1 (x), ˜ x˜ ∈ X. distinguished points in the fiber S to the set of distinguished points in the respective ˜ The total space X of the bundle F is holomorphically fiber of the bundle F. ˜ equivalent to the quotient of X×S by the following equivalence relation. Two points ˜ (x˜1 , ζ1 ) and (x˜2 , ζ2 ) in X × S are equivalent if P˜ Φ −1 (x˜1 , ζ1 ) = P˜ Φ −1 (x˜2 , ζ2 ), i.e. if P(x˜1 ) = P(x˜2 ) and (P˜ ϕx˜1 )(ζ1 ) = (P˜ ϕx˜2 )(ζ2 ). Put ϕx˜ = P˜ ϕ˜x˜ : S → P −1 (x) with x = P(x). ˜ If P(x1 ) = P(x2 ), the mapping ϕx−1 ˜2 ϕx˜1 is a holomorphic selfhomeomorphism of the fiber S. The set of such self-homeomorphims is finite. Let x˜ = σ (q˜0 ) for a covering transformation σ . The mapping ϕσ (q˜0 ) maps S to P −1 (q0 ) ∼ = S. For two covering transformations σ1 and σ2 the equality ϕσ1 σ2 (q˜0 ) = ϕσ2 (q˜0 ) ϕσ1 (q˜0 )

.

(8.12)

holds. As before (Isq˜0 )−1 denotes the isomorphism from the fundamental group to the group of covering transformations. Consider the set N of elements e ∈ π1 (X, q0 ) for which ϕ(Isq˜0 )−1 (e)(q˜0 ) is the identity. By (8.12) the set N is a normal subgroup of the fundamental group. It is of finite index, since two cosets e1 N and e2 N are equal if P˜ ϕ(Isq˜0 )−1 (e2 e−1 )(q˜0 ) = Id, and there are only finitely many distinct holomorphic 1

q˜0 −1 ∼ P −1 (q0 ). Hence, Xˆ def ˜ self-homeomorphisms of S = = X(Is ) (N ) is a finite unramified covering of X and the lift of the bundle F to Xˆ has the required property. Vice versa, if for a holomorphic (g, m)-bundle F there exists a finite unramified ˆ E, ˆ X) ˆ of F to Xˆ is covering Pˆ : Xˆ → X, such that the lift Fˆ = (Xˆ , P, holomorphically isomorphic to the trivial bundle, then F is locally holomorphically trivial. A smooth (holomorphic, respectively) bundle is called isotrivial, if it has a finite covering by the trivial bundle. If all monodromy mapping classes of a smooth bundle

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247

are periodic, then the bundle is isotopic (equivalently, smoothly isomorphic) to an isotrivial bundle. This can be seen by the same arguments as above.

8.3 (0, 4)-Bundles Over Genus 1 Surfaces with a Hole The .(0, n)-bundles over a manifold X are closely related to the separable quasipolynomials on X. In this section we will state a theorem that concerns .(0, 4)-bundles and is related to Theorem 7.3. We will prove it together with Theorem 7.3. Let X be a connected finite open Riemann surface (a connected oriented finite open smooth surface, respectively). By a holomorphic (smooth, respectively) .(0, n)bundle with a section over X we mean a holomorphic (smooth, respectively) .(0, n + 1)-bundle .(X , P, E, X), such that the complex manifold (smooth manifold, respectively) .E ⊂ X is the disjoint union of two complex manifolds (smooth ˚ and .s, where .E ˚ ⊂ X intersects each fiber .P −1 (x) manifolds, respectively) .E ˚ along a set .Ex of n points, and .s ⊂ X intersects each fiber .P −1 (x) along a single point .sx . In other words, the mapping .x → sx , x ∈ X, is a (holomorphic, smooth, ˚x in the respectively) section of the .(0, n)-bundle with set of distinguished points .E fiber over x. Two smooth .(0, n)-bundles with a section are called isomorphic if they are isomorphic as smooth .(0, n + 1)-bundles and the diffeomorphism between the total spaces takes the section of one bundle to the section of the other bundle. Two smooth .(0, n)-bundles with a section are called isotopic if they are isotopic as .(0, n+1)-bundles with an isotopy that joins the sections of the bundles. Two smooth .(0, n)-bundles with a section are isotopic iff they are isomorphic. We define a special smooth (holomorphic, respectively) .(0, n + 1)-bundle over X as a smooth (holomorphic, respectively) bundle over X of the form .(X × P1 , pr1 , E, X), where .pr1 : X × P1 → X is the projection onto the first factor, and ˚ ∪ s ∞ , where .E ˚ the smooth submanifold .E of .X × P1 is equal to the disjoint union .E 1 is a smooth submanifold of .X ×P that intersects each fiber along n finite points and ∞ is the smooth submanifold of .X × P1 that intersects each fiber .{x} × P1 along .s the point .{x} × {∞}. A special .(0, n + 1)-bundle is, in particular, a .(0, n)-bundle with a section. Each smooth mapping .f : X → Cn (C)Sn from X to the symmetrized configuration space defines a smooth special .(0, n + 1)-bundle over X. The smooth  submanifold .E of .X × P1 is equal to the union . x∈X x, f (x) ∪ {∞} . Vice versa, ˚ ∩ (pr1 )−1 (x) defines for each special .(0, n + 1)-bundle the mapping .X x → E a smooth separable quasi-polynomial f of degree n on X. The special .(0, n + 1)bundle is holomorphic if and only if the mapping f is holomorphic. Choose a base point .x0 ∈ X. Denote by .f∗ the mapping from .π1 (X, x0 ) to the fundamental group .π1 (Cn (C)Sn , f (x0 )) ∼ = Bn induced by f . Two smooth mappings .f1 and .f2 from X to .Cn (C)Sn define special .(0, n + 1)-bundles that are isotopic through special .(0, n+1)-bundles if and only if the quasipolynomials .f1 and .f2 are free isotopic. They define bundles that are isomorphic (equivalently, they are

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

isotopic through .(0, n)-bundles with a section) if and only if for a set of generators ej of the fundamental group .π1 (X, x0 ), for a braid .w ∈ Bn and integer numbers

.

2k

kj the equalities .(f1 )∗ (ej ) = w −1 (f2 )∗ (ej )wΔn j hold. Indeed, the bundles are isomorphic iff their monodromy homomorphisms are conjugate. The monodromy mapping classes of the bundles are elements of .M(P1 ; {∞}, f (x0 )) and the braid group on n strands modulo its center .Bn Zn is isomorphic to the mapping class group .M(P1 ; {∞}, f (x0 )). ˚ ∪ s, X) with a section over an Each smooth .(0, n)-bundle .F = (X , P, E oriented smooth finite open surface X is isomorphic (equivalently, isotopic through .(0, n)-bundles with a section) to a smooth special .(0, n + 1)-bundle. Indeed, put ˚= E ˚ ∩ P −1 (x0 ), .s = s ∩ P −1 (x0 ) for the base point .x0 ∈ X. A homeomorphism .E −1 (x ) → {x }×P1 with .ϕ(s) = (x , ∞) and .ϕ(E) ˚ = {x0 }× E ˚ for a set .E ˚ ⊂ .ϕ : P 0 0 0 −1 1 ˚ ˚ ). Cn (C)Cn induces an isomorphism .Is : M(P (x0 ); s, E) → M(P ; {∞}, E The bundle .F corresponds to a conjugacy class of homomorphisms .π1 (X, x0 ) → ˚ The isomorphism between mapping class groups gives us a M(P −1 (x0 ); s, E). ˚ ). There is conjugacy class of homomorphisms .π1 (X, x0 ) → M(P1 ; {∞}, E a special .(0, n + 1)-bundle that corresponds to the latter conjugacy class of homomorphisms. This bundle is isomorphic to .F. .

def ˚ ∪ s, X) with a holoLemma 8.3 Each holomorphic .(0, n)-bundle .F = (X , P, E morphic section over a connected finite open Riemann surface X is holomorphically isomorphic to a special holomorphic .(0, n + 1)-bundle.

Proof Choose D, .D˜ j , j = 0, . . . , and .U˜ as in the proof of Theorem 8.1 in Sect. 8.2 (see also Sect. 1.2). Lift the bundle .F to a bundle on the universal covering .X˜ and  ∪  of .E ˚ ˚ to .U˜ ˚ ˜ E restrict the lift to a bundle .F˜ = (P˜ −1 (U˜ ), P, s, U˜ ) on .U˜ . The lift .E   ˚ is the union of n connected components .E j  , j = 1, . . . , n, each intersecting each fiber along a single point. ˜ U˜ ) that is obtained from .F˜ Consider the holomorphic .(0, 0)-bundle .(P˜ −1 (U˜ ), P, by forgetting the distinguished points and the section. All fibers of the .(0, 0)-bundle ˜ −1 (U˜ ), P, ˜ U˜ ) are conformally equivalent compact Riemann surfaces. Hence, by .(P a theorem of Fischer and Grauert [26] the bundle is locally holomorphically trivial. This means that for each point .x˜ ∈ U˜ there exists a simply connected open subset ˜ −1 (Ux˜ ) → Ux˜ × P1 such ˜ containing .x, .Ux˜ ⊂ U ˜ and a biholomorphic map .ϕUx˜ : (P) that the diagram

.

commutes. For each .x˜  ∈ Ux˜ and .j  = 1, . . . , n, the mapping .ϕUx˜ takes the point   ∩(P) ˚ ˜ −1 (x˜  ) to a point denoted by .(x˜  , ζj  (x˜  )), and takes the point of the section .E j

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249

˜ −1 (x˜  ) to a point denoted by .(x˜  , ζn+1 (x˜  )). The .ζj  , j  = 1, . . . , n + 1, s ∩ (P) are holomorphic. Composing .ϕUx˜ with a holomorphic mapping from .Ux˜ × P1 onto itself which preserves each fiber, we may assume that .ϕUx˜ maps no point in  ∪ s) to .(x, ˚ ˜ −1 (x) .P ˜ ∩ (E ˜ ∞). Maybe, after shrinking the .Ux˜ , the mapping .

 (ζ2 (x˜  ) − ζ )(ζn+1 (x˜  ) − ζ1 (x˜  ))  Ux˜ × P1 (x˜  , ζ ) → wx˜ (x˜  , ζ ) = x˜  , −1 + 2 (ζ2 (x˜  ) − ζ1 (x˜  ))(ζn+1 (x˜  ) − ζ )

.

provides a biholomorphic map .Ux˜ × P1 ⊃, that takes for each .x˜  ∈ Ux˜ the point ζ1 (x˜  ) to 1, it maps .ζ2 (x˜  ) to .−1, and .ζn+1 (x˜  ) to .∞. Denote by .ψx˜ the mapping .wx˜ ◦ϕUx˜ from .P˜ −1 (Ux˜ ) onto .Ux˜ ×P1 . If for two points .x, ˜ y˜ ∈ U˜ the intersection .Ux˜ ∩Uy˜ is not empty and connected, then the mappings .ψx˜ and .ψy˜ coincide on .P˜ 1−1 (Ux˜ ∩ Uy˜ ), since in each fiber the ordered triple of points,

and the section, is mapped by each of the mappings to

, E ˚ ˚ corresponding to .E 1 2 the ordered triple .(−1, 1, ∞). We get a holomorphic isomorphism .ψ˜ from the total  to .U˜ ×{+1}, ˚ space .U˜ of the bundle .F˜ onto .U˜ ×P1 that maps the first component of .E ˜ ˜ the second to .U × {−1}, and the section to .U × {∞}. The isomorphism takes the set

 to a set .E ∞ = U ˚ ⊂ U˜ × P1 , and the section  ˚ ˜ × {∞}. The identity .E .s to the set  .s ˜ ˜ mapping from .U onto itself together with the mapping .ψ define a holomorphic

˚ ∪ isomorphism from the bundle .F˜ onto the bundle .(U˜ × P1 , pr1 , E s ∞ , U˜ ). ˜ Recall that the original bundle .F is obtained from the bundle .F by gluing for each ˜ D˜ j to the restriction .F| ˜ D˜ 0 using a holomorphic isomorphism .ϕj j the restriction .F| −1 −1 from .P˜ (D˜ 0 ) onto .P˜ (D˜ j ). A bundle that is holomorphically isomorphic to .F

˚ ∪  is obtained as follows. Take the bundle .(U˜ × P1 , pr1 , E s ∞ , U˜ ) and glue for each .j = 1, . . . , 2g + m − 1, its restrictions to .D˜ j and .D˜ 0 together using the def ˜ −1 | D˜ 0 × P1 between the subsets biholomorphic mapping .ψ˜ 0,j = ψ˜ ◦ ϕj ◦ (ψ)

˚ ∪ ˜ 0 × P1 and .D˜ 1 × P1 of the total space .U˜ × P1 of the bundle .(U˜ × P1 , pr1 , E .D ∞  s , U˜ ). ˜ over .(x˜0 , ζ ) ∈ D˜ 0 × P1 conformally onto the ˜ F) Since .ψ˜ 0,j maps the fiber of .ψ( fiber over .(x˜j , ζ ) ∈ D˜ j × P1 , the following equality holds .

  ψ˜ 0,j (x˜0 , ζ ) = x˜j , a˜ j (x˜0 ) + α˜ j (x˜0 )ζ , (x˜0 , ζ ) ∈ D˜ 0 × C ,

.

(8.13)

for holomorphic functions .a˜ j and .α˜ j with .α˜ j nowhere vanishing on .D˜ 0 . We want to find a biholomorphic fiber preserving holomorphic mapping ˜ × P1 ⊃ ˜ 0,j ◦ v −1 have the form .v : U  such that for each j the mappings .v ◦ ψ v ◦ ψ˜ 0,j ◦ v −1 (x˜0 , ζ ) = (x˜j , ζ ) , (x˜0 , ζ ) ∈ D˜ 0 × P1 ,

.

(8.14)

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

where .x˜j ∈ D˜ j is the point for which .P(x˜0 ) = P(x˜j ) for the projection .P : X˜ → X. ˜ provides a special holomorphic .(0, n + 1)-bundle that ˜ F) Then the bundle .(v ◦ ψ)( is holomorphically isomorphic to .F by gluing the fibers over .x˜j ∈ D˜ j to the fibers over .x˜0 ∈ D˜ 0 using the mapping .(x˜0 , ζ ) → (x˜j , ζ ). ˜ ζ ) ∈ U˜ × C we may write For .(x,   v −1 (x, ˜ ζ ) = x, ˜ a(x) ˜ + α(x) ˜ ζ , (x, ˜ ζ ) ∈ U˜ × C .

.

(8.15)

Condition (8.14) is by (8.13) and (8.15) equivalent to the following requirements on a and .α: α(x˜j ) = α˜ j (x˜0 ) · α(x˜0 ),

j = 1, . . . , 2g + m − 1 ,

(8.16)

a(x˜j ) = a˜ j (x˜0 ) + α˜ j (x˜0 ) a(x˜0 ) ,

j = 1, . . . , 2g + m − 1 .

(8.17)

.

.

The Eq. (8.16) leads to a Second Cousin Problem for .αj . If Eq. (8.16) is granted, Eq. (8.17) can be rewritten as .

a(x˜j ) a(x˜0 ) a˜ j (x˜0 ) = + . α(x˜j ) α(x˜0 ) α(x˜j )

(8.18)

For convenience of the reader we provide the reduction of Eq. (8.16) to the Second Cousin Problem. Let D, .Vj and .sj be the same objects as in Sect. 8.2. We consider an open cover of X as follows. Put .U0 = D. .U0 is an open topological disc in X that contains the base point .x0 . Cover each .Vj by two simply connected open sets .Uj + and .Uj − with connected and simply connected intersection, so that the .Uj ± are disjoint from the .Uk ± for .j = k, and for each .Uj ± its intersection with .U0 is connected and simply connected. We may also assume that the intersection of at least three of the sets is empty, the intersections of each .sj with .Uj + and .Uj − are connected, and each .sj is disjoint from .Uk + ∪ Uk − for .k = j . Label the .Uj ± so that walking on .sj \ U0 (which is contained in .Vj ) in the direction of the orientation of .sj we first meet .Uj − (see Fig. 8.1). ˜ Let .(U0 ∪j − Uj − ) be a lift of .U0 ∪j − We consider the following subsets of .X. ˜ Let .(x˜0 )0 be the point in the lifted set that projects to .x0 . For each .j ≥ 1 Uj − to .X. we consider a point .qj ∈ Uj − ∩ Uj + and the point .q˜j in .(U0 ∪j − Uj − ) that projects to .qj . For each j we denote by .(Uj + ∪ U0 ) the lift of .Uj + ∪ U0 to .X˜ that  j+ contains .q˜j . Let .U0 be the subset of .(Uj + ∪ U0 ) that projects to .U0 .

8.3 (0, 4)-Bundles Over Genus 1 Surfaces with a Hole

251

Fig. 8.1 The open cover of X and its lift to .X˜

The sets .U0 , .Uj ± cover X. The intersection of three distinct sets of the cover is empty. On the intersection of pairs of covering sets we define a non-vanishing holomorphic function as follows. Put .

α0,j − = ( αj − ,0 )−1 = 1

on U0 ∩ Uj −

j = 1, . . . , 2g + m − 1 ,

αj − ,j + = (αj + ,j − )−1 = 1

on Uj − ∩ Uj +

j = 1, . . . , 2g + m − 1 ,

αj + ,0 = ( α0,j + )−1 = α˜ j,0

on Uj + ∩ U0

j = 1, . . . , 2g + m − 1 , (8.19)

where for .Uj + ∩ U0 , the function .α˜ j,0 is defined as .α˜ j,0 (x) = α˜ j (x˜0 ) for the lift .x˜0  j+ of the point .x ∈ U0 to .U0 . Since there are no triple intersections, the Eq. (8.19) define a Cousin II distribution. A second Cousin distribution on a complex manifold defines a holomorphic line bundle. The Second Cousin Problem has a solution if and only if the line bundle is holomorphically trivial. Since on an open Riemann surface each holomorphic line bundle is holomorphically trivial the Cousin Problem has a solution (see e.g. [27]).

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

The solution of the Cousin Problem for the Cousin distribution (8.19) provides non-vanishing holomorphic functions .α0 on .U0 and .αj ± on .Uj ± such that α0

= αj −

on U0 ∩ Uj − ,

αj −

= αj +

on Uj − ∩ Uj + ,

αj + α0−1 = α˜ j,0

on Uj + ∩ U0 .

.

(8.20)

Put ⎧ ⎪ α0 (x) ⎪ ⎪ ⎪ ⎨α − (x) j .α(x) ˜ = ⎪ α ⎪ j + (x) ⎪ ⎪ ⎩ α˜ j,0 (x) · α0 (x)

x˜ ∈ U˜ 0 x˜ ∈ U˜ j − x˜ ∈ U˜ j +  j+ x˜ ∈ U0 .

(8.21)

Here .x = P(x) ˜ for the projection .P : X˜ → X. The function .α is a well-defined holomorphic function on .U˜ that satisfies (8.16). a(x˜ ) The reduction of (8.18) to a First Cousin Problem for the . α(x˜jj ) , .j = 0, . . . 2g + m−1, is similar. The First Cousin Problem is solvable since open Riemann surfaces are Stein manifolds (see e.g. [27] and [40], and also Appendix A.1).



8.4 (0, 4)-Bundles Over Tori with a Hole. Proof of Theorem 7.4 We will consider now deformations of a smooth .(0, 3)-bundle with a section over a connected finite open Riemann surface X to a holomorphic bundle. Statement .(1) of Theorem 8.2 below is the analog of Theorem 7.3 for .(0, 3)-bundles with a section. Definition 8.4 We will say that a smooth .(g, m)-bundle .F over a connected smooth finite open oriented surface X is isotopic to a holomorphic bundle for the conformal structure .ω : X → ω(X), if the pushed forward bundle .Fω (see Eq. (8.1)) is isotopic to a holomorphic bundle on .ω(X). If the bundle is isotopic to a holomorphic bundle for each conformal structure with only thick ends on X, then .F is said to have the Gromov-Oka property. A .(0, n)-bundle on X with a section, in particular, a special .(0, n + 1)-bundle on X, is said to have the Gromov-Oka property, if and only if for each conformal structure on X with only thick ends the bundle is isotopic through .(0, n)-bundles with a section to a holomorphic bundle. In the following theorem we start with a smooth special .(0, 4)-bundle, since each smooth .(0, 3)-bundle with a section is isotopic to a special .(0, 4)-bundle. Recall

8.4 (0, 4)-Bundles Over Tori with a Hole. Proof of Theorem 7.4

253

that each connected finite open Riemann surface .X is conformally equivalent to a domain in a connected closed Riemann surface .X c . Notice that a special .(0, 4)bundle is reducible if and only if all monodromies are powers of a single conjugate of .σ1 Z3 . Theorem 8.2 (1) Let X be a connected smooth oriented surface of genus one with a hole with base point .x0 , and with a chosen set .E = {e1 , e2 } of generators of .π1 (X, x0 ). Define the set .E0 = {e1 , e2 , e1 e2−1 , e1 e2−2 , e1 e2 e1−1 e2−1 } as in Theorem 7.3. Consider a smooth special .(0, 4)-bundle .F over X. Suppose for each .e ∈ E0 the restriction of the bundle .F to an annulus in X representing .eˆ has the GromovOka property. Then the bundle over X has the Gromov-Oka property. (2) If a bundle .F as in .(1) is irreducible, then it is isotopic to an isotrivial bundle, and, hence, for any conformal structure .ω on X the bundle .Fω is isotopic to a bundle that extends to a holomorphic bundle on .ω(X)c . In particular, the bundle is isotopic to a holomorphic bundle for any conformal structure on X (including conformal structures of first kind). (3) Let .F be any smooth reducible special .(0, 4)-bundle over X (without any further requirement). Then each irreducible bundle component of . F is isotopic to an isotrivial bundle. There exists a Dehn twist in the fiber over the base point such that the bundle .F can be recovered from the irreducible bundle components up to composing each monodromy with a power of this Dehn twist. The bundle .F itself is isotopic to a holomorphic bundle for each conformal structure of second kind on X. (4) Any reducible holomorphic .(0, 3)-bundle with a holomorphic section over a .punctured Riemann surface is holomorphically trivial. Recall that a bundle is called isotrivial if its lift to a finite covering of X is the trivial bundle. We will prove now Theorems 7.3 and 8.2 using Theorem 7.4. Proof of Theorem 8.2 The special .(0, 4)-bundle .F defines a smooth quasipolynomial f on X. Under the conditions of Statement (1) for each .e ∈ E0 the restriction of the bundle to an annulus .Aeˆ in X representing .eˆ has the Gromov-Oka property. that for a conformal structure .ω(Aeˆ ) of conformal By Lemma 8.3 we may assume √ module bigger than . π2 log( 3+2 5 )−1 on .Aeˆ the bundle .(F | Aeˆ )ω is isotopic (through .(0, 3)-bundles with a section on .ω(Aeˆ )) to a special holomorphic .(0, 4)-bundle .Feˆ (not merely to a holomorphic .(0, 3)-bundle with a section). The bundle .Feˆ defines a holomorphic quasipolynomial .feˆ on .ω(Aeˆ ). We may assume that .ω(Aeˆ ) has the form .{z ∈ C : 1r < |z| < r}. The monodromies of the isomorphic bundles .(F | Aeˆ )ω and .Feˆ differ by conjugation (after identification of the mapping class groups on the fiber over the base point by an isomorphism). Hence, there exists an integer number −1 | ω(A ) corresponding to .(F | A ) is .ke , such that the quasipolynomial .f ◦ ω eˆ eˆ ω (free) isotopic to the holomorphic quasipolynomial .z → e2π ke z feˆ (z), z ∈ ω(Aeˆ ). This holds for each .e ∈ E0 . We proved that the quasipolynomial f satisfies the conditions of Theorem 7.4.

254

8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

The monodromy mapping classes of the quasipolynomial f are elements of the braid group .B3 , the monodromy mapping classes of the bundle .F can be identified with elements of .B3 Z3 . In terms of the .(0, 4)-bundle .F the Theorem 7.4 implies the following. The isotopy class of the special .(0, 4)-bundle .F on X corresponds to the conjugacy class of a homomorphism .Φ : π1 (X, x0 ) → Γ ⊂ B3 Z3 , where .Γ is generated either by .σ1 σ2 Z3 , or by .Δ3 Z3 , or by .σ1 Z3 . The group .Γ is Abelian. Let .ω be any conformal structure on X (maybe, of first kind). The fundamental group .π1 ( ω(X)c , ω(x0 )) of the closed torus . ω(X)c (that contains a conformal copy of .ω(X)) is the Abelianization of .π1 (ω(X), ω(x0 )) ∼ = π1 ( X, x0 ) . Hence . Φ defines a homomorphism from . π1 ( ω(X)c , ω(x0 )) to .Γ which we also denote by .Φ. Consider first the case when the generator of .Γ is either .σ1 σ2 Z3 , or .Δ3 Z3 . By Lemma 6.4 this is exactly the case when the bundle .F is irreducible. The gener˚3 ), ator of .Γ corresponds to an element of the mapping class group .M(P1 ; {∞}, E 1 that is represented by a periodic mapping .ζ → θ · ζ, ζ ∈ P (see [11, 53]). In 2π i ˚3 consists of 3 equidistributed points on a circle with the first case .θ = e 3 and .E 2π i ˚3 consists of the origin and two center zero, in the second case .θ = e 2 and .E equidistributed points on a circle with center zero. Consider the homomorphism from .Γ to the group of complex linear transformations of .P1 , which assigns to the generator .b ∈ Γ the complex linear transformation .ζ → θ · ζ , .ζ ∈ P1 . Composing .Φ with this homomorphism we obtain a homomorphism .Ξ from .π1 ( ω(X)c , ω(x0 )) to the group of complex linear transformations of .P1 . Represent the closed torus . ω(X)c as quotient . ω(X)c = CΛ for a lattice .Λ = {λ1 k1 + λ2 k2 : k1 , k2 ∈ Z} with complex numbers .λ1 , λ2 ∈ C that are linearly independent over .R. Denote by .p1 the covering map .p1 : C → CΛ. Each element .λ ∈ Λ defines a covering transformation .z → z + λ, z ∈ C. We identify .Λ with the group of covering transformations. Recall that the fundamental group .π1 ( ω(X)c , ω(x0 )) is isomorphic to the group of covering transformations of c . ω(X) ∼ = CΛ. The group .Λ acts on .C × P1 as follows. Associate to .λ ∈ Λ the mapping λ

(z, ζ ) −→ (z + λ, Ξ (λ)(ζ )) ,

.

(z, ζ ) ∈ C × P1 .

(8.22)

˚3 to itself and fixes the set .C × {∞}. The thus defined mapping .λ takes the set .C × E The action is free and properly discontinuous. Hence the quotient .(C × P1 )Λ is a complex manifold. For each element of the quotient its projection to .CΛ is well-defined. Indeed, the following diagram commutes:

.

8.4 (0, 4)-Bundles Over Tori with a Hole. Proof of Theorem 7.4

255

The equivalence relation in the upper line gives the quotient .(C × P1 )Λ, the relation in the lower line gives .CΛ.   Denote by .P the projection .P : C×P1 Λ → CΛ. We obtain a holomorphic .(0, 3)-bundle def

Fω =

.



     ˚ (C × P )Λ, P, (C × E3 )Λ ∪ (C × {∞})Λ , CΛ 1

with a holomorphic section over the closed torus .ω(X)c . (See also (8.1) for notation.) By construction the restriction of this bundle to a representative of each generator .ej of .π1 (ω(X)c , x0 ) gives the mapping torus corresponding to .Ξ (ej ). Hence, the monodromy mapping classes of the bundle are equal to .Φ(ej ). We proved that in the irreducible case of Theorem 8.2 for any conformal structure .ω on X the bundle .Fω is smoothly isomorphic (equivalently, isotopic) to a holomorphic bundle that extends to a holomorphic bundle .(Fω )c on the closed torus .ω(X)c . Since the monodromy mapping classes of the bundle .Fω are periodic, the lift of the bundle .(Fω )c to a finite covering of X is isotopic to an isotrivial bundle. Statement (2) is proved and also statement (1) in the irreducible case. To prove Statement (3), we suppose .F is an arbitrary reducible smooth special ˚3 be the set of finite distinguished points in the fiber over the .(0, 4)-bundle. Let .E ˚3 ∪ {∞}) that reduces each base point. There is an admissible curve .γ ⊂ P1 \ (E monodromy mapping class of the bundle. Hence, by (the proof of) Lemma 6.4 each monodromy mapping class of the bundle is a non-trivial power of a single conjugate ˚3 ) that corresponds to .σ1 Z3 . We of the mapping class .m(σ1 Z3 ) in .M(P1 ; ∞, E ˚3 as in the proof are in the case when .Γ is generated by .σ1 Z3 . Label the points of .E of Lemma 6.4 so that the simple closed curve .γ separates .{ζ1 , ζ2 } from .{ζ3 , ∞}. Let .C1 be the connected component of the complement of .γ that contains .{ζ1 , ζ2 }, and let .C2 be the connected component of the complement of .γ that contains .{ζ3 , ∞}. There are two irreducible components of the mapping class .m(σ1 Z3 ). The irreducible component that corresponds to .C1 is described as follows. The connected component .C1 is homeomorphic to .P1 \ {∞} with set of distinguished points 1 .{ζ1 , ζ2 }. Each self-homeomorphism of .P \ {∞} with these two distinguished points extends to a self-homeomorphism of .P1 that maps the set .{ζ1 , ζ2 , ∞} of three distinguished points to itself. The irreducible component of the mapping class .m(σ1 Z3 ) corresponding to .C1 is an element of the mapping class group 1 .M(P ; {∞}, {ζ1 , ζ2 }). This mapping class group is generated by the element .σ = σ Z2 which corresponds to the braid .σ in the braid group .B2 on two strands. The square of .σ is the identity. We put .Γ 1 = M(P1 ; {∞}, {ζ1 , ζ2 }). To describe the irreducible component corresponding to .C2 of the mapping class 1 .m(σ1 Z3 ), we notice that .C2 is homeomorphic to .P \ {ζ4 } with distinguished 1 points .{ζ3 , ∞} for a point .ζ4 ∈ P , ζ4 = ζ3 , ∞. Each self-homeomorphism of 1 1 .P \ {ζ4 } with distinguished points .{ζ3 , ∞} extends to a self-homeomorphism of .P with set of distinguished points .{ζ3 , ζ4 , ∞}. The respective irreducible component of .m(σ1 Z3 ) is the identity in the group .M(P1 ; {ζ4 , ∞}, z3 ).

256

8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

All monodromy mapping classes of a bundle that is isomorphic to .F are contained in .Γ , hence are powers of .m(σ1 Z3 ). This implies that there are two irreducible bundle components, corresponding to .C1 and to .C2 , respectively. They are described as follows. The irreducible bundle component of the bundle .F, corresponding to .C2 , is associated to the trivial homomorphism. Hence, this irreducible bundle component is smoothly isomorphic to the trivial .(0, 3)-bundle over X. The irreducible bundle component of . F related to .C1 corresponds to the conjugacy class of the homomorphism . Φ 1 : π1 (X, x0 ) → Γ 1 induced by the homomorphism .Φ. The same proof as in the case, when .Γ is generated either by .σ1 σ2 Z3 , or by .Δ3 Z3 , shows that for any conformal structure .ω on X there is a holomorphic .(0, 2)-bundle .(F1 )ω with a holomorphic section on the closed torus .ω(X)c whose isomorphism class corresponds to the conjugacy class of the homomorphism .π1 (ω(X)c , ω(x0 )) → Γ 1 that is induced by .Φ 1 . The restriction of .(F1 )ω to a punctured torus .ω(X) , .ω(X)  ω(X) , is isomorphic to a special holomorphic .(0, 3)-bundle .(F1 )ω with set of distinguished points .{ζ1 (x), ζ2 (x), ∞} in the fiber .{ω(x)} × P1 over .ω(x). For .ω(x) ∈ ω(X)  ω(X) the points .ζ1 (x), ζ2 (x) are contained in a large closed disc in .C centered at the origin. this closed disc. Consider the special holomorphic Take a point .ζ3 ∈ C outside  def ω ˚ 3 ∪ s ∞ , X , where the set of finite .(0, 4)-bundle .(F) = X × P1 , ω ◦ pr1 , E ˚ 3 ∩ ({ω(x)} × P1 ) in the fiber over .ω(x) equals .{ω(x)} × distinguished points .E {ζ1 (x), ζ2 (x), ζ3 } and .s ∞ ∩ ({ω(x)} × P1 ) = (ω(x), ∞). The monodromy mapping class of the bundle .(F)ω along .ej differs from the monodromy mapping class .Fω , which is the push-forward to .ω(X) of the bundle .F, by the .kj -th power of a Dehn twist about the curve .γ . Consider again the punctured torus .ω(X) ⊃ ω(X), and assume that .ω(X) = (C \ Λ)Λ with .Λ = {λ1 n1 + λ2 n2 , n1 , n2 ∈ Z}, and λ1 +λ2 λ1 +λ2 2 .ω(x0 ) = + [0, λ1 ] and . λ1 +λ + [0, λ2 ] are 2 Λ, and that the segments . 2 2  lifts to .C \ Λ of curves representing a pair of generators of .π1 (ω(X) , ω(x0 )). For all integers .1 and .2 there is a holomorphic function F on .C\Λ such that F (y + λ1 ) = 1

.

and

F (y + λ2 ) = 2 , y = ω(x) ∈ C \ Λ .

(8.23)

This follows from the proof of Lemma 7.4. A quick way to see this for .2 = 0 and any .1 ∈ Z is the following. For each .y ∈ C there are uniquely determined real numbers .t1 (y) and .t2 (y) depending smoothly on y such that .y = λ1 t1 (y) + λ2 t2 (y). Consider a function .χ0 on the real axis which vanishes near zero such that def

χ0 (t +1) = χ0 (t)+1 , .t ∈ R. Consider the function .χ (y) = χ0 (t1 (y)), y ∈ C. The 1-form .∂χ descends to a smooth 1-form .δ on the torus .CΛ. Let ˚ .g be a solution of g = δ on the punctured torus .C\ΛΛ. Let g be the lift of ˚ .g to .C\Λ. the equation .∂ ˚ Put .F = χ − g. Let F be the holomorphic function on .C\Λ that satisfies Eq. (8.23) for the integers .1 = −k1 and .2 = −k2 . Then the monodromy mapping classes along the .ej of the special holomorphic .(0, 4)-bundle .(F )ω over .ω(X) with set of distinguished points .{ζ1 (x), ζ2 (x), ζ3 e2π iF (ω(x)) , ∞} in each fiber .{ω(x)} × P1 ,

.

8.4 (0, 4)-Bundles Over Tori with a Hole. Proof of Theorem 7.4

257

x ∈ X, are the same as the monodromy mapping classes of the bundle .Fω . Statement (3) and, hence, statement (1) in the reducible case are proved. It remains to prove Statement (4), namely the fact that any reducible holomorphic .(0, 3)-bundle .F with a holomorphic section over a punctured Riemann surface X is holomorphically isomorphic to the trivial bundle. The respective statement is known in more general situations, but the proof of the present statement is a simple reduction to Picard’s Theorem and we include it. By Lemma 8.3 we may assume that the bundle is equal to a special .(0, 4)-bundle 1 ˚ ∪ s ∞ , X). There is a finite unramified covering .Xˆ of the .F = (X × P , pr1 , E   ∪ sˆ ∞ , Xˆ  of ˚ punctured Riemann surface X so that for the lift .Fˆ = Xˆ × P1 , pr1 , E  ∪ sˆ ∞ is the union of four disjoint complex curves each ˚ the bundle .F to .Xˆ the set .E intersecting each fiber along a single point. After a holomorphic isomorphism (see  are ˚ the proof of Lemma 8.3) we may assume that the connected components of .E 1 ˆ × {−1}, .Xˆ × {1}, and a component that intersects each fiber .xˆ × P along a point .X .(x, ˆ g( ˆ x)) ˆ for a holomorphic function .gˆ on .Xˆ that omits the values .−1, 1, and .∞. Since the bundle .F is reducible, the bundle .Fˆ is reducible. Hence, .gˆ : Xˆ → C \ {−1, 1} is free homotopic to a mapping whose image is contained in a punctured disc around one of the points .−1, 1, or .∞. We may assume, after applying to each fiber the same conformal self-mapping of .P1 , that this point is .−1. By Picard’s Theorem the mapping .gˆ extends to a meromorphic mapping .gˆ c from the closure .Xˆ c of .Xˆ to .P1 . Then the meromorphic extension .gˆ c omits the value 1. ˆ then .gˆ would map a simple Indeed, if .gˆ c was equal to 1 at some puncture of .X, closed curve on .Xˆ that surrounds the puncture positively to a loop in .C \ {−1, 1} that surrounds 1 with positive winding number. This contradicts the fact that .gˆ is homotopic to a mapping into a disc punctured at .−1 and contained in .C \ {−1, 1}. Hence, .gˆ c is a meromorphic function on a compact Riemann surface that omits a value, and, hence .gˆ is constant. This means that the bundle .Fˆ over .Xˆ is holomorphically isomorphic to the trivial bundle. Hence, the monodromy mappings of the original bundle .F are periodic. By Lemma 6.4 this is possible for a reducible bundle only if the monodromies are equal ˆ to the identity. Repeat the same reasoning as above for the bundle .F instead of .F, we see that the bundle .F is holomorphically trivial. Theorem 8.2 is proved.

.

Proof of Theorem 7.3 Consider the special .(0, 4)-bundle .F that is associated to the quasipolynomial. By the conditions of Theorem 7.3 the bundle .F satisfies the assumptions of Statement (1) of Theorem 8.2. Theorem 8.2 implies, that for any a priori given conformal structure .ω of second kind on X the isotopy class (with respect to isotopies through .(0, 3)-bundles with a section) of the bundle .Fω on .ω(X) contains a holomorphic .(0, 3)-bundle with a section. Hence, by Lemma 8.3, the isotopy class contains a special holomorphic .(0, 4)-bundle .Fω on .ω(X). The monodromies of the two bundles .Fω and .Fω differ by conjugation by a selfhomeomorphism of .P1 that fixes the set of distinguished points setwise and fixes  .{∞}. There is an isotopy through special .(0, 4)-bundles that joins the bundle .Fω ω with a special .(0, 4)-bundle .F whose monodromies coincide with those of .Fω .

258

8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

The special holomorphic .(0, 4)-bundle .Fω determines a holomorphic quasipolynomial .f ω for this conformal structure by assigning to each .x ∈ ω(X) the triple of finite distinguished points in the fiber .{x} × P1 and associating to it the monic polynomial with this set of zeros. The monodromies of the quasipolynomial .f ω differ from the monodromies of the push forward .fω of the original quasipolynomial by powers of .Δ23 . Using the function F that satisfies (8.23) for suitable integers .1 and .2 we obtain a holomorphic quasipolynomial .f ω (x)e2π iF (x) , x ∈ ω(X), that is isotopic to .fω . Theorem 7.3 is proved.



8.5 Smooth Elliptic Fiber Bundles and Differentiable Families of Complex Manifolds Recall that Kodaira (see [54]) defines a complex analytic family (also called holomorphic family) .Mt , t ∈ B, of compact complex manifolds over the base .B as a triple .(M, P, B), where .M and .B are complex manifolds, and .P is a proper holomorphic submersion with .P −1 (t) = Mt . Thus, each holomorphic genus .g fiber bundle over a Riemann surface is a complex analytic family of compact Riemann surfaces of genus .g. We will consider holomorphic .(g, m)-bundles as complex analytic families of Riemann surfaces of genus .g equipped with .m distinguished points, for short, as complex analytic (or holomorphic) families of Riemann surfaces of type .(g, m), or holomorphic .(g, m)-families. Kodaira (see [54]) defines a differentiable family of compact complex manifolds as a smooth fiber bundle with compact fibers and the following additional structure. The fibers are equipped with complex structures that depend smoothly on the parameter and induce on each fiber the smooth fiber structure. The formal definition of a differentiable family of Riemann surfaces of type .(g, m) is the following. Definition 8.5 A differentiable family of Riemann surfaces .Mt , t ∈ B, of type (g, m) is a tuple .(M, P, E, B), where .M and .B are oriented .C ∞ manifolds, and .P is a smooth proper orientation preserving submersion with the following property. For each .t ∈ B the fiber .P −1 (t), denoted by .Mt , is equipped with the structure of a compact Riemann surface of genus .g, such that the complex structure induces the differentiable structure defined on .P −1 (t) as a submanifold of .M. .E ⊂ M is a smooth submanifold of .M that intersects each fiber .Mt along a set .Et of .m distinguished points. Moreover, the complex structures of the .Mt depend smoothly on the parameter t, more precisely, there is an open locally finite cover .Uj , j = j j 1, 2, . . ., of .M and complex valued .C ∞ functions .(z1 , . . . , zn ) on each .Uj (with n the complex dimension of .Mt ) such that for each j and t these functions define holomorphic coordinates on .Uj ∩ Mt . .

We will also call such families smooth families of Riemann surfaces of type .(g, m), or smooth .(g, m)-families.

8.5 Smooth Elliptic Fiber Bundles and Differentiable Families of Complex. . .

259

Notice that each smooth special .(0, n + 1)-bundle over a finite oriented smooth surface carries automatically the structure of a smooth family of Riemann surfaces of type .(0, n + 1). Two smooth families of Riemann surfaces of type .(g, m) will be called isomorphic (as families of Riemann surfaces) if they are isomorphic as smooth bundles (smoothly isomorphic, for short) and there is a bundle isomorphism that is holomorphic on each fiber. We will show in this section that each smooth elliptic fiber bundle with a section is smoothly isomorphic (equivalently, isotopic) to a bundle that carries the structure of a differentiable family of Riemann surfaces of type .(1, 1), also called a differentiable family of closed Riemann surfaces of genus 1 with a smooth section. Recall that each compact Riemann surface of genus 1 is conformally equivalent to the quotient .CΛ for a lattice .Λ = aZ + bZ where a and b are two complex numbers that are independent over the real numbers. A torus given in the form .CΛ will be called here a canonical torus. The torus .C(Z + iZ) will be called standard. A family of lattices .Ω x → Λ(x) on a smooth manifold .Ω is called smooth if for each .x0 ∈ X there is a neighbourhood .U (x0 ) ⊂ Ω of .x0 such that the lattices .Λ(x) can be written as Λ(x) = a(x)Z + b(x)Z .

.

(8.24)

for smooth functions a and b on .U (x0 ). If .Ω is a complex manifold, the family is called holomorphic if it can be locally represented by (8.24) with holomorphic functions a and b.    = Λ(x), with .Λ(x) = a (x)Z If on .U (x0 ) we have .Λ(x)  +  b(x)Z for other αβ smooth functions  .a and . b, then there is a matrix .A = ∈ SL2 (Z) not γ δ depending on x, such that  .a (x) = α a(x) + γ b(x),  b(x) = β a(x) + δ b(x) .

(8.25)

 Indeed, .Λ(x) and .Λ(x) are equal lattices and .(a(x), b(x)) and .( a (x),  b(x)) are   Re a(x) Re b(x)  pairs of generators of the lattice. Put .B(x) = and .B(x) = Im a(x) Imb(x)   Re a (x) Re  b(x)  . Then the real linear self-map of .C, defined by .B(x) (by .B(x), Im a (x) Im  b(x) .a (x) and i to . b(x), respectively). respectively) takes 1 to .a(x) and i to .b(x) (1 to   Hence, both maps take the standard lattice .Z + iZ onto .Λ(x) = Λ(x). Then −1 (x) ◦ B(x)  .B maps the standard lattice onto itself, hence, it is in .SL2 (Z). Since   .B(x) and .B(x) depend continuously on x, the matrices .B−1 (x)  ◦ B(x) depend αβ continuously on x and, hence, are equal to a matrix .A = ∈ SL2 (Z) that γ δ does not depend on x. Equation (8.25) is satisfied for the entries of this matrix. Two smooth families of lattices .Λ0 (x) and .Λ1 (x) depending on a parameter x in a smooth manifold .Ω are called isotopic, if for an interval .I ⊃ [0, 1] there is a

260

8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

smooth family of lattices .Λ(x, t), .(x, t) ∈ Ω × I , such that .Λ(x, 0) = Λ0 (x) and Λ(x, 1) = Λ1 (x) for all .x ∈ Ω. Let X be a finite open Riemann surface and .Λ(x), x ∈ X, a smooth family of lattices on X. Take an open subset U of X on which there are smooth function .a(x) and .b(x) such that .Λ(x) = a(x)Z+b(x)Z, x ∈ U . Then the family .Λ = Λ(x), x ∈ U, defines a free and properly discontinuous group action (see e.g. [58], III, 3K)

.

  U × C (x, ζ ) → x, ζ + a(x)n + b(x)m , n, m ∈ Z ,

.

(8.26)

on .U × C. The quotient of .U × C by this action depends only on .Λ, not on the choice of the generators .a(x) and .b(x) of the lattice. Denote by .(U × C)Λ the quotient of .U × C by this action (8.26). Let PU,Λ : (U × C)Λ → U

.

(8.27)

be the mapping whose value at the equivalence class of .(x, ζ ) equals x. Consider −1 in each fiber .PΛ (x) the distinguished point . sΛ,x = (x, 0)Λ(x) . The mapping def

x → sΛ,x , x ∈ U, is smooth, hence defines a smooth section. Put . s Λ,U =  def  ∪x∈U {sΛ,x }. Then the tuple .FΛ,U = (U × C)Λ, PU,Λ , s Λ,U , U defines a smooth .(1, 1)-bundle over U . Moreover, .FΛ,U is equipped with the structure of a differentiable family of Riemann surfaces of type .(1, 1). If the family .Λ of lattices on X is holomorphic, then .FΛ,U is a holomorphic .(1, 1)-bundle, equivalently, a holomorphic family of Riemann surfaces of type .(1, 1). The bundle .FΛ,U depends only on .Λ | U , not on the choice of the generators .a(x) and .b(x) of the lattice .Λ(x), .x ∈ U . Take an open cover of X by sets .Uj on which there are smooth functions .aj (x) and.bj (x) such that .Λ(x) = Λj (x)  = aj (x)Z + bj (x)Z, x ∈ Uj . Consider the   set . x, ζ Λ(x), x ∈ X, ζ ∈ C . The family .(Uj × C)Λj provides a system of smooth coodinates on this set that are holomorphic in .ζ for fixed value of x. We   denote the set . (x, ζ )Λ(x), x ∈ X, ζ ∈ C with the thus obtained structure by .XΛ = (X × C)Λ. Define a mapping .PΛ by the equalities .PΛ = PUj ,Λ −1 on .Uj , and define .s Λ by .s Λ ∩ (PΛ (Uj )) = s Λ,Uj for each j . We obtain a smooth family of Riemann surfaces of type .(1, 1) on X, which we denote by .FΛ,X = (XΛ , PΛ , s Λ , X). If the family .Λ of lattices on X is holomorphic, then .FΛ,X is a holomorphic .(1, 1)-bundle, equivalently, a holomorphic family of Riemann surfaces of type .(1, 1). The following lemma holds. .

8.5 Smooth Elliptic Fiber Bundles and Differentiable Families of Complex. . .

261

Lemma 8.4 Each smooth .(1, 1)-bundle .F over a smooth finite open oriented surface X is smoothly isomorphic (equivalently, isotopic) to a bundle of the form FΛ,X = (XΛ , PΛ , s Λ , X) .

.

(8.28)

Recall that the bundle (8.28) carries the structure of a smooth family of Riemann surfaces of type .(1, 1). Proof We need to find a smooth bundle of the form (8.28) whose monodromy homomorphism is conjugate to that of the original bundle .F. This can be done as in Sect. 8.2. We represent X as the union of an open disc D containing the base point of X, and a collection of . attached bands .Vj that are relatively closed in X and correspond to the generators of the fundamental group of X. Let as in Sect. 8.2 the set .U˜ be a simply connected domain on the universal covering of X such that each point of D is covered . + 1 times and each other point of X is covered once. Label the preimages of D under the projection .U˜ → X by .D˜ j , j = 0, 1, . . . , so that the preimage .V˜j of the band .Vj , j = 1, . . . , , is attached to .D˜ 0 and .D˜ j . Denote by .mj the monodromy of .F along the generator .ej of the fundamental group of X that corresponds to the j -th band .Vj . Map the fiber of .F over the base point diffeomorphically onto the standard torus .C(Z + iZ) so that the distinguished point is mapped to .0(Z + iZ). By an isomorphism between mapping class groups we identify each monodromy mapping class .mj , j = 1, . . . , , of .F with a mapping class on the standard torus .C(Z+iZ) with distinguished point .0(Z+iZ). We denote the new mapping class by the same letter .mj . Represent the new class .mj by a mapping .ϕj−1 : C(Z + iZ)⊃  , such that .ϕj lifts to a real linear self-mapping . ϕj of .C that maps the lattice .Z + iZ onto itself. In other words, . ϕj corresponds to a .2 × 2 matrix .Aj with integer entries and   x determinant 1, . ϕj (x + iy) = Aj , .Aj ∈ SL2 (Z). y j def = D˜ 0 ∪ V˜j ∪ D˜ j (which is a simply connected domain) Consider on each set .Ω j , of the complex plane .C such a smooth family of real linear self-maps .( ϕj )z , .z ∈ Ω that ( ϕj )z = Id for z ∈ D˜ 0 and ( ϕj )z =  ϕj for z ∈ D˜ j ,

.

(8.29)

and put j . Λj (z) = ( ϕj )z (Z + iZ), z ∈ Ω

.

(8.30)

For points .z˜ j ∈ D˜ j , j = 0, . . . , that project to the same point in D, the lattices .Λj (˜zj ) coincide. Hence, there exists a well defined smooth family of lattices j . We obtain a bundle .Λ(x), x ∈ X, that lifts to the family of lattices (8.30) on .Ω   .FΛ,X = (X × C)Λ , PΛ , s Λ , X over X, which is isomorphic (as a smooth bundle) to the original bundle over X, since the monodromy homomorphisms of the two bundles are conjugate to each other. The lemma is proved.



262

8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

By Lemma 8.4 the Problem 8.1 can be reformulated as follows. Problem 8.1. Let X be a finite open Riemann surface. Is a given smooth family of Riemann surfaces of type .(1, 1) on X isotopic to a complex analytic family of Riemann surfaces of type .(1, 1)?

8.6 Complex Analytic Families of Canonical Tori Let now .F = (X , P, X) be a holomorphic elliptic fiber bundle over a finite open Riemann surface X. The fiber bundle is, in particular, a smooth elliptic fiber bundle. For each disc .Δ ⊂ X there is a smooth family of diffeomorphisms .ϕt : S → P −1 (t), t ∈ Δ, from the reference Riemann surface .S = C(Z + iZ) of genus one onto the fiber over t. Consider the Teichmüller class .[ϕt ], t ∈ Δ. The following Lemma 8.5 states that these Teichmüller classes depend holomorphically on the parameter. Lemma 8.5 Let .F be a holomorphic elliptic fiber bundle over a Riemann surface X. For each small enough disc .Δ ⊂ X there is a holomorphic map .z → τ (z), .z ∈ Δ, into the Teichmüller space .T (1, 0) of the standard torus, such that each fiber −1 (z) is conformally equivalent to .C(Z + τ (z)Z). .P For convenience of the reader we will provide a proof. The key ingredient is a lemma of Kodaira which we formulate now. Let .X and X be complex manifolds and let .P : X → X be a proper holomorphic submersion such that the fibers .Xz = P −1 (z), .z ∈ X, are compact complex manifolds of complex dimension n. For each .z ∈ X we denote by .Θz the sheaf of germs of holomorphic tangent vector fields of the complex manifold .Xz . Denote by .H 0 (Xz , Θz ) the space of global sections of the sheaf. def

Lemma 8.6 (Kodaira, [54], Lemma 4.1, p. 204.) If the dimension .d = dim(H 0 (Xz , Θz )) is independent of z then for any small enough (topological) ball .Δ in X there is for each .z ∈ Δ a basis .(v1 (z), . . . , vd (z)) of .H 0 (Xz , Θz ) such that .(v1 (z), . . . , vd (z)) depends holomorphically on .z ∈ Δ. For a proof we refer to [54]. Let m be the dimension of the complex manifold X in the statement of Kodaira’s Lemma. The condition that the .vj (z), .j = 1, . . . , d, depend holomorphically on z, means the following. Let .Uα ⊂ X be a small open subset of .X on which there are local holomorphic coordinates α α .(ζ , . . . , ζn , z1 , . . . , zm ), where .z = (z1 , . . . , zm ) ∈ Δ and for fixed .z = 1 (z1 , . . . , zm ) the .ζ α = (ζ1α , . . . , ζnα ) are local holomorphic coordinates on the fiber over z. In these local coordinates the vector field .vj , .j = 1, . . . , n, can be written as vj (z) =

n 

.

k=1

vjα k (ζ α , z)

∂ , ∂ζkα

(8.31)

8.6 Complex Analytic Families of Canonical Tori

263

where the .vjα k are holomorphic in .(ζ α , z). The condition does not depend on the choice of local coordinates with the described properties. We prepare now the proof of Lemma 8.5. Kodaira’s Lemma applies in the situation of Lemma 8.5. Indeed, in the situation of Lemma 8.5 the base X has complex dimension one and the fibers are compact Riemann surfaces of genus 1. The space of holomorphic sections .H 0 (Xz , Θz ) of the sheaf of germs of holomorphic tangent vector fields of each fiber .Xz has complex dimension one. Indeed, let .T be a compact Riemann surface of genus 1. The complex structure of .T may be given by a system of local holomorphic coordinates with transition functions .ζ (z) = z + c for complex constants c (so called flat coordinates). A holomorphic vector field (equivalently, a holomorphic section in the holomorphic tangent bundle) on .T assigns to each chart .C ⊃ Uj → Vj ⊂ T a function .vj (zj ), zj ∈ Uj , such that dzk = vk (zk ). Since the complex structure on .T is if .Vj ∩ Vk = ∅ then .vj (zj (zk )) dz j dzk given by a system of flat local coordinates, the equality . dz ≡ 1 holds for each .j, k j with .Vj ∩ Vk = ∅. Hence, the .vj (zj ) define a holomorphic function v on .T. Each holomorphic function on a compact complex manifold is constant. Hence, the space of holomorphic tangent vector fields to any compact Riemann surface of genus one has complex dimension 1. In particular, for each .z ∈ X the space .H 0 (Xz , Θz ) is generated by a single holomorphic tangent vector field .v(z) on .Xz . By Kodaira’s Lemma .v(z) may be chosen to depend holomorphically on .z ∈ Δ for each small def

enough disc .Δ in X. We obtain a holomorphic vector field v on .XΔ = P −1 (Δ) whose restriction to each fiber .Xz equals .v(z). Δ of .XΔ . Denote by p the Consider the (holomorphic) universal covering space .X   covering map, .p : XΔ → XΔ . Consider the triple .(XΔ , P ◦ p, Δ). Δ , P ◦ p, Δ) is holomorphically isomorphic to the trivial holomorLemma 8.7 .(X phic fiber bundle over .Δ with fiber .C. Proof of Lemma 8.7 For each .z ∈ Δ the fiber .P −1 (z) of the bundle .(XΔ , P, Δ) is a compact Riemann surface of genus 1. Its preimage under p is the set .(P ◦ p)−1 (z) Δ , P ◦ p, Δ) over z. The set .P −1 (z) is a complex which is the fiber of the bundle .(X one-dimensional submanifold of .XΔ which is the zero set of the holomorphic function .P − z. The set .(P ◦ p)−1 (z) is a complex one-dimensional submanifold of Δ which is the zero set of the holomorphic function .P ◦p−z. Hence, the restriction .X −1 (z) : (P ◦p)−1 (z) → P −1 (z) defines a holomorphic covering. Indeed, .p | (P ◦p)  since .p : XΔ → XΔ is a covering, each point .z of .P −1 (z) has a neighbourhood −1 (U  ) is the disjoint union of open sets .U  and p ˜ k ⊂ X .Uz ⊂ XΔ such that .p z z k k maps each .U˜ z biholomorphically onto .Uz . Then p maps each .U˜ z ∩ (P ◦ p)−1 (z) conformally onto .Uz ∩ P −1 (z). Each fiber .(P ◦ p)−1 (z) is simply connected, since any loop in the fiber .(P ◦ −1 Δ , and the bundle .(X Δ , P ◦p, Δ) is smoothly isomorphic p) (z) is contractible in .X −1 to the trivial bundle .(Δ × (P ◦ p) (z), pr1 , Δ). Since for each .z ∈ Δ the covered manifold .P −1 (z) is a Riemann surface of genus 1, the covering manifold .(P ◦ p)−1 (z) is conformally equivalent to the complex

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

Δ , P ◦ p, Δ) is a holomorphic fiber bundle with fiber plane .C. Hence, the triple .(X C. P ◦p Δ −−−−→ Δ. It Take a holomorphic section .s(z), z ∈ Δ, of the mapping .X exists after, perhaps, shrinking .Δ. Let v be the holomorphic vector field on .XΔ from Δ . Define a mapping Kodaira’s Lemma, and  .v its lift to the universal covering .X Δ by .G : Δ × C → X .

G(z, ζ ) = γs(z) (ζ ) ,

.

(z, ζ ) ∈ Δ × C ,

(8.32)

where for each z the mapping .γs(z) is the solution of the holomorphic differential equation  γs(z) (ζ ) =  v (γs(z) (ζ )) ,

.

γs(z) (0) = s(z) ∈ (P ◦ p)−1 (z) ,

ζ ∈ C.

(8.33)

Since  .v is tangential to the fibers we have the inclusion γs(z) (C) ⊂ (P ◦ p)−1 (z) ∼ =C

.

for each

z ∈ Δ.

(8.34)

For each z the solution exists for all .ζ ∈ C since the restriction of  .v to the fiber over z is the lift of a vector field on a closed torus. The mapping (z, ζ ) → γs(z) (ζ ) ,

.

z ∈ Δ,

ζ ∈ C,

(8.35)

is a local holomorphic diffeomorphism. By the Poincaré-Bendixson Theorem for each z it maps .C one-to-one onto an open subset of the fiber .(P ◦ p)−1 (z) ∼ = C, hence, it maps .C onto .(P ◦ p)−1 (z). Hence, .G defines a holomorphic isomorphism Δ , P ◦ p, Δ). Lemma 8.7 of the trivial bundle .(Δ × C, pr1 , Δ) onto the bundle .(X is proved.

Δ → Proof of Lemma 8.5 Consider the covering transformations of the covering .X XΔ . In terms of the isomorphic bundle .(Δ × C, pr1 , C) the restrictions of the covering transformations to each fiber .{z} × C, are translations, hence, the covering transformations have the form   ψn,m (z, ζ ) = z, ζ + n a(z) + m b(z) ,

.

(z, ζ ) ∈ Δ × C ,

(8.36)

for integral numbers n and m. Here .a(z) and .b(z) are complex numbers which are linearly independent over .R and depend on z. Since the covering transformations .ψ1,0 and .ψ0,1 are holomorphic, the numbers .a(z) and .b(z) can be taken to depend holomorphically on .z ∈ Δ. After perhaps interchanging .a(z) and .b(z) we may a(z) assume that .Re a(z) b(z) > 0 . Hence, .τ (z) = b(z) depends holomorphically on .z ∈ Δ and can be interpreted as a holomorphic mapping from .Δ into the Teichmüller space (which is identified with the upper half-plane .C+ ). Lemma 8.5 is proved.



8.6 Complex Analytic Families of Canonical Tori

265

Corollary 8.1 Let .F be a holomorphic elliptic fiber bundle over a connected simply connected Riemann surface X, or let .F be a holomorphic elliptic fiber bundle over a connected finite open Riemann surface X, such that all monodromies are equal to the identity. Then there is a holomorphic map .z → τ (z), .z ∈ X, into the Teichmüller space .T (1, 0), such that each fiber .P −1 (z) is conformally equivalent to .C(Z + τ (z)Z). Proof Let first X be simply connected. Consider a smooth family of diffeomorphisms .ϕz : S → P −1 (z), z ∈ X, where .S = C(Z + iZ) is the standard def

torus. Then the Teichmüller classes .τ (z) = [ϕz ], z ∈ X, depend smoothly on z and .P −1 (z) is conformally equivalent to .C(Z + [ϕz ]Z). By Lemma 8.5 .[ϕz ] depends holomorphically on z. Indeed, on any small enough disc .Δ ⊂ X there is a holomorphic map .τΔ to the Teichmüller space such that .P −1 (z) is conformally ∗ (τ (z)), z ∈ Δ, for a modular equivalent to .C(Z + τΔ (z)Z). Hence, .[ϕz ] = ϕΔ Δ ∗ transformation .ϕΔ of the Teichmüller space. The corollary follows from the fact that modular transformations are biholomorphic self-maps of the Teichmüller space. Let now X be a finite open Riemann surface, and let all monodromies of the bundle .F be equal to the identity. Consider a lift .F˜ of the bundle to the universal P

covering .X˜ −−→ X. There exists a holomorphic mapping .τ˜ : X˜ → T (1, 0) such that for each .x˜ ∈ X˜ the value .τ˜ (x) ˜ represents the conformal class of the fiber of .F˜ over .x˜ (which is equal to the conformal class of the fiber of .F over .P(x)). ˜ Let e be an element of the fundamental group of X with base point .x0 . We let .ϕe be a selfhomeomorphism of the fiber of .F over .x0 , that represents the monodromy mapping class of the bundle along e. Denote by .ϕe∗ the modular transformation on .T (1, 0) corresponding to .ϕe . Then .τ˜ (e(x)) ˜ = ϕe∗ (τ˜ (x)) ˜ for each e. Since all monodromies of the bundle .F are trivial, each .ϕe∗ is equal to the identity, and therefore .τ˜ descends to a well-defined mapping .τ on X with the required property.

The following proposition holds. Proposition 8.1 Each holomorphic .(1, 1)-bundle .(X , P, s, X) over a finite open Riemann surface X is holomorphically isomorphic to a holomorphic bundle of the form (XΛ , PΛ , s Λ , X) .

(8.37)

.

for a holomorphic family of lattices .Λ = {Λ(x)}x∈X . Here as before, .XΛ = (X × C)Λ, and .PΛ assigns to each class .(x, ζ )Λ in the quotient the point .x ∈ X. The value at the point .x ∈ X of the holomorphic section .s Λ of the bundle (8.37) is the class in the quotient that contains .(x, 0). Proof For each .x ∈ X we let .Δx be a topological disc, .x ∈ Δx ⊂ X, for which Δx → XΔx be the Kodaira’s Lemma holds. As in Lemma 8.7 we let .pΔx : X def

universal covering of .XΔx = P −1 (Δx ). Let .s˜ Δx be a lift of .s Δx = s Λ ∩ XΔx

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

Δx . By Lemma 8.7 there exists (after perhaps shrinking .Δx ) a holomorphic to .X bundle isomorphism  .

Δx , (P ◦ pΔx )|X Δx , s˜ Δx , Δx X



  → Δx × C, pr1 , Δx × {0}, Δx .

(8.38)

The bundle isomorphism is given by the identity mapping on .Δx and a holomorphic Δx to .Δx × C, which maps the fiber of the first bundle over mapping .Φ˜ Δx from .X each point .x  ∈ Δx to the fiber of the second bundle over the same point .x  and maps the section of the first bundle to the section of the second bundle. Cover X by a locally finite set of such discs .Δj . For each .Δj we consider Δj → XΔj . The bundle the covering transformations of the projection .pΔj : X isomorphism conjugates the group of these covering transformations to a group of fiber preserving transformations of .Δj × C with free and properly discontinuous action. The conjugated group acts on each fiber .{x} × C as a lattice .Λj (x), and the lattices depend holomorphically on .x ∈ Δj . Taking the quotient, we obtain for each j a holomorphic bundle isomorphism     XΔj , PΔj , s Δj , Δj → (Δj ×C)Λj , pr1 , (Δj ×{0})Λj , Δj .

.

(8.39)

The bundle isomorphism (8.39) is given by the identity mapping from .Δj to itself and by a biholomorphic mapping Φj : XΔj → (Δj × C)Λj

.

(8.40)

that maps the fiber of the first bundle over any point .x ∈ Δj to the fiber of the second bundle over the same point x and maps the section of the first bundle to the section of the second bundle. Take j and k so that .Δj ∩ Δk = ∅. Then .Φj ◦ (Φk )−1 is a fiber preserving biholomorphic mapping from .((Δj ∩ Δk ) × C)Λk onto .((Δj ∩ Δk ) × C)Λj . Moreover, .Φj ◦ (Φk )−1 maps .((Δj ∩ Δk ) × {0})Λk onto .((Δj ∩ Δk ) × {0})Λj . The mapping .Φj ◦ (Φk )−1 lifts to a fiber preserving biholomorphic mapping .Aj,k of .(Δj ∩ Δk ) × C onto itself, that takes .(Δj ∩ Δk ) × {0} onto itself. Then Aj,k (x, ζ ) = (x, αj,k (x) · ζ ), (x, ζ ) ∈ (Δj ∩ Δk ) × C

.

(8.41)

for a nowhere vanishing holomorphic function .αj,k on .(Δj ∩ Δk ). Moreover, the mapping .Aj,k takes each .{x} × Λk (x) ⊂ {x} × C onto .{x} × Λj (x) ⊂ {x} × C. The .αj,k form a Cousin II cocycle on X (see [40] and Appendix A.1). Since X is a finite open Riemann surface, the Cousin II problem is solvable (see [27]), i.e. there exist holomorphic nowhere vanishing functions .αj on .Δj , such that on .(Δj ∩ Δk ) the equality αj,k = αj · αk−1

.

8.6 Complex Analytic Families of Canonical Tori

267

holds. Then for each .j, k with .Δj ∩ Δk = ∅ the equality .(αj (x))−1 Λj (x) = (αk (x))−1 Λk (x), x ∈ Δj ∩Δk holds. Hence, we obtain a well-defined holomorphic −1 family of lattices .Λ on X, .Λ(x)  = (αj (x)) Λj (x), x ∈ Δj . For this family of lattices we consider the bundle . (X × C)Λ, PΛ , s Λ , X of the form (8.37). Let .cj−1 : (Δj × C)Λj → (Δj × C)Λ be the biholomorphic mapping, that lifts to the biholomorphic self-mapping .(x, ζ ) → (x, (αj (x))−1 ζ ) of .Δj × C, which maps each .{x} × Λj (x) onto .{x} × (αj (x))−1 Λj (x) = {x} × Λ(x). Put .Φj = cj−1 Φj on .XΔj . Then on each non-empty intersection .(Δj ∩ Δk ) the equality .Φj (Φk )−1 = cj−1 Φj (ck−1 Φk )−1 = cj−1 (Φj (Φk )−1 )ck holds. The latter mapping lifts to (x, ζ ) → (x, αj (x)−1 αj,k (x)αk (x)ζ ) = (x, ζ ),

.

(x, ζ ) ∈ (Δj ∩ Δk ) × C .

Hence, we obtain .Φj = Φk on .XΔj ∩XΔk . The mapping .Φ  : X → (X×C)Λ, for which .Φ  (x) = Φj (x) for .x ∈ XΔj , is well defined and determines a holomorphic isomorphism from the original bundle to a bundle of the form (8.37). Proposition 8.1 is proved.

Similar arguments as used in the proof of Proposition 8.1 give the following statement. Proposition 8.1. Each holomorphic elliptic fiber bundle over a finite open Riemann surface X is holomorphically isomorphic to a holomorphic bundle that admits a holomorphic section. Proof Cover X by small discs .Δj on which a holomorphic section .s Δj of the original bundle can be chosen. For each j we consider the holomorphic bundle isomorphism (8.39), and obtain a holomorphic isomorphism .Φj of the total space of the bundle on the left onto the total space of the bundle on the right of (8.39), that maps fibers of the first bundle to fibers of the second one and the chosen section .s Δj of the first bundle to the section .(Δj × {0})Λj of the second one. The isomorphism .Φj : XΔj → (Δj × C)Λj lifts to a holomorphic Δj → Δj × C, that maps a lift .s˜ Δj of .s Δj to .Δj × {0}. For isomorphism .Φ˜ j : X   −1 on . (Δ ∩ Δ ) × {C} Λ lifts to .Δj ∩ Δk = ∅ the mapping .Φj,k = Φj ◦ (Φk ) j k k a fiber preserving biholomorphic mapping .Φ˜ j,k = Φ˜ j ◦ Φ˜ k−1 of .(Δj ∩ Δk ) × C onto itself. In contrast to the situation in Proposition 8.1 we have no control about the   image of . (Δj ∩ Δk ) × {0} Λk under the mapping .Φj,k = Φj ◦ (Φk )−1 . Instead of Eq. (8.41) we obtain the equation Φ˜ j,k (x, ζ ) = (x, aj,k (x) + αj,k (x) · ζ ), (x, ζ ) ∈ (Δj ∩ Δk ) × C

.

(8.42)

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

for a holomorphic function .aj,k and a nowhere vanishing holomorphic function .αj,k on .Δj ∩ Δk . Since .Φ˜ j,k Φ˜ k,i Φ˜ i,j = Id on .Δj ∩ Δk ∩ Δi , we obtain 

 aj,k (x) + αj,k (x)ak,i (x) + αj,k (x)αk,i (x)ai,j (x)   + αj,k (x)αk,i (x)αi,j (x) · ζ = ζ, x ∈ Δj ∩ Δk ∩ Δi , ζ ∈ C . .

(8.43)

Hence, .αj,k (x)αk,i (x)αi,j (x) ≡ 1, in other words, the .αj,k form a Cousin II cocycle. Since X is an open Riemann surface the Cousin Problem has a solution, i.e. there exist nowhere vanishing holomorphic functions .αj on .Δj such that for .k = j the α equality .αj,k = αjk holds on .(Δj ∩ Δk ). Then by Eq. (8.43) .

ai,j aj,k ak,i + + = 0. αj αk αi

a

In other words, the . αj,k form a Cousin I cocycle. Since X is a Stein manifold the j Cousin Problem is solvable, i.e. there exist holomorphic functions .aj on .Δj such that .

aj aj,k ak = − αj αj αk

(8.44)

on .(Δj ∩ Δk ). Consider for all j the mapping .(x, ζ ) → Aj (x, ζ ) = (x, aj (x) +   aj (x) 1 αj (x)ζ ). The inverse mapping has the form .A−1 j (x, ζ ) = x, − αj (x) + αj (x) ζ . Then by Eq. (8.44) we obtain  ζ  ak (x) + ) (Aj A−1 k )(x, ζ ) = x, aj (x) + αj (x)(− αk (x) αk (x)   = x, aj,k (x) + αj,k (x)ζ = Φ˜ j,k (x, ζ ) .

.

(8.45)

Consider the biholomorphic mapping .Φ˜ j∗ : X˜Δj → Δj ×C which is the composition ˜ ∗ = A−1 ◦ Φ˜ j of the mapping .A−1 : Δj × C ⊃ with .Φ˜ j . By Eq. (8.45) we obtain .Φ j

j

j

−1 ˜ −1 ˜ ˜ −1 Φ˜ j∗ (Φ˜ k∗ )−1 (x, ζ ) = A−1 j Φj Φk Ak (x, ζ ) = Aj Φj,k (Ak Aj )Aj (x, ζ ) = (x, ζ ) .

.

We found biholomorphic mappings .Φ˜ j∗ : X˜Δj → Δj × C, such that on .Δj ∩ Δk the mappings .Φ˜ ∗ and .Φ˜ ∗ coincide. j

k

Each .XΔj is the quotient of .X˜Δj by the group of covering transformations of the covering .pΔj : X˜Δj → XΔj . Conjugating for each j the group of covering transformations by the mapping .Φ˜ j∗ , we obtain a group acting on .Δj × C which can be identified with a holomorphic family of lattices .Λj on .Δj . The mapping

8.7 Special (0, 4)-Bundles and Double Branched Coverings

269

Φ˜ j∗ descends to a biholomorphic mapping .Φj∗ : XΔj → (Δj × C)Λj . Since on .Δj ∩ Δk the mappings .Φ˜ j∗ and .Φ˜ k∗ coincide, the lattices .Λj and .Λk coincide on .Δj ∩ Δk , and .Φj∗ = Φk∗ on .Δj ∩ Δk . For the lattice .Λ on X that equals .Λj on .Δj we obtain a bundle .((X × C)Λ, PΛ , s Λ , X) of the form (8.37), and the mappings .Φj define a holomorphic bundle isomorphism from the original bundle to .((X × C)Λ, PΛ , X). The proposition is proved.

.

8.7 Special (0, 4)-Bundles and Double Branched Coverings In Sect. 8.9 we will study the Gromov-Oka Principle for elliptic fiber bundles, or, in other words, for families of Riemann surfaces of type .(1, 1) (see Problems 8.1 and Problem 8.1. ). This will be done by a reduction to the Gromov-Oka Principle in the case of special .(0, 4)-bundles, which we will prepare now. We will represent tori with a distinguished point as double branched coverings over .P1 with a set of four branch points. This is done as follows. ˚ ∪ {∞}. The ˚ = {z1 , z2 , z3 } ∈ C3 (C)S3 , and put .E def Take any element .E = E set   ˚E def .Y = (z, w) ∈ C2 : w 2 = 4 (z − z1 )(z − z2 )(z − z3 ) (8.46) ˚E has a neighis a one-dimensional complex submanifold of .C2 . Each point of .Y ˚ bourhood in .YE on which one of the functions, z or w, defines local holomorphic coordinates. The mapping ˚E (z, w) → z ∈ C Y

(8.47)

.

˚ is a branched holomorphic covering of .C with branch locus .E. ˚ Consider the 1-point compactification .YE of .YE . The complex structure on it is ˚ r be the subset of .Y ˚E where .|z| > r for a large positive obtained as follows. Let .Y E z number r. On this set .| w | is small. Put .(˜z, w) ˜ = ( 1z , wz ) on a small neighbourhood ˚ r . In these coordinates the equation for .Y ˚ r becomes in .C2 of .Y E E 1 , w˜ 2 = z˜ 3 4 j =1 (1 − zj z˜ )

(8.48)

.

˚ r can be identified with the subset and .Y E  .

1 1 (˜z, w) ˜ ∈ C , 0 < |˜z| < : w˜ 2 = z˜ 3 r 4 j =1 (1 − zj z˜ ) 2

 (8.49)

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

˚ r we obtain a complex manifold of .C2 . Adding the point .(˜z, w) ˜ = (0, 0) ∈ C2 to .Y E r r and .Y ˚E form an open cover of the desired compact .Y . The two manifolds .Y E E complex manifold .YE . Denote the point .(0, 0) in coordinates .(˜z, w) ˜ on .YEr by .s ∞ . Each point of .YE has a neighbourhood where one of the functions z, w, .z˜ , or .w˜ defines local holomorphic coordinates. The holomorphic projection .YE → P1 is ˚E , and .(˜z, w) correctly defined by .(z, w) → z, (z, w) ∈ Y ˜ → z˜ , (˜z, w) ˜ ∈ YEr . We 1 1 obtain a double branched covering .YE → P over .P with branch locus equal to def ˚ ∪ {∞}. The manifold .YE is a closed Riemann surface of genus 1. We will .E = E consider it as a closed Riemann surface of genus 1 with distinguished point being ˚ (considered as subset the preimage .s ∞ of .∞ under the branched covering. The set .E of .C) will be called the finite branch locus of the covering .YE → P1 . Let Y be a closed Riemann surface of genus 1 with distinguished point s and let 1 ˚ ∪ {∞} for .E ˚ ⊂ C3 (C)S3 . .Y1 be equal to .P with set of distinguished points .E ˚∪ ∞ Suppose .Pr : Y → Y1 is a double branched covering with branch locus .E and .Pr(s) = ∞. A mapping class .m ∈ M(Y ; s, ∅) is called a lift of a mapping class ˚ if there are representing homeomorphisms .ϕ ∈ m and .ϕ1 ∈ m1 , .m1 ∈ M(Y1 ; ∞, E) such that .ϕ lifts .ϕ1 , i.e. .ϕ1 (Pr(ζ )) = Pr(ϕ(ζ )), .ζ ∈ Y . We define double branched coverings of smooth families of Riemann surfaces with distinguished points as follows. Definition 8.6 Let X be  an oriented  smooth surface (a Riemann surface, respectively). Suppose .F = X , P, s, X is a smooth (complex analytic, respectively) family of Riemann surfaces of type .(1, 1) over X. Let .E ⊂ X × P1 be a smooth (complex, respectively) submanifold of .X × P1 , that intersects each fiber .{x} × P1 ˚x ∪ {∞}) with .E ˚x ⊂ C3 (C)S3 . along a set of distinguished points .Ex = {x} × (E The family .F is called a double branched covering of the   special smooth (holomorphic, respectively) .(0, 4)-bundle . X × P1 , pr1 , E, X if there exists a smooth (holomorphic, respectively) mapping .Pr : X → X×P1 that maps each fiber −1 (x) of the .(1, 1)-family .F onto the fiber .{x} × P1 of the .(0, 4)-bundle over the .P same point x, such that the restriction .Pr : P −1 (x) → {x} × P1 is a holomorphic ˚x ∪ {∞}) of double branched covering with branch locus being the set .{x} × (E 1 distinguished points in the fiber .{x} × P , and .Pr maps the distinguished point −1 (x) in the fiber .P −1 (x) over x to the point .{x} × {∞} in the fiber .sx = s ∩ P 1 .{x} × P of the special .(0, 4)-bundle. We will also write .(X × P1 , pr1 , E, X) = Pr((X , P, s, X)), and call the family 1 .(X , P, s, X) a lift of .(X × P , pr1 , E, X).   Lemma 8.8 Each special holomorphic .(0, 4)-bundle . X × P1 , pr1 , E, X over a Riemann surface X admits a double branched covering by a complex analytic family of Riemann surfaces of type .(1, 1). Each special smooth .(0, 4)-bundle over an oriented differentiable manifold X has a double branched covering by a differentiable family of Riemann surfaces of type .(1, 1). Notice that the statement for the smooth case is true also for families of the mentioned complex manifolds over products .X × I where X is an oriented surface

8.7 Special (0, 4)-Bundles and Double Branched Coverings

271

and I is an interval, in other words, it is true for isotopies of families of such complex manifolds over Riemann surfaces and for isotopies of families of such complex manifolds over orientable smooth surfaces. Proof We prove the statement for the holomorphic case. In the smooth case the dependence on the variable x is only smooth, otherwise the smooth case is treated ˚x , x ∈ X, (with .E ∩ ({x} × similarly as the holomorphic case. Assume the sets .E ˚x ∪ {∞})) are uniformly bounded in .C3 (C)S3 . Consider the set P1 ) = {x} × (E

˚E = Y

.

⎧ ⎪ ⎨ ⎪ ⎩

(x, z, w) ∈ X × C2 : w 2 = 4

 ˚x zj ∈E

(z − zj )

⎫ ⎪ ⎬ ⎪ ⎭

(8.50)

,

equipped with the structure of an embedded complex hypersurface in .X × C2 . ˚E with .z0 ∈ ˚x the pair .(x, w) In a neighbourhood of a point .(x0 , z0 , w0 ) on .Y / E 0 ˚x the pair .(x, z) defines holomorphic defines holomorphic coordinates. If .z0 ∈ E 0 ˚E . The projection .P : Y ˚E → X, coordinates in a neighbourhood of .(x0 , z0 , w0 ) on .Y .P(x, z, w) = x is holomorphic. For some large positive number r we may define the set  r def .YE =

1 1 (x, z˜ , w) ˜ ∈ X × C2 , |˜z| < : w˜ 2 = z˜  r 4 zj ∈E ˚x (1 − zj z˜ )

 .

(8.51)

It is a complex hypersurface of complex dimension two of .X × C2 . Each of its points has a neighbourhood on which either .(x, z˜ ) or .(x, w) ˜ defines holomorphic ˚ r def ˜ → x is holomorphic. The part .Y coordinates. The mapping .(x, z˜ , w) E =    r r ˚E : (x, z˜ , w) ˜ ∈ YE : z˜ = 0 of .YE can be identified with the subset . (x, z, w) ∈ Y  1 ˚E using the transition functions .(x, z, w) → (x, z˜ , w) |z| > r of .Y ˜ = (x, z , wz ). ˚E and .Y r form an open cover of a complex manifold denoted by .YE The sets .Y E equipped with a proper holomorphic submersion .P : YE → X, such that .P −1 (x) is a torus for each .x ∈ X. We proved that .(YE , P, X) is a holomorphic elliptic fiber bundle. Let .s ∞ be the submanifold of .YE that intersects each fiber .P −1 (x) along the distinguished point .sx∞ ∈ YEr that is written in coordinates on .YEr as .(x, 0, 0). Let .Pr : YE → X × P1 be the map that assigns to .(x, z, w) the point .(x, z) (and to .(x, z˜ , w) ˜ the point .(x, z˜ )). By the construction it is clear that the obtained ∞ .(1, 1)-bundle .(YE , P, s , X) is a double branched covering of the .(0, 4)-bundle   1 . X × P , pr1 , E, X whose set of finite distinguished points in the fiber over x ˚x . The double branched covering map is equal to the finite branch locus .{x} × E 1 ∞ ∈ P −1 (x) to the point .Pr : YE → X × P maps the distinguished point .sx .{x} × ∞. ˚x , x ∈ X, are In the general case (i.e. without the assumption that the sets .E uniformly bounded) the statement is proved by considering an exhaustion of X by relatively compact open sets.



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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

  Let . X × P1 , pr1 , E j , X , .j = 0, 1, be two special holomorphic (smooth, respectively) . (0, 4)-bundles that are isotopic through smooth special . (0, 4)-bundles. Then the families .(YE j , Pj , s ∞ j , X), j = 0, 1, of Riemann surfaces of type .(1, 1) are isotopic. Indeed, let I be an open interval containing .[0, 1]. The isotopy of the special .(0, 4)-bundles is given by a bundle   (I × X) × P1 , pr1 , E, I × X

.

over .I × X. Here .E is a smooth submanifold of .(I × X) × P1 such that for each (t, x) ∈ I ×X the intersection of the set .E with the fiber over .(t, x) equals .{(t, x)}× E(t, x) for subsets .E(t, x) that are the union of the point .∞ with an element of 1 .C3 (CS3 ). Moreover, .E ∩ (({j } × X) × P ) = E j , j = 0, 1. The smooth special   1 .(0, 4)-bundle . (I × X) × P , pr1 , E, I × X has a double branched covering by a smooth family of Riemann surfaces of type .(1, 1) which defines the required isotopy for the families .(YE j , Pj , s ∞ j , X), j = 0, 1, of Riemann surfaces of type .(1, 1). Notice that for a non-contractible oriented finite open smooth surface X and  a smooth special .(0, 4)-bundle . X × P1 , pr1 , E, X over X the obtained double branched covering .(YE , P, s ∞ , X) is not the only double branched covering of the ∞ .(0, 4)-bundle (see Sect. 8.8). We will call the bundle .(YE , P, s , X) the canonical   double branched covering of the special .(0, 4)-bundle . X × P1 , pr1 , E, X . .

8.8 Lattices and Double Branched Coverings The Weierstraß .℘-Function Consider the quotient .CΛ of the complex plane by a lattice .Λ. We want to associate to the quotient a double branched covering over the Riemann sphere with covering space being conformally equivalent to .CΛ. The def

standard tool for this purpose is the Weierstraß .℘-function .℘Λ . Put .Λ = aZ + bZ, where a and b are real linearly independent complex numbers. The Weierstraß .℘function,    1 1 1 , ζ ∈ C \ Λ, .℘Λ (ζ ) = + − ζ2 (ζ − a n − b m)2 (a n + b m)2 2 (n,m)∈Z (n,m)=(0,0)

(8.52) is meromorphic on .C, has poles of second order at points of .Λ and is holomorphic on .C \ Λ. It has periods a and b and principal part .ζ → ζ12 at 0. The summation is over all non-zero elements of the lattice. Hence, the function depends only on the lattice, not on the choice of the generators a and b of the lattice. The Weierstraß .℘-function satisfies the following differential equation  2 (℘Λ ) (ζ ) = 4(℘Λ )3 (ζ ) − g2 (Λ)(℘Λ )(ζ ) − g3 (Λ)

.

(8.53)

8.8 Lattices and Double Branched Coverings

273

for complex numbers .g2 (Λ) and .g3 (Λ) that depend only on the lattice. This can be  and taking a proved using the Laurent series expansion near zero of .℘Λ and .℘Λ linear combination that has no pole and is therefore constant. For details see [5]. The numbers .g2 (Λ) and .g3 (Λ) are called the elliptic invariants of .Λ. Denote the zeros of the equation . 4t 3 − g2 (Λ)t − g3 (Λ) = 0 by .e1 (Λ), e2 (Λ), e3 (Λ). It follows by symmetry considerations that the zeros of this equation are the values of .℘Λ at the half-periods . a2 , . b2 , . a+b 2 . Indeed, since the doubly  is odd and has no pole at its half-periods, it vanishes at its periodic function .℘Λ  is of order 3, these are all its zeros in a period-parallelogram. half-periods. Since .℘Λ An argument using that the order of .℘Λ is 2, shows that all roots of .℘Λ are distinct. The differential equation for .℘Λ becomes  2 (℘Λ ) (ζ ) = 4(℘Λ (ζ ) − e1 (Λ))(℘Λ (ζ ) − e2 (Λ))(℘Λ (ζ ) − e3 (Λ)) ,

.

(8.54)

where e1 (Λ) = ℘Λ

.

a  2

,

  b e2 (Λ) = ℘Λ , 2

 e3 (Λ) = ℘Λ

a+b 2

 (8.55)

are the values of .℘Λ at points which are contained in the lattice . a2 Z + b2 Z but are not contained in the lattice .aZ + b Z. The half-periods can be found from the elliptic invariants. The integral  .

∞

1

(4t 3 − g2 t − g3 )− 2 dt

(8.56)

y

with constants .g2 = g2 (Λ) and .g3 = g3 (Λ) defines a holomorphic function in y on any simply connected domain in the complex plane which does not contain a zero of the function .t → 4t 3 − g2 t − g3 . The derivative in z of the function defined by the integral (8.56) with .y = ℘Λ (z) equals .

 (z) ℘Λ

 1 , 4℘Λ (z)3 − g2 (Λ)℘Λ (z) − g3 (Λ) 2

(8.57)

which equals either 1 or .−1 on each suitable domain for z. This shows that (after multiplying by .±1) the integral (8.56) provides an inverse of .℘Λ . Knowing .e1 , e2 , and .e3 we obtain the half-periods. Put for instance .e1 = 1, e2 = −1 and .e3 = 0. Then .g2 = 4, .g3 = 0. The integral ∞ 3   − 21 . dt defines a holomorphic function on .C \ i[0, ∞) ∪ [−1, 1] . y (4t − 4t) Its continuous extension .ω1 to 1 is positive, and its continuous extension .ω2 to .−1 equals .iω1 . Hence, the Weierstrass .℘-function corresponding to the periods .2ω1 and .2ω2 = i2ω1 takes the values 1, .−1 and 0 at the points .ω1 , .ω2 , .ω1 + ω2 , respectively.

274

8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles def

The lattice .Λτ = Z + τ Z often plays a special role. If the pair 1 and .τ generates the lattice, then also the pair 1 and .−τ generates it. Hence, we may assume that def

Imτ > 0. We will write .℘τ = ℘Λτ ,

.

1 .℘τ (ζ ) = + ζ2



 (n,m)∈Z2 (n,m)=(0,0)

1 1 − (ζ − n − m τ )2 ( n + τ m)2

 ,

ζ ∈ C \ Λτ . (8.58)

The function .℘τ satisfies the differential equation (℘τ )2 (ζ ) = 4(℘τ (ζ ) − e1 (τ ))(℘τ (ζ ) − e2 (τ ))(℘τ (ζ ) − e3 (τ )) ,

.

(8.59)

where   1 .e1 (τ ) = ℘τ , 2

e2 (τ ) = ℘τ

τ  2

 ,

e3 (τ ) = ℘τ

1+τ 2

 (8.60)

are the values of .℘τ at points which are contained in the lattice . 12 Z + τ2 Z but are not contained in the lattice .Z + τ Z. An arbitrary lattice   .Λ can be written as .Λ = α(Z + τ Z). By (8.52) the equality ζ −2 ℘ .℘Λ (ζ ) = α τ α holds, hence {e1 (Λ), e2 (Λ), e3 (Λ)} = {α −2 e1 (τ ), α −2 e2 (τ ), α −2 e3 (τ )} .

.

(8.61)

The Double Branched Covering Defined by the Weierstraß ℘-Function Let Λ   ζ  (ζ ) = −2 be an arbitrary lattice. Put z(ζ ) = ℘Λ (ζ ) = α ℘τ α and w(ζ ) = ℘Λ   α −3 ℘τ αζ . The mapping      C\Λ ζ → z(ζ ), w(ζ ) = ℘Λ (ζ ), ℘Λ (ζ ) ∈ C2

.

(8.62)

descends to a conformal mapping ωΛ from the punctured torus C\ΛΛ onto the ˚(Λ) of C2 , complex hypersurface Y def

˚(Λ) = ωΛ : (C \ Λ)Λ → Y

.

3    (z, w) ∈ C2 : w 2 = 4 (z − ej (Λ)) . j =1

(8.63) To see this we notice first that the mapping ωΛ is one-to-one. Indeed, the mapping ℘Λ descends to a holomorphic mapping pΛ : (C \ Λ)Λ → C, such that ℘Λ = pΛ ◦ p .

.

(8.64)

8.8 Lattices and Double Branched Coverings

275

Here p denotes the projection p : C \ Λ → (C \ Λ)Λ. The mapping pΛ is 2 to 1. Further, ℘Λ (ζ ) = ℘Λ (−ζ ). Hence, the preimage under ℘Λ of each point in C \ {0}  (ζ ) = −℘  (−ζ ), the equals ({ζ } + Λ) ∪ ({−ζ } + Λ) for some ζ ∈ C. Since, ℘Λ Λ mapping ωΛ is one-to-one. Moreover, the mapping (8.62) is locally conformal. Indeed, if ℘Λ (ζ0 ) =  (ζ ) = 0 and the mapping ζ → ℘ (ζ ) is conformal in a ej (Λ), j = 1, 2, 3, then ℘Λ 0 Λ neighbourhood of ζ0 . Suppose ℘Λ (ζ0 ) = ej (Λ). The differential Eq. (8.54) implies   3  ℘  = 2℘   that ℘Λ =1 k= (℘Λ − ek (Λ)). Hence, ℘Λ (ζ0 ) = 2 k=j (ej (Λ) − Λ Λ  (ζ ) is conformal in a ek (Λ)) = 0. Hence, in this case the mapping ζ → ℘Λ neighbourhood of ζ0 . ˚(Λ) (see (8.63)) is the covering space of the double branched covering The set Y ˚(Λ) (z, w) → z ∈ C Y

(8.65)

 def  ˚ = e1 (Λ), e2 (Λ), e3 (Λ) . BL(Λ)

(8.66)

.

of C with branch locus .

def ˚ ˚(Λ) = Y ˚BL(Λ) holds. With BL(Λ) = BL(Λ) ∪ {∞} the equality Y For a family of lattices Λ(z) depending holomorphically on a complex parameter ˚ z the sets BL(Λ(z)) (see (8.66)) depend holomorphically on z. Indeed, the ej (Λ) are the values of ℘Λ at the half-periods of the lattice. The half-periods of the lattice depend holomorphically on z, and also the Weierstraß function ℘Λ depends holomorphically on z (see Eq. (8.52)). If the family of lattices is merely smooth, then the set of finite branch points in the fiber over x depends smoothly on the point x ∈ X. ˚BL(Λ) (see Sect. 8.7) is the covering The one-point compactification YBL(Λ) of Y manifold of a double branched covering of the Riemann sphere P1 with branch locus ˚ ˚ BL = BL(Λ) ∪ {∞} and finite branch locus BL(Λ). We associate to the Riemann surface CΛ the conformally equivalent Riemann surface YBL(Λ) which we also denote by Y (Λ). Notice that ωΛ extends to a conformal mapping between the closed Riemann surfaces CΛ and Y (Λ).

Lifts of Mappings to the Double Branched Covering and the Involution For each lattice Λ the mapping C ζ → −ζ takes Λ onto itself. This mapping descends to an involution of CΛ, i.e. to a self-homeomorphism ιΛ of CΛ such that ι2Λ = id. Formula (8.52) implies the following equality .

      ℘Λ (−ζ ) , ℘Λ (−ζ ) = ℘Λ (ζ ) , −℘Λ (ζ ) .

(8.67)

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

−1 Conjugate the restriction ιΛ | (C \ Λ)Λ by the conformal mapping ωΛ . ˚ We obtain a self-homeomorphism of Y (Λ) which we denote by ι. The mapping ι satisfies the equality

ι(z, w) = (z, −w) .

.

(8.68)

The involution ι extends to an involution of Y (Λ), denoted also by ι. By (8.68) the involution ι fixes the projection to P1 of each point of the double branched covering space Y (Λ), and interchanges the sheets over each point. Hence, it fixes each of the three finite branch points, it also fixes ∞, and it does not fix any other point. Let  ϕ be any real linear self-homeomorphism of C that maps Λ onto itself, and let ϕ be the induced mapping on CΛ. Then ϕ commutes with ιΛ . Indeed,  ϕ (−ζ ) = − ϕ (ζ ), ζ ∈ C. Each mapping class m in M(CΛ; 0Λ, ∅) can be represented by a selfhomeomorphism of CΛ which commutes with ιΛ . Indeed, M(CΛ; 0Λ, ∅) contains a mapping ϕ that lifts to a real linear self-map  ϕ of C which maps Λ onto itself. This mapping ϕ commutes with ιΛ . As a corollary, each mapping class m in M(Y (Λ); s ∞ , ∅) ∼ = M(CΛ; 0Λ, ∅) 1 ˚ ˚ def ˚ = BL(Λ) ⊂ is a lift of a mapping class m1 ∈ M(P ; {∞}, E) for the set E  C3 (C)S3 . This can be seen as follows. If ψ represents a mapping class −1 m ∈ M(Y (Λ); s ∞ , ∅), then ωΛ ◦ ψ  ◦ ωΛ represents a mapping class m in M(CΛ; 0Λ, ∅). Let ϕ be the homemorphism in this class that commutes with −1 ιΛ . Then ωΛ ◦ ϕ ◦ ωΛ ∈ m commutes with ι. Suppose ψ represents a mapping class m ∈ M(Y (Λ); s ∞ , ∅) and commutes ˚. In coordinates (z, w) on Y ˚(Λ) we ˚ the restriction of ψ to Y with ι. Denote by ψ write ˚2 (z, w)) . ˚ w) = (ψ ˚1 (z, w), ψ ψ(z,

.

(8.69)

˚ commutes with ι we obtain by (8.68) Since ψ ˚1 (z, −w), ψ ˚2 (z, −w)) = ψ ˚ ◦ ι(z, w) = ι ◦ ψ(z, ˚ w) = (ψ ˚1 (z, w), −ψ ˚2 (z, w)) . (ψ (8.70) ˚(Λ) is determined ˚1 (z, w) = ψ ˚1 (z, −w). Since each point in (z, w) ∈ Y Hence ψ ˚1 (z, w) depends only on the coordinate by z and the sign of w, this means that ψ z ∈ C, not on the sheet (determined by w). Further, if ι fixes (z, w), then w = 0, ˚2 (z, w) = ψ ˚2 (z, −w) = −ψ ˚2 (z, w). In other words, and by (8.70) for w = 0, ψ ˚ ˚ if ι fixes (z, w), then it also fixes ψ(z, w). Hence, ψ maps the set of finite branch ˚ points (the preimage of BL(Λ) under the branched covering map) onto itself, and its extension ψ maps the preimage s ∞ of ∞ to itself. We saw, that ψ induces ˚ a self-homeomorphism Pr(ψ) of P1 in the class M(P1 ; {∞}, BL(Λ)). For this 2 ˚ ˚ ˚ class Pr(ψ)|C = Pr(ψ), Pr(ψ)(z) = ψ1 (z, w), (z, w) ∈ C . We call Pr(ψ) the projection of ψ. The mapping class m is a lift of the mapping class m1 of ψ1 . .

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277

Notice that the provided arguments imply also the following fact. For two mapping classes m, m ∈ M(Y (Λ); s ∞ , ∅) the equality Pr(m m ) = Pr(m)Pr(m ) holds. Indeed, take representing maps ψ 1 and ψ 2 , that commute with the involution ˚2 depend only on the first variable. Since ˚1 and ψ ι. Then ψ 1 1    1 2  ˚2 (z, w), ψ ˚ (z, w), ψ ˚2 (z, w) = ψ ˚2 (z, w) , ψ ˚ ψ ˚2 (z, w) , ˚1 ◦ ψ ˚1 (ψ ψ 1 1 2 2 1 2

.

the first component of ψ 1 ◦ ψ 2 depends only on the first variable z. The equality Pr(ψ 1 ◦ ψ 2 ) = Pr(ψ 1 ) ◦ Pr(ψ 2 ) follows. ˚ be an element of C3 (C)S3 and E = E ˚ ∪ {∞}. Any mapping Let again E 1 ˚ class m1 ∈ M(P ; {∞}, E) has exactly two lifts m± ∈ M(YE ; s ∞ , ∅). Here YE is the double branched covering of P1 with branch locus E, and s ∞ is the branch point over ∞. Indeed, for any representative ψ1 of m1 there are exactly two selfhomeomorphisms of the double branched covering that lift ψ1 . They are obtained as follows. Cut P1 along two disjoint simple curves γ1 and γ2 , that join disjoint pairs of points of E. ψ1 maps the pair of curves γ1 and γ2 to another pair of curves γ1 and γ2 joining (maybe different) disjoint pairs of points of E. The double branched covering over P1 is obtained in two different ways, either gluing two sheets of P1 \ (γ1 ∪γ2 ) crosswise together, or gluing two sheets of P1 \(γ1 ∪γ2 ) crosswise together. ψ1 maps P1 \(γ1 ∪γ2 ) homeomorphically onto P1 \(γ1 ∪γ2 ). Consider the mapping that takes the first sheet of P1 \ (γ1 ∪ γ2 ) onto the first sheet of P1 \ (γ1 ∪ γ2 ) and the second sheet of P1 \ (γ1 ∪ γ2 ) onto the second sheet of P1 \ (γ1 ∪ γ2 ) and lifts ψ1 | P1 \ (γ1 ∪ γ2 ). This mapping extends to a self-homeomorphism ψ of the double branched covering that lifts ψ1 . There is exactly one more self-homeomorphism of the double branched covering that lifts ψ1 . This self-homeomorphism maps the first sheet of P1 \ (γ1 ∪ γ2 ) onto the second sheet of P1 \ (γ1 ∪ γ2 ) and the second sheet of P1 \ (γ1 ∪ γ2 ) onto the first sheet of P1 \ (γ1 ∪ γ2 ). In other words, there are two lifts of ψ1 to the double branched covering of P1 with branch locus E, and they differ by involution. We proved the following lemma. Lemma 8.9 Each mapping class m ∈ M(CΛ; 0Λ, ∅) is the lift of a map˚ Vice versa, each m1 ∈ ping class m1 = Pr(m) ∈ M(P1 ; {∞}, BL(Λ)). 1 ˚ M(P ; {∞}, BL(Λ)) has two lifts m± ∈ M(CΛ; 0Λ, ∅). The lifts m± differ by involution. For two mapping classes m, m ∈ M(CΛ; 0Λ, ∅) the equality Pr(mm ) = Pr(m)Pr(m ) holds. Lifts of Special (0, 4)-Bundles to Elliptic Fiber Bundles with a Section The following proposition relates Theorem 8.2 to the respective Theorem 8.3 for elliptic bundles that will be formulated below. Proposition 8.2 Let X be a connected Riemann surface (connected oriented smooth surface, respectively) of genus g with m ≥ 1 holes with base point x0 and curves denoted by γj that represent a standard system of generators ej ∈ π1 (X, x0 ).

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

(1) Each complex analytic (differentiable, respectively) family of Riemann surfaces of type (1, 1) over X is holomorphically (smoothly, respectively) isomorphic to the canonical double branched covering F of a special holomorphic (smooth, respectively) (0, 4)-bundle F1 over X. The monodromy of the bundle F along each γj is a lift of the respective monodromy of the bundle F1 . (2) Vice versa, for each special holomorphic (special smooth, respectively) (0, 4)bundle over X and each collection mj of lifts of the 2g + m − 1 monodromy j mapping classes m1 of the bundle along the γj there exists a double branched covering by a complex analytic (differentiable, respectively) family of Riemann surfaces of type (1, 1) with collection of monodromy mapping classes equal to the mj . For each given holomorphic (smooth, respectively) special (0, 4)bundle over X there are up to holomorphic (smooth, respectively) isomorphisms exactly 22g +m−1 holomorphic (smooth, respectively) families of Riemann surfaces of type (1, 1) that lift the (0, 4)-bundle. (3) A lift of a special (0, 4)-bundle is reducible if and only if the special (0, 4)bundle is reducible. Proof We start with the proof of the first statement of the proposition. Consider a complex analytic (smooth, respectively) family of Riemann surfaces of type (1, 1) over X. By Lemma 8.4 and Proposition 8.1 we may assume that the family has the form (XΛ , PΛ , s Λ , X) for a holomorphic (smooth, respectively) family of lattices Λ(x), x ∈ X. ˚ We consider the complex (smooth, respectively) submanifold BL(Λ) of X × C ˚ that intersects each fiber {x} × C along the set {x} × BL(Λ(x)) (see Eqs. (8.55) and (8.66)) and the complex (smooth, respectively) submanifold BL(Λ) of X × P1 that intersects each fiber {x} × P1 along the set {x} × BL(Λ(x)) with BL(Λ(x)) = ˚ BL(Λ(x)) ∪ {∞}. We prove first that the holomorphic (smooth, respectively) family (XΛ , PΛ , s Λ , X)

.

is holomorphically isomorphic (isomorphic as a smooth family of Riemann surfaces, respectively) to the canonical double branched covering (YBL(Λ) , P, s ∞ , X)

.

(see Sect. 8.7) of the special (0, 4)-bundle (X × P1 , pr1 , BL(Λ), X). To see this we ˚BL(Λ ) is a surrecall that for each x ∈ X the mapping ωΛx : (C \ Λx )Λx → Y x jective conformal mapping that depends holomorphically (smoothly, respectively) on x (see equalities (8.62) and (8.63)). Hence, the mapping defines a holomorphic ˚BL(Λ) , that maps the (smooth, respectively) homeomorphism from XΛ \ s Λ onto Y fiber over x of the first bundle conformally onto the fiber over x of the second bundle. The set YBL(Λ) is obtained as in Sect. 8.7 by “adding a point to each fiber”. As in Sect. 8.7 the mapping extends to a holomorphic (smooth, respectively) homeomorphism XΛ → YBL(Λ) , that maps the fiber over x conformally onto the fiber over

8.8 Lattices and Double Branched Coverings

279

x. The extension maps s Λ to s ∞ . We proved that the bundles (XΛ , PΛ , s Λ , X) and (YBL(Λ) , P, s ∞ , X) are holomorphically isomorphic (isomorphic as smooth families of Riemann surfaces, respectively). Hence, each holomorphic (smooth, respectively) (1, 1)-family over a finite open Riemann surface is holomorphically isomorphic (smoothly isomorphic), and in particular, isotopic, to the canonical double branched covering of a holomorphic (smooth, respectively) (0, 4)-bundle. We identify the mapping class groups in the fiber over the base point x0 of the bundles (XΛ , PΛ , s Λ , X) and (YBL(Λ) , P, s ∞ , X) using the extension of the isomorphism ωΛ0 to the closed fiber CΛ0 . Having in mind this identification, we prove now that the monodromy mapping class along each γj of the (1, 1)-bundle (XΛ , PΛ , s Λ , X) is a lift of the monodromy mapping class along γj of the special (0, 4)-bundle (X × P1 , pr1 , BL(Λ), X). Parameterise γj by the unit interval [0, 1]. def

Write Λt = Λ(γj (t)) = a(t)Z + b(t)Z, t ∈ [0, 1], for smooth functions a and b on [0, 1]. For t ∈ [0, 1] we denote by  ϕ t the real linear self-homeomorphism of C that maps a(0) to a(t) and b(0) to b(t). Let ϕ t be the homeomorphism from the fiber over γj (0) onto the fiber over γj (t) of the (1, 1)-bundle that lifts to  ϕt . 0 1 −1 Then ϕ = Id and (ϕ ) represents the monodromy mapping class of the (1, 1)˚(Λ0 ) → Y ˚(Λt ) be obtained from the ˚t : Y bundle (XΛ , PΛ , s Λ , X) along γj . Let ψ commutative diagram

.

where ˚ ϕ t = ϕ t |(C \ Λ0 )Λ0 . ˚(Λ0 ) of the coordinates (z, w) on C2 . Let ι0 be the We use the restriction to Y ˚ ˚(Λt ). Choose respective coordinates involution on Y (Λ0 ) and ιt the involution on Y t t ˚ ˚(Λ0 ). The ˚ ˚ ˚t (z, w)), (z, w) ∈ Y on Y (Λt ) and write ψ (z, w) = (ψ1 (z, w), ψ 2 t t t ϕ t . From equality  ϕ (−ζ ) = − ϕ (ζ ), ζ ∈ C, can be written as ˚ ϕ ιΛ0 = ιΛt ˚ −1 −1 the diagram we obtain with ιt = ωΛt ιΛt (ωΛt ) , ι0 = ωΛ0 ιΛ0 (ωΛ0 ) ˚t ι0 = ιt ψ ˚t . ψ

.

(8.71)

˚t (z, −w), ψ ˚t (z, −w)) = (ψ ˚t (z, w), −ψ ˚t (z, w)). In coordinates (z, w) this means (ψ 1 2 1 2 ˚t (z, −w), so that ψ ˚t depends only on the coordinate ˚t (z, w) = ψ Hence, ψ 1 1 1 ˚ z ∈ C. Moreover, if ι0 fixes (z, w) (equivalently, if z ∈ BL(Λ 0 ) ), then w = 0, t t t ˚ ˚ ˚ ˚t (z, w) (equivalently, hence ψ2 (z, w) = ψ2 (z, −w) = −ψ2 (z, w), i.e. ιt fixes ψ ˚ ˚t of C maps the set ˚t (z, w) ∈ BL(Λ ψ t )). We saw that the self-homeomorphism ψ 1 1 ˚ ˚ ˚t extends to a self-homeomorphism ψ t of BL(Λ 0 ) onto the set BL(Λt ). Each ψ 1 1 def ˚ P1 that maps the set of distinguished points BL(Λ0 ) = BL(Λ 0 ) ∪ {∞} onto the def 1 ˚ set of distinguished points BL(Λt ) = BL(Λ t ) ∪ {∞}. Hence, ψ1 represents the

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

monodromy mapping class along γj of the special (0, 4)-bundle with set of finite ˚ ˚1 = (ψ ˚1 , ψ ˚1 ) distinguished points BL(Λ t ) in the fiber over γj (t). The mapping ψ 1 2 −1 is equal to ωΛ0 ◦ ˚ ϕ 1 ◦ ωΛ0 . Identifying the mapping class groups of CΛ0 with the mapping class group of YBL(Λ0 ) by the isomorphism induced by ωΛ0 , we may identify the monodromy of the bundle (XΛ , PΛ , s Λ , X) along γj with the mapping ˚1 = (ψ ˚1 , ψ ˚1 ). The mapping ψ 1 is the projection class of the extension ψ 1 of ψ 1 2 1 1 of ψ under the double branched covering from CΛ0 with distinguished point 0Λ0 onto P1 with distinguished points BL(Λ0 ) ∪ {∞}. We proved that the monodromy mapping class of the bundle (XΛ , PΛ , s Λ , X) along each γj is a lift of the respective monodromy mapping class of the bundle (X × P1 , pr1 , BL(Λ), X). The first statement is proved. We prove now the second statement. Consider the special holomorphic, or  def  smooth, respectively, (0, 4)-bundle F1 = X × P1 , pr1 , E, X . The complex or smooth, respectively, submanifold E of X × P1 intersects each fiber {x} × P1 along ˚x ∪ {∞}), where E ˚x ⊂ C3 (C)S3 . In Sect. 8.7 we the set {x} × Ex , with Ex = (E obtained the canonical lift (YE , P, s ∞ , X). Denote the monodromy mapping class j j along γj of the canonical lift by m+ . Recall that each mapping class m1 has two lifts j and they differ by involution. Let m− be the mapping class whose representatives j differ from those of m+ by composition with the involution ι of the fiber over the base point. Take any subset J of {1, 2, . . . , 2g + m − 1}. The lift of the (0, 4)-bundle whose j j monodromy mapping class along γj equals m+ if j ∈ / J , and m− if j ∈ J , is P

obtained as follows. Let U˜ be the subset of the universal covering X˜ −−→ X that was used in Sect. 8.2 in the proof of Theorem 8.1, and let D, and D˜ j , j = 0, . . . , be as in Sect. 8.2. Consider the lift of the bundle (YE , P, s ∞ , X) to X˜ and restrict the ˜ For each point x ∈ D we lifted bundle to U˜ . Denote the obtained bundle on U˜ by F. let x˜j ∈ D˜ j , j = 0, . . . , be the points with P(x˜j ) = x. We identify the fiber of F˜ over x˜j , j = 0, . . . , with the fiber Yx , x = P(x˜j ), of the canonical double branched ˚x is obtained from Yx by removing the point of s ∞ covering of the (0, 4)-bundle. Y from Yx . Glue for each x ∈ D and each j = 1, . . . , the fiber of F˜ over x˜j to the fiber of F˜ over x˜0 using the identity if j ∈ / J and the mapping ι if j ∈ J . More detailed, the gluing mapping of the punctured fibers in the case j ∈ J equals ˚x . (x˜j , z, w) → (x˜0 , z, −w), x ∈ D, (z, w) ∈ Y

.

(8.72)

Since the gluing mappings are holomorphic (smooth, respectively), we obtain a complex analytic (smooth, respectively) family over X of double branched coverings of C. Extend the family to a complex analytic (smooth, respectively) family of double branched coverings of P1 over X. This can be done as in Sect. 8.7 by considering an exhausting sequence of relatively compact open subsets Xn of X and constructing the required extension for the restriction of the bundle to

8.8 Lattices and Double Branched Coverings

281 j

each Xn . We obtained for each collection of lifts of the m1 a complex analytic (smooth, respectively) family of Riemann surfaces of type (1, 1) over X with given fiber over the base point and with monodromy mapping classes equal to this collection. It is easy to see that two holomorphic (smooth, respectively) (1, 1)bundles that lift a given (0, 4)-bundle over a finite open Riemann surface (smooth oriented surface, respectively) are holomorphically isomorphic (smoothly isomorphic, respectively), if and only if their monodromies coincide. Hence, for each given holomorphic (smooth, respectively) special (0, 4)-bundle over X there are up to holomorphic (smooth, respectively) isomorphisms exactly 22g +m−1 holomorphic (smooth, respectively) families of Riemann surfaces of type (1, 1) that lift the (0, 4)bundle. The second statement is proved. It remains to prove the third statement. Let Pr : (X , P, s, X) → (X × P1 , pr1 , E, X)

.

be a double branched covering of a (0, 4)-bundle by a (1, 1)-family of Riemann surfaces. If the special (0, 4)-bundle is reducible, then there is a simple closed curve γ that divides the fiber P1 over the base point x0 into two connected components, each of which contains two distinguished points, such that the following holds. The curve γ is mapped by each monodromy mapping class of the (0, 4)-bundle to a curve that is free homotopic to γ . The preimage of the closed curve γ under the covering map consists of two simple closed curves γ˜1 and γ˜2 , that are homotopic to each other in P −1 (x) \ s, and each of the two curves cuts the torus into an annulus. Each monodromy mapping class of the (1, 1)-bundle, being a lift of the respective monodromy mapping class of the (0, 4)-bundle, takes γ˜1 to a curve that is free homotopic to γ˜1 (and is also free homotopic to γ˜2 ). Hence, the (1, 1)-bundle is also reducible. Suppose now that a double branched covering F = (X , P, s, X) of a special (0, 4)-bundle Pr(F) = (X × P1 , pr1 , E, X) is reducible and prove that the (0, 4)bundle is reducible. Let x0 ∈ X be the base point of X, and let the fiber of the bundle F over x0 be CΛ with distinguished point 0Λ. Any admissible system of curves on the torus CΛ with a distinguished point contains exactly one curve. Let γ0 be a simple closed curve in (C \ Λ)Λ that reduces all monodromy mapping classes of the bundle F. The curve γ0 is free isotopic in the closed torus CΛ to a curve γ with base point 0Λ that lifts under the covering map P : C → CΛ to the straight line segment that joins 0 with another lattice point λ in Λ. Since γ0 is free isotopic in CΛ to a simple closed curve on the torus, λ is a primitive element of the lattice. Indeed, with Λ = {na + mb : n, m ∈ Z} for real linearly independent complex numbers a and b we get λ = n(λ)a + m(λ)b, where n(λ) and m(λ) are relatively prime integer numbers. Then there is another element λ = n(λ )a + m(λ )b ∈ Λ such that .

! ! ! n(λ) n(λ ) ! ! ! !m(λ) m(λ )! = 1 .

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

The two elements λ and λ generate Λ. Multiplying Λ by a non-zero complex number and changing perhaps λ to −λ , we may assume that λ = 1 and λ = τ with Imτ > 0. After similar changes of all fibers we obtain an isomorphic bundle for which the fiber over x0 equals CΛτ with distinguished point 0Λτ . After a free isotopy in (C \ Λτ )Λτ we may assume that the reducing curve γ0 lifts under the covering map P : C → CΛτ to the segment 1+τ 2 + [0, 1] ⊂ C. Denote by γ the curve in the closed torus CΛτ that lifts to [0, 1], and by γ  the curve on the closed torus that lifts to [0, τ ]. γ and γ  represent a pair of generators of the fundamental group of the closed torus with base point 0Λτ . The complement (CΛτ ) \ γ0 of γ0 in the fiber over x0 is a topological annulus with distinguished point 0Λτ . Since γ0 reduces all monodromy mapping classes of the (1, 1)-family F, each monodromy mapping class has a representative that fixes γ0 pointwise. These representatives map the topological annulus (CΛτ ) \ γ0 homeomorphically onto itself, fixing the distinguished point and fixing the boundary pointwise, or fixing the boundary pointwise after an involution. Hence, each monodromy is a power of a Dehn twist about γ0 , maybe, composed with an involution. The Dehn twist on CΛτ about γ0 can be represented by a selfhomeomorphism ψτ of CΛτ which lifts under P to the real linear self-map τ of C which maps 1 to 1 and τ to 1 + τ . This can be seen by looking at the ψ τ on the lifts of the curves γ and γ  . ψ τ takes  action of ψ γ = [0, 1] to itself, and  γ = [0, τ ] to a curve that is isotopic to [0, τ + 1] through simple closed arcs in C  with fixed endpoints, the interiors arcs  avoiding Λτ . Note that the real 2 × 2  of the −1 1 (Im τ ) τ is matrix corresponding to ψ . 0 1 We consider the double branched covering pΛτ : (C\Λτ )Λτ → C (see (8.64)) ˚ with branch locus BL(Λ τ ) = {e1 (τ ), e2 (τ ), e3 (τ )} determined by the Weierstraß ℘function, ℘Λτ = pΛτ ◦ p (with p being the projection p : C \ Λτ → (C \ Λτ )Λτ ). Consider the continuous extension pcΛτ : CΛτ → P1 of pΛτ . The Weierstrass ℘function extends to a mapping, also denoted by ℘Λτ , ℘Λτ : C → P1 , that takes 0 1 τ 1 1+τ to ∞, 12 to e1 , τ2 to e2 , and 1+τ 2 to e3 . The line segments [0, 2 ], [0, 2 ], [ 2 , 2 ], τ 1+τ and [ 2 , 2 ] are mapped under ℘Λτ to simple curves γ∞,e1 , γ∞,e2 , γe1 ,e3 , and γe2 ,e3 in P1 , each of which joins the first-mentioned point with the last-mentioned point. (See Fig. 8.2.) The union of the four curves (each with suitable orientation) is a closed curve in P1 (the union of the real axis with the point ∞ on Fig. 8.2) that divides P1 into two connected components C1 and C2 . The Weierstrass ℘-function ℘Λτ maps the open parallelogram R in the complex plane with vertices 0, 1, τ2 , 1+ τ2 conformally onto its image. Indeed, the “double” of this parallelogram, i.e. the parallelogram with vertices 0, 1, τ , τ +1, is the interior of a fundamental polygon for the covering C → CΛτ and the equality ℘Λτ (ζ ) = ℘Λτ (−ζ +n+mτ ), n, m ∈ Z holds. By the last equality we get in particular ℘Λ τ (

.

1 τ +1 τ +1 + t) = ℘Λτ ( − t) , t ∈ [0, ] . 2 2 2

(8.73)

8.8 Lattices and Double Branched Coverings

283

α2 γe 1 ,e2 α1 α2

γe2 ,e3

γe1 ,e3

γe2 ,∞ e2

×

× γe3 ,e2

e1 ×

γ∞,e1

e3 α1

γe3 ,∞

Fig. 8.2 The semi-conjugation of a Dehn twist to a half-twist. In the figure we consider the torus CΛ for Λ = 2α(Z + iZ) with a real number α such that e1 = ℘Λ (α) = 1, e2 = ℘Λ (iα) = −1, e3 = ℘Λ ((1 + i)α) = 0

After perhaps relabeling C1 and C2 , we may assume that ℘Λτ maps the open parallelogram R− with vertices 0, 12 , τ2 , 1+τ 2 , (the “left half” of R) conformally onto C1 (the lower half-plane (shadowed) in the case of Fig. 8.2), and then it maps the τ parallelogram R+ with vertices 12 , 1, 1+τ 2 , 1+ 2 , (the “right half” of R) conformally onto C2 (the upper half-plane in the case of Fig. 8.2). The mapping ψ˜ τ fixes the segment [0, 12 ], and translates the segment [ τ2 , 1+τ 2 ] by 1 1 ˜ 2 . It maps R− to the parallelogram ψτ (R− ) ⊂ R whose horizontal sides are [0, 2 ] 1+τ 2+τ and [ 2 , 2 ]. The Weierstraß ℘-function ℘Λτ maps [ 12 + τ2 , 1 + τ2 ] to γe3 ,e2 which equals γe2 ,e3 with inverted orientation (see (8.73)), and takes the segment [0, 12 ] to a curve γ∞,e1 that joins ∞ with e1 . It maps the segment [0, 1+τ 2 ] to a curve γ∞,e3 which is contained in C1 except of its endpoints and joins ∞ and e3 . The segment [ 12 , 1 + τ2 ] is mapped to a curve γe1 ,e2 with initial point e1 and terminal point e2 , which is

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

contained in C2 except its endpoints, so that the union of the curves γ∞,e1 , γe1 ,e2 , γe2 ,e3 , and γe3 ,∞ is the oriented boundary of a domain C1 , i.e. the domain C1 is “on the left” when walking along the oriented curve. (In Fig. 8.2 C1 is the dotted domain.) Here γe3 ,∞ equals γ∞,e3 with inverted orientation. Hence, ℘Λτ maps ψ˜ τ (R− ) onto C1 . def The mapping ℘Λτ ◦ ψ˜ τ takes R− to the domain C1 . The mapping ϕτ,1 = ℘Λτ ◦ ψ˜ τ ◦ (℘Λτ |R− )−1 takes C1 onto C1 . It extends continuously to a homeomorphism between closures. The extension fixes γ∞,e1 pointwise, it takes γe1 ,e3 to γe1 ,e2 fixing e1 , it takes γe3 ,e2 to γe2 ,e3 mapping e2 to e3 and e3 to e2 , and it takes γe2 ,∞ to γe3 ,∞ fixing ∞. A simple arc α1 in the closure C1 (i.e. an arc without self-intersection) with initial point on γ∞,e1 and terminal point on γe3 ,e2 is mapped by the extension of ϕτ,1 to a simple arc α1 in C1 with initial point on γ∞,e1 and terminal point on γe2 ,e3 . A similar argument for the parallelogram R replaced by the parallelogram with vertices ± 12 ,

τ 2

±

1 2

def

and the domain C1 replaced by C2 shows, that with R+ − 1 = def

{z − 1 : z ∈ R+ } the mapping ϕτ,2 = ℘Λτ ◦ ψ˜ τ ◦ (℘Λτ |R+ − 1)−1 takes C2 homeomorphically onto C2 and extends to a homeomorphism between closures. Here C2 is the domain that is bounded by the oriented curve γe2 ,e1 ∪ γe1 ,∞ ∪ γ∞,e3 ∪ γe3 ,e2 . Moreover, by the periodicity properties of ℘Λτ (see equality (8.73)) the extensions of ϕτ,1 and ϕτ,2 coincide on the boundary ∂C1 = ∂C2 . Similarly as def before the extension of the mapping ϕτ,2 = ℘Λτ ◦ ψ˜ τ ◦ (℘Λτ |R+ − 1)−1 to the closure of C2 takes simple arcs α2 in C2 with initial point on γe3 ,e2 and terminal point on γ∞,e1 to simple arcs α2 in C2 with initial point on γe2 ,e3 and terminal point on γ∞,e1 . It follows that the mapping ϕτ that is equal to ϕτ,j on Cj extends to a selfhomeomorphism of P1 , denoted again by ϕτ , whose mapping class in P1 with distinguished points e1 , e2 , e3 , ∞, is conjugate to mσ1 . Indeed, ϕτ takes a simple closed curve α1 ∪ α2 with α1 and α2 as before to the simple closed curve α1 ∪ α2 and takes a simple closed curve that surrounds γ∞,e1 and no other point in the branch locus to a simple closed curve that surrounds γ∞,e1 and no other point in the branch locus. The mapping class, that is represented by a half-twist around the interval [e2 , e3 ] (see Sect. 1.5), is determined by the following properties. It takes the homotopy class of α1 ∪ α2 to the homotopy class of α1 ∪ α2 , and it takes the homotopy class of a simple closed curve on P1 , that surrounds γ∞,e1 and no other point in the branch locus, to a curve that is isotopic to this curve on C \ {e1 , e2 , e3 }. Since ψ˜ τ |R− = (p|R)−1 ◦ ψτ ◦ p|R− and ℘Λτ ◦ ψ˜ τ |R− = ϕτ,1 ◦ ℘Λτ |R− we obtain ℘Λτ ◦ (p|R)−1 ◦ ψτ = ϕτ,1 ◦ ℘Λτ ◦ (p|R− )−1 , hence, since ℘Λτ = pΛτ ◦ p (see (8.64)), the equality pΛτ ◦ ψτ = ϕτ,1 ◦ pΛτ = ϕτ ◦ pΛτ holds on p(R− ). By the same reason the equality pΛτ ◦ ψτ = ϕτ,2 ◦ pΛτ = ϕτ ◦ pΛτ holds on p(R+ ). Since p(R− ∪ R+ ) is dense in CΛτ , the equality pΛτ ◦ ψτ = ϕτ ◦ pΛτ holds on CΛτ . We proved that the projection of ψτ (see Sect. 8.7) is in the homotopy class of a conjugate of mσ1 , which is a reducible mapping class. We proved that the (0, 4)-bundle is reducible. Proposition 8.2 is proved.



8.9 The Gromov-Oka Principle for (1, 1)-Bundles over Tori with a Hole

285

Let  F be an isomorphism class of smooth (1, 1)-bundles over the circle. By the analog of Proposition 8.2 for bundles over the circle each smooth (1, 1)-bundle F over ∂D is isomorphic to the canonical double branched covering of a smooth special (0, 4)-bundle F1 = Pr(F) over ∂D (the projection of F). Moreover, by Lemma 8.9 the projections of the monodromy homomorphisms of isomorphic bundles over ∂D are conjugate, hence the projections of isomorphic bundles are isomorphic. Vice versa, isomorphic special (0, 4)-bundles over ∂D lift to isomorphic (1, 1)-bundles. An isomorphism class of special (0, 4)-bundles over ∂D corresponds 3 . By Corollary 2.2 for any braid to a conjugacy class of 3-braids modulo center bZ ˆ = b that is associated to a bundle representing the class Pr( F) the equality M(b)   M(bΔ2k 3 ) = M(mb,∞ ) = M(Pr(F)) = M(Pr(F)) holds. Proposition 8.2 implies the following corollary. 



Corollary 8.2 ˆ . M( F) = M(b)

.

Proof M( F) is the supremum of the conformal modules of annuli on which there exists a holomorphic (1, 1)-bundle that represents  F. M(Pr( F)) is the supremum of the conformal modules of annuli on which there exists a special holomorphic (0, 4)bundle that represents Pr( F). Statement (1) of Proposition 8.2 implies the inequality M( F) ≤ M(Pr( F)), Statement (2) implies the opposite inequality. The corollary is proved.



8.9 The Gromov-Oka Principle for (1, 1)-Bundles over Tori with a Hole In the following Theorem 8.3 we consider isotopies of smooth .(1, 1)-families over tori with a hole to complex analytic .(1, 1)-families. Theorem 8.3 (1) Let X be a smooth surface of genus one with a hole with base point .x0 , and with a chosen set .E = {e1 , e2 } of generators of .π1 (X, x0 ). Define the set −1 −2 −1 −1 .E0 = {e1 , e2 , e1 e 2 , e1 e2 , e1 e2 e1 e2 } as in Theorem 7.3. Consider a smooth .(1, 1)-bundle .F on X. Suppose for each .e ∈ E0 the restriction of the bundle .F to an annulus representing .eˆ has the Gromov-Oka property. Then the bundle .F has the Gromov-Oka property on X. (2) If a bundle .F as in .(1) is irreducible, then it is isotopic to an isotrivial bundle, and hence, for each conformal structure .ω on X the bundle .Fω is isotopic to a bundle that extends to a holomorphic .(1, 1)-bundle on the closed torus .ω(X)c . In particular, .F is isotopic to a holomorphic bundle for any conformal structure .ω on X (maybe, of first kind).

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8 Gromov’s Oka Principle for .(g, m)-Fiber Bundles

(3) Any smooth reducible bundle .F on X has a single irreducible bundle component. This irreducible bundle component is isotopic to an isotrivial bundle. There is a Dehn twist in the fiber over the base point such that the .(1, 1)-bundle .F can be recovered from the irreducible bundle component up to composing each monodromy by a power of the Dehn twist. Any smooth reducible bundle on X is isotopic to a holomorphic .(1, 1)-bundle for each conformal structure of second kind on X. (4) A reducible holomorphic .(1, 1)-bundle over a punctured Riemann surface is locally holomorphically trivial. Proof By Proposition 8.2, Statement (1), we may assume that .F is the canonical double branched covering of a smooth special .(0, 4)-bundle, denoted by .Pr(F). Suppose that for each .e ∈ E0 the restriction of the smooth .(1, 1)-bundle .F to an annulus .Aeˆ representing .eˆ has the Gromov-Oka property. Then for each .e ∈ E0 there is a conformal structure .ω : Aeˆ → ωe (Aeˆ ) such that .ωe (Aeˆ ) has conformal √e 3+ 5 −1 π module bigger than . 2 log( 2 ) and the pushed forward bundle .(F|Aeˆ )ωe is isotopic to a holomorphic bundle .Fe on .ωe (Aeˆ ). By Proposition 8.1 we may assume that .Fe is of the form (8.37), and by Proposition 8.2 we may assume that .Fe is the canonical double branched covering of a holomorphic special .(0, 4)-bundle on .ωe (Aeˆ ), denoted by .Pr(Fe ). Since the monodromy homomorphisms of .F|Aeˆ and .Fe are conjugate, the monodromy homomorphisms of .Pr(F|Aeˆ ) and .Pr(Fe ) are also conjugate. Hence, .Pr(F|Aeˆ ) and .Pr(Fe ) are isotopic. We saw that for each .e ∈ E the restriction of the special .(0, 4)-bundle .Pr(F) to an annulus representing .e ˆ is isotopic to a holomorphic bundle for a conformal structure on the annulus of √ conformal module bigger than . π2 log( 3+2 5 )−1 . By Theorem 8.2 the special .(0, 4)bundle .Pr(F) is isotopic to a holomorphic special .(0, 4)-bundle for any conformal structure of second kind on X. Hence, the canonical double branched covering of .Pr(F) is isotopic to a holomorphic .(1, 1)-bundle for each conformal structure of second kind on X. Therefore, .F is isotopic to a holomorphic .(1, 1)-bundle for each conformal structure of second kind on X. Statement (1) is proved. Suppose a bundle .F as in (1) is irreducible. Then by Proposition 8.2 the special .(0, 4)-bundle .P(F) is also irreducible. Moreover, by Theorem 8.2, Statement (1), the .(0, 4)-bundle .P(F) is isotopic to an isotrivial bundle, i.e. its lift to a finite covering ˆ is isotopic to the trivial bundle, and therefore, for each conformal structure .ω .X on X (maybe, of first kind) the .(0, 4)-bundle .P(F)ω is isotopic to a bundle that extends holomorphically to the closed torus .ω(X)c . Then also the canonical double branched covering of the .(0, 4)-bundle is isotopic to an isotrivial bundle. Hence, the bundle .F is isotopic to an isotrivial bundle and extends to a holomorphic bundle on the closed torus. In particular, for each conformal structure (maybe, of first kind) on X the bundle .Fω is isotopic to a holomorphic bundle on .ω(X). This gives statement (2). Suppose now .F is any smooth reducible bundle on X. Then by Proposition 8.2 the bundle .Pr(F) is reducible. By Theorem 8.2 the special .(0, 4)-bundle .Pr(F) is isotopic to a holomorphic special .(0, 4)-bundle for any conformal structure of second kind on X. Hence, for any conformal structure of second kind on X the

8.9 The Gromov-Oka Principle for (1, 1)-Bundles over Tori with a Hole

287

bundle .F is isotopic to a holomorphic .(1, 1)-bundle which is a double branched covering. Each admissible set of curves on a punctured torus consists of exactly one curve. Hence, there is an admissible closed curve .γ in the punctured fiber .P −1 (x0 ) \ s ∞ over the base point .x0 , that is mapped by the monodromy mapping class along each curve .γj representing an element .ej ∈ E to a curve that is isotopic to .γ in the punctured fiber over .x0 . Choose for each monodromy mapping class a representative that maps the curve .γ onto itself. The complement of .γ on the punctured torus .P −1 (x0 ) \ s ∞ is connected and homeomorphic to the thrice punctured Riemann sphere. This implies first that there is a single irreducible component of .F. Further, the restrictions of suitable representatives of the monodromy mapping classes to the complement of .γ on the punctured torus are conjugate to self-homeomorphisms of the thrice punctured Riemann sphere. The conjugated homeomorphisms extend to self-homeomorphisms of the Riemann sphere with three distinguished points. The self-homeomorphisms may interchange the two points that come from different edges of .γ , but each of them fixes the third distinguished point. Except the identity there is only one isotopy class of such mappings and its square is the identity (in other words, the mapping class is the class of an involution). Hence, the monodromy mapping classes of the irreducible bundle component are powers of a single periodic mapping class. Therefore the irreducible bundle component is isotopic to an isotrivial bundle. The monodromy mapping classes of the original bundle .F are powers of Dehn twists about .γ , maybe, composed with an involution, and up to powers of Dehn twists about .γ in the fiber .P −1 (x0 ) the bundle can be recovered from the irreducible bundle component. Statement (3) is proved. We prove now statement (4). Consider a reducible holomorphic (1,1)-bundle .F over a punctured Riemann surface of genus 1. By Proposition 8.2 it is holomorphically isomorphic to a double branched covering of a holomorphic special .(0, 4)-bundle .Pr(F) over a punctured Riemann surface. By Proposition 8.2 the special holomorphic.(0, 4)-bundle .Pr(F) is also reducible. By Theorem 8.2 the bundle .Pr(F) is holomorphically trivial. Then the double branched covering .F of .Pr(F) is locally holomorphically trivial. Theorem 8.3 is proved.



Chapter 9

Fundamental Groups and Bounds for the Extremal Length

In Chap. 3 we computed several versions of the extremal length of certain elements of the fundamental group of the twice punctured complex plane. It becomes impracticable to compute the versions of extremal length for all elements regardless of their “complexity”. In this chapter we give effective upper and lower bounds for the extremal length with totally real horizontal boundary values of any element of the fundamental group of the twice punctured complex plane. The bounds differ by a multiplicative constant not depending on the element. The estimates are provided in terms of a natural syllable decomposition of the reduced word representing the element. Estimates for the extremal length of conjugacy classes of elements of the fundamental group (i.e. of the invariants mentioned by Gromov) are also obtained. The extremal length with totally real horizontal boundary values is capable to give more subtle information regarding Gromov’s Oka Principle and limitations of its validity than the respective invariant of conjugacy classes. In the last section (Sect. 9.7) we give estimates for the extremal length with totally real horizontal boundary values of 3-braids, and estimates of the extremal length of conjugacy classes of 3-braids (and, hence, of the entropy of 3-braids).

9.1 The Fundamental Group and Extremal Length. Two Theorems It will be convenient here to normalize the twice punctured complex plane as def

C\{−1, 1}. With this normalization the fundamental group .π1 = π1 (C\{−1, 1}, 0) is a free group in two generators .aj , .j = 1, 2, where .a1 is represented by simple closed curves that surround .−1 counterclockwise, and .a2 is represented by simple closed curves that surround 1 counterclockwise. Recall that there is a     def canonical isomorphism .π1 C \ {−1, 1}, 0 → π1tr = π1 C \ {−1, 1}, (−1, 1) .

.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025 B. Jöricke, Braids, Conformal Module, Entropy, and Gromov’s Oka Principle, Lecture Notes in Mathematics 2360, https://doi.org/10.1007/978-3-031-67288-0_9

289

290

9 Fundamental Groups and Bounds for the Extremal Length

  The elements of the relative fundamental group .π1 C \ {−1, 1}, (−1, 1) are homotopy classes of arcs in .C \ {−1, 1}, each of whose endpoints is contained in .(−1, 1). The isomorphism .π1 (C \ {−1, 1}, 0) → π1tr assigns to the element .e ∈ π1 (C \ {−1, 1}, 0), that is represented by a curve .γ with base point 0, the   element .etr ∈ π1tr = π1 C\{−1, 1}, (−1, 1) , that is represented by .γ . The extremal length .Λ(etr ) of the homotopy class .etr in the sense of Definition 3.2 is called the extremal lengths of e with totally real horizontal boundary values and is sometimes denoted by .Λtr (e). Recall that .Λ(etr ) is the infimum of the extremal length of all rectangles that admit a holomorphic mapping into .C \ {−1, 1} whose restriction to each maximal vertical segment represents .etr . The name “totally real” refers to the notion for 3-braids which is related to the present notion. The conformal module −1 is also denoted by .M (e). .M(etr ) = (Λ(etr )) tr pb def

We will also consider the relative fundamental group .π1 = π1 (C\{−1, 1}, iR) whose elements are homotopy classes of arcs in .C \ {−1, 1} with endpoints on the pb def

imaginary axis .iR. The canonical group isomorphism .π1 (C\{−1, 1}, 0) → π1 = π1 (C \ {−1, 1}, iR) assigns to each element e of .π1 (C \ {−1, 1}, 0) represented by a curve .γ with base point 0 the element .epb of .π1 (C \ {−1, 1}, iR) represented by .γ . The extremal length .Λ(epb ) in the sense of Definition 3.2 is called the extremal length with pb boundary values of e and is also denoted by .Λpb (e). The conformal module .M(epb ) = (Λ(epb ))−1 is also denoted by .Mpb (e). ("pb" stands for “perpendicular bisector”. In fact, the imaginary axis is the perpendicular bisector of the line segment .(−1, 1).) More generally, it will be sometimes convenient to use the fact that .π1 (C \ {−1, 1}, 0) is canonically isomorphic to the relative fundamental group   + def .π 1 = π1 C \ {−1, 1}, iR ∪ (−1, 1) whose elements are homotopy classes of arcs in .C \ {−1, 1} with endpoints in .iR ∪ (−1, 1)). The group isomorphism assigns to each element e of .π1 (C \ {−1, 1}, 0), represented by a closed curve .γ , the  element .e+ of .π1 C \ {−1, 1}, iR ∪ (−1, 1) , represented by .γ . For each element .e ∈ π1 (C \ {−1, 1}, 0) all its representatives are contained in both classes, .etr and .epb , associated to e, and all representatives of .etr and .epb are contained in the element .e+ of .π1+ that is associated to e. We will identify each element of .π1 (C \ {−1, 1}, 0) with the reduced word in the generators representing it. A word w in the generators is reduced if it is the identity or it has the form .w = w1n1 · w2n2 · . . . , where the .nj are non-zero integers and the nj .wj are alternately equal to either .a1 or .a2 . We refer to the .w j as the terms of the word. Any reduced word . w in . π1 (C \ {−1, 1}, 0) can be uniquely decomposed into syllables. They are defined as follows. Definition 9.1 The syllables of any reduced word .w ∈ π1 (C \ {−1, 1}, 0) are all its terms .ajkii with .|ki | ≥ 2 (called syllables of form .(1)), and all maximal sequences

of consecutive terms .ajkii for which .|ki | = 1 and all .ki have the same sign. A syllable of the latter form is called a syllable of form .(2), if it contains more than one term and is called a singleton or a syllable of form .(3) if it consists of a single term.

9.1 The Fundamental Group and Extremal Length. Two Theorems

291

(See also [44, 45]). Define the degree of a syllable .deg(syllable) to be the sum of the absolute values of the powers of terms entering the syllable. For example, the syllables of the word .a2−1 a12 a2−3 a1−1 a2−1 a1−1 a2 a1−1 from left to right are the singleton .a2−1 , the syllable .a12 of form .(1) and degree 2, the syllable .a2−3 of form (1) and degree 3, the syllable .a1−1 a2−1 a1−1 of form .(2) and of degree 3, the singleton .a2 and the singleton .a1−1 . We call words consisting of a single syllable elementary words. Label the syllables of a word from left to right by consecutive integral numbers .j = 1, 2, . . . . Let .dj be the degree of the j -th syllable .sj . (We consider each syllable as a reduced word in the elements of the fundamental group.) Put def

L(w) =



.

log(2dj +

 4dj2 − 1) ,

(9.1)

j

where the sum runs over the degrees of all syllables of w. If w is the identity we put .L(w) = 0. We want to point out that .L(w −1 ) = L(w). Noticethat for the word

consisting of the single syllable .sj we have .L(sj ) = log(2dj + 4dj2 − 1). Thus,  .L(w) = L(sj ) where the sum runs over the syllables of w. In this chapter we will prove the following theorems. The present proof of the upper bound is shorter and less technical than the proof in [44], but the proof in [44] gives a better constant. Theorem 9.1 For any element .w ∈ π1 (C\{−1, 1}, 0) the following estimates hold. .

1 1 L(w) ≤ Λtr (b) = ≤ 214 · L(w), 2π Mtr (w)

(9.2)

except in the following cases: .w = a1n or .w = a2n for an integer n. In these cases .Λtr (b) = 0, i.e .Mtr (b) = ∞. Moreover, .

1 1 L(w) ≤ Λpb (w) = ≤ 213 · L(w), 2π Mpb (w)

(9.3)

except in the following case: each term in the reduced word w has the same power, which equals either .+1 or .−1. In these cases .Λpb (w) = 0, i.e. .Mpb (w) = ∞. pb

We will also consider the relative fundamental groups .pb π1tr and .tr π1 with mixed horizontal boundary values. Here the elements of .pb π1tr are the homotopy pb classes .iR h(−1,1) and the elements of .tr π1 are the homotopy classes .(−1,1) hiR in the space .X = C \ {−1, 1} . There is a canonical isomorphism from .π1 onto .pb π1tr pb (onto .tr π1 , respectively) that assigns to an element .e ∈ π1 the element .pb etr that contains all representatives of e (the element .tr epb , respectively, that contains all representatives of e).

292

9 Fundamental Groups and Bounds for the Extremal Length

The statement for mixed boundary values is given in the following Theorem 9.2. Note that for mixed boundary values the estimate of the extremal length holds always while for pb or tr boundary values there are exceptional cases. Theorem 9.2 For all .w ∈ π1 the following inequalities hold .

1 L(w) ≤ Λ(tr wpb ) ≤ 214 · L(w), . 2π 1 L(w) ≤ Λ(pb wtr ) ≤ 214 · L(w). 2π

(9.4) (9.5)

Recall that the conjugacy class .wˆ of an element .w ∈ π1 (C \ {−1, 1}, 0) can be identified with the free homotopy class of closed curves that contains w. Definition 9.2 A word .w ∈ π1 (C \ {−1, 1}, 0) is called cyclically reduced, if either the word consists of a single term, or it has at least two terms and the first and the last term of the word are powers of different generators. Each reduced word which is not the identity is conjugate to a cyclically reduced word. The following theorem gives upper and lower bounds of the invariant .M(w) ˆ of conjugacy classes of elements of the twice punctured complex plane, that was first mentioned in Gromov’s seminal paper [35] in connection with obstructions for the Gromov-Oka principle. Each element of the fundamental group .π1 (C \ {−1, 1}, 0) corresponds to an element of .PB3 Z3 and, equivalently, to a mapping class of the twice punctured complex plane. By the Main Theorem the conformal module .M(w) ˆ can be interpreted as a multiple of the inverse of the entropy of the conjugay class .w. ˆ Theorem 9.3 Let .wˆ be a conjugacy class of elements of .π1 (C \ {−1, 1}, 0), and let w be a cyclically reduced word representing the conjugacy class .w. ˆ Then .

1 · L(w) ≤ Λ(w) ˆ ≤ 213 · L(w), 2π

n  with the following exceptions: .wˆ = a1n , .wˆ = a2n and .wˆ = (a 1 a2 ) . In the exceptional cases .Λ(w) ˆ = 0 and .M(w) ˆ = ∞.

9.2 Coverings of C \ {−1, 1} and Slalom Curves For the proof of the theorems it will be convenient to use instead of the universal covering of .C \ {−1, 1} two different coverings of .C \ {−1, 1} by .C \ iZ. Though the universal covering is well studied, factorizing the universal covering through two different coverings by .C \ iZ has the advantage to make the contribution of the syllables to the extremal length transparent.

9.2 Coverings of C \ {−1, 1} and Slalom Curves

293

To obtain the first covering .C \ iZ → C \ {−1, 1} we take the universal covering of the twice punctured Riemann sphere .P1 \ {−1, 1} (the logarithmic covering) and remove all preimages of .∞ under the covering map. Geometrically the logarithmic covering of .P1 \ {−1, 1} can be described as follows. Take copies of .P1 \ [−1, 1] labeled by the set .Z of integer numbers. Attach to each copy two copies of .(−1, 1), the .+-edge (the set of accumulation points contained in .(−1, 1) of the upper halfplane) and the .−-edge (the set of accumulation points contained in .(−1, 1) of the lower half-plane). For each .k ∈ Z we glue the .+-edge of the k-th copy to the .−-edge of the .k +1-st copy (using the identity mapping on .(−1, 1) to identify points on different edges). Denote by .Ulog the set obtained from the described covering by removing all preimages of .∞. Choose curves .αj (t), t ∈ [0, 1], j = 1, 2, in .C \ {−1, 1} that represent the generators .aj , j = 1, 2, of the fundamental group .π1 (C \ {−1, 1}, 0 }) and have the following properties. The initial point and the terminal point of each of the curves are the only points of the curve on the interval .(−1, 1) and are also the only points of the curve on the imaginary axis. The following proposition holds. Proposition 9.1 The set .Ulog is conformally equivalent to .C \ i Z. The mapping eπ z −1 1 1 .f1 ◦ f2 , .f2 (z) = π z e +1 , z ∈ C \ i Z, .f1 (w) = 2 (w + w ), w ∈ C \ {−1, 0, 1}, is a covering map from .C \ i Z to .C \ {−1, 1}. It has period i and takes the punctured strip .{z ∈ C : − 12 < Imz < 12 } \ {0} conformally onto .C \ [−1, 1]. Points of the strip contained in the upper half-plane are mapped to the lower half-plane and points of the strip contained in the lower half-plane are mapped to the upper half-plane. The mapping .f1 ◦ f2 takes the set .(− 2i , 2i ) \ {0} onto .iR \ {0}, and the line .R + 2i onto .(−1, 1). For each .k ∈ Z the lift of .α1 with initial point . −i 2 + ik is a curve which joins −i −i . + ik with . + i(k + 1) and is contained in the closed left half-plane. The only 2 2 points on the imaginary axis are the endpoints. −i The lift of .α2 with initial point . −i 2 + ik is a curve which joins . 2 + ik with −i . 2 + i(k − 1) and is contained in the closed right half-plane. The only points on the imaginary axis are the endpoints. Figure 9.1 shows the curves .α1 and .α2 which represent the generators of the fundamental group .π1 (C \ {−1, 1}, 0) and their lifts under the covering maps .f1 and .f2 ◦ f1 . For .j = 1, 2 the curves .αj and .αj are the two lifts of .αj under the double covering .f1 : C \ {−1, 0, 1} → C \ {−1, 1}. The curve .αˆ 1 is the lift of .α1 under the mapping .f2 with initial point . −i ˆ 1 is the lift of .α1 under the 2 , the curve .α i mapping .f2 with initial point . 2 , the curve .αˆ 2 lifts .α2 and has initial point . 2i , and the curve .αˆ 2 lifts .α2 and has initial point .− 2i . Proof The mapping .f1 is the restriction of the Zhukovsky function to .C\{−1, 0, 1}. The Zhukovski function defines a double branched covering of the Riemann sphere 1 .P with branch locus .{−1, 1}. In particular, the Zhukovsky function provides a conformal mapping from the unit disc .D onto .P1 \ [−1, 1]. It maps .−1 to .−1, 1 to 1

294

9 Fundamental Groups and Bounds for the Extremal Length ζ∈

\ {−1, 1} α1

−1 ×

1 ×

α2

ζ = f1 (w) = 12 (w +

w∈

1 w)

i

\ {−1, 0, 1}

α1 α2 × 0 α1

α2 −i

w = f2 (z) =

z∈

eπz −1 eπz +1

w = f˜2 (z) = eπz

\i

z∈ α ˆ 1 α ˆ 1

α ˜ 1

× × ×

\i

α ˆ 2

α ˆ 2

+

i 2

−

i 2

α ˜ 2

×2i ×i ×0

α ˜ 1

α ˜ 2

×−i

Fig. 9.1 The generators of .π1 (C \ {−1, 1}, 0) and their lifts under two coverings by .C \ iZ

and 0 to .∞. The upper half-circle is mapped onto the .−-edge, the upper half-disc is mapped onto the lower half-plane, the lower half-circle is mapped onto the .+-edge and the lower half-disc is mapped onto the upper half-plane. Similarly, it provides a conformal mapping of the exterior of the closed unit disc onto .P1 \ [−1, 1] which preserves the upper half-plane and also preserves the lower half-plane. The mapping .f2 extends through .iZ to an infinite covering of .P1 \ {−1, 1} by .C. By an abuse of notation we denote this extension also by .f2 . (The extension of) 1 1 0 .f2 provides a conformal mapping .f 2 from the strip .{z ∈ C : − 2 < Im z < 2 } onto the unit disc, which takes the real axis onto the segment .(−1, 1) , such that i i .limx∈R,x→−∞ = −1, .limx∈R,x→+∞ = 1. Further, .f2 maps . to i, .− to .−i, and 0 2 2

9.2 Coverings of C \ {−1, 1} and Slalom Curves

295

to 0. The line .{z ∈ C : Im z = 12 } is mapped onto the upper half-circle, the upper half-strip .{z ∈ C : 0 < Im z < 12 } is mapped onto the upper half-disc, the line 1 .{z ∈ C : Im z = − } is mapped onto the lower half-circle and the lower half-strip 2 is mapped onto the lower half-disc. It follows that .f1 ◦ f2 takes the punctured strip 1 1 .{z ∈ (C \ {0}) : − < Im z < } conformally onto .C \ [−1, 1]. 2 2 The mapping .f = f1 ◦f2 has period i. Indeed, .f2 has period 2i, .f2 (z+i) = f21(z) , and .f1 ( w1 ) = f1 (w). The statements concerning the lift of curves under .f1 ◦ f2 are now clear. To see that .Ulog is conformally equivalent to .C\i Z we identify the set .C\[−1, 1] with the sheet of .Ulog labeled by 0. The conformal mapping .f | {z ∈ C \ {0} : − 12 < Im z < 12 } (whose image is .C \ [−1, 1] ) extends by Schwarz’s Reflection Principle through the line .{z ∈ C : Im z = 12 } which is mapped onto the .−-edge of 1 3 .C \ [−1, 1]. The reflected mapping takes .{z ∈ C \ {i} : 2 < Im z < 2 } conformally onto .C \ [−1, 1]. We identify the image of the punctured strip .{z ∈ C \ {i} : 12 < Im z < 32 } with the sheet of .Ulog labeled by .−1. Induction on reflection through the lines .{z ∈ C : Im z = 12 + j }, j ∈ Z, gives the conformal mapping from .C \ iZ onto .Ulog . 

The remaining statements are easy to see. The second covering is given by the mapping .f1 ◦ f˜2 : C \ iZ → C \ {−1, 1}, where .f1 is as before and .f˜2 is the exponential map, .f˜2 (z) = eπ z . Recall that each curve .αj , j = 1, 2, has two lifts .αj and .αj under .f1 . For an illustration of the following proposition see the right part of Fig. 9.1. Proposition 9.2 The mapping .f˜2 takes .C \ iZ to .C \ {−1, 0, 1}, and .f1 takes the latter set to .C \ {−1, 1}. The composition .f˜2 ◦ f1 is a covering of .C \ {−1, 1} by .C\{−1, 0, 1} and has period 2i. For each integer k it takes the interval .(ki, (k +1)i) to .(−1, 1), it maps . 2i + R to .i(0, ∞), and maps .− 2i + R to .i(−∞, 0). The lifts .α˜ j and .α˜ j , .j = 1, 2, of .αj and .αj under .f˜2 have the following properties. The lifts .α˜ j , j = 1, 2, are contained in the closed left half-plane and are directed downwards (i.e in the direction of decreasing y), the lifts .α˜ j , j = 1, 2, are contained in the closed right half-plane and directed upwards. The initial point of .α˜ 1 is .i + 12 i, the initial point of .α˜ 1 and .α˜ 2 is . 12 i, the initial point of .α˜ 2 is .− 12 i. All other lifts are obtained by translation by an integer multiple of 2i. The straightforward proof is left to the reader. Consider the curve .α1n , n ∈ Z \ {0}. It runs n times along the curve .α1 if .n > 0, and .|n| times along the curve .α1 with inverted orientation if .n < 0. It is homotopic in .C \ {−1, 1} with base point 0 to a curve whose interior (i.e. the complement of its endpoints) is contained in the open left half-plane. We call a representative of .a1n whose interior is in the open left half-plane a standard representative of .a1n . In the same way we define standard representatives of .a2n . For each .k ∈ Z the curve .α1n lifts under .f1 ◦ f2 to a curve with initial point . −i 2 + ik and terminal point −i . + ik + in which is contained in the closed left half-plane and omits the points 2

296

9 Fundamental Groups and Bounds for the Extremal Length

in .iZ. Respectively, .α2n , n ∈ Z \ {0}, lifts under .f1 ◦ f2 to a curve with initial point +i +i . 2 + ik and terminal point . 2 + ik − in which is contained in the closed right halfplane and omits the points in .iZ. The mentioned lifts are homotopic through curves in .C \ iZ with endpoints on .i R \ iZ to curves with interior contained either in the open right half-plane or in the open left half-plane. Standard representatives of .a1n and .a2n lift to such curves. Definition 9.3 A simple curve in .C \ iZ with endpoints on different connected components of .iR\iZ is called an elementary slalom curve if its interior is contained in one of the open half-planes .{z ∈ C : Re z > 0} or .{z ∈ C : Re z < 0}. A curve in .C\iZ is called an elementary half slalom curve if one of the endpoints is contained in a horizontal line .{z ∈ C : Im z = k+ 12 } for an integer k and the union of the curve with its (suitably oriented) reflection in the line .{z ∈ C : Im z = k + 12 } is an elementary slalom curve. A slalom curve in .C \ iZ is a curve which can be divided into a finite number of elementary slalom curves so that consecutive elementary slalom curves are contained in different half-planes. A curve which is homotopic to a slalom curve (elementary slalom curve, respectively) in .C\iZ through curves with endpoints in .iR\iZ is called a homotopy slalom curve (elementary homotopy slalom curve, respectively). A curve which is homotopic to an elementary half-slalom curve in .C \ iZ through curves with one endpoint in .iR \ iZ and the other endpoint on the line .{z ∈ C : Im z = k + 12 } for an integer k is called an elementary homotopy half-slalom curve. We call an elementary slalom curve non-trivial if its endpoints are contained in intervals .(ik, i(k + 1)) and .(i, i( + 1)) with .|k − | ≥ 2. Note that the union of an elementary half-slalom curve with its reflection in the horizontal line that contains one endpoint is always a non-trivial elementary slalom curve (i.e. an elementary half-slalom curve is “half” of a non-trivial elementary slalom curve). We saw that pb the lifts under .f1 ◦ f2 of representatives of terms .pb (ajn )pb ∈ π1 with .|n| ≥ 2 are non-trivial elementary homotopy slalom curves. The lifts of representatives of elements of .π1 under .f1 ◦ f˜2 look different. The representatives of .(ajn )tr , .|n| ≥ 1, lift to curves which make .|n| half-turns around a point in .iZ (positive half-turns if .n > 0, and negative half-turns if .n < 0). Each such representative lifts under .f1 ◦ f˜2 to the composition of .|n| trivial elementary homotopy slalom curves. Take any syllable of from (2), i.e. any maximal sequence of at least two consecutive terms of the word which enter with equal power being either 1 or .−1. Recall that d denotes the sum of the absolute values of the powers of the terms of the syllable. There is a representing curve that makes d half-turns around the interval .[−1, 1] (positive half-turns, if the exponents of terms in the syllable are 1, and negative half-turns otherwise). The lift under .f1 ◦ f˜2 of this representative is a non-trivial elementary slalom curve. We will call homotopy classes of homotopy slalom curves for short slalom classes.

9.2 Coverings of C \ {−1, 1} and Slalom Curves Fig. 9.2 Lifts of a curve in \ {−1, 1} under two different coverings by .C \ iZ

297

×

.C

× a−1 1

×5i −1 −1 a−1 1 a2 a1

×4i ×3i

a−1 1

a21

4i× a2

×2i

a−3 2

×0

a−2 2 a−1 2

a2

3i×

−1 −1 −1 a−1 2 a1 a2 a1

×i

−i×

5i×

2i× i× 0×

a−1 2

a21 −i×

×

×

a

b

Figure 9.2 shows two slalom curves. The curve in Fig. 9.2a is a lift under .f1 ◦ f2 of a curve in .C \ {−1, 1} with initial point and terminal point equal to .0 ∈ iR representing the word .a2−1 a12 a2−3 a1−1 a2−1 a1−1 a2 a1−1 in the relative fundamental group .π1 (C \ {−1, 1}, iR). The curve in Fig. 9.2b is a lift under .f1 ◦ f˜2 of a curve with initial and terminal point in .iR \ {0} representing the same word. For each elementary slalom curve or half-slalom curve in Fig. 9.2 we indicate the element of .π1 which lifts under the considered mapping to the respective elementary slalom class or half-slalom class. Non-trivial elementary slalom classes and elementary half-slalom classes have positive extremal length (in the sense of Definition 3.2) which can be effectively estimated from above and from below. In this sense homotopy classes of curves in .C \ {−1, 1} whose lifts under .f1 ◦ f2 or .f1 ◦ f˜2 are non-trivial elementary slalom classes or elementary half-slalom classes serve as building blocks. We obtained the following fact. For an element of a relative fundamental group of the twice punctured complex plane the pieces representing syllables of form (1) with pb boundary values lift to non-trivial elementary slalom classes under .f1 ◦f2 , while the pieces representing syllables of form (2) with tr boundary values lift to non-trivial elementary slalom classes under .f1 ◦ f˜2 . For a word in .π1 containing a syllable that is a singleton we may select a piece of a representing curve that lifts under .f1 ◦ f2 to a non-trivial elementary homotopy half-slalom curve. These facts will be used to obtain a lower bound for the extremal length of elements of the relative fundamental groups. Recall that the extremal length of a homotopy class of curves is equal to the extremal length of the class of their lifts under a holomorphic covering.

298

9 Fundamental Groups and Bounds for the Extremal Length

The method to obtain the upper bound is roughly to patch together in a quasiconformal way the holomorphic mappings of rectangles representing syllables and to perturb the obtained quasiconformal mapping to a holomorphic mapping. We conclude this section with relating the two explicitly given coverings of .C \ {−1, 1} by .C \ iZ to the universal covering of the twice punctured plane. Denote by .P : U → C \ iZ the universal covering of .C \ iZ. Geometrically the set U is obtained in the following way. Consider the left half-plane .C and call it the Riemann surface of generation 0. The first step is the following. For each integer k we take a copy of the right half-plane .Cr and glue it to .C along the interval .(ki, (k + 1)i) (using the identity mapping for gluing). We obtain a Riemann surface with a natural projection to .C \ iZ called the Riemann surface of first generation. At the second step we consider the Riemann surface of first generation and proceed similarly by gluing the left half-plane along each copy of intervals .(ki, (k + 1)i) which is an end of the Riemann surface of first generation. By induction we obtain the universal covering of .C \ iZ. C the lift under .P of the left half-plane to the first sheet of U over Denote by . Cl be the closure of .C in .C \ iZ, let .  CCl .C . Let .C    be the closure of .C in the Cl C be the lift of the intervals universal covering of .C \ iZ and let .(ki, (k + 1)i)⊂  .(ki, (k + 1)i). For each k we denote by .Dk the half-disc .{z ∈ C : |z − i(k + 12 )| < 12 } with diameter .(ik, i(k + 1)) which is contained in the left half-plane and by .ρk the respective open half-circle .{z ∈ C : |z − i(k + 12 )| = 12 }. We call the .ρk half-circles of generation 0. C onto .C \ Lemma 9.1 There is a conformal mapping .ϕ : U → C that takes . ∞  so that for each k the set .(ki, (k + 1)i)is mapped onto .ρ under the D k k=−∞ k continuous extension of .ϕ. def

Proof Consider the half-strip .H0 = {z ∈ C : 0 < Im z < 1}. By a theorem of Caratheodory ([32], Chapter II.3, Theorem 4 and Theorem .4 , and also [62] , Theorem 2.24 and Theorem 2.25) each conformal mapping that takes the lift . H0 of .H0 onto the set .H0 \ D0 , extends continuously to a homeomorphism between closures. Let .ϕ0 be the conformal mapping from . H0 onto the set .H0 \ D0 whose extension to the boundary takes the point .0˜ over 0 to 0, the point .1˜ to 1 and the point ˜ (considered as prime end of . .∞ H0 ) to .∞. The extension of the conformal mapping to the boundary takes the lift .{z ∈ C : Im z = 1} of .{z ∈ C : Im z = 1} onto .{z ∈ C : Im z = 1}, it takes the lift .{z ∈ C : Im z = 0}of .{z ∈ C : Im z = 0} onto .{z ∈ C : Im z = 0}, and maps the lift .(0, i)of .(0, i) onto .ρ0 . By induction we extend .ϕ0 by Schwarz’s Reflection Principle across the halflines .{z ∈ C : Im z = k}which are the lifts of the respective half-lines in .C . We  C onto .C \ ∞ obtain a conformal mapping of . k=−∞ Dk , denoted again by .ϕ0 , Cl whose extension to . C takes for each integer number k the segment .(ik, i(k + 1)) onto the half-circle .ρk of generation 0. (See Fig. 9.3.)

9.2 Coverings of C \ {−1, 1} and Slalom Curves

299

Fig. 9.3 A conformal image of the Riemann surface of generation 2

ϕ( ˜ )

ρk+1

ρk

 k+1

 k

Schwarz’s Reflection Principle across each segment .(ik, i(k + 1))provides an extension of the conformal mapping .ϕ0 to the Riemann surface of first generation. Note that for each .k0 the image of the half-circles .ρk , .k = k0 , under reflection of ∞  .C \ k=−∞ Dk across .ρk0 are half-circles with diameter on the imaginary axis. We call them half-circles of the first generation. We obtained a conformal mapping of the Riemann surface of first generation to the unbounded connected component of the left half-plane with the half-circles of first generation removed. Apply the reflection principle by induction to the Riemann surface of generation n and all copies of intervals .(ik, i(k + 1)) in its “boundary”. We obtain the Riemann surface of generation .n+1, half-circles of generation .n+1 and a conformal mapping of this Riemann surface onto the unbounded connected component of the left halfplane with the half-circles of generation .n + 1 removed. The supremum of the diameters of the half-circles of generation n does not exceed .2−n . Indeed, the half-circles of generation n are obtained as follows. Take a half-circle .ρ of generation .n − 1 and reflect all other half-circles of generation .n − 1 across it. Each half-circle of generation .n − 1 different from .ρ has diameter on one side of the diameter of .ρ. Hence, the image of each half-circle of generation .n − 1 different from .ρ under reflection across .ρ is contained in a quarter disc, hence has diameter not exceeding half of the diameter of .ρ. The statement is obtained by

300

9 Fundamental Groups and Bounds for the Extremal Length

induction. Since the supremum of the diameters of the half-circles of generation n tends to zero for .n → ∞, we obtain in the limit a conformal mapping .ϕ from U onto .C . 

∞ Consider the inverse of the mapping .ϕ0 :  C → C \ k=−∞ Dk . By construction −1 the composition .P ◦ ϕ0 of .P with this inverse has the following property: the  restriction to .(H0 + ik) \ (D0 + ik) of .P ◦ ϕ0−1 : C \ ∞ k=−∞ Dk → C is a conformal mapping onto .H0 + ik that takes ik to ik, .i(k + 1) to .i(k + 1), and .∞ to −1 .∞. Hence, .P ◦ ϕ 0 commutes with translation by i, i.e. P ◦ ϕ0−1 (z + i) = P ◦ ϕ0−1 (z) + i for z ∈ C \

∞

.

Dk .

(9.6)

k=−∞

9.3 Building Blocks and Their Extremal Length Consider an elementary slalom curve in the left half-plane with initial point in the interval .(ik, i(k + 1)) and with terminal point in the interval .(ij, i(j + 1)). Suppose that .|k − j | ≥ 2, i.e. the slalom curve is non-trivial. After a translation by a (halfinteger or integer) multiple of i we obtain a curve that has endpoints in the intervals |k−j |−1 .(−i(M + 1), −iM) and .(iM, i(M + 1)) with .M = ≥ 12 . The homotopy 2 class of the slalom curve has the same extremal length as the translated class. Let M be any positive number. Denote a curve with interior in the open left halfplane, with initial point in .(−i(M +1), −iM) and terminal point in .(iM, i(M +1)), −1 by .γ,M , and the curve with inverted orientation by .γ,M . Notice that with the afore mentioned value .M = |k−j2|−1 the curve .γ,M is not an elementary slalom curve for even .|k − j |, but the curve .γ,M + iM is always an elementary slalom curve. It will be convenient to work with the normalized curve .γ,M for computing the extremal length. For any integer or half-integer number .M ≥ 12 we assign to the curve .γ,M the ∗ of curves which are homotopic to .γ class .γ,M ,M through curves in .C \ i(Z − M) with initial point in .(−i(M + 1), −iM) and terminal point in .(iM, i(M + 1)). The −1 ∗ )−1 . Respectively, let .γ in this way is denoted by .(γ,M class assigned to .γ,M r,M be a curve with interior contained in the right half-plane .Cr with initial point in the interval .(−i(M + 1), −iM) and terminal point in .(iM, i(M + 1). Consider the class ∗ .γ r,M of curves which are homotopic to .γr,M through curves in .C \ i(Z − M) with initial point in .(−i(M + 1), −iM) and terminal point in .(iM, i(M + 1)). We will call a homotopy class of elementary slalom curves in .C \ iZ an elementary slalom class with integer or half-integer parameter M, if after a ∗ )±1 translation by an imaginary integer or half-integer it becomes equal to .(γ,M ∗ ±1 or .(γr,M ) . A homotopy class of elementary half-slalom curves in .C \ iZ with endpoints in a real line . 2i + ik + R (.k ∈ Z) and a component of .iR \ iZ is called an

9.3 Building Blocks and Their Extremal Length

301

elementary half-slalom class with parameter M, if for a representative .γ the union (γ ∪ γ refl ) represents an elementary slalom class with parameter M. Here .γrefl γ ∈Γ is the reflection of .γ through the real line . 2i + ik + R. Notice that for a given M ∗ )±1 or .(γ ∗ )±1 are equal. The same the extremal lengths of all four classes .(γ,M r,M remark concerns the respective half-slalom classes.

.

Lemma 9.2 The extremal length of an elementary slalom class .γ ∗ with parameter M equals Λ(γ ∗ ) =

.

√ √ 4 log( M + M + 1) . π

(9.7)

The extremal length of an elementary half-slalom class .γ˜ ∗ with parameter M equals √ √ 2 log( M + M + 1) . (9.8) π √ √ ∗ ) = 4 log( M + M + 1) . Let .ϕ : Proof It is enough to prove that . Λ(γ,M π U → C be the mapping considered in the previous section. For a rectangle R we ∗ + iM. Consider take a holomorphic mapping .f : R → C \ iZ, that represents .γ,M its lift .f˜ : R → U to a mapping to the universal covering U of .C \ iZ. Define the mapping .G(ζ ) = −M − iϕ(ζ ), ζ ∈ U, from U onto the upper half-plane. The composition .G ◦ f˜ maps R into .C+ , and its extension to the closure of R takes the horizontal sides to the circles .{|z ± (M + 12 )| = 12 }. Lemmas 3.2 and 3.4 imply the √ √ inequality .λ(R) ≥ π4 log( M + M + 1). (Compare also with the proof of the statement in Example 3.4 of Sect. 3.2.) √ Take the conformal mapping .f0 of a rectangle .R0 with .λ(R0 ) = π4 log( M + √ M + 1) onto .C+ \ {|z ± (M + 12 )| ≤ 12 } that maps the horizontal sides to the two boundary circles .{|z ± (M + 12 )| = 12 }. Precompose this mapping with the inverse of G, and project the obtained mapping to .C \ iZ. We obtain√ a mapping √ that represents ∗ + iM. Hence, .Λ(γ ∗ ) = Λ(γ ∗ + iM) = 4 log( M + M + 1). .γ ,M ,M ,M π 

The statement for the half-slalom class .γ˜ ∗ follows in the same way. Λ(γ˜ ∗ ) =

.

Lemma 9.3 The extremal length of an element of each of the relative fundamental pb pb groups .π1 , .π1tr , .tr π1 , and .pb π1tr is realized on a locally conformal mapping of a rectangle representing the element. The extremal mapping extends locally conformally across the open horizontal sides of the rectangle. The extremal length of a conjugacy class of elements of the fundamental group of the twice punctured plane is realized on a locally conformal mapping of an annulus into the twice punctured plane. Proof of Lemma 9.3 For this proof it is more convenient to consider the fundamental group of the complex plane punctured at 0 and 1 rather than at .−1 and 1, and to consider the upper half plane .C+ as universal covering of .C \ {0, 1}.

302

9 Fundamental Groups and Bounds for the Extremal Length

Each holomorphic mapping of a rectangle into .C \ {0, 1} that represents an element of one of the relative fundamental groups lifts to the universal covering. The lift takes the open horizontal sides of the rectangle to certain half-circles with diameter on the real axis (maybe, after a conformal self-map of the half-plane) and represents the class of curves that are contained in the half-plane and join the two half-circles. As in the proof of Lemma 3.4 the extremal length is realized on a conformal mapping of a rectangle onto the half-plane with two deleted halfdiscs such that the horizontal sides are mapped onto the half-circles. Composing with the covering map we obtain a locally conformal mapping that extends locally conformally across the open horizontal sides of the rectangle. We will now prove the statement concerning conjugacy classes of elements of the fundamental group of .C \ {0, 1} with base point. Recall that each element of the fundamental group corresponds to a covering transformation. Each covering transformation is a holomorphic selfhomeomorphism of the universal covering .C+ that extends to a Möbius transformation .T (z) = az+b cz+d of the Riemann sphere with integer coefficients .a, b, c, d , such that .ad − bc = 1. Moreover, T is either parabolic (i.e. T has one fixed point and is conjugate to the mapping .z → z + b for a constant .b ), or T is hyperbolic (i.e., T has two fixed points and is conjugate to .z → κz for a positive real number .κ), or T is elliptic (i.e., T has two complex fixed points symmetric with respect to the imaginary axis and is conjugate to .z → eiθ z for a real number .θ ). See [58], Chapter II, 9D and 9E. Let .aˆ be a conjugacy class of elements of the fundamental group of .C \ {0, 1} and let a be an element of the fundamental group that represents .a. ˆ Denote by .Ta the covering transformation corresponding to a, and by .Ta  the subgroup of the group of covering transformations generated by .Ta . Then the quotient .C+ Ta  is an annulus. It has extremal length 0 if .Ta is parabolic or elliptic and has positive extremal length if .Ta is hyperbolic. If .f : A → C \ {0, 1} is a holomorphic mapping of an annulus A to .C \ {0, 1} that represents .a, ˆ then f lifts to a holomorphic map of A into .C+ Ta . The lift represents the class of a generator of the fundamental group of .C+ Ta . The corollary follows from Lemma 3.3. 

The following proposition considers elements of the fundamental group .π1 , that are represented by elementary words (i.e. by words consisting of a single syllable), and gives upper and lower bounds for the extremal length of the corresponding classes of curves in the relative fundamental groups with pb, tr or mixed boundary values. Proposition 9.3 The following statements hold for elements of the relative fundamental groups that are represented by elementary words. 1. Each syllable .s of form .(1) and degree .d ≥ 2 with pb horizontal boundary values lifts under .f1 ◦ f2 to an elementary slalom class with parameter .M = d−1 2 . The following equality for the extremal length holds Λ(pb (s)pb ) =

.

 2 log(d + d 2 − 1) . π

9.3 Building Blocks and Their Extremal Length

303

2. Each syllable .s of form .(2) and degree .d ≥ 2 with tr horizontal boundary values lifts under .f1 ◦ f˜2 to an elementary slalom class with parameter .M = d−1 2 . The following equality for the extremal length holds Λ(tr (s)tr ) =

.

 2 log(d + d 2 − 1) . π

3. Any syllable .s of degree .d ≥ 1 with mixed boundary values lifts under .f1 ◦ f2 to an elementary half-slalom class with parameter .M = d − 12 . For the extremal length the equalities  1 log(2d + 4d 2 − 1) , π  1 Λ(tr (s)pb ) = log(2d + 4d 2 − 1) π

Λ(pb (s)tr ) =

.

hold. Proof of Statement 1 Assume first that .n = d > 0. For any integer number k the class .pb (a1n )pb lifts under .f1 ◦ f2 to an elementary homotopy slalom class with initial point in (.i(k − 1), ik) and terminal point in .i(k − 1 + d), i(k + d)). There is a representative in the closed left half-plane. The slalom class has parameter .M = −d d−1 2 . A lift of the class .pb (a1 )pb is obtained by inverting the orientation. For .n = n class, d the class .pb (a2 )pb lifts under .f1 ◦ f2 to an elementary homotopy slalom  represented by a curve in the closed right half-plane with initial point in . i(k−1), ik   and terminal point in . i(k − 1 − d), i(k − d) . The class has parameter .M = d−1 2 .A −d lift of the class .pb (a2 )pb is obtained by inverting the orientation. See Fig. 9.4a for an elementary slalom curve with .d = 3 that is the lift under .f1 ◦f2 of a representative of .pb (a2−3 )pb with initial point not equal to 0. The estimate for the extremal length follows from formula (9.7), since √ √ 2 log( M + M + 1)2 = π 2 d −1 d +1 · )= = log(d + 2 π 2 2 .

 2 log(M + M + 1 + 2 M(M + 1)) π  2 log(d + d 2 − 1) . (9.9) π

Proof of Statement 3 for Syllables of the Form .ajn , .|n| ≥ 1, with Mixed Boundary Values Let .n = d > 0. For each .k ∈ Z the class .pb (a1n )tr ∈ pb π1tr lifts under the covering .f1 ◦ f2 to a class of elementary homotopy half-slalom curves with initial point in 1 .(ik, i(k + 1)) and terminal point in .i(k + d + ) + R. The half-slalom class has 2 1 parameter .M = d − 2 , and can be represented by a curve contained in the closed left half-plane. In the remaining cases (.n < 0, or .a1 replaced by .a2 , or both) a lift of the curve can be obtained from the present one by suitable choices of inverting

304

9 Fundamental Groups and Bounds for the Extremal Length

Fig. 9.4 Some elementary slalom curves and their projections under different coverings

orientation or reflection in the imaginary line or both. See Fig. 9.4b for .pb (a2−1 )tr with .d = 1. The estimate for the extremal length follows by Lemma 9.2, (9.8). The case .tr (ajn )pb is treated symmetrically.

9.4 The Extremal Length of Words in π1 . The Lower Bound

305

Proof of Statement 3 for Syllables of Form (2) with Mixed Boundary Values Recall that for each integer k the mapping .f1 ◦ f˜2 takes .(ik, i(k + 1)) onto i .(−1, 1), and . 2 + ki + R onto .i(0, ∞), if k is even and onto .−i(0, ∞), if k is odd. Hence, for each .k ∈ Z the class .tr (s)pb = tr (a1 a2 . . .)pb lifts under .f1 ◦ f˜2 to an elementary homotopy half-slalom class in the closed left half-plane with initial point in .((2k − 1)i, 2ki) and terminal point in .i(2k − d − 12 ) + R, and it lifts also under .f1 ◦ f˜2 to an elementary homotopy half-slalom class in the closed right halfplane with initial point in .((2ki, (2k + 1)i) and terminal point in .i(2k + d + 12 ) + R. Here d is the number of letters in .s. See also Proposition 9.2, as well as Fig. 9.1. The class has parameter .M = d − 12 . For each .k ∈ Z the class .tr (s)pb = tr (a2 a1 . . .)pb lifts under .f1 ◦ f˜2 to an elementary homotopy half-slalom class in the closed right half-plane with initial point in .((2k − 1)i, 2ki) and with terminal point in .i(2k + d − 12 ) + R, and it also lifts under .f1 ◦f˜2 to an elementary homotopy half-slalom class in the closed left halfplane with initial point in .(2ki, (2k+1)i) and with terminal point in .i(2k−d+ 12 )+R. Again d is the degree of the elementary word. See Fig. 9.4c for . tr (a2 a1 a2 a1 a2 )pb with number of half-turns equal to .d = 5. If all generators enter with power .−1 the orientation is reversed. In all cases we obtain elementary half-slalom curves with parameter .M = d − 12 . The estimate of the extremal length follows from (9.8). Proof of Statement 2 (Syllables of Form (2) with tr Boundary Values) The proof of Statement 2 is related to the proof of Statement 1 in the same way as the proof of Statement 3 for syllables of form (2) is related to the proof of Statement 3 for syllables of form (1). We leave it to the reader. The proposition is proved. 

9.4 The Extremal Length of Words in π1 . The Lower Bound Take an element of a relative fundamental group of the twice punctured complex plane. We will break representing curves into elementary pieces. The pieces will be chosen so that we have a good lower bound of the extremal length of the homotopy class of each piece. Ahlfors’ Theorem B and Corollary 3.1 will give a lower bound for the extremal length of the element of the relative fundamental group by the sum of the extremal lengths of the classes of the elementary pieces. We will use the following terminology. Let R be a rectangle. By a curvilinear subrectangle .R of R we mean a simply connected domain in R whose boundary looks as follows. It consist of two vertical segments, one in each vertical side of R, and either two disjoint simple arcs with interior contained in R and endpoints on opposite open vertical sides of R, or one such arc and a horizontal side of the rectangle R. The rectangle R itself may also be considered as a curvilinear subrectangle of R. The vertical segments in the boundary of the curvilinear rectangle

306

9 Fundamental Groups and Bounds for the Extremal Length

R are considered as its vertical curvilinear sides, the remaining curvilinear sides are considered as the horizontal curvilinear sides of .R . Recall that each curvilinear rectangle admits a conformal mapping onto a true rectangle whose continuous extension to the boundary takes curvilinear horizontal sides to horizontal sides and curvilinear vertical sides to vertical sides. We will use the symbol .# for the boundary values if we are free to choose either pb or tr boundary values. Let .v1 and .v2 be words in .π1 . For the word .# (v)# = # # # (v1 v2 )# ∈ π1 we say that there is a sign change of exponents for the pair .(v1 , v2 ) if the sign of the exponent of the last term of .v1 is different from the sign of the exponent of the first term of .v2 . The following lemma is a key part for the proof of the lower bound of the extremal length of slalom classes, respectively, of elements of the relative fundamental group of .C \ {−1, 1}.

.

Lemma 9.4 Consider a reduced word .w ∈ π1 . Take a locally conformal mapping g from a neighbourhood of the closure .R¯ of a rectangle R into the twice punctured plane .C \ {−1, 1}, such that the restriction .g|R represents the word w with pb, tr or mixed boundary values. The following statements hold. 1. If w has at least two terms, then R can be divided into a collection of pairwise disjoint curvilinear subrectangles .Rj of R which are in bijection to the terms of w in such a way that the restriction of g to .Rj represents the j -th term with pb boundary values if the term is not at the (right or left) end of the word and, possibly, with mixed boundary values if the term is at the end of the word. 2. If w has at least two syllables and among them there is at least one syllable of form .(1) then R contains a collection of mutually disjoint curvilinear subrectangles .Rj of R, .j ∈ J, which are in bijective correspondence to the collection of syllables .sj of w that are not of the form .(1) such that either the restriction of g to .Rj represents the respective syllable .sj with mixed boundary values, or the restriction represents an elementary word of form .(2) with one more letter than .sj and mixed boundary values. 3. If the word has at least two syllables and all syllables .sj , j = 1, . . . , N, of the word w are of form .(2) or .(3) then there is a division of R into curvilinear subrectangles .Rj of R, .j = 1, . . . , N, such that for .j = 1, . . . , N − 1, the restriction .g | Rj represents the j -th syllable with mixed boundary values. In the same way there is a decomposition of R into curvilinear subrectangles .Rj of R, .j = 1, . . . , N, such that for .j = 2, . . . , N, the restriction .g | R represents the j j -th syllable with mixed boundary values. For the proof of Statements 2 and 3 of Lemma 9.4 and we need Statements 2 and 3 of the following lemma. Lemma 9.5 1. Let .wˆ be a conjugacy class of elements of .π1 , .wˆ = ajk for .j = 1 or .j = 2 and integers k. Take a cyclically reduced word w that represents the conjugacy class

9.4 The Extremal Length of Words in π1 . The Lower Bound

307

w, ˆ and a mapping .g : A → C \ {−1, 1} of an annulus A to .C \ {−1, 1}, that extends to a locally conformal mapping on a neighbourhood of the closure .A¯ and represents .w. ˆ Then there exists a smooth arc .L0 ⊂ A¯ with endpoints on different boundary components of A, such that .g | A \ L0 represents .pb wpb . 2. Suppose .g : R → C \ {−1, 1} extends to a locally conformal mapping in a neighbourhood of the closure .R¯ of the rectangle R and represents .# (ajn11 ajn22 )# , where .ajk are different standard generators of .π1 , and the integer numbers .n1 and 0 ¯ with endpoints on .n2 have different sign. Then there exists a smooth arc .L ⊂ R 0 different open vertical sides of R, such that .R \ L = R1 ∪ R2 for two disjoint open curvilinear rectangles, the restriction .g | R1 represents .# (ajn11 )tr , and the restriction .g | R2 represents .tr (ajn22 )# . 3. Suppose .g : R → C \ {−1, 1} extends to a locally conformal mapping in a neighbourhood of the closure .R¯ of the rectangle R and represents .# (ajn )# for a standard generator .aj and an integer number n with .|n| > 1. Let .n1 and .n2 be non-zero integer numbers of the same sign, such that .n = n1 + n2 . Then there is a smooth arc .L0 ⊂ R¯ with endpoints on different open vertical sides of R, such that .R \ L0 = R1 ∪ R2 for two disjoint open curvilinear rectangles, .g | R1 represents .# (ajn1 )tr and .g | R2 represents .tr (ajn2 )# . .

We prove Lemma 9.5 after Lemma 9.4 is proved. Proof of Lemma 9.4 We start with the proof of Statement 1. Notice first that for the reduced word .w = w1n1 w2n2 . . . wknk ∈ π1 the corresponding element .# (w)# can be represented by a smooth curve .β that intersects .R and .iR transversally and lifts under .f1 ◦ f2 to a slalom curve (not merely to a homotopy slalom curve) in the case of pb horizontal boundary values, or to the union of a slalom curve with some elementary half-slalom curve(s) if some horizontal boundary values of w are tr. The intersection points of .β with the imaginary axis divide .β into pieces .βj n

that represent .wj j , .j = 1, . . . k. The .βj represent elements .# (w1n1 )pb if .j = 1, n

(wj j )pb for .j = 2, . . . , k − 1, and .pb (wknk )# if .j = k. Consider the mapping g of the lemma. By our condition 0 is a regular value of ¯ Normalize both, .Re g(z) and .Im g(z) in a neighbourhood of the closed rectangle .R. the rectangle so that we have .R = {z ∈ C : x ∈ (0, 1), y ∈ (0, a)}. Consider the mentioned curve .β . Take a homotopy that joins the mentioned curve .β with the restriction .β 0 of g

.pb

def

to the left vertical side of R, i.e. we consider a smooth mapping h from .R = {z ∈ C : x ∈ [−1, 0], y ∈ [0, a]} to .C \ {−1, 1}, whose restriction to the left side of .R equals .β and whose restriction to the right side of .R equals .β0 . Since .β and 0 .β are smooth curves that intersect .(−1, 1) ∪ iR transversely we may assume that zero is a regular value of both, .Re h and .Im h. Indeed, there is a neighbourhood U in .R of the two closed vertical sides of the rectangle such that 0 is a regular value of both, .Re h and .Im h, on U , and we may assume, by choosing the homotopy and extending it to a neighbourhood of .R suitably, that in a neighbourhood .U in .R of the closed horizontal sides this is so. Let .Ψ be a non-negative smooth function in a neighbourhood of the closed rectangle .R which equals 1 outside .U ∪ U and

308

9 Fundamental Groups and Bounds for the Extremal Length

equals zero near the boundary of the rectangle. If .ε is a sufficiently small complex value such that .Re ε is a regular value of .Re h on the closed rectangle, and .Im ε is a regular value of .Im h on the closed rectangle, then .h − εΨ is another smooth homotopy joining .β 0 with .β 1 , and zero is a regular value of both, the real part and the imaginary part of this mapping on the closed rectangle. We may choose the homotopy h so that we obtain a smooth mapping .g˜ from a neighbourhood of the closed rectangle .R ∪ R¯ to .C \ {−1, 1}, which equals g on .R¯ and h on .R such that ¯ 0 is a regular value of both, .Re g˜ and .Im g. ˜ We defined .R˜ as the interior of .R ∪ R. def Each connected component of the level set .{L0 = z ∈ R˜ : Re h(z) = 0} on the rectangle .R˜ is either an open arc (i.e. it is non-compact), and both its end points are ˜ or it is a circle (i.e. it is compact). Consider the closures of on the boundary of .R, all non-compact connected components of .L0 . This is a collection of closed arcs. The endpoints of these arcs that are contained in the left side of the rectangle divide the left side into connected components. The restrictions of h to these components are the .βj . The closure of each non-compact component of .L0 has at most one endpoint on the open left side of the rectangle. Indeed, assume the contrary. Then the restriction of .β to the interval I on the left side of the rectangle between the two endpoints of the arc is equal to the product of at least one or more successive curves .pb (β )pb and, hence is not homotopic to a constant curve through curves with pb j boundary values. On the other hand, the existence of an arc in the level set .Re h = 0 joining the two endpoints of I would provide a relative homotopy in .C \ {−1, 1} (with endpoints in the imaginary axis) joining the restriction of h to the interval I with a constant curve contained in the imaginary axis, which is a contradiction. The same reasoning shows that the closure of each non-compact component of .L0 with one endpoint on the open left side of the rectangle cannot have its other endpoint on a closed horizontal side of R. Indeed, otherwise the restriction of h to the interval on the vertical side between the endpoint of the arc and the vertex of the rectangle belonging to the closed horizontal side would be homotopic to a constant through curves with endpoints in .(−1, 1) ∪ iR, which is impossible. We may ignore the non-compact components of .L0 whose closures have no endpoint on the open left side of R, and will also ignore the circles contained in .L0 . We consider the connected components of .L0 whose closures have one endpoint on the open left side of R, and the other endpoint on the open right side. These arcs divide the rectangle into curvilinear rectangles which are in bijective correspondence to the intervals of division on the left side. We call them dividing arcs. Take the curvilinear rectangle whose left side corresponds to .βj . The restriction of h to this curvilinear rectangle provides a homotopy with boundary values in the imaginary axes (in .(−1, 1) ∪ iR, respectively), that joins .βj to the restriction .βj1 of ˜ Since the division of the rectangle h to the right side of the curvilinear rectangle .R. ˜ into curvilinear rectangles induces a division of the right side of the rectangle into .R intervals, the curve .β 1 , which is the restriction of g to the right side of R, is the composition of the curves .βj1 . Denote by E the subset of the right side of R that consists of the endpoints of the mentioned intervals.

9.4 The Extremal Length of Words in π1 . The Lower Bound

309

Consider all connected components of the level set .{z ∈ R : Re g = 0} in the original rectangle R that have an endpoint on the right side of R contained in E. ˜ Hence, each such component Each such component is a part of a dividing arc for .R. has its other endpoint on the left side of the original rectangle R. We call these components the dividing arcs for R. The dividing arcs for R provide a division of the rectangle R into curvilinear rectangles .Rj . The right side of each .Rj is a connected component .Ij of the complement of E in the right side of R and the restriction of n g to .Ij equals .βj1 . Since .βj1 represents the term .wj j with the required horizontal boundary conditions, we obtained the required collection of curvilinear rectangles .Rj . Statement 1 is proved. Proof of Statement 2 Choose a syllable .sk of form (1). Suppose there are syllables on the left of .sk which are not of form (1). Consider them in the order from left to right. If the left boundary values of w are pb then we consider the most left syllable .sj1 with .j1 < k which is not of form (1). For the curvilinear rectangle .Rj1 associated to .sj1 the restriction .g | Rj1 has pb left boundary values. Suppose the next syllable .sj1 +1 to the right of .sj1 is not of form (1). Then .j1 + 1 < k and there is a sign change of exponents for the pair .(sj1 , sj1 +1 ) of consecutive syllables. For the curvilinear rectangles .Rj of Statement 1 the restriction of g to def

Rj1 ,j1 +1 = Int(R j1 ∪ R j1 +1 ) represents .pb (sj1 sj1 +1 )pb . (Recall that .IntX denotes the interior of a subset X of a topological space.) By Lemma 9.5, Statement 2 the curvilinear rectangle .Rj1 ,j1 +1 can be divided into two curvilinear rectangles such that the restriction of g to them represents the syllables .pb (sj1 )tr and .tr (sj1 +1 )pb . We obtained the required representation of the two syllables .sj1 and .sj1 +1 which are both not of form (1). Suppose the next syllable .sj1 +1 to the right of .sj1 is of form (1), i.e it equals n .w j1 +1 for an integer n, .|n| ≥ 2, with .wj1 +1 being a standard generator of .π1 . We include here the case when .j1 + 1 = k. If there is a sign change of exponents for the pair .(sj1 , wjn1 +1 ) the preceding argument applies. Suppose there is no sign change. By Lemma 9.5 Statement 3 the curvilinear rectangle .Rj1 ,j1 +1 can be divided into two curvilinear rectangles such that the restrictions of g to them represent the sgn(n) n−sgn(n) syllables .pb (sj1 wj1 +1 )tr and .tr (wj1 +1 )pb with mixed boundary values. Since .

sgn(n)

there is no sign change, the word .sj1 wj1 +1 is of form .(2). If the left boundary values of w are tr then .g | R1 represents .s1 with mixed boundary values. Consider the most left syllable .sj1 with .1 < j1 < k which is not of form (1) (if there is any) and proceed as in the previous case. The inductive procedure. We obtained the following facts. If there is a syllable .sj1 not of form (1) on the left of .sk then Statement 2 holds for all syllables not of form (1) with label .j ≤ j1 + 1. The disjoint curvilinear rectangles contained in R used for representing these syllables (or syllables with one letter more) are contained in the union of the closure of the rectangles .Rj , 1 ≤ j ≤ j1 + 1, of Statement 1 j +1 of the lemma. Notice that the restriction .g | Int( j1=1 Rj ) has pb right boundary values.

310

9 Fundamental Groups and Bounds for the Extremal Length

We proceed by induction as follows. Suppose for some .l < k we achieved the following. We found disjoint curvilinear rectangles contained in . j ≤l R j which are in bijective correspondence to all syllables .sj with .j ≤ l that are not of form (1) such that the restrictions of g to these rectangles represent the syllable, or a syllable with one more letter, with mixed boundary values. Moreover, the restriction l .g | Int( j =1 Rj ) has pb right boundary values. Consider the first syllable .sj2 with .j2 < k on the right of .sl with .sj2 not of form (1) (if there is any). If .j2 + 1 < k we proceed with .sj2 in the same way as it was done for .sj1 in the case .j1 + 1 < k and continue the process. If there is no such syllable we stop the process. Make the same procedure from right to left until each syllable not of form (1) on the right of .sk is represented in the desired way. The curvilinear rectangles obtained by the construction which starts from the left do not intersect the curvilinear rectangles obtained by the construction which starts from the right because .sk = wknk with .|nk | ≥ 2. Statement 2 is proved. Proof of Statement 3 Under the conditions of Statement 3 there is a sign change of exponents for any pair of consecutive syllables. If the left boundary values of w are pb we may consider curvilinear rectangles .R2j −1,2j , j = 1, 2, . . . , such that the restriction of g to .R2j −1,2j , j = 1, 2, . . . , represents .pb (s2j −1 s2j )pb . By Lemma 9.5 Statement 2 there exists a smooth arc that divides .R2j −1,2j into two curvilinear rectangles such that the restriction of g to the first curvilinear rectangle .R2j −1 represents the syllable .s2j −1 with right tr boundary represents the values and the restriction of g to the second curvilinear rectangle .R2j syllable with left tr boundary values. In this way each syllable except, maybe, the last one is represented with mixed boundary values by restricting g to a member of the collection of the obtained pairwise disjoint curvilinear rectangles. If the left boundary values of w are tr then .g | R1 represents .s1 with mixed boundary values and we consider instead the rectangles .R2j,2j +1 , j = 1, . . .. We obtained the collection .Rj for both cases of the left boundary values. Repeating the procedure from right to left gives the rectangles .Rj . This finishes the proof of Statement 3. 

Proof of Lemma 9.5, Statement 1 We may assume that the annulus in the Statement 1 equals .A = {z ∈ C : 2 < |z| < r} for a positive number .r > 2. ˆ intersects Take a smooth closed curve .β : ∂D → C \ {−1, 1} that represents .w, .iR transversally, and has minimal possible number of intersection points with the imaginary axis. Then the intersection points of .β with the imaginary axis divide the curve .β into curves .pb (βl )pb whose interiors do not intersect .iR. No .pb (βl )pb is homotopic with endpoints in .iR to a constant curve, since otherwise the curve .β would be homotopic to a curve with smaller number of intersection points with .iR. Then the .pb (βl )pb represent words of the form .pb (ajl )kpb , and neighbouring .pb (βl )pb on the closed curve .β are powers of different generators .ajkl among the .a1 and .a2 . Since the word w is cyclically reduced, we may suppose (after perhaps relabeling . the .βl ) that .w = β1 β2 . . . βN

9.4 The Extremal Length of Words in π1 . The Lower Bound

311

The proof follows now along the same lines as the proof of Statement 1 of Lemma 9.4. We consider a smooth homotopy that joins the curves .β and .g | {|z| = 2}, more precisely, we consider a smooth mapping .h : {1 ≤ |z| ≤ r} → C \ {−1, 1} for which zero is a regular value of the real part, whose restriction to .{|z| = 1} coincides with .β , and whose restriction to .A¯ equals g. Similarly as in the proof of Lemma 9.4 the division of .β into .pb (βl )pb induces a division of the annulus A into curvilinear rectangles .Rl by smooth arcs in .A, each with endpoints on different boundary circles of A and contained in the preimage .g −1 (iR). The restriction .g | Rl represents .pb (βl )pb . The arc that is the common part of the boundaries of .R1 and .R N satisfies the requirement of Statement 1 of the lemma. Proof of Statement 2. Since there is a sign change of exponents for the two terms of the word, we may represent .# (ajn11 ajn22 )# by a smooth curve that intersects the real axis transversally and equals the product .# (β1 )tr tr (β2 )tr . . .tr (βn 1 +n2 )# of curves, sgn(n1 )

where each of the first .j1 curves represents .aj1

and the remaining curves

sgn(n2 ) .a , j2

represent and the interior of each curve does not intersect .R. The proof follows now along the lines of proof of Statement 1 of Lemma 9.4. Proof of Statement 3 We choose a representative .β of .# (ajn )# that has minimal number of intersection points with the real axis. The curve .β can be written as .# (β )tr . . .tr (β 1 n−1 )# for curves .βj that represent .# (aj )# whose interior does not intersect .(−1, 1). The proof now follows along the same lines of arguments as in the proof of Statement 1 of Lemma 9.4. 

The lemma is proved. We will now give the proof of the lower bound in Theorem 9.1 and in Theorem 9.2. The plan is the following. We consider a locally conformal mapping in a pb neighbourhood of a closed rectangle R, that represents an element .w ∈ π1 . The rectangle will be covered by the closures of curvilinear subrectangles for which the following holds. Each point of R is contained in at most two subrectangles. The restriction of the mapping to each subrectangle .Rj represents a syllable .sj for which the horizontal boundary values are chosen so that Proposition 9.3 provides a suitable lower bound of the extremal length of the subrectangle in terms of .L(sj ). Figure 9.5 illustrates a mapping g from a rectangle R to .C \ iZ which represents a lift under .f1 ◦ f2 of an element .w ∈ π1 with pb boundary values. The rectangle R is covered by relatively closed curvilinear rectangles. For each curvilinear rectangle we indicate the word with suitable horizontal boundary conditions (the “building block”) whose lift under .f1 ◦ f2 is represented by the restriction of g to the curvilinear rectangle. Notice that for some choices of the boundary values the curvilinear rectangles may intersect, but each point of R is contained in no more than two relatively closed curvilinear rectangles. The right part of the figure shows the image .g(R) of R in .C \ iZ and a slalom curve that lifts a representative of w. For the elementary pieces of the slalom curve we indicate the element of .π1 a lift of which the piece represents.

312

9 Fundamental Groups and Bounds for the Extremal Length

Fig. 9.5 A representative of a lift to .C \ iZ of an element of .π1 and building blocks

Proof of the Lower Bound in Theorem 9.1 and in Theorem 9.2 By Lemma 9.3 the extremal length of an element of a relative fundamental group of .C \ {−1, 1} is attained on a locally conformal mapping g from an open rectangle R with sides parallel to the axes into .C \ {−1, 1}, which represents the word w in the relative fundamental group of the twice punctured complex plane with tr, pb, or mixed horizontal boundary values. Since the continuous extension of g maps each horizontal side to the real axis or to the imaginary axis, we may extend g holomorphically through the horizontal sides. Shrinking the rectangle slightly in the horizontal direction (and hence, increasing the extremal length slightly), we may assume, that the mapping g is locally conformal in a neighbourhood of the closed ¯ rectangle .R. We assume first that the word has at least two syllables, and the word has at least one syllable of form (1). Let I be the set of the natural numbers j for which the j -th term of w is a syllable of form (1) and let .Rj , j ∈ I, be the curvilinear rectangles of Statement (1) of Lemma 9.4.

9.4 The Extremal Length of Words in π1 . The Lower Bound

313

By Corollary 3.1  .

λ(Rj ) ≤ λ(R) .

j ∈I

The rectangle .Rj admits the holomorphic mapping .g | Rj to .C \ {−1, 1} which represents the syllable corresponding to .Rj with pb boundary values if the syllable is not at the end of the word, and, possibly, with mixed boundary values if the By syllable is at the end of the word. Let .dj be the degree of the j -th syllable. 

Proposition 9.3 the extremal length of .Rj is not smaller than . π1 log(2dj + 4dj2 − 1) if .g | Rj has mixed boundary values and .dj ≥ 1, and it is not smaller than .

  2 1 log(dj + dj2 − 1) ≥ log(2dj + 4dj2 − 1) π π

(9.10)

if .g | Rj has pb boundary values and .dj ≥ 2. We obtain .

 1 1 1 1 λ(Rj ) ≥ λ(R) ≥ log(2dj + 4dj2 − 1) . 2 2 2 π j ∈I

(9.11)

j ∈I

Suppose as before that w contains at least one syllable of form (1). Denote by J the set of all natural numbers j for which .sj is not of form (1). For .j ∈ J we denote by .R the curvilinear rectangle of Statement 2 of Lemma 9.4 corresponding to .sj (or j to an elementary word of form .(2) with one more letter than .sj ). Since .g | Rj has mixed boundary values, Proposition 9.3 implies that the  extremal length of .Rj is not smaller than . π1 log(2dj + 4dj2 − 1). (If .Rj represents an elementary word of form .(2) with one letter more than .sj then the lower bound is even bigger, namely, .dj may be replaced by .dj + 1.) As before .dj ≥ 1 is the degree of the syllable .sj . Hence, .

  1 1 1 λ(R) ≥ log(2dj + 4dj2 − 1) . λ(Rj ) ≥ 2 2 2π j ∈J

(9.12)

j ∈J

For the case when w contains syllables of form (1) and syllables not of form (1) we add the two inequalities (9.11) and (9.12). We obtain λ(R) ≥

.

   1  1 log(2dj + 4dj2 − 1) + log(2dj + 4dj2 − 1) 2π 2π j ∈J

=

 1 log(2dj + 2π s j



j ∈I

4dj2 − 1) .

(9.13)

314

9 Fundamental Groups and Bounds for the Extremal Length

The last sum is extended over all syllables .sj of the word w, and .dj is the degree of sj . Suppose that the word does not contain syllables of form (1) and the syllables are labeled from left to right by .j = 1, . . . , N . Let .Rj be the rectangles from Statement 3 of Lemma 9.4. Since .g | Rj represents .sj with mixed boundary values we obtain

.

λ(R) ≥

N −1 

.

N −1 

λ(Rj ) ≥

j =1

j =1

 1 log(2dj + 4dj2 − 1) . π

(9.14)

On the other hand, with the curvilinear rectangles .Rj from Statement 3 of Lemma 9.4 we obtain the inequality λ(R) ≥

N 

.

N   1 log(2dj + 4dj2 − 1) . π

λ(Rj ) ≥

j =2

(9.15)

j =2

It follows that for any word .w ∈ π1 with at least two syllables for the respective family .Γ the inequality λ(R) ≥

.

  1 log(2dj + 4dj2 − 1) 2π s

(9.16)

j

holds. If the word consists of a single syllable, Proposition 9.3 implies (9.16) in the case of mixed boundary values as well as in the non-exceptional cases with both boundary values being tr or pb. Consider the exceptional cases. If .w = a1n with .n > 0 the mapping .ζ → −1 + ζ e , ζ ∈ R, with .R = {ξ + iη, ξ ∈ (−∞, 0), η ∈ (0, 2π n)}, represents .wtr . Hence, .Λ(wtr ) = 0. If .w = a1 a2 . . . and has degree .d ≥ 2 then the mapping .ζ → eζ , ζ ∈ R, with π π .R = {ξ + iη, ξ ∈ (1, ∞), η ∈ ( , 2 2 + π d)}, represents .wpb . Hence, .Λ(wpb ) = 0. The other exceptional cases are similar. The lower bound in Theorem 9.1 and in 

Theorem 9.2 is proved. It will be useful to have in mind the following corollary. Corollary 9.1 Let .wˆ be a conjugacy class of elements of .π1 that is not among the exceptional cases of Theorem 9.3. Then for any cyclically reduced representative .w ∈ π1 of .wˆ the inequality Λ(wpb ) ≤ Λ(w) ˆ

.

holds. The inequality may be strict.

(9.17)

9.5 The Upper Bound of the Extremal Length for Words Whose Syllables. . .

315

Further, let w be any representative of .w, ˆ that is obtained by a cyclic permutation of a cyclically reduced representative of .wˆ and has one of the following properties. Either the first and the last term of w are powers of the same sign of the same standard generator of .π , or the first and the last term of w are powers of different sign of different standard generators of .π1 . Then Λ(wtr ) ≤ Λ(w) ˆ .

.

(9.18)

The inequality may be strict. Proof Let .wˆ = ajk and .gˆ a locally conformal mapping from an annulus A into ˆ We may assume the twice punctured complex plane .C \ {−1, 1} that represents .w. that .gˆ is locally conformal in a neighbourhood of .A¯ and .λ(A) < Λ(w) ˆ − ε for an a priori given small positive number .ε . By Statement 1 of Lemma 9.5 there is a smooth arc .L0 ⊂ A¯ with endpoints on different boundary circles of A, such that .g ˆ | A \ L0 represents the cyclically reduced word w with pb boundary values. Hence, by Corollary 3.1 the inequality .Λ(wpb ) ≤ λ(A \ L0 ) ≤ λ(A) ≤ Λ(w) ˆ −ε holds. This proves inequality (9.17). Inequality (9.18) is proved in the same way using Corollary 3.1 and the analog of Statement 2 or Statement 3 of Lemma 9.5 for annuli instead of rectangles. 

Examples 3.4 and 3.5 of Section 4.2 show that the inequality may be strict. Proof of the Lower Bound of .Λ(w) ˆ in Theorem 9.3 The lower bound of .Λ(w) ˆ in Theorem 9.3 for the non-exceptional case follows immediately from Lemma 9.5, Statement 1, Theorem 9.1, and Corollary 3.1. Similarly as in the proof of the lower bound in Theorem 9.1 we see that in the 

exceptional cases the extremal length of .bˆ equals zero.

9.5 The Upper Bound of the Extremal Length for Words Whose Syllables Have Degree 2 We first obtain the upper bound of the extremal length of reduced words of the form w = a1±2 a2±2 . . .

.

(9.19)

with at least two syllables and pb, tr, or mixed horizontal boundary values, because the proof in this case avoids technical details and gives a better estimate than in the general case. We will use this estimate in Chap. 11. The proof in the general situation is given in Sect. 9.6. We first represent the lift under .f1 ◦ f2 of the elementary word .pb (a2−2 )pb by a holomorphic mapping .ga −2 of a rectangle .Ra −2 into .C \ iZ. 2

2

316

9 Fundamental Groups and Bounds for the Extremal Length

Consider the rectangle .Ra −2 = R in the plane with vertices .± 15 ± i π2 and the 2 mapping i + eζ , ζ ∈ R . 2

ga −2 (ζ ) =

.

2

(9.20)

The mapping .ga −2 takes the rectangle .R conformally onto a half-annulus in the 2

right half-plane. It maps the point .− π2i to .− 2i , and maps − 15

.

πi 2

to

1 5

.

3i 2.

The upper

side of the rectangle is mapped to the interval + i(e , e ). This interval is contained in .(i, 2i), and has distance to the endpoints of this interval equal to 1 1 3 −1 5 .min{e 5 − 2 , 2 − e } ≥ 0.25. The lower side is mapped to the interval i . 2

1

1

− i(e− 5 , e 5 ), that is contained in .(−i, 0) and has distance from the endpoints of .(−i, 0) at least equal to .0.25. Hence, the mapping represents a lift of .pb (a2−2 )pb under .f1 ◦ f2 . Its image has distance at least . 14 to .iZ. Moreover, .g −2 (− π2i ) = −i,

.

i 2

πi .g −2 ( 2 ) a2

= i, .vsl(R) = π , .hsl(R) =

2 5.

a2

The mapping .ga −2 extends holomorphically 2

across the horizontal sides of .R. The extension is denoted again by .ga −2 . 2

A lift of the elementary word

.pb

(a22 )pb under .f1 ◦ f2 is represented by the

def

mapping .ga 2 (ζ ) = ga −2 (−ζ ) , .ζ ∈ R, whose derivatives at .± π2i coincide with 2

2

those of .ga −2 at these points, and lifts of the words .pb (a1±2 )pb under .f1 ◦ f2 are 2

def

represented by the mappings .ga ±2 (ζ ) = −ga 2 (±ζ ), .ζ ∈ R, whose derivatives at 2

1

± π2i are .∓i. A lift under .f1 ◦ f2 of the syllable .tr (a2−2 )pb with mixed horizontal boundary

.

ζ

values can be represented by the mapping .gtr (a −2 )pb (ζ ) = − 2i +2e 2 on the rectangle 2

def

Rtr (a −2 )pb = Rmix = {ζ ∈ C : Imζ ∈ (0, π ), |Reζ |
1 such the following holds.

.

9.5 The Upper Bound of the Extremal Length for Words Whose Syllables. . .

317

For each .j = 1, 2, each choice of the sign, and each integer number k the compositions . f1 ◦ f2 ◦ (ga ±2 + ik) | R , . f1 ◦ f2 ◦ (tr (ga ±2 )pb + ik) | Rmix , j

and . f1 ◦ f2 ◦ (pb (ga ±2 )tr + ik) | Rmix |z| < C , |z ± 1| >

j

j

have its image in the domain .{z ∈ C :

1 C }. Indeed, .gaj±2 (R), .pb (gaj±2 )tr (Rmix ) and .tr (gaj±2 )pb (Rmix ) have distance at least . 14 from .C \ iZ, hence the statement is true for .k = 0. For arbitrary integer numbers k it follows from the periodicity of .f1 ◦ f2 . Consider the word .pb wpb =pb (a1±2 a2±2 . . . )pb with .N ≥ 2 syllables. Denote the

j -th syllable by .sj . Represent the lift under .f1 ◦ f2 of each .pb (sj )pb by the mapping gsj : R → C \ iZ described above.

.

def

Put .Rj = R + π icj for integer numbers .cj , so that for the midpoints .ξj− and .ξj+ , respectively, of the lower and upper side of .Rj the equalities .ξj+ = ξj−+1 , j = 1, . . . N − 1, hold, where N is the number of syllables of the word def

w. Take .gj (ζ ) = gsj (ζ − π icj ) + π ibj , ζ ∈ Rj , for integer numbers .bj such that + − .gj (ξ ) = gj +1 (ξ j j +1 ) . def

The rectangle .Rw = Int(∪Rj ) has vertical side length .vsl(Rw ) = π N and horizontal side length . 25 . The mappings .gj : Rj → C \ iZ represent .pb (sj )pb and extend by the Reflection Principle holomorphically across the horizontal sides to the rectangle of thrice the vertical side length, with the same horizontal side length and the same center as .Rj . The extended mappings are denoted again by .gj . The distance of the images of the extended mappings from .iZ is the same as that of the original mappings. We will perform quasiconformal gluing of the .gj using the fact that .gj (ξj+ ) = gj +1 (ξj−+1 ) and .gj (ξj+ ) = gj +1 (ξj−+1 ). The latter statement follows from the fact that one of two consecutive syllables is a power of .a1 and the other is a power of .a2 .

t Consider the .C 1 -function .χ0 on the interval .[0, 1], .χ0 (t) = 6 0 τ (1−τ )dτ . Then 3 5 .χ0 (0) = 0, .χ0 (1) = 1, and .0 ≤ χ (t) = 6t (1 − t) ≤ 0 2 . Put .χ (t) = χ0 ( 2 t). Define a function .gw on .Rw as follows. Each point .ξ in .Rw , for which .|Im (ξ − ξj+ )| > 15 for all .j < N, belongs to a single rectangle .Rj (depending on .ξ ) and we put def

gw (ξ ) = gj (ξ ) for such a point.

.

def

Fix a number .j < N and consider the set .Qj = {ξ ∈ Rw : |Im (ξ − ξj+ )| ≤ 15 }. def

Put .χj (ξ ) = χ (Im (ξ − ξj+ ) + 15 ) for .ξ ∈ Qj . Let .gw = (1 − χj )gj + χj gj +1 on .Qj . For .{ξ ∈ Qj : Im ξ = ξj+ − 15 } the equalities .χj (ξ ) = χ0 (0) = 0 and 1 .χ (ξ ) = χ (0) = 0 hold. Hence, the function .gw is .C smooth near such points j 0 + 1 .ξ . Further, for .{ξ ∈ Qj : Im ξ = ξ j + 5 } the equalities .χj (ξ ) = χ0 (1) = 1 and 1 .χ (ξ ) = χ (1) = 0 hold, hence, the function .gw is .C smooth near such .ξ . j 0

318

9 Fundamental Groups and Bounds for the Extremal Length

Since .gj +1 − gj and .gj +1 − gj vanish at .ξj+ = ξj−+1 , and the absolute value of the 1

second derivative of .gj and of .gj +1 on .Qj does not exceed .e 5 < 1.222, the estimate 1

|(gj +1 − gj )(ξ )| ≤ 2e 5

.

|ξ − ξj+ |2 2

1

≤ e5

1 2 < 25 10

(9.21)

holds on .Qj . Make the same definition for all but the last number j . We obtain a smooth mapping .gw from the rectangle .Rw to .C \ iZ which represents a lift of .pb wpb under .f1 ◦ f2 . Since the distance of the image of each mapping .gj from .iZ is not smaller than . 14 , by inequality (9.21) the distance of the image .gw (Qj ) from .iZ is not smaller than .0.15. It follows that the image .f1 ◦ f2 ◦ gw (Rw ) is contained in the closure of a domain .{z ∈ C : |z| < C, |z ± 1| > C1 } with another universal constant C. Lemma 9.6 The mapping .gw is a quasiconformal mapping from .Rw onto its image. The Beltramy differential .μgw of .gw has absolute value .|μgw | < 25 . Proof Put .ξ = u + iv. If .ξ ∈ Rw , .|Im (ξ − ξj+ )| ≤ the Beltrami differential at .ξ equals

μgw (ζ ) =

.

∂ g (ζ ) ∂ζ w ∂ ∂ζ gw (ζ )

=

i 2

−i ∂ 2 ∂v χj



∂ ∂v χj

1 5

for some .j, 1 ≤ j < N, then

 · (gj +1 − gj ) (ζ )

 . · (gj +1 − gj ) + (1 − χj ) · gj + χj · gj +1 (ζ ) (9.22)

On the rest of the rectangle .Rw the function is analytic. Notice that |

.

∂ 3 5 χj | ≤ · ∂v 2 2

on .Qj . By inequality (9.21) the numerator in the right hand side of (9.22) has 1

absolute value smaller than .e 5 · Since .max |gj |

1 5

3 20

< 0.1833.

≤ e on .Qj , the inequality √

max{|gj − gj (ξj+ )|, |gj +1 − gj (ξj+ )|} < e 5 · 1

.

√

2 5

holds on the . 52 -neighbourhood of .ξj+ . Since .gj (ξj+ ) = gj +1 (ξj−+1 ) has absolute value 1, the denominator of the right hand side of (9.21) is not smaller than √ 1 1 3 2 5 .1 − e 5 · 20 − e · 5 > 1 − 0.53 = 0.47.

9.5 The Upper Bound of the Extremal Length for Words Whose Syllables. . .

319

We obtain kw = sup |μgw (ζ )|
C }. We obtained the inequality λ(Rw ) ≤ Kw

.

Since .L(w) = N · log(4 +

√

πN 2 5

15) and .

≤

35 7 5 · πN = πN 3 2 6

35π√ 6 log(4+ 15)

< 9 we obtain

Λpb (w) < 9L(w) .

.

The case of words of the form (9.19) with tr or mixed boundary values is treated in the same way. For instance, the first syllable of the word may have totally real left boundary values. But it has always pb right boundary values. We described a holomorphic mapping from a rectangle of vertical side length .π and horizontal side length . 25 , that represents the lift of such a syllable under .f1 ◦ f2 . This representing mapping has the same properties near the upper side of the rectangle as the chosen representing mapping of the lift of the syllable with pb boundary values. Therefore the gluing procedure is the same as in the case of pb boundary values. The argument concerning the last syllable is the same with left and right boundary values interchanged. We proved the following

320

9 Fundamental Groups and Bounds for the Extremal Length

Proposition 9.4 If .w ∈ π1 is a word with .N ≥ 2 syllables, all of the form .a1±2 or ±2 .a 2 , then Λpb (w) ≤

.

35 π N < 9L(w) . 6

The same estimate holds for the extremal length with tr and with mixed boundary values. The following proposition is obtained in the same way. As in the proof of Proposition 9.4 we represent lifts under .f1 ◦ f2 of the terms of a cyclically reduced word w, that represents a class .w, ˆ by holomorphic mappings with pb horizontal boundary values from rectangles .Rj to .C \ Z. We glue the rectangles together so that we obtain an annulus, and perform quasiconformal glueing of the representing holomorphic mappings. We compose the solution of a Beltrami equation on an annulus with a conformal mapping from a round annulus to a curvilinear annulus to obtain a holomorphic mapping on a round annulus that represents .w. ˆ Proposition 9.5 Let w be a cyclically reduced word with .N ≥ 2 syllables, all of the form .a1±2 or .a2±2 (in particular, w has an even number of syllables). Then the upper bound of the extremal length of the free homotopy class .wˆ is given by the following inequality Λ(w) ˆ ≤

.

35 π N < 9L(w) ˆ . 6

The proofs of the propositions gave the following slightly more comprehensive statement that will be used later. Remark 9.1 Let R be a rectangle in the complex plane with horizontal side length .b and vertical side length .a. Then for any word w of the form (9.19) with N terms and a .6Nπ < b there exists a holomorphic map .gw : R → C \ {−1, 1} that represents w with pb horizontal boundary values and has its image in .{z ∈ C : |z| < C, |z ± 1| > C1 } for a universal constant .C > 1. Moreover, .gw takes the value zero at the midpoints of the horizontal sides of the rectangle. def Further, for positive numbers . α and . δ we consider the annulus . Aα,δ = {z ∈ δ α α,δ C : |Rez| < 2 }(z ∼ z + iα) . If .λ(A ) = δ > 6Nπ for an even number N, then for each word w of the form (9.19) with N terms there exists a holomorphic mapping .gw : (Aα,δ , 0(z ∼ z + iα)) → (C \ {−1, 1}, 0) from .Aα,δ with base point .0(z ∼ z + iα) to .C \ {−1, 1} with base point 0, that represents .w. ˆ Moreover, the image of .gw is contained in the domain .{z ∈ C : |z| < C, |z ± 1| > C1 } for a constant .C > 1. Indeed, for instance to obtain the second statement we consider the holomorphic mapping .gw ◦ ωw −1 ◦ ψw that represents a lift of .wˆ and was constructed in the proofs of Propositions 9.4 and 9.5. The mapping is defined on an annulus of the

9.6 The Extremal Length of Arbitrary Words in π1 . The Upper Bound

321

form .Aα,δ with .λ(Aα,δ ) = αδ > 6Nπ . The image .f1 ◦ f2 ◦ gw ◦ ωw −1 ◦ ψw (Aα,δ ) under the composition is contained in .{z ∈ C : |z| < C, |z ± 1| > C1 }. Perhaps after precomposing with a rotation of the annulus .Aα,δ it maps .0(z ∼ z + iα) to 0.

9.6 The Extremal Length of Arbitrary Words in π1 . The Upper Bound We take an arbitrary word w with at least two syllables and write .# w# as product of its syllables .# w# =# (s1 )pb . . .pb (sN )# . For each elementary word .pb wpb with pb horizontal boundary conditions (or for the respective word with other horizontal boundary conditions) we will represent its lift under .f1 ◦ f2 by a holomorphic mapping to .C \ iZ from a rectangle of extremal length not exceeding a constant times .L(w). The properties of the representing mappings will allow quasiconformal gluing. The building block is the following lemma. pb

Lemma 9.7 For each elementary word .pb wpb ∈ π1 there exists a rectangle .Rw with horizontal side length .hsl(Rw ) ≥ 1√ , and vertical side length .vsl(Rw ) ≤ 16 2 π L(w), and a holomorphic mapping .gw : Rw → C \ iZ with the following properties. There are two points .ξw+ and .ξw− in the boundary of .Rw with .Reξw+ = Reξw− , such that the restriction of .gw to the (vertical) line segment that joins .ξw− with + ± .ξw represents a lift of .pb wpb under .f1 ◦ f2 . The values of .gw at the points .ξw are imaginary half-integers that are not imaginary integers. The equalities for the derivatives at these points are .gw (ξw− ) = i if the first letter of the word is .a1±1 and ±1 ±1 + .−i if the first letter is .a 2 , and .gw (ξw ) = −i if the last letter of the word is .a1 and i if the last letter is .a2±1 . Further, the mapping .gw extends holomorphically to the 1 1 discs .|ζ − ξw± | < 16 of radius . 16 around .ξw± and the inequality .|gw (ζ )| ≤ 28 holds 1 . for the extended function if .|ζ − ξw± | < 16 For each elementary word .tr wpb or .pb wtr of the form .(1) or .(3) with mixed horizontal boundary values there is a lift under .f1 ◦ f2 that can be represented by a holomorphic mapping from a rectangle with horizontal side length .hsl(Rw ) ≥ 1√ , 16 2 and vertical side length .vsl(Rw ) ≤ π L(w). Moreover, the horizontal side, that is mapped to .iR, contains a point for which the previous statements hold. Proof of Lemma 9.7 1. Syllables of Form (1) with pb Horizontal Boundary Conditions We first consider syllables .pb wpb of the form .pb (a1d )pb , .d ≥ 2. Let .M = d−1 2 . Consider the holomorphic mapping .T a,b from the proof of Lemma 3.4 (see Example 3.4, Sect. 3.2) with by .TM . Let √ .a = M and .b = M + 1. Denote the obtained mapping −1 .tM = −M + M(M + 1) (compare with equality (3.12)). The inverse .T M provides 1 1 a conformal mapping from the half-annulus with center . 2 and radii . 2 and . 12 − tM to

322

9 Fundamental Groups and Bounds for the Extremal Length

1 {z ∈ C : Imz > 0, Rez > 0, |z− M+1 2 | > 2 }, that takes the half-circle with diameter .(0, 1) to the imaginary half-axis and the half-circle with diameter .(tM , 1 − tM ) to the half-circle with diameter .(M, M + 1). The mapping takes 0 to 0, 1 to .∞, .tM to M and .1 − tM to .M + 1. The first two conditions imply that the mapping .TM−1 has z the form .z → ac z−1 . The last two conditions imply that .( ac )2 = M(M + 1) and a . c < 0. This implies that .

TM−1 (z) =



.

M(M + 1)

z z . , and TM (z) = √ 1−z M(M + 1) + z

(9.23)

Further, the entire mapping ω(z) =

.

1 1 iz + e 2 2

(9.24)

˚+ with vertices 0, .π , .π + i log 1 , .i log 1 conformally takes the rectangle .R M 1−2tM 1−2tM onto the half-annulus with center . 12 and radii . 12 and . 12 − tM . The lower side of the rectangle is mapped onto the larger half-circle, the upper side is mapped onto the ˚+ be the rectangle that is symmetric with respect ˚M ⊃ R smaller half-circle. Let .R M ˚M ) = hsl(R ˚+ ) = π , and to the real axis, has the same horizontal side length .hsl(R M (see Eqs. (3.13), (9.9) and (9.10)) the double vertical side length √ √ 1 = 2 log( M + M + 1)2 1 − 2tM   =2 log(d + d 2 − 1) ≤ 2 log(2d + 4d 2 − 1).

˚M ) =2vsl(R ˚+ ) = 2 log vsl(R M

.

(9.25)

˚M has extremal length The rectangle .R ˚M ) = .λ(R

1 2 log 1−2t M

π

≤

 2 2 log(2d + 4d 2 − 1) = L(w) . π π

(9.26)

˚M onto the half-annulus in .C+ with center . 1 and radii of the The mapping .ω takes .R 2 1 circles equal to . 2 − tM and . 1 1 . 2 −tM

Denote by .I the mapping, that acts by multiplication with the imaginary unit, −1 −1 ˚M conformally .I(z) = iz, z ∈ C. The mapping .I ◦ T takes .R M ◦ ω = iTM ◦ ω   1 1 1 1 onto .C \ {|ζ − i(M + 2 )| ≤ 2 } ∪ {|ζ + i(M + 2 )| ≤ 2 } . ˜  is the lift of the left half-plane .C to the first sheet of the universal Recall that .C ˜  onto .C \ covering U of .C \ iZ. We consider the conformal mapping .ϕ0 from .C ∞ 1 1 {|z−i(k+ )| ≤ }, whose continuous extension to the boundary takes each k=−∞ 2 2 1 1 interval .(ki, (k + 1)i)to the half-circle .C ∩ {|z − i(k + 2 )| = 2 } (see Sect. 9.2). Let .ϕ be the extension of .ϕ0 to a conformal mapping of the universal covering U of −1 be its inverse. Denote by .P the covering .C\iZ onto .C (see Lemma 9.1), and let .ϕ

9.6 The Extremal Length of Arbitrary Words in π1 . The Upper Bound

323

map from the universal covering U of .C \ iZ to the set .C \ iZ. The composition (Pϕ −1 ) of .P with the inverse .ϕ −1 of .ϕ maps .C locally conformally to .C \ iZ.

.

def

Moreover, it maps the subset .H0 \ {|z − 2i | < 12 } of the half-strip .H0 = {z ∈ C : 0 < Imz < 1} conformally onto .H0 . Hence, by reflection in the half-circle i 1 −1 ) .{z ∈ C : |z − | < } and in the interval .(0, i), respectively, it follows that .(Pϕ 2 2  def maps .Q = H0 \ {|z − 4i | < 14 } ∪ {|z − 3i4 | < 14 } conformally onto the strip −1 | Q takes . i−1 to . i . .H = {z ∈ C : 0 < Imz < 1}. The mapping .P ◦ ϕ 2 2 def

The mapping .(Pϕ −1 )M , (Pϕ −1 )M (z) = Pϕ −1 (z − iM) + iM takes 1 1 .C \{|z±i(M+ )| < } locally conformally into .C\(iZ+iM) so that its continuous 2 2 extension takes the half-circle .C ∩ {|z − i(M + 12 )| = 12 } to .(iM, i(M + 1)), and the half-circle .C ∩ {|z + i(M + 12 )| = 12 } to .(−i(M + 1), −iM). The composition def

GM = (P ◦ ϕ −1 )M ◦ I ◦ TM−1 ◦ ω

.

(9.27)

˚M locally conformally into .C \ (iZ + iM), and its continuous extension maps .R ˚M to .(−(M + 1)i, −Mi) and the upper side to the interval takes the lower side of .R .(iM, i(M + 1)). See Fig. 9.6. ˚M → C \ iZ represents a lift under .F def = f1 ◦ f2 The mapping .GM − iM : R d of .pb (a1 )pb (see Sect. 9.2, and also Fig. 9.1). In other words, .F ◦ (GM − iM) ˚M be the rectangle of vertical side length represents .pb (a1d )pb . Let .RM ⊃ R ˚ ˚M ) which is symmetric with respect .3 · vsl(RM ) and horizontal side length .hsl(R to the real axis. By the reflection principle .GM extends holomorphically through ˚ to a holomorphic mapping from the rectangle .R each of the horizontal sides of .R M into .C \ iZ. The extension is denoted again by .GM . (see Fig. 9.6.) def

± ± = ± 2M+1 + 2i . Notice that .iqM is the midpoint of the half-circle in We put .qM 2 the left half-plane with diameter .(iM, i(M + 1) (.(−i(M + 1), −iM), respectively). ± ± Let .ξ˜M and .pM be the points for which ω

I◦T −1

M ± ± ± −−−→ pM −−−−→ iqM . ξ˜M

.

We will consider the inverse of .I ◦ TM−1 ◦ ω, .(I ◦ TM−1 ◦ ω)−1 = ω−1 ◦ TM ◦ I−1 . By equality (9.23), and since for the branch of the logarithm on .C+ with imaginary part in .(0, π ) the equality .ω−1 (z) = 1i log(2z − 1) holds, we obtain ω

.

−1

√ − M(M + 1) + z 1 1 . ◦ TM (z) = log(2TM (z) − 1) = log √ i i + M(M + 1) + z

(9.28)

324

9 Fundamental Groups and Bounds for the Extremal Length

Fig. 9.6 A conformal mapping representing a lift of a syllable of form (1)

For the derivative we get (ω

.

−1



1 1 − √ √ − M(M + 1) + z + M(M + 1) + z √ 2 M(M + 1) . =i M(M + 1) − z2

1 ◦ TM ) (z) = i



(9.29)

9.6 The Extremal Length of Arbitrary Words in π1 . The Upper Bound

325

Hence, 1

+ (TM−1 ◦ ω) (ξ˜M )=

.

−1 (ωM

◦ TM

) (M

+

1 2

+ 2i )

1 M(M + 1) − (M + 12 + 2i )2 1 M + 12 =− √ . √ 2i 2 M(M + 1) M(M + 1)

=

By Eq. (9.6) the equality .(P ◦ ϕ0−1 ) (z + i) = (P ◦ ϕ0−1 ) (z) holds on ∞ 1 1 −1 −1 .C \ k=−∞ {|z − i(k + 2 )| ≤ 2 }, hence .(P ◦ ϕ ) (z + i) = (P ◦ ϕ ) (z) holds for .z ∈ C . This implies that   + + (GM ) (ξ˜M ) = (P ◦ ϕ −1 )M ◦ (iTM−1 ) ◦ ω (ξ˜M )=

.

+ + ) · (iTM−1 ◦ ω) (ξ˜M )= (P ◦ ϕ −1 ) M (iqM   M + 12 i −1 i − 1 )· − √ (P ◦ ϕ ) ( . 2 2 M(M + 1)

(9.30)

+ We used that .iqM + iM = − 12 + i(M + 12 ) + iM differs by an imaginary integer 1 1 from .− 2 + 2 i. The derivative .(P ◦ ϕ −1 ) ( i−1 2 ) is a positive real number. Indeed, the composition  −1 maps .Q = H \ {|z − i | < 1 } ∪ {|z − 3i | < 1 } conformally onto .H , and .P ◦ ϕ 0 0 4 4 4 4 by the symmetry properties of the source and the target it maps the horizontal line 1 .{Imz = 2 } to itself, and on this line its real part is increasing in .x = Rez. Hence, ∂ −1 )( i−1 ) = ∂ (P ◦ ϕ −1 )( i−1 ) is a positive real number .α that does not . (P ◦ ϕ ϕ ∂z 2 ∂x 2 depend on M. We will prove now, that .αϕ ≤ π8 . The set Q contains the disc of radius . 14 around −π z , is a conformal mapping from the the point . 2i − 12 . The mapping ., (z) = ee−π z i−1 i+1 i strip .H onto the unit disc. It maps . 2 to 0. Hence the composition .◦P◦ϕ −1 : Q → D maps the disc of radius . 14 around the point . 2i − 12 conformally onto a subset of the unit disc. By Cauchy’s formula the derivative of the composition . ◦ P ◦ ϕ −1 at . 2i − 12 does not exceed 4. The derivative of . at . 2i equals . ( 2i ) = − π2 . Since .( ◦ P ◦ ϕ −1 ) ( 2i − 1 1 1 8 i −1 i −1 i 2 ) =  ( 2 ) · (P ◦ ϕ ) ( 2 − 2 ), the inequality .|(P ◦ ϕ ) ( 2 − 2 )| ≤ π holds. We proved that

M + 12 αϕ √ 2 M(M + 1)

(9.31)

M + 12 αϕ . √ 2 M(M + 1)

(9.32)

+ (GM ) (ξ˜M ) = −i ·

.

with .0 < αϕ ≤

8 π.

By symmetry reasons − (GM ) (ξ˜M )=i·

.

326

9 Fundamental Groups and Bounds for the Extremal Length

Since M + 12 4 1 1 ≤ for M ≥ , )2 = 1 + (√ 4M(M + 1) 3 2 M(M + 1)

.

we get 8 ± |(GM ) (ξ˜M )| ≤ √ . 3π

.

(9.33)

We put + ˚M , RM =|(GM ) (ξ˜M )| · R

.

± + ± ξM =|(GM ) (ξ˜M )| · ξ˜M ,

gM (ζ ) =(GM )



ζ + |(GM ) (ξ˜M )|

 , ζ ∈ RM .

(9.34)

(ξ ± ) = ∓i holds. By Eqs. (9.31) and (9.32) the equality .gM M We will prove now that .gM extends holomorphically to a disc of radius . 18 around ± 8 .ξ , and for the extended function the inequality .|g (ζ )| < 2 holds on a disc of M M ± 1 radius . 16 around .ξM . Let .RM,+ be the rectangle of twice the vertical side length and the same horizontal side length as .RM , that is symmetric with respect to the upper side of .RM . The mapping .gM takes .RM locally conformally to .C \ (iZ + iM) and its continuous extension to the boundary maps the upper side to the segment .(iM, i(M + 1)). Hence, .gM extends by reflection through the upper side of the rectangle and the segment .(iM, i(M + 1)), respectively, to a locally conformal mapping from .RM,+ into .C \ (iZ + iM). Denote the extension again by .gM . Put ˜ + )| · (I ◦ T −1 ◦ ω)−1 (Q). Note that .V ⊂ R −1 | Q is .V = |(GM ) (ξ M,+ . Since .P ◦ ϕ M M −1 a conformal mapping, and .GM = P ◦ ϕ −1 ◦ (I ◦ TM ◦ ω − iM) + iM, the restriction .gM | V is by (9.34) a conformal mapping onto .H + iM. The image of the mapping 1 1 .gM | V contains the disc of radius . 2 around .i(M + 2 ). Let .fM be the inverse of .gM | V . The restriction of .fM to the disc of radius . 12 around .i(M + 12 ) is a conformal ζ mapping with image in .RM,+ . The mapping .ζ → 2fM (i M+1 2 + 2 ) takes the unit + disc .D conformally onto its image. It takes 0 to .ξM , and has derivative of absolute value equal to 1 at 0. By Koebe’s . 14 Covering Theorem (see A.2) the image of the + unit disc under this mapping contains the disc of radius . 14 around .ξM . Hence, the + M+1 1 1 image .fM ({|z − i 2 | < 2 }) covers the disc of radius . 8 around .ξM . In particular, + is contained in .RM,+ , hence .hsl(RM ) ≥ 14 . Moreover, the disc of radius . 18 around .ξM + into the mapping .gM , being the inverse of .fM , takes a disc of radius . 18 around .ξM 1 a disc of radius . 2 . By Cauchy’s formula the second derivative of .gM on the disc of + 1 (ζ )| ≤ 28 . radius . 16 around .ξM is estimated by the inequality .|gM

9.6 The Extremal Length of Arbitrary Words in π1 . The Upper Bound

327

− + + The same arguments apply to the point .ξM instead of .ξM obtained from .ξM by reflection in the real axis. We obtain ± |gM (ζ )| ≤ 28 for |ζ − ξM |
2 , .L(w) = .(g0 ) (ξ ) = ±i. Moreover, .|g (ζ )| ≤ 4, .hsl(R 0 √ 0 √ 3 16 π √ ˚0 ) = π = log(2d + 4d 2 − 1) = log(2 + 3), and .vsl(R L(w) < .

2

2 log(2+ 3)

˚0 , .ξ ± = ξ ± , and .gw = g0 . The points .ξ ± are For .w = a2−1 we put .Rw = R w w 0 the midpoints of the horizontal sides of .Rw . The lemma is proved for .pb (a2−1 )pb . For the syllables .a2 and .a1±1 with pb horizontal boundary values the statement is obtained by the symmetry arguments used for syllables of form .(1). Consider a singleton with mixed boundary values. For instance, a lift under .f1 ◦f2 of the word .tr (a2−1 )pb is represented by the mapping .g(ζ ˜ ) = − 2i + eζ , ζ ∈ R0mixed , π 2 L(w).

def

where .R0mixed = {ζ ∈ C : Imζ ∈ (0, π2 ), |Reζ | < 15 }. Indeed, the lower side of the rectangle is mapped to .− 2i + R, and the upper side is mapped to .(0, i). Further, 1

√

g ( π2 i) = i, .|g | < e 5 < 32 , .hsl(R0mixed ) = 25 > 162 , and .vsl(R0mixed ) = π2 . The other cases with mixed horizontal boundary values follow by symmetry arguments.

.

4. Syllables of Form (2) with pb Horizontal Boundary Values of Degree At Least 2 We give the proof for the syllables .w =pb (a2−1 a1−1 . . .)pb . For other syllables of form (2) and degree d the statement follows by symmetry reasons. Put d+1 .M = 2 , where .d ≥ 2 is the degree, i.e. the number of letters of the word. The holomorphic function gM (z) =

.

1 exp(π(z + i(M − 1)), z ∈ C , 2

(9.36)

takes the set .{z ∈ C : Rez < logπ 2 , |Imz| < M} to the punctured unit disc .{0 < |ζ | < 1} ⊂ C \ iZ. The mapping .gM takes .{Imz = −(M − 12 ), Rez < logπ 2 } to the line segment .(−i, 0). Further, .gM takes .{Imz = M − 12 , Rez < logπ 2 } to the line segment .(0, i) if .d = 2M − 1 is odd, and takes it to .(−i, 0) if .d = 2M − 1 is even. The restriction of .gM to .(−i(M − 12 ), i(M − 12 )) is the curve .θ → 12 · eπ iθ , θ ∈ (− 12 , d − 12 ). The initial point of this curve is .− 2i and the curve makes d half-turns around 0. The curve .θ → 12 · eπ iθ , θ ∈ (− 12 , d − 12 ), represents a lift under .f1 f2 of the syllable .a2−1 a1−1 . . . of degree d with pb boundary values, and the restriction log 2 −1 −1 1 .gM | {z ∈ C : Rez < π , |Imz| < M − 2 } represents a lift of .pb (a2 a1 . . .)pb under .f1 f2 . The derivative of the mapping .gM at .−i(M − 12 ) equals πi 1 1 (gM ) (−i(M − )) = π gM (−i(M − )) = − . 2 2 2

.

(9.37)

9.6 The Extremal Length of Arbitrary Words in π1 . The Upper Bound

329

and  1 + π2i , if d = 2M − 1 is odd , )) = .(gM ) (+i(M − 2 − π2i , if d = 2M − 1 is even . For each natural number k the set .{z ∈ C : Rez
0

1 4

+ 2t (M − 12 )i. Hence, (see

√ 2 M(M + 1)(−2M − t 2 + 14 ) ∂ 1 1 log(2TM (M − + it) − 1) = .Re < 0, ∂t i 2 (−2M − t 2 + 14 )2 + (2t (M − 12 ))2 √ −2 M(M + 1)2t (M − 12 ) ∂ 1 1 Im log(2TM (M − + it) − 1) = < 0. ∂t i 2 (−2M − t 2 + 14 )2 + (2t (M − 12 ))2 (9.42) We obtain for .t > 0 .

For .0 ≤ t ≤

.

Im ∂t∂ 1i log(2TM (M − Re ∂t∂ 1i

log(2TM (M −

1 2 1 2

+ it) − 1) + it) − 1)

=

2t (M − 12 ) 2M + t 2 −

1 4

> 0.

1 2

Im ∂t∂ 1i log(2TM (M − Re ∂t∂ 1i log(2TM (M −

1 2 1 2

+ it) − 1) + it) − 1)

≤

M− 2M

1 2 − 14




4 . 5

(9.44)

By inequality (9.44) for .t = 12 the point . 1i log(2TM (M − 12 + it) − 1) lies in .{z ∈ C : + ∈ {Rez = π }. Inequality Rez < π − 25 }. Recall that . 1i log(2TM (M − 12 ) − 1) = η˜ M ± (9.43) shows that the part of the curve .ΓM that is contained in .{z ∈ C : π − 25 < + ) > Re(z − π )}. A similar Rez < π }, is contained in the set .{z ∈ C : Im(z − η˜ M − fact holds for the curve .ΓM . Hence, by the global behaviour of .ΓM± (see (9.42)) the set ω−1 ◦ TM

.



z∈C:

0 < Imz
2 5 8 > 12 . It follows that 2

Note that . M+18 < for M increasing,

√

.

π π (2) ± < |(GM ) (η˜ M )| < . 4 2

(9.48)

is increasing

(9.49)

Put def

(2) ± rM = |(GM ) (η˜ M )| .

.

(9.50)

9.6 The Extremal Length of Arbitrary Words in π1 . The Upper Bound

333

We define (2)

gM,2 (ζ ) =(GM )(

.

± ηM

ζ ) , ζ ∈ rM RM,2 , rM

± =rM · η˜ M .

(9.51)

We will prove now that .rM RM,2 (see (9.39) for the definition) contains a disc of ± 1 radius . 8 around .ηM and prove the required estimate of the second derive of .gM,2 in ± 1 around .ηM . This will be done similarly as for elementary words a disc of radius . 16 def

log 2 1 of form (1). Consider the half-strip .S− M = {z ∈ C : Rez < π , |Imz + M − 2 | < log 2 − 1 2 } ⊂ {z ∈ C : Rez < π , |Imz| < M}. The restriction .gM | SM is a conformal mapping onto its image. The image is the set

  1 log 2 1 · exp π(−i(M − ) + ζ ) + π i(M − 1) : Reζ < , |Imζ | < 2 2 π 2 1  π  log 2 1 = exp − i + π ζ : Reζ < , |Imζ | < 2 2 π 2   = reiϕ : 0 < r < 1, −π < ϕ < 0 .

1 .

− i 1 −1 This set contains the disc .{|ζ + 2i | < 12 }. Put .VM = (gM | S− M ) ({|ζ + 2 | < 2 }). ˜ − = (I ◦ T −1 ◦ ω)−1 (V − ). We have the following sequence of conformal Let .Q M M M mappings −1

I◦TM ◦ω gM 1 − η˜ M −−−−−−−→ − (M − )i −−−−−→ 2

1 − i 2

.

− VM

∈

∈

∈ I◦T −1 ◦ω

˜ − −−−−M −−−→ Q M

gM 1 i −−−−−→ {|ζ + | < } . 2 2

(9.52)

(2) ˜ − conformally onto .{|ζ + i | < 1 }. The composition .GM = gM ◦I◦TM−1 ◦ω maps .Q M 2 2 def (2) ˜ − conformally onto The mapping .gM,2 , gM,2 (ζ ) = G ( ζ ) takes .Q− = rM Q M

rM

M

M

{|ζ + 2i | < 12 }. Moreover, the derivative of .gM,2 at .η− has absolute value equal to one (see equality (9.50)). Consider the inverse of the restriction of .gM,2 to .Q− M . In the same way as for syllables of form (1) we see that .Q− ⊂ r R contains a disc M M,2 M − 1 of radius . 8 around .ηM , and the absolute value of the second derivative of .gM,2 in a − + 1 around .ηM does not exceed .28 . The same arguments apply to .ηM disc of radius . 16 − instead of .ηM .

.

334

9 Fundamental Groups and Bounds for the Extremal Length

M , and by (9.40) and inequality (9.49) The set .rM RM,2 also contains the set .rM R  the set .rM RM contains the set

2 π + {z ∈ C : − · < Re(z − π rM ) < 0, Imz − ηM < Re(z − π rM ), 5 2

.

− > −Re(z − π rM )} . Imz − ηM

(9.53)

1 1 Since . 16 = 0.0625 < 0.628 < 2π 10 , the union of the discs of radius . 16 around ± .η M with the set (9.53) contains the rectangle of horizontal side length equal to + − 1 . √ whose right vertices are .η M and .ηM , respectively. For the syllable .pb wpb = 16 2 −1 −1 pb (a2 a1 . . .)pb

of degree d and .M = d+1 2 we let .Rw be equal to this rectangle. ± ± = ηM = rM η˜ M . The points .ξw± are the right vertices of the Further, we put ˚M . By inequalities (9.25) and (9.49), and by rectangle .Rw . Hence, .Rw ⊂ rM R equality (9.50) the inequalities ± .ξw

 ˚M ) ≤ π · 2 log(2d + (4d 2 − 1)) = π L(w) . vsl(Rw ) ≤ rM vsl(R 2

.

hold. Put .gw = gM,2 . The conditions of the lemma concerning the first derivatives ± .gw (ξw ) are satisfied. Lemma 9.7 is proved for syllables of the form .pb wpb =pb (a2−1 a1−1 . . .)pb of degree .d ≥ 2. For the other elementary words of form .(2) with pb horizontal boundary values and degree .d ≥ 2 the Lemma is obtained by symmetry arguments. The points .ξw± are either the right vertices or the left vertices of the rectangle .Rw . 

Proof of the Upper Bound in Theorems 9.1, 9.2, and 9.3 Take any word w with pb horizontal boundary values. Write it as product of syllables .pb (wj )pb with pb horizontal boundary values. By Lemma 9.7 each .pb (wj )pb can be represented by a holomorphic mapping .gwj from the rectangle .Rwj into .C \ {−1, 1}. Recall that the vertical side length of .Rwj does not exceed .π L(wj ), the horizontal side length of .Rwj is at least . 1√ . We shrink the rectangles .Rwj in the horizontal direction, 16 2

so that its horizontal side length is exactly equal to . 1√ , and the points .ξw±j are on 16 2 the boundary of the shrinked rectangle .Rwj . Moreover, they are midpoints of the open horizontal sides of the shrinked rectangles, if they were midpoints of the open horizontal sides of the original rectangle, and they are on the open right (left) side of the shrinked rectangle, if they were on the open right (left) side of the original rectangle. 1 We may assume that .gwj extends holomorphically to discs of radius . 16 around the points .ξw±j . Take complex numbers .aj , , j = 1, . . . , d, such that for the translated rectangles def

def

Rj = Rwj + aj and the translated points .ξj± = ξw±j + aj the equalities .ξj+ = ξj−+1

.

9.6 The Extremal Length of Arbitrary Words in π1 . The Upper Bound

335

hold for .j ≤ N − 1. Choose complex numbers .bj so that for the translated functions def

gj (ζ ) = bj + gwj (ζ − aj ) the equalities .gj (ξj+ ) = gj +1 (ξj−+1 ), j = 1, . . . , N − 1, hold. We will perform quasiconformal gluing of the .gj using the fact that each .gj 1 extends holomorphically to a disc of radius . 16 around .ξj± . Denote the extended 1 around .ξj+ contains the square .Qj of functions again by .gj . The disc of radius . 16

.

√

side length . 162 with center .ξj+ . We define a smooth mapping g on ⎛ def .Rw =

Int ⎝

N

⎞ Rj ⎠ ∪

j =1

N

−1

Qj

j =1

as follows.

t Consider the .C 1 -function .χ0 on the interval .[0, 1], .χ0 (t) = 6 0 τ (1 − τ )dτ . Then .χ0 (0) = 0, .χ0 (1) = 1, and .0 ≤ χ0 (t) ≤ 6t (1 − t) ≤ 32 . Let .a < 2−5 be a positive number that will be chosen in a moment. If .ζ ∈ R w and .|Imζ − ξj+ | ≤ a2 for some .j = 1, . . . , N − 1, then .ζ ∈ Qj , and .gj and .gj +1 are defined and holomorphic in .Qj . In this case we put on .Qj g(ζ ) = (1 − χj )(Imζ ) · gj (ζ ) + χj (Imζ ) · gj +1 (ζ ) .

.

t−Imξj+ + a2 Here .χj (t) = χ0 ( ). If .|Imζ − ξj± | > a2 for all .j = 1, . . . , N − 1, then .ζ a is contained in a single rectangle .Rj , and we put .g = gj . The mapping g is of class + a 1 .C . If .|Imζ − ξ | ≤ j 2 for some .j = 1, . . . , N − 1, then for the Beltrami coefficient we obtain with .ζ = u + iv

μg (ζ ) =

.

∂ g(ζ ) ∂ζ ∂ ∂ζ g(ζ )

=

∂ ( −i 2 ∂v χj

i ∂ 2 ( ∂v χj

· (gj +1 − gj ))(ζ )

· (gj +1 − gj ) + (1 − χj ) · gj + χj · gj +1 )(ζ )

.

At all other points the Beltrami coefficient equals zero. For .ζ in the square of side length a with center .ξj+ the inequalities .|gj (ζ ) − gj (ξj+ )| ≤ 28 |ζ − ξj+ | and + + + + + 8 .|g j +1 (ζ ) − gj +1 (ξj )| ≤ 2 |ζ − ξj | hold. Since .gj (ξj ) = gj +1 (ξj ), and .gj (ξj ) = gj +1 (ξj+ ), we obtain on this square |gj − gj +1 | ≤ 2 ·

.

1 8 a2 ·2 · = 27 a 2 , 2 2

and √ (1 − χj ) · |gj − gj (ξj+ )| + χj · |gj +1 − (gj +1 )(ξj+ )| ≤ 27 2a .

.

336

9 Fundamental Groups and Bounds for the Extremal Length

Since .gj (ξj+ ) = gj +1 (ξj+ ) has absolute value equal to 1 and .|χj | ≤ |μg (ξ )| ≤

.

1−

Put .a = 13 2−8 . Since . 34 +

1 3 7 2 · 2 ·2 a √ 1 3 7 7 2a 2 · 2 ·2 a−2

=

3 2

· a1 , we obtain

3 7 42 a

1 − ( 34 +

√

2)27 a

.

√ 2 < 3,we obtain the inequality

|μg (ζ )| ≤

.

3 4

1 − ( 34

·

+

1 6 √

2) ·

1 6


0, or in the upper half-plane if .k < 0, and joins a point in .(−∞, −1) with a point in .(−1, 1), and a curve representing the element .C∗ (ϑ(b)tr ) ∈ π1tr . It can be proved along the same lines as the proof of Statement 3 of Lemma 9.5, that the rectangle R contains a curvilinear rectangle such that the restriction of the mapping .C(g) ˜ to it represents 2k .C∗ (ϑ(b)tr ). Therefore .Λ(btr ) ≥ Λ(ϑ(b)tr ). If .ϑ(b) = σ 1 for an integer .k , in other words, if .ϑ(b) is not among the exceptional cases of Theorem 9.1, the lower bound holds. Suppose .|k| = 1. If .b = σ1±1 then .btr can be represented by a curve .γ for whose lift .γ˜ with initial point in .{(x1 , x2 , x3 ) ∈ R3 : x2 < x1 < x3 } the curve .C(γ˜ ) is the union of the following two curves: a quarter-circle .Γ0 in .C \ {−1, 1} in the upper or lower half-plane which joins a point in .(−∞, −1) with a point in .iR, and a curve .Γ1 that represents .C∗ (pb ϑ(b)tr ). Following along the lines of proof of Statement 3 of Lemma 9.5, we see that the rectangle R contains a curvilinear rectangle such that the restriction of the mapping .C(g) ˜ to it represents .C∗ (pb ϑ(b)tr ). The lower bound for this case follows from Theorem 9.2. The proof for the case .b = σ2k b1 (with .b1 a reduced word in .σ12 and .σ22 ) is similar and is left to the reader. In this case the lift to .C3 (C) of a representing mapping for .btr is chosen with initial point in .{x1 < x3 < x2 }. The lower bound of Theorem 9.4 is proved. 

Proof of the Upper Bound of Theorem 9.4 We prove the upper bound for 3-braids that are not pure and not among the exceptional cases. We write again such a braid in the form (9.54), and assume that the braid equals .b = σ1k b1 for an odd natural number k and .b1 a pure braid as in Lemma 9.8 which is not the identity. We want to find a holomorphic mapping .g2 from a rectangle .C \ {−1, 1}, such that the mapping .(−1, g2 , 1) from the rectangle into .C3 (C) projects under .Psym to a holomorphic maping that represents .btr . If .k ≥ 3 (or .k ≤ −3, respectively) we first represent a lift under .f1 ◦ f2 of the homotopy class of a half-circle with center .−1 in the lower half-plane (in the upper half-plane, respectively), that joins .(−∞, −1) with .(−1, 1), by a suitable holomorphic mapping g from a rectangle to .C \ iZ. Then we represent a lift of .C∗ (tr ϑ(b)tr ) under .f1 ◦ f2 by a suitable holomorphic mapping from a rectangle to .C \ iZ. Quasiconformal gluing of the two mappings yields a

9.7 The Extremal Length of 3-Braids

341

holomorphic mapping from a rectangle to .C\iZ. Denote the projection under .f1 ◦f2 of this mapping by .g2 . The mapping .(−1, g2 , 1) defines a holomorphic mapping of a rectangle into .C3 (C) whose projection under .Psym represents .btr . If .k = ±1 we represent a lift under .f1 ◦ f2 of the homotopy class of a quartercircle with center .−1 that joins .(−∞, −1) with the imaginary axis by a suitable holomorphic mapping from a rectangle to .C\iZ, and represent a lift of .C∗ (pb ϑ(b)tr ) under .f1 ◦ f2 by a suitable holomorphic mapping from a rectangle to .C \ iZ. After quasiconformal gluing of the two mappings we finish the proof of the upper bound as in the previous case. We leave the details to the reader. 

The space .C3 (RS3 ) has a disadvantage especially when we want to deal with conjugacy classes of 3-braids. Namely, the codimension of .C3 (RS3 ) in .C3 (CS3 ) is bigger than one, hence, each curve in generic position avoids this space. It is convenient to consider the set def

H = {{z1 , z2 , z3 } ∈ C3 (C)S3 : the three points z1 , z2 , z3

.

are contained in a real line in the complex plane} .

(9.55)

The set .H is a smooth real hypersurface of .C3 (C)S3 . Indeed, let .{z10 , z20 , z30 } be a point of the symmetrized configuration space. Introduce coordinates near this point by lifting a neighbourhood of the point to a connected open set in .C3 (C) equipped def

1 , z ∈ C, maps with coordinates .(z1 , z2 , z3 ). Since the linear map .M(z) = zz−z 3 −z1 the points .z1 and .z3 to the real axis, the three points .z1 , z2 , and .z3 lie on a real 1 line in the complex plane iff the imaginary part of .z2 = M(z2 ) = zz23 −z −z1 vanishes. z2 −z1 The equation .Im z3 −z1 = 0 in local coordinates .(z1 , z2 , z3 ) defines a local piece of a smooth real hypersurface. The set .H is not simply connected. Nevertheless we may define homotopy classes of curves in .C3 (C)S3 with endpoints in this set. For later use we need the following lemmas.

def

Lemma 9.9 Let .γ be a loop in .H with base point .{−1, 0, 1} ∈ H. Then for some k ∈ Z the loop .γ represents the braid .Δk3 with this base point.

.

Proof Consider the lift .γ˜ = (γ˜1 , γ˜2 , γ˜3 ) : [0, 1] → C3 (C) of .γ with .γ˜ (0) = (−1, 0, 1). By a homotopy of .γ in .H with fixed endpoints we may assume that for the lift .γ˜ = (γ˜1 , γ˜2 , γ˜3 ) for all .t ∈ [0, 1] the equality .γ˜2 (t) = 0 holds and the segment .[γ˜1 (t), γ˜3 (t)] has length 2. Then for each point .γ˜ (t) there is a unique complex number .eiα(t) such that .α(0) = 0 and .γ˜ (t) = eiα(t) (−1, 0, 1) and the .α(t) depend continuously on .t ∈ [0, 1]. Since .γ˜ (1) is equal to .(−1, 0, 1) or to iα(1) = ±1 holds. The curve .t → eiα(t) , t ∈ [0, 1], .(1, 0, −1), the equality .e is homotopic as a curve in the unit circle with fixed endpoints to the curve π kti , t ∈ [0, 1]. Hence, .γ˜ is homotopic with fixed endpoints to the curve .t → e π kti (−1, 0, 1), t ∈ [0, 1] for an integer number k. .t → e

342

9 Fundamental Groups and Bounds for the Extremal Length

After applying the projection .Psym we obtain the curve .t → eπ kti {−1, 0, 1}, .t ∈ [0, 1], in .C3 (C)S3 that is homotopic with fixed base point to .γ and represents k .Δ . 

3 Lemma 9.10 Suppose for a conjugacy class .bˆ there exists a representing loop that avoids .H. Then .bˆ is the conjugacy class of a periodic braid of the form .(σ1 σ2 )k for an integer k. Proof Let .γ be a loop that represents .bˆ and avoids .H. Choose a base point on .γ , so that .γ represents a braid b. For some natural number k the power .bk of b is a def

pure braid, and the curve .γ k = γ ◦ . . . γ represents the pure braid .bk and does not   ! k γ2k −γ1k is a loop in .C \ {−1, 1} that avoids .R. Here .γjk γ3k −γ1k of .γ k . This means that .C(γ k ) is contractible, hence .γ k

meet .H. Then .C(γ k ) = 1 − 2

are the coordinate functions represents .Δ2 for an integer ., and hence .γ represents a periodic braid. If a representative .γ : [0, 1] → C3 (C)S3 , γ (0) = γ (1), of a 3-braid b avoids .H, then the associated permutation .τ3 (b) cannot be a transposition. Indeed, assume the contrary. Then there is a lift .γ˜ of .γ to .C3 (C), for which .(γ˜1 (1), γ˜2 (1), γ˜3 (1)) = (γ˜3 (0), γ˜2 (0), γ˜1 (0)). Let .Lt be the line in .C that contains .γ˜1 (t) and .γ˜3 (t), and is oriented so that running along .Lt in positive direction we meet first .γ˜1 (t) and then .γ˜3 (t). The point .γ˜2 (0) is not on .L0 . Assume without loss of generality, that it is on the left of .L0 with the chosen orientation of .L0 . Since for each .t ∈ [0, 1] the three points .γ˜1 (t), γ˜2 (t) and .γ˜3 (t) in .C are not on a real line, the point .γ˜2 (t) is on the left of .Lt with the chosen orientation. But the unoriented lines .L0 and .L1 coincide, and their orientation is opposite. This implies .γ˜2 (1) = γ˜2 (0), which is a contradiction. We proved that the braid b is periodic with period 3 or a power of .Δ2 

3 . Lemma 9.11 Let .A = {z ∈ C : r1 < |z| < r2 } be an annulus and .g : A → C3 (C)S3 a holomorphic mapping that extends holomorphically to a neighbourhood of the closure .A¯ and represents a conjugacy class .bˆ of braids. Suppose there exists a smooth arc in a neighbourhood of .A¯ that intersects .A along an arc .L0 , with endpoints on different boundary circles, such that .g(L0 ) ⊂ H. Take any point .q ∈ L0 and a positively oriented dividing curve .α in A with initial and terminal points equal to q that does not intersect .L0 at any other point. Let .A be a complex linear mapping for which .A(q) = {0, x, 1} ∈ C3 (R)S3 with .x ∈ (0, 1) . Denote by .btr the element of .π1 (C3 (C)S3 , C3 (R)S3 ) represented by .A(g) | α, and by b the respective braid. Then .b ∈ bˆ and .Λ(btr ) ≤ λ(A). If .bˆ is represented by non-periodic braids and g intersects .H transversally, then an arc .L0 as above exists. Proof Since the mapping g is free homotopic to the mapping .A(g) for any complex affine mapping .A, we may assume that for the mapping g itself .g(q) ∈ C3 (R)S3 , and, moreover, .g(q) = {0, x, 1}. The set .A \ L0 can be considered as curvilinear rectangle with curvilinear horizontal sides being copies of .L0 . Let .ω(z) : A \ L0 → R be the conformal mapping of the curvilinear rectangle onto a rectangle of the

9.7 The Extremal Length of 3-Braids

343

form .R = {z ∈ C : Rez ∈ (0, 1), Imz ∈ (0, a)}, that takes the curvilinear side of A \ L0 , that is attained by moving in .A \ L0 clockwise towards .L0 , to the lower side of R. (Note that the number .a is uniquely defined by .A \ L0 .) Consider the restriction .g | A \ L0 and take the lift .g˜ = (g˜1 , g˜2 , g˜3 ) : A \ L0 → C3 (C) of g to a mapping from .A\L0 to the configuration space .C3 (C), for which the clockwise continous extension to q equals .(0, x, 1) with .x ∈ (0, 1). For .z ∈ A \ L0 we consider the complex affine mapping

.

def

Az (ζ ) = a(z)ζ + b(z) =

.

ζ − g˜1 (z) , ζ ∈ C. g˜3 (z) − g˜1 (z)

We define the holomorphic mapping def

g(z) ˆ = Az (g(z)) ˜ = (0,

.

g˜2 (z) − g˜1 (z) , 1), z ∈ A \ L0 , g˜3 (z) − g˜1 (z)

g˜ 1 (z) and put .gˆsym (z) = Psym (g)(z) ˆ = Az (g(z)) = {0, gg˜˜ 23 (z)− (z)−g˜ 1 (z) , 1}, z ∈ A \ L0 . The mapping .gˆsym extends continuously to the horizontal sides of .A \ L0 . Since 2 −Z1 for a triple of complex numbers .{Z1 , Z2 , Z2 } the ratio . Z Z3 −Z1 is real if and only if the points .Z1 , Z2 , and .Z3 lie on a real line and g maps the horizontal sides of .A \ L0 to .H, the mapping .g ˆsym takes the horizontal sides of .A\L0 into .C3 (R)S3 . Hence, the ∈ π (C (C)S , C (R)S ). mapping .gˆsym on .A \ L0 represents an element .btr 1 3 3 3 3 We obtain .Λ(btr ) ≤ λ(A \ L0 ) . By Corollary 3.1 def

Λ(btr ) ≤ λ(A) .

.

By construction the mappings .t → g(α(t)) and .t → gˆsym (α(t)), .t ∈ [0, 1], take the same value at 0. Moreover, for each .t ∈ (0, 1) the value .gˆsym (α(t)) is obtained by applying to the value .g(α(t)) a complex linear mapping, i.e. .gˆsym (α(t)) = a(α(t)) · g(α(t)) + b(α(t)), t ∈ [0, 1], for continuous functions .a and .b with .a nowhere vanishing, .b(α(0)) = 0, .a(α(0)) = 1, and .b(α(1)) and .a(α(1)) real valued. The function .b ◦ α : [0, 1] → C is homotopic with endpoints in .R to the function that is identically equal to zero. The mapping .a ◦ α : [0, 1] → C \ {0} is homotopic with a(α) a◦α endpoints in .R to . |a◦α| . Hence, the mappings .gˆsym (α) and . |a(α)| g(α) from .[0, 1] to .C3 (C)S3 are homotopic with endpoints in .C3 (R)S. a◦α(t) There is a continous function .β on .[0, 1] such that . |a◦α(t)| = eiβ(t) and .β(0) = 0. π kiz

− a Then .β(1) = kπ for an integer number k. Put ˚ .g(z) = e · gˆsym (z), z ∈ R. a◦α(t) iβ(t) , t ∈ [0, 1], are homotopic with fixed Since the curves .t → |a◦α(t)| and .t → e .g | α and .g | α are homotopic. Hence, the equality endpoints, the restrictions of ˚ −k ) = b holds, and by Lemma 3.5 .(b Δ tr 3 tr

Λ(btr ) ≤ λ(A) .

.

344

9 Fundamental Groups and Bounds for the Extremal Length

To prove the last statement of the lemma we notice that the set .{z ∈ A : g(z) ∈ H} is ¯ If it did not contain a smooth manifold of codimension 1 in a neighbourhood of .A. an arc that joins the two boundary circles there would exist a simple closed dividing curve for the annulus whose image under g avoids .H. This would contradict Lemma 9.10. The lemma is proved. 

Chapter 10

Counting Functions

In this chapter we prove that the number of elements of .B3 Z3 with positive .Λtr not exceeding a positive number Y grows exponentially. As a corollary we give an alternative proof of the exponential growth of the number of conjugacy classes of elements of .B3 Z3 that have positive entropy not exceeding Y . The corollary is a particular case of results of Veech and Eskin-Mirzakhani. Our proof does not use deep techniques from Teichmüller theory. The first result that states exponential growth of the entropy counting function is due to Veech [80]. More precisely, he considered conjugacy classes of pseudoAnosov elements of the mapping class group of a closed Riemann surface S, maybe with distinguished points .En , with hyperbolic universal covering of .S \ En , and proved that the number of classes with entropy not exceeding a positive number Y , grows exponentially in Y . Notice that the 3-braids with positive entropy are exactly the 3-braids that correspond to pseudo-Anosov elements of the mapping class group of . P1 with fixed point .∞ and three other distinguished points. The precise asymptotic for the entropy counting function is given by Eskin and Mirzakhani [20] for closed Riemann surfaces of genus at least 2. Both papers, [80] and [20], use deep techniques of Teichmüller theory, in particular, these papers are based on the study of the Teichmüller flow. We give special attention to the growth estimates for the counting function related to the extremal length .Λtr of braids. The reason is that the extremal length with totally real boudary values is more flexible for applications to Gromov’s Oka Principle than the entropy (equivalently the extremal length of conjugacy classes of braids). In particular, estimates of the counting function related to the extremal length .Λtr allow to obtain effective finiteness theorems in the spirit of the Geometric Shafarevich Conjecture for the case when the base manifold is of second kind ([46]). We will address this application of the concept of extremal length in Chap. 11. In this chapter we obtain as a corollary of the results on the counting function related to .Λtr an alternative proof of the exponential growth of the number of conjugacy

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025 B. Jöricke, Braids, Conformal Module, Entropy, and Gromov’s Oka Principle, Lecture Notes in Mathematics 2360, https://doi.org/10.1007/978-3-031-67288-0_10

345

346

10 Counting Functions

classes of elements of .B3 Z3 that have positive entropy not exceeding Y . Our proof follows [48] and does not use deep techniques from Teichmüller theory. Notice that the bounds in the theorems of the present chapter can be improved. For the sake of simplicity of the proof we restrict ourselves to these estimates. The first theorem of this chapter treats the pure braid group modulo its center, 2 .PB3 Z3 . Recall that .PB3 Z3 is a free group in two generators .σ Z3 , .j = 1, 2, j which is isomorphic to the fundamental group .π1 (C \ {−1, 1}, 0) with standard generators .aj (see Lemma 1.2). The isomorphism takes the generators .σj2 Z3 to the generators .aj . By Lemma 3.5 for each element .b ∈ PB 3 Z3 the quantities .Λtr (b) and ˆ are well defined by .Λtr (b) = Λtr (b) and .Λ(b) ˆ = Λ(b) ˆ for any 3-braid b .Λ(b) representing the class .b. Λ (Y ), Y ∈ (0, ∞), is defined as follows. For each The counting function .NPB 3 Λ (Y ) is equal to the number of elements positive parameter Y the value of .NPB 3 .b ∈ PB3 Z3 with .0 < Λtr (b) ≤ Y . Note that here we count elements of .PB3 Z3 rather than conjugacy classes of such elements. We wish to point out that (with the mentioned choice of the generators) the condition .Λtr (b) > 0 excludes exactly the classes in .PB3 Z3 of the even powers .σj2k of the standard generators of the Λ Λ braid group .B3 (see Theorem 9.1 in Sect. 9.1). Notice that .NPB = NC\{−1,1} 3 Λ for the counting function .NC\{−1,1} (Y ), that counts the number of elements of the fundamental group .π1 (C \ {−1, 1}, 0) with positive .Λtr smaller than Y . Theorem 10.1 For all positive numbers .Y ≥ 12π the inequality .

log 2 1 1 Λ exp( Y ) ≤ NPB (Y ) ≤ e6π Y 3 2 6π 2

(10.1)

holds. The upper bound is true for all .Y > 0. Consider arbitrary 3-braids. The counting function .NBΛ3 (Y ), Y ∈ (0, ∞), is defined as the number of elements .b ∈ B3 Z3 with .0 < Λtr (b) ≤ Y . Note that the condition .Λtr (b) > 0 excludes exactly the elements .b ∈ B3 Z3 that are represented by .b = σjk Δ3 for .j = 1 or 2, and . = 0 or 1. (See Lemma 9.8 and Theorem 9.4.) Theorem 10.2 For any positive number .Y ≥ 12π the inequality .

log 2 1 exp( Y ) ≤ NBΛ3 (Y ) ≤ 4e6π Y 2 6π

(10.2)

holds. The upper bound is true for all .Y > 0. The entropy counting function .NBentr (Y ), Y > 0, for n-braids is defined as n the number of conjugacy classes of pseudo-Anosov elements of .Bn Zn with (Y ), Y > 0, is also equal entropy not exceeding Y . For 3-braids the value .NBentr 3 to the number of conjugacy classes of elements of .B3 Z3 with positive entropy not exceeding Y . Indeed, by Lemma 6.4 the reducible elements of .B3 Z3 are exactly the conjugates of powers of .σ1 Z3 . By the remarks following the proof

10 Counting Functions

347

in Example 3.8 of Sect. 3.2 their entropy equals zero. Since periodic elements of B3 Z3 have zero entropy, the pseudo-Anosov 3-braids are exactly the 3-braids of positive entropy. The following theorem will be obtained as a corollary of the Main Theorem and Theorem 10.2.

.

Theorem 10.3 For any number .Y ≥ .

70 2 3 π

the estimate

1 log22 Y e 6π ≤ NBentr (Y ) ≤ 4e12Y 3 8

(10.3)

holds. The upper bound holds for all positive Y . We will prove now Theorem 10.1 using the isomorphism from .PB3 Z3 onto π1 (C \ {−1, 1}, 0) that takes for .j = 1, 2 the generator .σj2 Z3 to the generator .aj . An element of .b ∈ PB3 Z3 ∼ = π1 (C \ {−1, 1}, 0) will often be identified with the reduced word in the .aj ∼ = σj2 Z3 that represents .b. The proof of Theorem 10.1 uses the syllable decomposition of words .w ∈ π1 (C\{−1, 1}, 0) (see Definition 9.1). Recall that for a word .w ∈ π1 (C\{−1, 1}, 0) in reduced form .w = ajk11 ajk22 . . . each term .ajkii with .|ki | ≥ 2 is a syllable of form

.

(1). Further, any maximal sequence of consecutive terms .ajkii , for which .|ki | = 1 and all .ki have the same sign, is a syllable of form (2) (if the number of terms is bigger than 1) or of form .(3) (if the number of terms is equal to 1). This gives a uniquely defined decomposition into syllables. Recall that the degree or length of the syllable is the sum of absolute values of the exponents of terms appearing in the syllable. We recall also the convention that the number of syllables of the identity equals zero.

.

def



Recall (see (9.1))  that we associated to each word w in the .aj the value .L(w) =

log(2dj + 4dj2 − 1), where the sum runs over the degrees .dj of all syllables of the word. For convenience we estimate the value .L(w) from below by a slightly ∼ simpler value. More precisely, for a non-trivial word .w ∈ π1 (C \ {−1,  1}, 0) =  def  PB3 Z3 we put .L− (w) = log(3dk ) ≤ L(w) = log(2dk + 4dk2 − 1), where each sum runs over the degrees .dk of all syllables of w. The main ingredient of the proof is the following lemma. j

L

− Lemma 10.1 Let .NPB be the function whose value at any .Y > 0 is the number 3 of reduced words .w ∈ π1 (C \ {−1, 1}, 0), .w = Id, for which .L− (w) ≤ Y . The following inequality

L

− NPB (Y ) ≤ 3

.

1 3Y e 2

(10.4)

holds. We need some preparation for the proof of Lemma 10.1. Consider all finite tuples (d1 , . . . , dj ), where .j ≥ 1 is any natural number (depending on the tuple) and the .dk ≥ 1 are natural numbers. Before proving Lemma 10.1 we estimate for .Y > 0 the number .N ∗ (eY ) of different ordered tuples .(d1 , d2 , . . . , dj ) (with varying j ) that .

348

10 Counting Functions

may serve as the degrees of the syllables of words (counted from left to right) with j Y ∗  1 log(3dk ) ≤ Y . Put .X = e . Then .N (X) is the number of distinct tuples with (3dk ) ≤ X. . Fix a natural number j . Denote by .Nj∗ (X), X ≥ 1, the number of tuples j .(d1 , . . . , dj ) for which . 1 (3dk ) ≤ X. The number is not zero if and only if log X .j ≤ log 3 . For .X ≥ 1 the equality .



N ∗ (X) =

.

Nj∗ (X)

(10.5)

X j ∈Z: 1≤j ≤ log log 3

holds. Notice first that .N1∗ (X) = [ X3 ], where .[x] denotes the largest integer not exceeding the positive number x. Indeed we are looking for the number of .d1 ’s for which .1 ≤ d1 ≤ X3 . The value of .Nj∗ for .j ≥ 2 is estimated by the following lemma. Notice that the def

lemma holds also for .j = 1 if in the inequality (10.6) we define .0! = 1. Lemma 10.2 Let .j ≥ 2. Then .Nj∗ (X) = 0 for .X < 3j . For .X ≥ 3j Nj∗ (X) ≤

.

1 2  j −1 1 2 j −1  1 ( ) X log ( )j −1 X . (j − 1)! 3 3 3 3

(10.6)

Proof The number .N2∗ (X) is the number of tuples .(d1 , d2 ) for which .3d1 · 3d2 ≤ X. Since .d1 d2 ≥ 1, .N2∗ (X) = 0 for .X < 32 . If .X ≥ 32 the inequality .d1 ≤ X9 holds, X . and for given .d1 the number .d2 runs through all natural numbers with .1 ≤ d2 ≤ 9d 1 Hence, for .X ≥ 32 N2∗ (X) ≤



.

k∈Z: 1≤k≤ X2 3

Put .a =

X 32 inequality . k1

N2∗ (X) ≤

and .k =

k ≤ α1 k −α

k a. dx x



.

k ∈ a1 Z: a1 ≤k ≤1

X . 32 k

(10.7)

Since for positive numbers .k and .α with .k > α the holds, we obtain

1 ≤ 2a k



1 1 2a

1 2 1 2 dx = 2a log(2a) = · X · log( · X). x 3 3 3 3 (10.8)

10 Counting Functions

349

For .j ≥ 3 we provide induction using the following fact. Let .0 < x < 1. Then j −1 x)j for any positive integer j the value .( (− log ) = − (− logx 2x) (j −log x) is negative. x Hence, for .k ∈ (0, 1) and .0 < α < k 1 1 . (− log(k ))j ≤ k α



k k −α

(− log x)j dx . x

(10.9)

We saw that the lemma is true for .j = 2. Suppose it is true for j . Prove that then it holds for .j + 1. The number .Nj∗+1 (X) is the number of tuples .(d1 , . . . , dj +1 ) for which .3d1 · . . . · 3dj +1 ≤ X. Hence, .Nj∗+1 (X) = 0 if .X < 3j +1 . If .X ≥ 3j +1 , then X .d1 ≤ and for given .d1 the tuples .(d2 , . . . , dj +1 ) run through all tuples with 3j +1 X j +1 .3d2 · . . . · 3dj +1 ≤ 3d1 . Hence, for .X ≥ 3 Nj∗+1 (X) =



Nj∗ (

.

k∈Z: 1≤k≤

≤

X 3j +1

 k∈Z: 1≤k≤

X 3j +1

X ) 3k

1 2 1 2 j −1 X  1 X  j −1 log ( )j −1 . ( ) (j − 1)! 3 3 3k 3 3 3k (10.10)

Put .a = 19 ( 23 )j −1 X and .k = ak . Then Nj∗+1 (X) ≤

.

1 (j − 1)!

 k ∈ a1 Z: a1 ≤k ≤

1 2a ≤ (j − 1)! ≤

1 2a (j − 1)!

=(−1)j −1

 

1 2j −1 1 2a

1 1 2a

1 2j −1

1 1 j −1 log( ) k k

1 (− log x)j −1 dx x

1 (− log x)j −1 dx x

1 1 2a (log x)j |11 . (j − 1)! j 2a

(10.11)

We obtain Nj∗+1 (X) ≤

.

j 1 2  j 1  11 2 j  2a log(2a) = ( ) X log ( )j X . j! j! 3 3 3 3

Lemma 10.2 is proved. Lemma 10.2 implies the following upper bound for .N ∗ .

(10.12) 

350

10 Counting Functions

Lemma 10.3 For .X < 3 the function .N ∗ (X) vanishes. Moreover, for any positive number X N ∗ (X) ≤ (

.

X 5 )3 . 3

(10.13)

Proof of Lemma 10.3 Since all .Nj∗ (X) vanish for .X < 3 the value .N ∗ (X) vanishes for such X. For .X ≥ 3 the equality (10.5) and Lemma 10.2 imply 

N ∗ (X) ≤

.

X j ∈Z: 1≤j ≤ log log 3

≤

≤

1 X 3

1 2  j −1 1 2 j −1  1 ( ) X log ( )j −1 X (j − 1)! 3 3 3 3

 X j ∈Z: 1≤j ≤ log log 3

2 1 j −1 1 log( X) (j − 1)! 3 3

2 1 1  X 5 X exp log( X) = ( ) 3 . 3 3 3 3

(10.14) 

Lemma 10.3 is proved. Proof of Lemma 10.1 We assume that

Y .[ log 3 ]

≥ 1. Otherwise

∗ Y .N (e )

vanishes,

L− and therefore .NPB (Y ) = 0, and the inequality is satisfied. We will use the notation 3 L− .N j (Y ), j ≥ 1, for the number of different reduced words w in .π1 (C \ {−1, 1}, 0) j that consist of j syllables and satisfy the inequality .L− (w) = k=1 log(3dk ) ≤ Y . j0 def L− L− L− Y Then .NPB (Y ) = j =1 Nj (Y ) with .j0 = [ log 3 ]. We will estimate .Nj (Y ) 3 by .Nj∗ (X) with .X = eY . Recall that .N ∗ (eY ) is the number of different tuples j Y .(d1 , . . . , dj ) with .dk ≥ 1 for which . k=1 (3dk ) ≤ e .

For this purpose we take a tuple .(d1 , . . . , dj ) and estimate the number of different reduced words with tuple of lengths of syllables (from left to right) equal to .(d1 , . . . , dj ). The first syllable is completely determined by its first letter (which may be .a1±1 or .a2±1 ), its length, and the fact which of the following two options hold: either it is of form (1), or of form (2) or (3). Hence, there are at most 8 different choices for the first syllable if we require the syllable to have exactly degree .d1 . For all other syllables the first letter of the syllable cannot be .ai±1 if the last letter in the preceding syllable is .ai±1 for the same .ai . Hence, for all but the first syllable there are at most 4 choices given the degree of the syllable and the preceding syllable. We showed that for all .j = 1, . . . , j0 = [ logY 3 ], and each tuple .(d1 , . . . , dj ), there are at most .2 · 4j different reduced words with tuple of lengths of syllables L− equal to .(d1 , . . . , dj ). Hence, for .Y ≥ log 3 the number .NPB (Y ) of reduced words 3 j0 .w ∈ π1 (C \ {−1, 1}, 0), .w = Id, with . 1 (3dk ) ≤ exp(Y ) equals L

− NPB (Y ) = 3

.

j0  j =1

L

Nj − (Y ) ≤

j0  j =1

2 · 4j0 Nj∗ (eY ) = 2 · 4j0 · N ∗ (eY ).

(10.15)

10 Counting Functions

351

Using the inequality .j0 ≤ logY 3 and Lemma 10.3 with .X = eY we obtain the requested estimate by the value 5

2 · 3− 3 exp((

.

1 log 4 5 + )Y ) < exp(3Y ). log 3 3 2

(10.16)

5

4 5 −3 We used the inequalities .( log < 0.65 < 1. Lemma 10.1 log 3 + ( 3 )) < 2.93 and .4 · 3 is proved. 

1 1 Proof of Theorem 10.1 By Theorem 9.1 the inequality . 2π L− (w) ≤ 2π L(w) ≤ Λtr (w) holds for all reduced words w representing elements in .PB3 Z3 ∼ = π1 (C \ {−1, 1}, 0) that are not equal to a power of .a1 or of .a2 or to the identity (equivalently, for which .Λtr (w) > 0). This inequality implies the inclusion .{w : 0 < Λtr (w) ≤ Y } ⊂ {w = Id : L− (w) ≤ 2π Y }. We obtain the inequality L− Λ .N PB3 (Y ) ≤ NPB3 (2π Y ) and by Lemma 10.1 the right hand side of this inequality

does not exceed . 12 e6π Y . This gives the upper bound. The lower bound is obtained as follows. Consider all reduced words in .π1 (C \ {−1, 1}, 0) of the form a12k1 a22k2 . . .

.

(10.17)

where each .ki is equal to 1 or .−1. If j is the number of syllables (i.e the number of the .aiki ) of a word w of the form (10.17), then by Proposition 9.4 the inequality 35 .Λtr (w) ≤ 6 πj holds. Consider the words of the mentioned form for which .j = def

Y j0 = [ 6π ]. Since .j0 must be at least equal to 2 we get the condition .Y ≥ 12π . For the chosen .j0 the extremal length of the considered words does not exceed Y . The Y Y −1)) = number of different words of such kind equals .2j0 = 2[ 6π ] ≥ exp(log 2·( 6π log 2 1 

2 exp( 6π Y ). Theorem 10.1 is proved.

Consider now arbitrary elements of the braid group modulo its center .B3 Z3 and their extremal length with totally real horizontal boundary values. Proof of Theorem 10.2 We will use the fact that by Lemma 9.8 any .b ∈ B3 which is not a power of .Δ3 can be uniquely written in the form .σjk b1 Δ3 , where k and . are integers, .b1 is a word in .σ12 and .σ22 in reduced form whose first term is a non-zero q(k) power of .σj = σj if .b1 = id. For a braid in this form we defined .ϑ(b) = σj b1 where .q(j ) is that even integer neighbour of j (including perhaps j itself), which is closest to zero. For each element .b of .B3 Z3 the value .ϑ(b) is well defined by putting it equal to .ϑ(b) for any .b ∈ B3 that represents .b. Take any element .b of .PB3 Z3 . Choose its unique representative that can be written as a reduced word w in .σ12 and .σ22 . We describe now all elements .b of .B3 Z3 with .ϑ(b) = w. If .w = Id these are the elements represented by the following braids. If the first term of w is .σj2k with .k = 0, then the possibilities

352

10 Counting Functions

 are .b = wΔ3 with . = 0 or 1, .b = σj wΔ3 with . = 0 or 1, or .b = σj±1 wΔ3 with . = 0 or 1 and .σj = σj . Hence, for .w = Id there are 8 possible choices of elements .b ∈ B3 Z3 with .ϑ(b) = w. By Theorem 9.4 the set of .b ∈ B3 Z3 with .0 < Λtr (b) ≤ Y is contained in the set of .b ∈ B3 Z3 with .ϑ(b) = w = Id, .L− (w) ≤ 2π Y . We obtain sgnk

L

− NBΛ3 (Y ) ≤ 8NPB (2π Y ). 3

.

(10.18)

By Lemma 10.1 we obtain .NBΛ3 (Y ) ≤ 4e6π Y . Since each pure 3-braid is also an element of the braid group .B3 the lower bound of Theorem 10.1 provides also a lower bound for Theorem 10.2. Theorem 10.2 is proved. 

Proof of the Upper Bound of Theorem 10.3 For each conjugacy class .bˆ of eleˆ > 0 and each positive number .ε we will find a braid b ments of .B3 Z3 with .h(b) ˆ  for which .bZ3 = b and Λtr (b) ≤

.

2 ˆ h(b) + ε . π

(10.19)

For this purpose we represent a conjugacy class .bˆ that represents .bˆ by a holomorphic map .g : A → C3 (C)S3 from an annulus A of extremal length λ(A)
0, the representing braids of .bˆ are non-periodic and the conditions of Lemma .h(b) 9.11 are satisfied. Hence, there exists a braid b for which .bZ3 represents .bˆ such that Λ(btr ) ≤ λ(A) .

.

ˆ By inequality (10.20) we obtain .Λtr (b) ≤ π2 h(b)+ε. Using also Lemma 3.5, we see ˆ that the number of conjugacy classes .b of .B3 Z3 of positive entropy not exceeding Y does not exceed the number of elements .b ∈ B3 Z3 with .Λtr (b) < π2 Y + ε. In (Y ) ≤ NBΛ3 ( π2 Y + ε). Since for .ε we may take any a priory given other words, .NBentr 3 positive number, Theorem 10.2 implies 2 NBentr (Y ) ≤ NBΛ3 ( Y ) ≤ 4e12Y . 3 π

.

We obtained the upper bound.

(10.21) 

10 Counting Functions

353

For obtaining the lower bound we need the following preparations. Lemma 10.4 Suppose .b1 and .b2 are elements of the free group .Fn in n generators, that are both represented by cyclically reduced words. Then .b1 and .b2 are conjugate in .Fn if and only if the word representing .b2 is obtained from the word representing .b1 by a cyclic permutation of terms. Proof It is enough to prove the following statement. If under the conditions of the lemma .b2 = w −1 b1 w for an element .w ∈ Fn , .w = Id, then .b2 = w −1 b1 w where .b is represented by a cyclically reduced word that is obtained from the reduced 1 word representing .b1 by a cyclic permutation of terms, and .w is represented by a reduced word which has less terms than the word representing w. This statement is proved as follows. Write .w = w1 w where .w1 ∈ Fn is represented by the first term of the word representing w, and .b1 = a B1 a where .a and .a are represented by the first and the last term, respectively, of the word that represents .b1 . Then .b2 = w −1 w1−1 a B1 a w1 w . If both relations .w1−1 a = Id and .a w1 = Id were true, then the first and the last term of the reduced word representing .b2 would be a power of the same generator of .Fn , which contradicts the fact that .b2 can be represented by a cyclically reduced word. If either .w1−1 a = Id def

or .a w1 = Id, then the reduced words representing .b1 = w1−1 a B1 a w1 and .b1 are cyclic permutations of each other. Hence, the statement is true.

 Lemma 10.5 The following equalities hold. Δ3 σ1 =σ2 Δ3 ,

.

Δ3 σ2 =σ1 Δ3 , σ1−1 (σ2−4 Δ43 )σ1 =σ22 σ12 σ22 σ12 , σ2−1 (σ1−4 Δ43 )σ2 =σ12 σ22 σ12 σ22 .

(10.22)

Proof The third equality is obtained as follows σ1−1 (σ2−4 Δ43 )σ1 = σ1−1 σ2−1 Δ23 σ2−2 Δ23 σ2−1 σ1

.

=(σ1−1 σ2−1 )(σ2 σ1 σ2 )(σ2 σ1 σ2 )σ2−1 σ2−1 (σ2 σ1 σ2 )(σ2 σ1 σ2 )σ2−1 σ1 =σ22 σ12 σ22 σ12 . The fourth equality is obtained by conjugating the third equality by .Δ3 .

(10.23) 

Consider two elements .b1 and .b2 of the pure braid group modulo center PB3 Z3 . In particular, .b1 and .b2 are elements of the full braid group modulo center .B3 Z3 . We write .a1 and .a2 for the generators of .PB3 Z3 . .

354

10 Counting Functions

Lemma 10.6 Suppose both elements .b1 and .b2 of . PB3 Z3 are of the form a1±2 a2±2 · · · a1±2 a2±2

or

.

a2±2 a1±2 · · · a2±2 a1±2 ,

(10.24)

with a positive number of terms, and the reduced word representing .b1 has at least four terms. Then .b2 cannot be conjugated to .b1 by an element .β of .B3 Z3 that can be represented by a braid .β = σj β1 Δ3 with some j and . and .β1 being a word in 2 2 .σ and .σ . 1 2 (See Lemma 9.8 for writing a 3-braid in a special form.) Proof of Lemma 10.6 Indeed, suppose the contrary, b1 = β −1 b2 β

.

(10.25)

with .β represented by .β = σj β1 Δ3 , where . = 0, 1, and .β1 ∈ PB3 is a word in 2 2 .σ and .σ . For .j = 1, 2, we let .bj be the representative of .b j which can be written 1 2 as reduced word in .σ12 and .σ22 . (In other words, for the representing braids .bj the first and third strand have linking number zero.) By Eq. (10.22) there is an integer def

number n such that the braid .b2 = σj−1 b2 σj Δ2n 3 is a product of a positive even

number of factors which either have alternately the form .σ1±4 and .(σ22 σ12 σ22 σ12 )±1 , or −1 they have alternately the form .σ2±4 and .(σ12 σ22 σ12 σ22 )±1 . The braid .β1 Δ3 b1 Δ− 3 β1 2 2 can also be written as reduced word in .σ1 and .σ2 . Hence, by Eq. (10.25) the two −1 braids .b2 and .β1 Δ3 b1 Δ− 3 β1 must be equal. The element .b2 Z3 is the product of at least two factors equal to .a1±2 and ±1 alternately, or it is the product of at least two factors equal to .a ±2 .(a1 a2 a1 a2 ) 2 and .(a2 a1 a2 a1 )±1 alternately. Hence, the reduced word in .a1 and .a2 representing .b2 contains at least 4 terms, and each sequence of four consecutive terms of this word contains at least two terms that appear with power .+1 or .−1. Indeed, put .A = a1 a2 a1 a2 . Replace in the product .A1 a1±2 A2 with .1 = 0, ±1,  .2 = 0, ±1, each factor .A 1 by the reduced word in .a1 , .a2 , representing it. We obtain a (possibly not reduced) word in .a1 , .a2 . If .1 = 0 or .1 = 1, and .2 = 0 or .−1, the word is already reduced. If .1 = −1, and .2 = 0 or .−1, the reduced word representing the product, consists of the first three letters of the word representing .A−1 = A1 , a non-trivial power of .a1 and all letters of the word representing .A2 . If .1 = 0 or .1 = 1, and .2 = 1, the reduced word representing the product, consists of all letters of the word representing .A1 , a non-trivial power of .a1 and the last three letters of the word representing .A = A2 . Finally , if .1 = −1 and .2 = 1 then the reduced word representing the product, consists of the first three letters of the word representing .A−1 = A1 , a non-trivial power of .a1 and the last three letters of the word representing .A = A2 . Hence, the reduced word representing .b2 Z3 contains for each factor .Aj in the whole product at least the two middle letters of the word representing .Aj . The case,

10 Counting Functions

355

when .b2 Z3 is the alternating product of .a2 a1 a2 a1 and .a22 is obtained by replacing the role of the generators .a1 and .a2 . Since the element .(Δ3 b1 Δ− 3 )Z3 can be represented by a word of the form (10.24) with at least four terms, we arrived at the following conjugation problem in the free group .F2 . We have an element .B1 ∈ F2 of the from (10.24) with at least 4 terms, and an element .B2 ∈ F2 written as reduced word with at least 4 terms, such that each sequence of four consecutive terms of .B2 contains at least two terms that appear with power .+1 or .−1. We have to prove that there is no element of .F2 that conjugates .B1 to .B2 . To prove this statement we assume the contrary. Suppose the reduced word k k k1 .a j1 . . . aj representing .B2 is not cyclically reduced. Identify each term .aj of the 

word with the element of .F2 represented by it. The conjugate .(ajk11 )−1 B2 ajk11 can be represented by a reduced word. If this reduced word is not cyclically reduced, then the consecutive sequence of all its terms is also a consecutive sequence of terms of the reduced word representing .B2 . Continue by induction in this way with the k .a j until we arrive at an element .B2 ∈ F2 that can be represented by a cyclically reduced word. At each inductive step the consecutive sequence of all terms of the new reduced but not cyclically reduced word is a consecutive sequence of terms of the previously obtained word and, hence, also a consecutive sequence of terms of .B2 . Since .B2 is not the identity, the reduced word representing .B is not the identity 2 and is cyclically reduced. The sequence of all its consecutive terms except the last one is also a sequence of consecutive terms of the reduced word representing .B2 . Since by our assumption .B2 is conjugate to .B1 by an element of .F2 , and both, .B and .B1 , are represented by cyclically reduced words, the representing words 2 have the same number of terms (see Lemma 10.4 ). By the assumption for .B1 the number of terms is at least 4. Hence, the words .B2 and .B2 have at least 3 consecutive terms in common. Therefore .B2 contains a term that appears with power .±1 which contradicts Lemma 10.4. The lemma is proved. 

Proof of the Lower Bound of Theorem 10.3 Consider the elements .b ∈ PB3 Z3 ⊂ B3 Z3 which can be represented by words of the form w = a1±2 a2±2 · · · a1±2 a2±2

.

(10.26)

in the generators .a1 ∼ = σ12 Z3 and .a2 ∼ = σ12 Z3 of .PB3 Z3 with at least 4 terms. We denote the number of syllables (in this case the number of the terms .aiki ) of the word (10.26) by 2j . Since each word of the form (10.26) is cyclically reduced, the ˆ of the conjugacy Main Theorem and Proposition 9.5 imply that the entropy .h(b) class of the element .b ∈ PB3 Z3 represented by w satisfies the inequality ˆ ≤ h(b)

.

π 35 · · 2πj 2 6

(10.27)

356

10 Counting Functions

ˆ ≤ Y holds. Take .j = j0 = [ 35Y 2 ]. For this choice of .j0 the inequality .h(b) def

6

π

Since we required that the number of syllables .2j0 is at least 4, we get the condition 70 2 2j0 .Y ≥ 3 π . The number of different words of such kind equals .2 . We prove now that the number of different conjugacy classes of .B3 Z3 that can be represented by elements in .PB3 corresponding to words (10.26) with .2j0 2j

syllables is not smaller than . 22j 0 . It is enough to prove the following claim. For each 0 element .b ∈ PB3 Z3 of form (10.26) the number of elements of .PB3 Z3 of form (10.26) that are conjugate to .b by an element of .B3 Z3 does not exceed .2j0 . Suppose two elements .b1 and .b2 of .PB3 Z3 ⊂ B3 Z3 are represented by a word of form (10.26) and belong to the same conjugacy class, i.e. .b2 = βb1 β −1 for an element .β ∈ B3 Z3 . Then by the Lemmas 9.8 and 10.6 for the element  2 2 .β the equality .β = β 1 Δ holds with .β 1 being a word in .σ Z3 and .σ Z3 , 3 1 2 .Δ3 = Δ3 Z3 and . = 0, 1. Put .b 1 = Δ3  b1 Δ3 − . Then .b 1 is represented by a word of form (10.24). The element .b2 is conjugate by an element of .PB3 Z3 either to .b1 or to .b 1 . By Lemma 10.4 the reduced word representing the element .b2 is obtained from the reduced word representing either .b1 or .b 1 by a cyclic permutation of terms. The number of cyclic permutations of .2j0 letters, equals .2j0 . Hence the number of different elements of .PB3 Z3 that can be represented by words of form (10.26) and are obtained from .b1 by conjugation with an element of .B3 Z3 does not exceed twice the number of cyclic permutations of .2j0 letters, i.e. it does not exceed .2 · 2j0 . (Notice, that e.g. by cyclically permuting the terms of a word with symmetries we may sometimes arrive at the same word.) We proved that the number of different conjugacy classes of .PB3 Z3 repre2j

2 0 sented by words of form (10.26) with .2j0 terms is not smaller than . 2·2j . 0 We obtain

NBentr (Y ) ≥ 3

.

22j0 −1 . 2 · 2j0

j

(10.28)

Notice that . 2j ≥ 2 for natural j . Indeed, the function .x → x x 2 = − x2 2 + 2 log , and .log 2 > 0.6) and x 2j 2 j 2 Hence, . 2j ≥ 2 . Hence, for .Y ≥ 70 3 π

x (since .( 2x )

equals 2).

j0 −2

NB3 (Y ) ≥ 2

.

entr

2x x

for .j = 1 and 2 the expression

Y

1 log 2 Y 1 35 2 ≥ 2 6 π ≥ e 6π 2 . 8 8

The lower bound of the theorem is proved.

increases for .x ≥ 2

(10.29) 

Chapter 11

Riemann Surfaces of Second Kind and Finiteness Theorems

This chapter presents our deepest application of the concept of conformal module and extremal length to Gromov’s Oka Principle. We address Problem 7.3 that proposes to obtain information on the set of homotopy classes of continuous mappings from a connected open Riemann surface X to a complex manifold Y . We consider the case when the target Y is the twice punctured complex plane and focus on the irreducible homotopy classes, since the reducible classes are well described and have the Gromov Oka property. The information we are interested in is the number of irreducible homotopy classes of mappings .X → Y that contain a holomorphic mapping. More specifically, for any connected finite open Riemann surface X (maybe, of second kind) we give an effective upper bound for the number of irreducible holomorphic mappings up to homotopy from X to the twice punctured complex plane, and an effective upper bound for the number of irreducible holomorphic torus bundles up to isotopy on such a Riemann surface. The bound depends on a conformal invariant of the Riemann surface that is expressed in terms of the conformal module of a finite number of annuli in X. On the other hand, in the proof we use instead of the conformal modules of conjugacy classes of elements of the fundamental group of the twice punctured complex plane the more powerful extremal length with totally real boundary values of elements of the fundamental group themselves. The estimates are in some sense asymptotically sharp: If .Xσ is the .σ neighbourhood of a skeleton of a connected finite open hyperbolic Riemann surface with a Kähler metric, then the number of irreducible holomorphic mappings up to homotopy from .Xσ to the twice punctured complex plane grows exponentially in . σ1 . These statements are analogs for Riemann surfaces of second kind of the Geometric Shafarevich Conjecture and the Theorem of de Franchis, that state the finiteness of the number of certain holomorphic objects on closed or punctured Riemann surfaces.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025 B. Jöricke, Braids, Conformal Module, Entropy, and Gromov’s Oka Principle, Lecture Notes in Mathematics 2360, https://doi.org/10.1007/978-3-031-67288-0_11

357

358

11 Finiteness Theorems

11.1 Riemann Surfaces of First and of Second Kind and Finiteness Theorems It seems that the oldest finiteness theorem for mappings between complex manifolds is the following theorem, which was published by de Franchis [19] in 1913. Theorem .A (de Franchis) For closed connected Riemann surfaces X and Y with Y of genus at least 2 there are at most finitely many non-constant holomorphic mappings from X to Y . There is a more comprehensive Theorem in this spirit. Theorem .B (de Franchis-Severi) For a closed connected Riemann surface X there are (up to isomorphism) only finitely many non-constant holomorphic mappings .f : X → Y where Y ranges over all closed Riemann surfaces of genus at least 2. A finiteness theorem which became more famous because of its relation to number theory was conjectured by Shafarevich [73]. Theorem .C (Geometric Shafarevich Conjecture) For a given compact or punctured Riemann surface X and given non-negative numbers .g and .m such that .2g − 2 + m > 0 there are only finitely many locally holomorphically non-trivial holomorphic fiber bundles over X with fiber of type .(g, m). Theorem .C was conjectured by Shafarevich [73] in the case of compact base and fibers of type .(g, 0). It was proved by Parshin [69] in the case of compact base and fibers of type .(g, 0), g ≥ 2, and by Arakelov [6] for punctured Riemann surfaces as base and fibers of type .(g, 0). Imayoshi and Shiga [41] gave a proof of the quoted version using Teichmüller theory. See also the subsequent research [18] and [37]. The statement of Theorem .C “almost” contains the so called Finiteness Theorem of Sections which is also called the Geometric Mordell Conjecture (see [66]), giving an important conceptual connection between geometry and number theory. For more details we refer to the surveys by C.McMullen [66] and B.Mazur [63]. (See also [21, 22], and [34].) Theorem .A is a consequence of Theorem .C, and Theorem .A has analogs for the source X and the target Y being punctured Riemann surfaces. Indeed, we may associate to any holomorphic mapping .f : X → Y of Theorem .A the bundle over X with fiber over .x ∈ X equal to Y with distinguished point .{f (x)}. Thus, the fibers are of type .(g, 1). A holomorphic self-isomorphism of a locally holomorphically non-trivial .(g, 1)-bundle may lead to a new holomorphic mapping from X to Y , but there are only finitely many different holomorphic self-isomorphisms. We will consider here analogs of Theorems .A and .C for the case when the base X is a Riemann surface of second kind. Notice that finite hyperbolic Riemann surfaces of second kind are interesting from the point of view of spectral theory of the Laplace operator with respect to the hyperbolic metric (see also [15]). There are

11.1 Riemann Surfaces of First and of Second Kind and Finiteness Theorems

359

interesting relations to scattering theory and (the Hausdoff dimension of) the limit set of the Fuchsian group defining X. The Theorems .A and .C do not hold literally if the base X is of second kind. If the base is a Riemann surface of second kind the problem to be considered is the finiteness of the number of irreducible isotopy classes (homotopy classes, respectively) containing holomorphic objects. In case the base is a punctured Riemann surface this is equivalent to the finiteness of the number of holomorphic objects. For more detail see Sects. 11.3 and 11.4. We will present here finiteness theorems with effective estimates for the case when the base is a Riemann surface of second kind. The estimates depend on a conformal invariant of the Riemann surface. We will now prepare the definition of this invariant. Let X be a connected open Riemann surface of genus .g ≥ 0 with .m + 1 holes, .m ≥ 0, equipped with a base point .q0 . Recall that the fundamental group .π1 (X, q0 ) of X is a free group in .2g +m generators. We describe now the conformal invariant of the Riemann surface X that will appear in the mentioned estimate. We fix a standard system of generators .E of .π1 (X, q0 ) that is associated to a standard bouquet of circles for X (see Sect. 1.2). The circles are denoted by .αj , .βj and .γj (see Fig. 1.1). The generators .E ⊂ π1 (X, q0 ) are labeled as follows. The elements .e2j −1,0 ∈ π1 (X, q0 ), .j = 1, . . . , g, are represented by the curves .αj , the elements .e2j,0 ∈ π1 (X, q0 ), j = 1, . . . , g, are represented by the curves .βj , and the elements .e2g +k,0 ∈ π1 (X, q0 ), k = 1, . . . , m, of .π1 (X, q0 ) are represented by the curves .γk . Let .X˜ be the universal covering of X. For each element .e0 ∈ π1 (X, q0 ) we consider the subgroup .e0  of .π1 (X, q0 ) generated by .e0 . Fix a point .q˜0 in .X˜ that projects to the base point .q0 ∈ X, let .(Isq˜0 )−1 (e0 ) be the covering transformation corresponding to .e0 (see Sect. 1.2), and .(Isq˜0 )−1 (e0 ) the group generated by q˜ −1 .(Is 0 ) (e0 ). Definition 11.1 Denote by .Ej , j = 2, . . . , 10, the set of primitive elements of π1 (X, q0 ) which can be written as product of at most j factors with each factor being either an element of .E or an element of .E −1 , the set of inverses of elements of .E. Define .λj = λj (X) as the maximum over .e0 ∈ Ej of the extremal length of the q˜0 −1 ˜ annulus .X(Is ) (e0 ). .

The quantity .λ7 (X) (for mappings to the twice punctured complex plane), or .λ10 (X) (for .(1, 1)-bundles) is the mentioned conformal invariant. In the following theorem we put .E = {−1, 1, ∞}. We will often refer to .P1 \ {−1, 1, ∞} as the thrice punctured Riemann sphere or the twice punctured complex plane .C \ {−1, 1}. Recall that a continuous mapping from a Riemann surface to the twice punctured complex plane is called reducible, iff it is homotopic to a mapping with image in a once punctured disc contained in .P1 \ E. (The puncture may be equal to .∞.) See Definition 7.4. For any connected finite open Riemann surface X with only thick ends and non-trivial fundamental group there are countably many non-homotopic reducible holomorphic mappings from X to the twice punctured complex plane (see Lemma 7.4). On the other hand the following theorem holds.

360

11 Finiteness Theorems

Theorem 11.1 For each open connected Riemann surface X of genus .g ≥ 0 with m + 1 ≥ 1 holes there are up to homotopy at most .3( 32 e36π λ7 (X) )2g +m irreducible

.

def

holomorphic mappings from X into .Y = P1 \ {−1, 1, ∞}. Notice that the Riemann surface X is allowed to be of second kind. If X is a torus with a hole, .λ7 (X) may be replaced by .λ3 (X). If X is a planar domain, .λ7 (X) may be replaced by .λ4 (X) Recall that a holomorphic .(1, 0)-bundle is also called a holomorphic torus bundle. A holomorphic torus bundle equipped with a holomorphic section is also considered as a holomorphic .(1, 1)-bundle. Moreover, a smooth .(1, 0)-bundle over a finite connected open smooth surface admits a smooth section. A holomorphic torus bundle over a finite connected open Riemann surface is (smoothly) isotopic to a holomorphic torus bundle that admits a holomorphic section. The following theorem for torus bundles holds. Theorem 11.2 Let X be an open connected Riemann surface of genus .g ≥ 0 with  2g +m m+1 ≥ 1 holes. Up to isotopy there are no more than . 2·156 ·exp(36π λ10 (X)) irreducible holomorphic .(1, 1)-bundles over X.

.

For the definition of irreducible .(g, m)-bundles see Sect. 8.2. Since on each finite open Riemann surface with only thick ends and non-trivial fundamental group there are countably many non-homotopic reducible holomorphic mappings with target being the twice punctured complex plane, there are also countably many non-isotopic reducible holomorphic .(1, 1)-bundles over each such Riemann surface. This follows from Proposition 8.2. Notice that Caporaso proved the existence of a uniform bound for the number of locally holomorphically non-trivial holomorphic fiber bundles over closed Riemann surfaces of genus g with .m ≥ 0 punctures with fibers being closed Riemann surfaces of genus .g ≥ 2. The bound depends only on the numbers g, .g and m. Heier gave effective uniform estimates, but the constants are huge and depend in a complicated way on the parameters. Theorems 11.1 and 11.2 imply effective estimates for the number of locally holomorphically non-trivial holomorphic .(1, 1)-bundles over punctured Riemann surfaces, however, the constants depend also on the conformal type of the base. More precisely, the following corollaries hold. Corollary 11.1 There are no more than .3( 32 e36π λ7 (X) )2g +m non-constant holomorphic mappings from a Riemann surface X of type .(g, m + 1) to .P1 \ {−1, 1, ∞}.  2g +m Corollary 11.2 There are no more than . 2 · 156 · exp(36π λ10 (X)) locally holomorphically non-trivial holomorphic .(1, 1)-bundles over a Riemann surface X of type .(g, m + 1). The following examples demonstrate the different nature of the problem in the two cases, the case when the base is a punctured Riemann surface, and when it is a Riemann surface of second kind.

11.1 Riemann Surfaces of First and of Second Kind and Finiteness Theorems

361

Example 11.1 There are no non-constant holomorphic mappings from a torus with one puncture to the twice punctured complex plane. Indeed, by Picard’s Theorem each such mapping extends to a meromorphic mapping from the closed torus to the Riemann sphere. This implies that the preimage of the set .{−1, 1, ∞} under the extended mapping must contain at least three points, which is impossible. The situation changes if X is a torus with a large enough hole. Let .α ≥ 1 and α,σ that is obtained from .C(Z + iαZ), .σ ∈ (0, 1). Consider the torus with a hole .T (with .α ≥ 1 being a real number) by removing a closed geometric rectangle of vertical side length .α − σ and horizontal side length .1 − σ (i.e. we remove a closed subset that lifts to such a closed rectangle in .C). A fundamental domain for this Riemann surface is “the golden cross on the Swedish flag” turned by . π2 with width of the laths being .σ and length of the laths being 1 and .α. 2α+1

Proposition 11.1 Up to homotopy there are at most .7e2 ·3 π σ irreducible holomorphic mappings from .T α,σ to the twice punctured complex plane. On the other hand, there are positive constants c, C, and .σ0 such that for any α positive number .σ < σ0 and any .α ≥ 1 there are at least .ceC σ non-homotopic α,σ to the twice punctured complex plane. holomorphic mappings from .T 5

2

Example 11.2 There are only finitely many holomorphic maps from a thrice punctured Riemann sphere to another thrice punctured Riemann sphere. Indeed, after normalizing both, the source and the target space, by a Möbius transformation we may assume that both are equal to .C\{−1, 1}. Each holomorphic map from .C \ {−1, 1} to itself extends to a meromorphic map from the Riemann sphere to itself, which maps the set .{−1, 1, ∞} to itself and maps no other point to this set. By the Riemann-Hurwitz formula the meromorphic map takes each value exactly once. Indeed, suppose it takes each value l times for a natural number l. Then each point in .{−1, 1, ∞} has ramification index l. Apply the Riemann Hurwitz formula for the branched covering .X = P1 → Y = P1 of multiplicity l χ (X) = l · χ (Y ) −



.

(ex − 1).

x∈Y

Here .ex is the ramification index at the  point x. For the Euler characteristic we have χ (P1 ) = 2, and . x∈Y (ex − 1) ≥ x=−1,1,∞ (ex − 1) = 3 (l − 1). We obtain .2 ≤ 2 l − 3 (l − 1) which is possible only if .l = 1. We saw that each non-constant holomorphic mapping from .C \ {−1, 1} to itself extends to a conformal mapping from the Riemann sphere to the Riemann sphere that maps the set .{−1, 1, ∞} to itself. There are only finitely may such maps, each a Möbius transformation permuting the three points. For Riemann surfaces of second kind the situation changes, as demonstrated in the following proposition. The proposition does not only concern the case when the Riemann surface equals .P1 with three holes. We consider an open Riemann surface X of genus g with .m ≥ 1 holes. .

362

11 Finiteness Theorems

Proposition 11.2 Let X be a connected finite open hyperbolic Riemann surface of genus g with .m + 1 holes, that is equipped with a Kähler metric. Suppose S is a standard bouquet of piecewise smooth circles for X with base point .q0 . We assume that .q0 is the only non-smooth point of the circles, and all tangent rays to circles in S at .q0 divide a disc in the tangent space into equal sectors. Let .Sσ be the .σ neighbourhood of S (in the Kähler metric on X). Then there exists a constant .σ0 > 0, and positive constants .C , .C

, .c , .c

, depending only on X, S and the Kähler metric, such that for each positive .σ < σ0 C\{−1,1} the number .NSσ of non-homotopic irreducible holomorphic mappings from .Sσ to the twice punctured complex plane satisfies the inequalities c

C\{−1,1}

c e σ ≤ NSσ

.

≤ C e

C

σ

.

(11.1)

The following proposition demonstrates the role of the conformal module (extremal length, respectively) with suitable horizontal boundary values for the homotopy problem of individual continuous mappings to holomorphic ones. The proposition concerns mappings from small neighbourhoods of standard bouquets of circles for open Riemann surfaces and uses the estimates of the extremal lengths in terms of the .L-invariant of the words representing the monodromies along the standard generators of the fundamental group. Proposition 11.3 Let X, .q0 , S and .Sσ be as in Proposition 11.2. For each .j = 1, . . . 2g + m we let .γj be the circle of the standard bouquet S for X that represents the standard generator .ej of .π1 (X, q0 ). Denote by . j the length of .γj in the Kähler metric. There exist positive constants .σ0 and .C1 , depending only on X, S, .q0 and the Kähler metric, such that for each positive .σ < σ0 any continuous mapping .f : Sσ → C \ {−1, 1} with .f (q0 ) = 0 is homotopic to a holomorphic mapping if the inequalities L(f∗ (ej )) ≤ C1

.

j σ

(11.2)

hold for all .j = 1, . . . 2g + m for the induced mapping .f∗ on fundamental groups with base points. Vive versa, there exists a constant .C2 depending only on X, S, .q0 and the Kähler metric, such that for each holomorphic mapping .f : Sσ → C\{−1, 1} with .f (q0 ) = 0 and each .j = 1, . . . 2g + m the inequality L(f∗ (ej )) ≤ C2

.

j σ

(11.3)

holds for the induced mapping .f∗ on fundamental groups with base points. Theorem 11.1 and Propositions 11.1, 11.2 and 11.3 contribute to Problem 7.3 on Gromov’s Oka Principle. Theorem 11.1 gives an upper bound for the number of homotopy classes of holomorphic mappings from connected finite open Riemann

11.2 Some Further Preliminaries on Mappings, Coverings, and Extremal Length

363

surfaces to the twice punctured complex plane. Theorem 11.2 gives an upper bound for the number of the isotopy classes of .(1, 1)-bundles on connected finite open Riemann surfaces that contain holomorphic bundles. Propositions 11.1 and 11.2 consider families of Riemann surfaces .Yσ , σ ∈ (0, σ0 ), obtained by continuously changing the conformal structure of a fixed Riemann surface, and determine the growth rate for .σ → 0 of the number of irreducible holomorphic mappings .Xσ → C \ {−1, 1} up to homotopy. In Proposition 11.1 the family of Riemann surfaces depends also on a second parameter .α, and the growth rate is determined in .α and .σ . The proof of the lower bound in both propositions uses solutions of a .∂problem. The solution in the case of Proposition 11.1 uses a simple explicit formula. Proposition 11.3 concerns the existence of homotopies of individual continuous mappings to holomorphic ones. For the proof of the upper bound in the theorems and propositions we have to understand what prevents a continuous map from a finite open Riemann surface X to .C\{−1, 1} to be homotopic to a holomorphic map. Proposition 11.5 below says that an irreducible map .X → C \ {−1, 1} can only be homotopic to a holomorphic map, if the “complexity” of the monodromies of the map are compatible with conformal invariants of the source manifold.

11.2 Some Further Preliminaries on Mappings, Coverings, and Extremal Length In this section we will prepare the proofs of the Theorems. Regular Zero Sets We will call a subset of a smooth manifold .X a simple relatively closed curve if it is a connected component of a regular level set of a smooth realvalued function on .X . Let .X be a connected finite open Riemann surface. Suppose the zero set L of a non-constant smooth real valued function on .X is regular. Each component of L is either a simple closed curve or it can be parameterized by a continuous mapping . : (−∞, ∞) → X . We call a component of the latter kind a simple relatively closed arc in .X . A relatively closed curve .γ in a connected finite open Riemann surface .X is said to be contractible to a hole of .X , if the following condition holds. Consider .X as domain .X c \ ∪Cj on a closed Riemann surface .X c . Here the .Cj are the holes, each is either a closed topological disc with smooth boundary or a point. The condition on .γ is the following. For each pair .U1 , .U2 of open subsets of .X c , .∪Cj ⊂ U1  U2 , there exists a homotopy of .γ that fixes .γ ∩ U1 and moves .γ into .U2 . Taking for .U2 small enough neighbourhoods of .∪Cj we see that the homotopy moves .γ into an annulus adjacent to one of the holes. For each relatively compact domain .X  X in .X there is a finite cover of .L∩X

by open subsets .Uk of .X such that each .L ∩ Uk is connected. Each set .L ∩ Uk is contained in a component of L. Hence, only finitely many connected components of

364

11 Finiteness Theorems

L intersect .X . Let .L0 be a connected component of L which is a simple relatively closed arc parameterized by . 0 : R → X . Since each set .L0 ∩ Uk is connected it is the image of an interval under . 0 . Take real numbers .t0− and .t0+ such that all these     intervals are contained in .(t0− , t0+ ). Then the images . (−∞, t0− ) and . (t0+ , +∞) are contained in .X \ X , maybe, in different components. Such parameters .t0− and +

.t 0 can be found for each relatively compact deformation retract .X of .X . Hence for each relatively closed arc .L0 ⊂ L the set of limit points .L+ 0 of . 0 (t) for .t → ∞ is contained in a boundary component of .X . Also, the set of limit points .L− 0 of . 0 (t) for .t → −∞ is contained in a boundary component of .X . The boundary components may be equal or different. Moreover, if .X  X is a relatively compact domain in .X which is a deformation retract of .X , and a connected component .L0 of L does not intersect .X then .L0 is contractible to a hole of .X . Indeed, .X \ X is the union of disjoint annuli, each of which is adjacent to a boundary component of .X , and the connected set .L0 must be contained in a single annulus. Further, denote by .L the union of all connected components of L that are simple relatively closed arcs. Consider those components .Lj of .L that intersect .X . There are finitely many such .Lj . Parameterize each .Lj by a mapping . j : R → X . For each j we let .[tj− , tj+ ] be a compact interval for which j (R \ [tj− , tj+ ]) ⊂ X \ X .

.

(11.4)

Let .X

, .X  X

 X , be a domain which is a deformation retract of .X such that − +





. j ([t , t ]) ⊂ X for each j . Then all connected components of .L ∩ X , that do j j − + not contain a set . j ([tj , tj ]), are contractible to a hole of .X

. Indeed, each such component is contained in the union of annuli .X

\ X .

Subgroups of Covering Transformations Let again .q0 be the base point of a Riemann surface X, and let .q˜0 be the base point in the universal covering .X˜ with .P(q ˜0 ) = q0 for the covering map .P : X˜ → X. Let N be a subgroup of .π1 (X, q0 ), and .(Isq˜0 )−1 (N ) the respective subgroup of covering transformations.. Denote by q˜0 −1 N : X ˜ → X(N ) ˜ .X(N ) the quotient .X(Is ) (N ). We obtain a covering .ωId with group of covering transformations isomorphic to N . The fundamental group def

N (q˜ ) can be identified with N. of .X(N ) with base point .(q0 )N = ωId 0 If .N1 and .N2 are subgroups of .π1 (X, q0 ) and .N1 is a subgroup of .N2 (we N2 q˜0 −1 ˜ write .N1 ≤ N2 ), then there is a covering map .ωN : X(Is ) (N1 ) → 1 N2 N1 N2 q˜0 −1 ˜ X(Is ) (N2 ), such that .ωN1 ◦ ωId = ωId . Moreover, the diagram Fig. 11.1 is commutative. q˜0 −1 ˜ Indeed, take any point .x1 ∈ X(Is ) (N1 ) and a preimage .x˜ of .x1 under N1 .ω . There exists a neighbourhood .V (x) ˜ of .x ˜ in .X˜ such that .V (x) ˜ ∩ σ (V (x)) ˜ =∅ Id ˜ for all covering transformations .σ ∈ Deck(X, X). Then for .j = 1, 2 the mapping N ,x˜ def

ωIdj

.

N

= ωIdj | V (x) ˜ is a homeomorphism from .V (x) ˜ onto its image denoted by

11.2 Some Further Preliminaries on Mappings, Coverings, and Extremal Length

ω N1 Id

Fig. 11.1 A commutative diagram related to subgroups of the group of covering transformations

X(N1 )

365 π (X,q0 )

ωN11

N2 ωN 1

˜ X

X(N2 )

ω N2 Id

X π (X,q0 )

ωN12

P

N2 ,x˜ q˜0 −1 ˜ Vj . Put .x2 = ωId (x). ˜ The set .Vj ⊂ X(Is ) (Nj ) is a neighbourhood of .xj for .j = 1, 2. N1 −1 For each preimage .x˜ ∈ (ωId ) (x1 ) there is a covering transformation .ϕx, ˜ x˜

q˜0 −1 in .(Is ) (N1 ) which maps a neighbourhood .V (x˜ ) of .x˜ conformally onto the N1 ,x˜

N1 ,x˜ ˜ of .x˜ so that on .V (x˜ ) the equality .ωId = ωId ◦ ϕx, neighbourhood .V (x) ˜ x˜

holds. N1 −1 Choose .x˜ ∈ (ωId ) (x1 ) and define

.

N2 N2 ,x˜ N2 ,x˜ N1 ,x˜ −1 ωN (y) = ωN (y) = ωId ((ωId ) (y)) for each y ∈ V1 . 1 1 def

(11.5)

.

We get a correctly defined mapping from .V1 onto .V2 . Indeed, since .N1 is a subgroup N1 −1 of .N2 , for another point .x˜ ∈ (ωId ) (x1 ) the covering transformation .ϕx, ˜ x˜ is

N2 ,x˜ N2 ,x˜ contained in .(Isq˜0 )−1 (N2 ), and we get the equality .ωId = ωId ◦ ϕx, ˜ x˜ . Hence, for .y ∈ V1 (x1 )



N2 ,x˜ N1 ,x˜ −1 N2 ,x˜ N1 ,x˜ −1 ωId ◦ (ωId ) (y) =(ωId ◦ ϕx, ◦ ϕx, ˜ x˜ ) ◦ (ωId ˜ x˜ ) (y)

.

N2 ,x˜ N1 ,x˜ −1 ◦ (ωId ) (y). =ωId

(11.6)

N ,x˜

Since each mapping .ωIdj , .j = 1, 2, is a homeomorphism from .V (x) ˜ onto its N2 is a homeomorphism from .V (x1 ) onto .V (x2 ). The same image, the mapping .ωN 1 N2 N2 holds for all preimages of .V (x2 ) under .ωN . Hence, .ωN is a covering map. The 1 1

N1 N2 commutativity of the part of the diagram that involves the mappings .ωId , .ωId , and N2 .ω N1 follows from Eq. (11.5). π (X,q0 )

The existence of .ωN11

π (X,q0 )

and the equality .P = ωN11

N1 ◦ ωId follows by π (X,q0 )

applying the above arguments with .N2 = π1 (X, q0 ). The equality .P = ωN12 N2 ωId follows in the same way. Since

π (X,q0 )

P =ωN12

.

π (X,q0 )

=ωN11

N2 N1 ◦ ωN ◦ ωId 1 N1 ◦ ωId ,

◦

366

11 Finiteness Theorems

we have π (X,q0 )

ωN12

.

π (X,q0 )

N2 ◦ ωN = ωN11 1

def

def

π (X,q )

1 0 N and .ω We will also use the notation .ωN = ωId for a subgroup N N = ωN of .π1 (X, q0 ). N2 Let again .N1 ≤ N2 be subgroups of .π1 (X, q0 ). Consider the covering .ωN : 1 q˜0 −1 q˜0 −1 ˜ ˜ ) (N1 ) → X(Is ) (N2 ). Let .β be a simple relatively closed curve in X(Is N2 −1 q˜0 −1 ˜ .X(Is ) (N2 ). Then .(ωN ) (β) is the union of simple relatively closed curves 1 q˜0 −1 ˜ ) (N1 ) and .ωN2 : (ωN2 )−1 (β) → β is a covering. Indeed, we cover in .X(Is

N1

N1

N2 q˜0 −1 ˜ β by small discs .Uk in .X(Is ) (N2 ) such that for each k the restriction of .ωN 1

.

N2 −1 to each connected component of .(ωN ) (Uk ) is a homeomorphism onto .Uk , and 1 .Uk intersects .β along a connected set. Take any k with .Uk ∩ β = ∅. Consider N2 −1 N2 the preimages .(ωN ) (Uk ). Restrict .ωN to the intersection of any preimage 1 1

N2 −1 N2 −1 (ωN ) (Uk ) with .(ωN ) (β). We obtain a homeomorphism onto .Uk ∩β. It follows 1 1

.

N2 N2 −1 that the map .(ωN ) is a covering from each connected component of .(ωN ) (β) 1 1 onto .β.

The Extremal Length of Monodromies Let as before X be a connected finite open Riemann surface with base point .q0 , and .q˜0 a point in the universal covering ˜ for which .P(q˜0 ) = q0 for the covering map .P : X˜ → X. .X Recall that for an arbitrary point .q ∈ X the free homotopy class of an element e of the fundamental group .π1 (X, q) can be identified with the conjugacy class of elements of .π1 (X, q) containing e and is denoted by  .e . Notice that for .e0 ∈ π1 (X, q0 ) and a curve .α in X with initial point .q0 and terminal point q the free homotopy classes of .e0 and of .e = Isα (e0 ) coincide, i.e.  .e =  e0 . Consider a simple smooth relatively closed curve L in X. We will say that a free homotopy class of curves .e0 intersects L if each representative of .e0 intersects L. Choose an orientation of L. The intersection number of .e0 with the oriented curve L is the intersection number with L of some (and, hence, of each) smooth loop representing .e 0 that intersects L transversally. The intersection number of such a loop with the relatively closed curve L is the sum of the intersection numbers over all intersection points. The intersection number at an intersection point equals .+1 if the orientation determined by the tangent vector to the loop followed by the tangent vector to L is the orientation of X, and equals .−1 otherwise. Let A be an annulus equipped with an orientation (called positive orientation) of simple closed dividing curves in A. Recall that a relatively closed curve .γ in a surface X is called dividing, if .X \ γ consists of two connected components. A continuous mapping .ω : A → X is said to represent a conjugacy class  .e of elements of the fundamental group .π1 (X, q) for a point .q ∈ X, if the composition .ω ◦ γ represents  .e for each positively oriented dividing curve .γ in A. Let A be an annulus with base point p with a chosen positive orientation of simple closed dividing curves in A. Let .ω be a continuous mapping from A to a finite

11.2 Some Further Preliminaries on Mappings, Coverings, and Extremal Length

367

Riemann surface .X with base point q such that .ω(p) = q. We write .ω : (A, p) → (X, q). The mapping is said to represent the element e of the fundamental group .π1 (X, q) if .ω ◦ γ represents e for some (and hence for each) positively oriented simple closed dividing curve .γ in A with base point q. Let .q0 be the base point of X chosen above and .q˜0 a point in the universal covering .X˜ with .P(q˜0 ) = q0 . We associate to each element .e0 ∈ π1 (X, q0 ) of q˜0 −1 ˜ the free group .π1 (X, q0 ) the annulus .X(e0 ) = X(Is ) (e0 ) with base point e 

def

π (X,q )

qe0  = ωId0 (q˜0 ) and the covering map .ωe0  = ωe10  0 : X(e0 ) → X. By the commutative diagram Fig. 11.1 the equality .ωe0  (qe0  ) = q0 holds. We choose q˜0 −1 ˜ ) (e0 ) the orientation of simple closed dividing curves in .X(e0 ) = X(Is so that for a curve .γ˜ in .X˜ with initial point .q˜0 and terminal point .Isq˜0 (e0 ) the

.

def

curve .γe0  = ωe0  (γ˜ ) is positively oriented. The locally conformal mapping π1 (X,q0 ) .ω : (X(e0 ), qe0  ) → (X, q0 ) represents .e0 . This follows from the equality e0  e  .ωe0  (γe0  ) = ωe0  (ω 0 (γ˜ )) = P(γ˜ ) = γ , since .P(γ˜ ) represents .e0 . Take a curve .α in X that joins .q0 and q, and the point .q˜ = q(α) ˜ ∈ X˜ such that .α and .q˜ are compatible, i.e. .Isq˜ = Isα ◦ Isq˜0 (see Eq. (1.2)). Put q˜ −1 q˜ −1 ˜ .e = Isα (e0 ). By Eq. (1.2) .(Is ) (e) = (Isq˜0 )−1 (e0 ), hence, .X(Is ) (e) = q ˜ −1 0) ˜ (e0 ) = X(e0 ). The locally conformal mapping .ωe0  : X(e0 ) → X(Is def

˜ to .q ∈ X. Moreover, .ωe0  : (X(e0 ), qe ) → X takes the point .qe = ωe0  (q) (X, q) represents .e ∈ π1 (X, q). This can be seen by repeating the previous arguments. Let .α be an arbitrary curve in X joining .q0 with q, and .q˜ ∈ P−1 (q) be arbitrary (i.e. .α and .q˜ are not required to be compatible). Let .e ∈ π1 (X, q). Denote the q˜ −1 q˜ −1 ˜ ˜ ) (e) by .ωe,q˜ , and the projection .X(Is ) (e) → projection .X˜ → X(Is e, q ˜ X by .ωe,q˜ . Put .qe,q˜ = ω (q). ˜ For any such choice we choose the orientation q˜ −1 ˜ of simple closed dividing curves on .X(Is ) (e) so that .ωe,q˜ maps any curve ˜ .γ˜ in .X with initial point .q ˜ and terminal point .(Isq˜ )−1 (e)(q) ˜ to a positively oriented dividing curve. We will call it the standard orientation of dividing curves   q˜ −1 q˜ −1 ˜ ˜ in .X(Is ) (e). The mapping .ωe,q˜ : X(Is ) (e), qe,q˜ → (X, q) represents e. Since the mapping . ωe0  : (X(e0 ), (q0 )e0  ) → (X, q0 ) represents . e0 , the mapping .ωe0  : X(e0 ) → X represents the free homotopy class .e0 . The following simple lemma will be useful. Lemma 11.1 The annulus .X(e0 ) has smallest extremal length among annuli which admit a holomorphic mapping to X, that represents the conjugacy class .e0 . In other words, .X(e0 ) is the “thickest” annulus with the property stated in Lemma 11.1. Proof Take an annulus A with a choice of positive orientation of simple closed ω

dividing curves. Suppose .A −−→ X is a holomorphic mapping that represents .e0 . The annulus A is conformally equivalent to a round annulus in the plane, hence, we

368

11 Finiteness Theorems

may assume that A has the form .A = {z ∈ C : r < |z| < R} for .0 ≤ r < R ≤ ∞ and the positive orientation of dividing curves is the counterclockwise one. Take a positively oriented simple closed dividing curve .γ A in A. Its image .ω◦γ A under .ω represents the class .e0 . Choose a point .q A in .γ A , and put .q = ω(q A ). Then A represents a generator of .π (A, q A ) and .γ = ω ◦ γ A represents an element .γ 1 e of .π1 (X, q) in the conjugacy class .e0 . Choose a curve .α in X with initial point .q0 and terminal point q, and a point .q ˜ in .X˜ so that .α and .q˜ are compatible, and, hence, for .e = Isα (e0 ) the equality .(Isq˜0 )−1 (e0 ) = (Isq˜ )−1 (e) holds. Let L be the relatively closed arc .{q A · r : r ∈ R} ∩ A in A that contains .q A . After a homotopy of .γ A with fixed base point, we may assume that its base point .q A is the only point of .γ A that is contained in L. The restriction .ω|(A \ L) lifts to a mapping ˜ that extends continuously to the two strands .L± of L. (We .ω ˜ : (A \ L) → X, A be assume that .L− is reached by moving clockwise from .A \ L towards L.) Let .q± A A the copies of .q on the two strands .L± . We choose the lift .ω˜ so that .ω(q ˜ − ) = q. ˜ A ) = σ (q) Since the mapping .(A, q A ) → (X, q) represents e, we obtain .ω(q ˜ + ˜ for q˜ −1 .σ = (Is ) (e). Then for each .z ∈ L the covering transformation .σ maps the point .z ˜ − ∈ ω(L ˜ − ) for which .P(˜z− ) = z to the point .z˜ + ∈ ω(L ˜ + ) for which .P(˜z+ ) = z. Hence .ω lifts to a holomorphic mapping .ι : A → X(e0 ). By Lemma 3.3 (see also Lemma 7 of [45]) .λ(A) ≥ λ(X(e0 )).   For each point .q ∈ X and each element .e ∈ π1 (X, q) we denote by .A( e) the conformal class of the “thickest” annulus that admits a holomorphic mapping q˜0 −1 ˜ .e . We saw that .λ(A( e0 )) = λ(X(Is ) (e0 )) for into X that represents 

˜ .e0 ∈ π1 (X, q0 ). By the same reasoning as before .λ(A( e )) = λ(X(Isq˜ )−1 (e)) for each .q˜ ∈ X˜ and each element .e ∈ π1 (X, P(q˜ )). Hence, if .e0 and e are free homotopic (equivalently, if they represent to the same conjugacy class,) then q˜0 −1 q˜ −1 ˜ ˜ .λ(X(Is ) (e0 )) = λ(X(Is ) (e)) for any .q˜0 ∈ P−1 (q0 ) ⊂ X˜ and any 

q˜ ∈ P−1 (q). Notice that .A(e−1 ) = A( e ) for each .e ∈ π1 (X, q), q ∈ X.

.

11.3 Holomorphic Mappings into the Twice Punctured Plane The following lemma will be crucial for the estimate of the .L− -invariant of the monodromies of holomorphic mappings from a finite open Riemann surface to .C \ {−1, 1}. For a point .q ∈ (−1, 1) we will identify .π1 (C \ {−1, 1}, q ) with .π1 (C \ {−1, 1}, 0) by the canonical isomorphism .Isαq for the curve .αq that runs along the line segment in .(−1, 1) joining 0 with .q . Lemma 11.2 Let .f : X → C\{−1, 1} be a non-contractible holomorphic mapping on a connected finite open Riemann surface X, such that 0 is a regular value of .Imf . Assume that .L0 is a simple relatively closed curve in X such that .f (L0 ) ⊂ (−1, 1). Let .q ∈ L0 and .q = f (q).

11.3 Holomorphic Mappings into the Twice Punctured Plane

369

If for an element .e ∈ π1 (X, q) the free homotopy class  .e intersects .L0 , then either the reduced word .f∗ (e) ∈ π1 (C \ {−1, 1}, q ) is a non-zero power of a standard generator of .π1 (C \ {−1, 1}, q ) or the inequality L− (f∗ (e)) ≤ 2π λ(A( e ))

(11.7)

.

holds. Notice that we make a normalization in the statement of the Lemma by requiring that f maps .L0 into the interval .(−1, 1), not merely into .R \ {−1, 1}. Lemma 11.2 will be a consequence of the following lemma. Lemma 11.3 Let X, f , .L0 , .q ∈ L0 be as in Lemma 11.2, and .e ∈ π1 (X, q). Let def q˜ −1 ˜ .q ˜ be an arbitrary point in .P−1 (q). Consider the annulus .A = X(Is ) (e) def

def

and the holomorphic projection .ωA = ωe,q˜ . Put .qA = ωe,q˜ (q). ˜ The mapping .ωA : (A, qA ) → (X, q) represents e. Let .LA ⊂ A be the connected component of .(ωA )−1 (L0 ) that contains .qA . If  .e intersects .L0 , then .LA is a relatively closed curve in A that has limit points on both boundary components of A, and the lift .f ◦ ωA is a holomorphic mapping from A to .C \ {−1, 1} that maps .LA into .(−1, 1). Proof of Lemma 11.3 Let .γ : [0, 1] → X be a curve with base point q in X that represents e, and let .γ˜ be the lift of .γ to .X˜ with initial point .γ˜ (0) equal to .q. ˜ Put def

σ = (Isq˜ )−1 (e). Then the terminal point .γ˜ (1) equals .σ (q). ˜ All connected components of .P−1 (L0 ) are relatively closed curves in .X˜ ∼ = C+ (where .C+ denotes the upper half-plane) with limit points on the boundary of ˜ Indeed, the lift .f ◦ P of f to .X˜ takes values in .(−1, 1) on .P−1 (L0 ). Hence, .X. −1 (L ). A compact connected component of .P−1 (L ) .| exp(± i f ◦ P)| = 1 on .P 0 0 would bound a relatively compact topological disc in .X˜ = C+ , and by the maximum principle .| exp(± i f ◦ P)| = 1 on the disc. This would imply that .f ◦ P is constant on .X˜ in contrary to the assumptions. ˜ The point .σ (q) ˜ Let .L˜ q˜ be the connected component of .P−1 (L0 ) that contains .q. cannot be contained in .L˜ q˜ . Indeed, assume the contrary. Then the arc .α˜ on .L˜ q˜ that joins .q˜ and .σ (q) ˜ is homotopic in .X˜ with fixed endpoints to .γ˜ . The projection .α = P(α) ˜ is contained in .L0 and is homotopic in X with fixed endpoints to .γ . Since .γ represents e and e is a primitive element of the fundamental group .π1 (X, q), this is possible only if .L0 is compact and (after orienting it) .L0 represents e. A small translation of .α to a side of .L0 gives a curve in X that does not intersect .L0 and represents the free homotopy class  .e of e. This contradicts the fact that  .e intersects .L0 . Take any other point .q˜ ∈ L˜ q˜ . If .σ (q˜ ) was in .L˜ q˜ , then the arc on .L˜ q˜ that joins .q ˜ with .q˜ would project to a loop that represents .eˆ and is contained in .L0 . By the arguments above this is not possible. We proved that the curves .L˜ q˜ and .σ (L˜ q˜ ) are disjoint. .

370

11 Finiteness Theorems

˜ Let .Ω be the Each of the two connected components .L˜ q˜ and .σ (L˜ q˜ ) divides .X. ˜ ˜ ˜ ˜ domain on .X that is bounded by .Lq˜ and .σ (Lq˜ ) and parts of the boundary of .X. After a homotopy of .γ˜ that fixes the endpoints we may assume that .γ˜ ((0, 1)) is contained in .Ω. Indeed, for each connected component of .γ˜ ((0, 1)) \ Ω there is a homotopy with fixed endpoints that moves the connected component to an arc on ˜ q˜ or .σ (L˜ q˜ ). A small perturbation yields a curve .γ˜ which is homotopic with fixed .L endpoints to .γ˜ and has interior contained in .Ω. Notice that by the same reasoning as above, .γ˜ ((0, 1)) does not meet any .σ k (L˜ q˜ ). The curve .ωe,q˜ (γ˜ ) is a closed curve on A that represents a generator of the fundamental group of A with base point .qA . Moreover, .ωA ◦ ωe,q˜ (γ˜ ) = ωe,q˜ ◦ ωe,q˜ (γ˜ ) = P(γ˜ ) represents e. Hence, the mapping .ωA : (A, qA ) → (X, q) represents e. The curve .ωe,q˜ (γ˜ ) intersects .Le = ωe,q˜ (L˜ q˜ ) exactly once. Hence, .Le has limit points on both boundary circles of A for otherwise .Le would intersect one of the components of .A \ ωe,q˜ (γ˜ ) along a set which is relatively compact in A, and

.γ˜ would have intersection number zero with .Le . It is clear that .f ◦ ωA (LA ) = f (L0 ) ⊂ (−1, 1). The lemma is proved.   Proof of Lemma 11.2 Let .ωA : (A, qA ) → (X, q) be the holomorphic mapping from Lemma 11.3 that represents e, and let .LA  qA be the relatively closed curve in A with limit set on both boundary components of A. Consider a positively oriented dividing curve .γA : [0, 1] → A with base point .γ (0) = γ (1) = qA such that .γA ((0, 1)) ⊂ A \ LA . The curve .γ = ωA (γA ) represents e. The mapping .f ◦ ωA is holomorphic on A and .f ◦ ωA (γA ) = f (γ ) represents .f∗ (e) ∈ π1 (C \ {−1, 1}, q ) with .q = f ◦ ωA (qA ) = f (q) ∈ (−1, 1). Hence, .f ◦ ωA (γA ) also represents the element .(f∗ (e))tr ∈ π1tr (C \ {−1, 1}) in the relative fundamental group .π1 (C \ {−1, 1}, (−1, 1)) = π1tr (C \ {−1, 1}) corresponding to .f∗ (e). We prove now that .Λtr (f∗ (e)) ≤ λ(A). Let .A0  A be any relatively compact annulus in A with smooth boundary such that .qA ∈ A0 . If .A0 is sufficiently large, then the connected component .LA0 of .LA ∩ A0 that contains .qA has endpoints on different boundary components of .A0 . The set .A0 \ LA0 is a curvilinear rectangle. The open horizontal curvilinear sides are the strands of the cut that are reachable from the curvilinear rectangle moving counterclockwise, or clockwise, respectively. The open vertical curvilinear sides are obtained from the boundary circles of .A0 by removing an endpoint of the arc .LA0 . Since .f ◦ ωA maps .LA to .(−1, 1), the restriction of .f ◦ ωA to .A0 \ LA0 represents .(f∗ (e))tr . Hence, Λtr (f∗ (e)) ≤ λ(A0 \ LA0 ) .

(11.8)

λ(A0 \ LA0 ) ≤ λ(A0 ) .

(11.9)

.

By Corollary 3.1 .

11.3 Holomorphic Mappings into the Twice Punctured Plane

371

We obtain the inequality .Λtr (f∗ (e)) ≤ λ(A0 ) for each annulus .A0  A, hence, since A belongs to the class .A( e ) of conformally equivalent annuli, Λtr (f∗ (e)) ≤ λ(A( e )) ,

(11.10)

.

 

and the Lemma follows from Theorem 9.1.

The Monodromies Along Two Generators In the following Lemma we combine the information on the monodromies along two generators of the fundamental group .π1 (X, q). We allow the situation when the monodromy along one generator or along each of the two generators of the fundamental group of X is a power of a standard generator of .π1 (C \ {−1, 1}, f (q)). Lemma 11.4 Let .f : X → C\{−1, 1} be a non-contractible holomorphic function on a connected open Riemann surface X such that 0 is a regular value of the imaginary part of f . Suppose f maps a simple relatively closed curve .L0 in X to (1) and .e(2) be primitive elements of .π (X, q). .(−1, 1), and q is a point in .L0 . Let .e 1 (1) Suppose that for each .e = e , e = e(2) , and .e = e(1) e(2) , the free homotopy class (j ) ), j = 1, 2, are (trivial or non-trivial) powers  .e intersects .L0 . Then either .f∗ (e of the same standard generator of .π1 (C \ {−1, 1}, q ) with .q = f (q) ∈ (−1, 1), or each of them is the product of at most two elements .w1 and .w2 of .π1 (C\{−1, 1}, q ) with L− (wj ) ≤ 2π λe(1) ,e(2) , j = 1, 2,

.

where def







λe(1) ,e(2) = max{λ(A(e(1) )), λ(A(e(2) )), λ(A(e(1) e(2) ))}.

.

Hence, L− (f∗ (e(j ) )) ≤ 4π λe(1) ,e(2) , j = 1, 2.

.

(11.11)

The .L− invariant was defined before Lemma 10.1. Proof If the monodromies .f∗ (e(1) ) and .f∗ (e(2) ) are not powers of a single standard generator (the identity is considered as zeroth power of a standard generator) we obtain the following. At most two of the elements, .f∗ (e(1) ), .f∗ (e(2) ), and (1) e(2) ) = f (e(1) ) f (e(2) ), are powers of a standard generator, and if two of .f∗ (e ∗ ∗ them are powers of a standard generator, then they are non-zero powers of different standard generators. If two of them are non-zero powers of different standard generators, then the third

has the form .a k a k with .a and .a being different generators and k and .k being non-zero integers. By Lemma 11.2 the .L− of the third element does not exceed k

.2π λe(1) ,e(2) . On the other hand it equals .log(3|k |) + log(3|k|). Hence, .L− (a ) =

k

log(3|k|) ≤ 2π λe(1) ,e(2) and .L− (a ) = log(3|k |) ≤ 2π λe(1) ,e(2) .

372

11 Finiteness Theorems

If two of the elements .f∗ (e(1) ), .f∗ (e(2) ), and .f∗ (e(1) e(2) ) = f∗ (e(1) ) f∗ (e(2) ), are not powers of a standard generator, then the .L− of each of the two elements does not exceed .2π λe(1) ,e(2) . Since the .L− of an element coincides with the .L− of its inverse, the third element is the product of two elements with .L− not exceeding



.2π λe(1) ,e(2) . Since for .x, x ≥ 2 the inequality .log(x + x ) ≤ log x + log x holds, the .L− of the product does not exceed the sum of the .L− of the factors. Hence the .L− of the third element does not exceed .4π λe(1) ,e(2) . Hence, inequality (11.11) holds.   The following proposition states the existence of suitable connected components of the zero set of the imaginary part of certain analytic functions on tori with a hole and on planar domains. As in Sect. 11.1 for a Riemann surface X we let .E be a standard system of generators of .π1 (X, q0 ) that is associated to a standard bouquet of circles for X. Recall that .Ej is the set of primitive elements of .π1 (X, q0 ) which can be written as product of at most j elements of .E ∪ (E)−1 . Here as before for any subset .E of .π1 (X; q0 ) we denote by .(E )−1 the set of all elements that are inverse to elements in .E . Proposition 11.4 Let X be a torus with a hole or a planar domain with base point q0 and fundamental group .π1 (X, q0 ), and let .E be a standard system of generators of .π1 (X, q0 ) that is associated to a standard bouquet of circles for X. Let .f : X → C \ {−1, 1} be a non-contractible holomorphic mapping such that 0 is a regular value of .Imf . Then there exist a simple relatively closed curve .L0 ⊂ X such that

.f (L0 ) ⊂ R \ {−1, 1}, and a set .E ⊂ E2 ⊂ π1 (X, q0 ) of primitive elements of 2 .π1 (X, q0 ), such that the following holds. Each element .ej,0 ∈ E ⊂ π1 (X, q0 ) is the product of at most two elements of .E2 ∪ (E2 )−1 . Moreover, for each .e0 ∈ π1 (X, q0 ) which is the product of one or two elements from .E2 the free homotopy class .e0 has positive intersection number with .L0 (after suitable orientation of .L0 ). If X is a torus with a hole or X equals .P1 with three holes, we may chose .E2

consisting of two elements, one of them contained in .E, the other is either contained in .E ∪ E −1 or is a product of two elements of .E. .

Notice the following facts. By Theorem 1.1 a mapping .f : X → C \ {−1, 1} is contractible if and only if for each .e0 ∈ π1 (X, q0 ) the monodromy .f∗ (e0 ) is equal to the identity. The mapping f is reducible if and only if the monodromy mapping .f∗ : π1 (X, q0 ) → π1 (C \ {−1, 1}, f (q0 )) is conjugate to a mapping into a subgroup .Γ of .π1 (C \ {−1, 1}, f (q0 )) that is generated by a single element that is represented by a curve which separates one of the points .1, −1 or .∞ from the other points. In other words, .Γ is (after identifying fundamental groups with different base point up to conjugacy) generated by a conjugate of one of the elements .a1 , .a2 or .a1 a2 of .π1 (C \ {−1, 1}, 0). Recall that for .q ∈ (−1, 1) the fundamental group

.π1 (C \ {−1, 1}, q ) is canonical isomorphic to .π1 (C \ {−1, 1}, 0) (by changing the base point along the segment in .(−1, 1) that joins .q and 0), and we will identify

.π1 (C \ {−1, 1}, q ) with .π1 (C \ {−1, 1}, 0). If f is irreducible, then it is not contractible, and, hence, the preimage .f −1 (R) is not empty.

11.3 Holomorphic Mappings into the Twice Punctured Plane

373

Denote by .M1 a Möbius transformation which permutes the points .−1, 1, ∞ and maps the interval .(−∞, −1) onto .(−1, 1), and let .M2 be a Möbius transformation which permutes the points .−1, 1, ∞ and maps the interval .(1, ∞) onto .(−1, 1). def

Let .M0 = Id. The main step for the proof of Theorem 11.1 is the following Proposition 11.5. Recall that the invariant .λj (X) was defined in Definition 11.1, Sect. 11.1. Since q˜0 −1 ˜ ) (e0 )) = λ(A( e0 )) holds, .λj (X) is for .e0 ∈ π1 (X, q0 ) the equality .λ(X(Is the maximum of .λ(A( e0 )) over .e0 ∈ Ej . Proposition 11.5 Let X be a connected finite open Riemann surface with base point q0 , and let .E be the same system of generators of .π1 (X, q0 ) as in Proposition 11.4. Suppose .f : X → C \ {−1, 1} is an irreducible holomorphic mapping, such that 0 is a regular value of . Imf . Then for one of the functions .Ml ◦ f, l = 0, 1, 2, which we denote by F , there exists a point .q ∈ X (depending on f ), such that the

.

def

point .q = F (q) is contained in .(−1, 1), and there exists a smooth curve .α in X joining .q0 with q, such that the following holds. For each element .ej ∈ Isα (E) the monodromy .F∗ (ej ) is the product of at most six elements of .π1 (C \ {−1, 1}, q ) (canonically identified with .π1 (C \ {−1, 1}, 0)) of .L− not exceeding .2π λ7 (X) and, hence, L− (F∗ (ej )) ≤ 12π λ7 (X) for each j.

.

(11.12)

If X is a torus with a hole the proposition holds with .λ7 (X) replaced by .λ3 (X). If X is a planar domain the proposition holds with .λ4 (X) instead of .λ7 (X). Notice, that all monodromies of contractible mappings are equal to the identity, hence the inequality (11.12) holds automatically for contractible mappings. We postpone the proof of the two propositions and prove first the Theorem 11.1. Proof of Theorem 11.1 Let X be a connected finite open Riemann surface (possibly of second kind) with base point .q0 . We need to estimate the number .NX of homotopy classes of irreducible mappings .X → C \ {−1, 1} that contain a holomorphic mapping. Take any relatively compact open subset .X0  X that is a deformation retract of X and contains .q0 . We claim that the number .NX does not exceed the number .NX∗ 0 of homotopy classes of irreducible mappings 0 → C \ {−1, 1} that contain a holomorphic mapping for which 0 is a regular .X value of the imaginary part. Notice first that homotopic mappings .X → C \ {−1, 1} restrict to homotopic mappings .X0 → C\{−1, 1}. Vice versa, if the restrictions .fj | X0 of two mappings .fj : X → C \ {−1, 1}, .j = 1, 2, are homotopic then the .fj are homotopic. Indeed, let .ϕt : X → X be a continuous family of mappings .ϕt : X → X, t ∈ [0, 1], that map X homeomorphically onto a domain .Xt in X such that .ϕ0 is the identity and 0 .X1 = X (see also Lemma 1.1). Then .fj is homotopic to .fj ◦ ϕ1 , .j = 1, 2, and .f1 ◦ ϕ1 is homotopic to .f2 ◦ ϕ1 .

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11 Finiteness Theorems

For each irreducible holomorphic mapping .f : X → C \ {−1, 1} and each small enough complex number .ε the holomorphic mapping .(f − iε) | X0 is homotopic to 0 .f | X , is irreducible and takes values in .C \ {−1, 1}. Hence, the homotopy class of the restriction to .X0 of any irreducible holomorphic mapping .X → C \ {−1, 1} contains an irreducible holomorphic mapping for which zero is a regular value of the imaginary part. This means that the number .NX does not exceed the number ∗ 0 .N X0 of irreducible homotopy classes .X → C \ {−1, 1} that contain a holomorphic mapping for which 0 is a regular value of the imaginary part. We estimate now the number .NX∗ 0 . Take any irreducible holomorphic mapping 0 .f : X → C \ {−1, 1} such that 0 is a regular value of the imaginary part .Im f and apply Proposition 11.5. We obtain the following objects: a Möbius transformation .Ml , that maps one of the components of .R \ {−1, 1} onto .(−1, 1); further, a point 0 0 .q ∈ X and a smooth curve .α in .X with initial point .q0 and terminal point q, such def

that for the mapping .F = Ml ◦ f the inclusion .q = F (q) ∈ (−1, 1) holds, and for def

the generators .ej = Isα (ej,0 ), ej,0 ∈ E, of .π1 (X0 , q) the inequalities (11.12) hold with X replaced by .X0 . Our goal is to find a smooth mapping .Fˆ : X0 → C\{−1, 1}, that is free homotopic to F , maps .q0 to .q , and satisfies .(Fˆ )∗ (e0 ) = F∗ (e). Then the inequalities (11.12)) will imply the inequality L− ((Fˆ )∗ (e0 )) ≤ 12π λ7 (X0 )

(11.13)

.

for each .ej,0 ∈ E. Write .e = Isα (e0 ) ∈ π1 (X, q) for each .e0 ∈ π1 (X, q0 ). Parameterise .α by the interval .[0, 1]. The image of .α under the mapping F is the curve .β = F ◦ α in

.C \ {−1, 1} with initial point .F (q0 ) and terminal point .F (q) = q . Then .F∗ (e0 ) = −1 (Isβ ) (F∗ (e)). The curve .β is parameterized by the interval .[0, 1]. Choose a free homotopy .Ft , t ∈ [0, 1], of mappings from .X0 to .C \ {−1, 1}, that changes the def

mappings only in a small neighbourhood of .β and joins the mapping .F0 = F with a (smooth) mapping .F1 denoted by .Fˆ , so that .Ft (q0 ) = β(t), t ∈ [0, 1]. The value .β(t) moves from the point .β(0) = F (q0 ) to .β(1) = q along the curve .β. Denote by .βt the curve that runs from .β(t) to .β(1) along .β. Then .β0 = β and def

β1 is a constant curve. Let .γ0 be a curve that represents .e0 . Then .γ = Isα (γ0 ) represents e. The base point of the curve .Ft (γ0 ) equals .Ft (q0 ) = β(t). Hence, we obtain a continuous family of curves .βt−1 Ft (γ0 )βt with base point .β(1) = F (q). For .t = 1 the curve is equal to .F1 (γ0 ) = Fˆ (γ0 ), for .t = 0 the curve is equal to −1 F (γ )β = F (α −1 γ α) = F (Is (γ )) = F (γ ). Since the two curves .F (γ ) .β 0 0 0 0 0 α 0 0 1 0 and .F0 (γ ) are homotopic and .F1 = Fˆ , .F0 = F , we obtain .Fˆ∗ (e0 ) = F∗ (e). We estimate now the number of free homotopy classes of mappings .X0 → C \ {−1, 1} whose associated conjugacy class of homomorphisms .π1 (X0 , q0 ) → π1 (C \ {−1, 1}, q ) contains a homomorphism that satisfies the inequalities (11.13). Recall that .π1 (C \ {−1, 1}, q ) is canonically isomorphic to .π1 (C \ {−1, 1}, 0) by the isomorphism .Isβ where the curve .β runs along the segment that joins .q and 0 0 and is contained in .(−1, 1). By Lemma 10.1 there are at most . 12 e36π λ7 (X ) + 1 ≤ .

11.3 Holomorphic Mappings into the Twice Punctured Plane

375

3 36π λ7 (X0 ) 2e

different reduced words .w ∈ π1 (C\{−1, 1}), 0) (including the identity) 0 with .L− (w) ≤ 12π λ7 (X0 ). Hence, there are at most .( 32 e36π λ7 (X ) )2g +m different homomorphisms .h : π1 (X, q0 ) → π1 (C\{−1, 1}, 0) with .L− (h(e0 )) ≤ 12π λ7 (X0 ) for each element .e0 of the set of generators .E of .π1 (X0 , q0 ). By Theorem 1.1 there 0 are at most .( 32 e36π λ7 (X ) )2g +m different free homotopy classes of mappings .X0 → C \ {−1, 1}, whose associated conjugacy class of homomorphisms .π1 (X0 , q0 ) → π1 (C \ {−1, 1}, 0) contains a homomorphism that satisfies the inequalities (11.13). 0 It follows that .NX∗ 0 ≤ ( 32 e36π λ7 (X ) )2g +m . Our arguments show that for each irreducible or contractible holomorphic mapping f on X there are arbitrarily small numbers .ε, such that one of the three 0 mappings .Ml (f −iε) | X0 , .l = 0, 1, 2, belongs to one of at most .( 32 e36π λ7 (X ) )2g +m free homotopy classes of mappings .X0 → C \ {−1, 1}. Since for small enough numbers .ε the mappings .Ml (f ) | X0 and .Ml (f − iε) | X0 are homotopic, the 0 mapping .Ml (f ) | X0 belongs to one of at most .( 32 e36π λ7 (X ) )2g +m free homotopy classes of mappings .X0 → C \ {−1, 1}. Compose the mappings of each class with each of the three Möbius transformations .Ml−1

. We obtain no more than

3( 32 e36π λ7 (X ) )2g +m free homotopy classes of mappings .X0 → C \ {−1, 1}, and −1 0 0 0 .f | X = M l (Ml (f )) | X belongs to one of them. Identifying .π1 (X , q0 ) with .π1 (X, q0 ), the homotopy classes of mappings .X → C \ {−1, 1} have the same associated conjugacy classes of homomorphism .π1 (X, q0 ) → π1 (C \ {−1, 1}, 0) as the homotopy classes of their restrictions. We see that there are no more than 3 36π λ7 (X0 ) 2g +m .3( e ) holomorphic mappings .X → C \ {−1, 1} that are irreducible 2 0 or contractible. Theorem 11.1 is proved with the upper bound .3( 32 e24π λ7 (X ) )2g +m for an arbitrary relatively compact domain .X0 ⊂ X that is a deformation retract of X. It remains to prove that 0

.

λ7 (X) = inf{λ7 (X0 ) : X0  X is a deformation retract of X } .

.

q˜0 −1  We have to prove that for each .e0 ∈ π1 (X, q0 ) the quantity .λ(X(Is ) (e0 )) q˜0 −1 0 0 is equal to the infimum of .λ(X (Is ) (e0 )) over all .X being open relatively 0 is the universal compact subsets of X which are deformation retracts of X. Here .X 0 0 0 (.X,  covering of .X , and the fundamental groups of X and .X are identified. .X 0 respectively) can be defined as set of homotopy classes of arcs in .X (in X, respectively) joining .q0 with a point .q ∈ X0 (in X respectively) equipped with the complex structure induced by the projection to the endpoint of the arcs, and the point .q˜0 corresponds to the class of the constant curve. The isomorphism .(Isq˜0 )−1 0 is defined in the from .π1 (X0 , q0 ) to the group of covering transformations on .X 0 same way as it was done for X instead of .X . These considerations imply that there q˜0 −1 0 (Isq˜0 )−1 (e0 ) into .X(Is  is a holomorphic mapping from .X ) (e0 ). Hence, the extremal length of the first set is not smaller than the extremal length of the q˜0 −1 0 Isq˜0 )−1 (e0 ) ≥ λ(X(Is  second set, .λ(X ) (e0 ).

376

11 Finiteness Theorems

Vice versa, take any annulus .A0 that is a relatively compact subset of q˜0 −1 q˜0 −1   .X(Is ) (e0 ) and is a deformation retract of .X(Is ) (e0 ). Its projection to X is relatively compact in X, hence, it is contained in a relatively compact open subset .X0 of X that is a deformation retract of X. Hence, .A0 can be considered as 0 (Isq˜0 )−1 (e0 )) ≤ λ(A0 ). Since 0 (Isq˜0 )−1 (e0 ), and, hence, .λ(X subset of .X q˜0 −1 q˜0 −1   λ(X(Is ) (e0 )) = inf{λ(A0 ) : A0  X(Is ) (e0 )

.

q˜0 −1  ) (e0 )} , is a deformation retract of X(Is q˜0 −1 0 Isq˜0 )−1 (e0 ) ≤ λ(X(Is  we obtain .λ(X ) (e0 ). We are done.

 

We proved a slightly stronger statement, namely, the number of homotopy classes of mappings .X → C \ {−1, 1} that contain a contractible holomorphic mapping or an irreducible holomorphic mapping does not exceed .3( 32 e36π λ7 (X) )2g +m . Proof of Proposition 11.4 Denote the zero set .{x ∈ X : Imf (x) = 0} by L. Since f is not contractible, .L = ∅. 1. A Torus with a Hole Assume first that X is a torus with a hole with base point q0 . For notational convenience we denote by .e0 and .e0 the two elements of the set of generators .E of .π1 (X, q0 ) that is associated to a standard bouquet of circles for X. We claim that there is a connected component .L0 of L such that (after suitable orientation) the intersection number of the free homotopy class of one of the elements of .E, say of .e0 , with .L0 is positive, and the intersection number with

.







one of the classes .e0 , or .(e0 )−1 , or .e0 e0 with .L0 is positive. The claim is easy to prove in the case when there is a component .L0 of L which is a simple closed curve that is not contractible and not contractible to the hole of X. Indeed, consider the inclusion of X into a closed torus .Xc and the homomorphism on fundamental groups .π1 (X, q0 ) → π1 (Xc , q0 ) induced by the inclusion. Denote by .e0c and .e0 c the images of .e0 and .e0 under this homomorphism. Notice that .e0c and

c .e 0 commute. The (image under the inclusion of the) curve .L0 is a simple closed non-contractible curve in .Xc . It represents the free homotopy class of an element c j c k .(e ) (e ) for some integers j and k which are not both equal to zero. Hence, .L0 0 0 is not null-homologous in .Xc , and by the Poincaré Duality Theorem for one of the generators, say for .e0c , the representatives of the free homotopy class .e0c have nonzero intersection number with .L0 . After possibly reorienting .L0 , we may assume that this intersection number is positive. There is a representative of the class .e0c which is contained in X, hence, .e0 has positive intersection number with .L0 . Suppose all compact connected components of L are contractible or contractible to the hole of X. Consider a relatively compact domain .X

 X in .X with smooth boundary which is a deformation retract of .X such that for each connected component of L at most one component of its intersection with .X

is not contractible to the hole of .X

. (See the paragraph on “Regular zero sets”.) There is at least one component of .L ∩ X

that is not contractible to the hole of .X

. Indeed, otherwise the free homotopy class of each element of .E could be represented by a 



11.3 Holomorphic Mappings into the Twice Punctured Plane

377

loop avoiding L, and, hence, the monodromy of f along each element of .E would be conjugate to the identity, and, hence, equal to the identity, i.e. contrary to the assumption, .f : X → C \ {−1, 1} would be free homotopic to a constant. Take a component .L

0 of .L∩X

that is not contractible to the hole of .X

. There is an arc of .∂X

between the endpoints of .L

0 such that the union .L˜ 0 of the component

.L with this arc is a closed curve in .X that is not contractible and not contractible 0 to the hole. Hence, for one of the elements of .E, say for .e0 , the intersection number of the free homotopy class .e0 with the closed curve .L˜ 0 is positive after orienting the curve .L˜ 0 suitably. We may take a representative .γ of .e0 that is contained in .X

. Then .γ has positive intersection number with .L

0 . Denote the connected component of L that contains .L

0 by .L0 . All components of .L0 ∩ X

different from .L

0 are contractible to the hole of .X

. Hence, .γ has intersection number zero with each of these components. Hence, .γ has positive intersection number with .L0 since .γ ⊂ X

. We proved that the class .e0 has positive intersection number with .L0 . If .e0 also has non-zero intersection number with .L0 we define .e0

= (e0 )±1 so that 





the intersection number of .e0

with .L0 is positive. If .e0 has zero intersection number 

with .L0 we put .e0

= e0 e0 . Then again the intersection number of .e0

with .L0 is 

def

positive. Also, the intersection number of .e0 e0

with .L0 is positive. The set .E2 = {e0 , e0

} satisfies the condition required in the proposition. We obtained Proposition 11.4 for a torus with a hole. 2. A Planar Domain Let X be a planar domain. The domain X is conformally equivalent to a disc with m smoothly bounded holes,

equivalently, to the Riemann sphere with .m + 1 smoothly bounded holes, .P1 \ jm+1 =1 Cj , where .Cm+1 contains the point .∞. As before the base point of X is denoted by .q0 , and for each .j = 1, . . . , m, the generator .ej,0 ∈ E ⊂ π1 (X, q0 ) is represented by a curve that surrounds .Cj once counterclockwise. Since f is not contractible, there must be a connected component of L that has limit points on some .Cj with .j ≤ m. Indeed, otherwise the free homotopy class of each generator could be represented by a curve that avoids L. This would imply that all monodromies are equal to the identity. We claim that there exists a component .L0 of L with limit points on the boundary of two components .∂Cj and .∂Cj

for some .j , j

∈ {1, . . . , m + 1} with .j

= j . Indeed, assume the contrary. Then, if a component of L has limit points on a component .∂Cj , j ≤ m, then all its limit points are on .∂Cj . Take a smoothly bounded simply connected domain .Cj  X ∪ Cj that contains the closed set .Cj , so that its boundary .∂Cj represents .ej,0 . Then all components .L k of .L \ Cj with an endpoint on .∂Cj have another endpoint on this circle. The two endpoints of each 

L k on .∂Cj divide .∂Cj into two connected components. The union of each .L k with

.

each of the two components of .∂Cj \ L k is a simple closed curve in .C, and, hence, by the Jordan Curve Theorem it bounds a relatively compact topological disc in

.C. One of these discs contains .C , the other does not. Assign to each component j





.L of .L \ C with both endpoints on .∂C the closed arc .αk in .∂C with the same k j j j endpoints as .L k , whose union with .L k bounds a relatively compact topological disc

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11 Finiteness Theorems

in .C that does not contain .Cj . These discs are partially ordered by inclusion, since the .L k are pairwise disjoint. Hence, the arcs .αk are partially ordered by inclusion. For an arc .αk which contains no other of the arcs (a minimal arc) the curve .f ◦ αk except its endpoints is contained in .C \ R. Moreover, the endpoints of .f ◦ αk lie on



.f (L ), which is contained in one connected components of .R \ {−1, 1}, since .L is k k connected. Hence, the curve .f ◦αk is homotopic in .C\{−1, 1} (with fixed endpoints) to a curve in .R \ {−1, 1}. The function f either maps all points on .∂Cj \ αk that are close to .αk to the open upper half-plane or maps them all to the open lower halfplane. (Recall, that zero is a regular value of .Imf .) Hence, for an open arc .αk ⊂ ∂Cj

that contains .αk the curve .f ◦ αk is homotopic in .C \ {−1, 1} (with fixed endpoints) to a curve in .C \ R. Consider the arcs .αk with the following property. For an open arc .αk in .∂Cj which contains the closed arc .αk the mapping .f ◦ αk is homotopic in .C \ {−1, 1} (with fixed endpoints) to a curve contained in .C \ R. Induction on the arcs by inclusion shows that this property is satisfied for all maximal arcs among the .αk and, hence,

.f | ∂C is contractible in .C \ {−1, 1}. Hence, if the claim was not true, then for each j hole .Cj , j ≤ m, whose boundary contains limit points of a connected component of L, the monodromy along the curve .Cj (with any base point contained in .Cj ) that represents .ej,0 would be trivial. Then all monodromies would be trivial, which contradicts the fact that the mapping is not contractible. The contradiction proves the claim. With .j and .j

being the numbers of the claim and .j ≤ m we consider the set .E2 ⊂ E2 which consists of the following primitive elements: .ej ,0 , the element −1 provided .j

= m + 1, and .e e



.(ej

,0 ) j ,0 j,0 for all .j = 1, . . . , m, j = j , j = j .

The free homotopy class of each element of .E2 has intersection number 1 with .L0 after suitable orientation of the curve .L0 . Each product of at most two different elements of .E2 is a primitive element of .π1 (X, q) and is contained in .E4 . Moreover, the intersection number with .L0 of the free homotopy class of each product of two different elements of .E2 equals 2. Each element of .E is the product of at most two elements of .E2 ∪ (E2 )−1 . The proposition is proved for the case of planar domains X.   

Proof of Proposition 11.5 1. A Torus with a Hole Suppose X is a torus with a hole. Consider the curve .L0 and the set .E2 ⊂ π1 (X, q0 ) obtained in Proposition 11.4. For one of the functions

.Ml ◦ f , denoted by F , the image .F (L0 ) is contained in .(−1, 1). Let .e0 , .e be the 0 two elements of .E. Move the base point .q0 to a point .q ∈ L0 along a curve .α in X, and consider the generators .e = Isα (e0 ) and .e = Isα (e0 ) of .π1 (X, q), and the set .Isα (E2 ) ⊂ π1 (X, q). Then e and .e are products of at most two elements of

.Isα (E ). Since the free homotopy class of an element of .π1 (X, q0 ) coincides with 2 the free homotopy class of the element of .π1 (X, q) obtained by applying .Isα , the free homotopy class of each product of one or two elements of .Isα (E2 ) intersects

.L0 . We may assume as in the proof of Proposition 11.4 that .Isα (E ) consists of 2 the elements e and .e

, where .e

is either equal to .e ±1 , or equals the product of

11.3 Holomorphic Mappings into the Twice Punctured Plane

379

e and .e . Lemma 11.4 applies to the pair e, .e

, the function F , and the curve .L0 . Since F is irreducible, the monodromies of F along e and .e

are not powers of a single standard generator of the fundamental group of .π1 (C \ {−1, 1}, q ). Hence, the monodromy along each of the e and .e

is the product of at most two elements of .L− not exceeding .2π λe,e

. Therefore, the monodromy of F along each of the e and .e

has .L− not exceeding .4π λe,e

. Notice that .λe,e

= λe0 e0

≤ λ3 (X), since .e0

is the product of at most two factors, each an element of .E ∪ E −1 . Since .e is the product of at most two different elements among the e and .e

and their inverses, we obtain Proposition 11.5 for e and .e , in particular .L− (F∗ (e)) and .L− (F∗ (e )) do not exceed .8π λ3 (X). Proposition 11.5 is proved for tori with a hole. 2. A Planar Domain Consider the curve .L0 and the set .E2 of Proposition 11.4. Move the base point .q0 along an arc .α to a point .q ∈ L0 . Then .f (q) ∈ R \ {−1, 1} and for one of the mappings f , .M1 ◦ f , or .M2 ◦ f , denoted by F , def

the inclusion .F (L0 ) ⊂ (−1, 1)) holds, hence, .q = F (q) is contained in .(−1, 1). Put .ej = Isα (ej,0 ) for each element .ej,0 ∈ E. The .ej form the basis .Isα (E) of

.π1 (X, q). The set .Isα (E ) consists of primitive elements of .π1 (X, q) such that the 2 free homotopy class of each product of one or two elements of .Isα (E2 ) intersects .L0 . Moreover, each element of .Isα (E) is the product of one or two elements of

−1 . .Isα (E ) ∪ (Isα (E )) 2 2 By the condition of the proposition not all monodromies .F∗ (e), e ∈ Isα (E2 ), are (trivial or non-trivial) powers of the same standard generator of .π1 (C \ {−1, 1}, q ). Apply Lemma 11.4 to all pairs of elements of .Isα (E2 ) whose monodromies are not (trivial or non-trivial) powers of the same standard generator of .π1 (C \ {−1, 1}, q ). Since the product of at most two different elements of .Isα (E2 ) is contained in .Isα (E4 ), Lemma 11.4 shows that the monodromy .F∗ (e) along each element .e ∈ Isα (E2 ) is the product of at most two factors, each with .L− not exceeding .2π λ4 (X). Since each element of .Isα (E) is a product of at most two factors in .E2 ∪ (E2 )−1 , the monodromy .F∗ (ej ) along each generator .ej of .π1 (X, q) is the product of at most 4 factors of .L− not exceeding .2π λ4 (X), and, hence, each monodromy .F∗ (ej ) has .L− not exceeding .8π λ4 (X). Proposition 11.5 is proved for planar domains. 3.1. The General Case. Diagrams of Coverings We will use diagrams of coverings to reduce this case to the case of a torus with a hole or to the case of the Riemann sphere with three holes. Let as before .q˜0 be the point in .X˜ with .P(q˜0 ) = q0 chosen in Sect. 1.2. Let N be a q˜0 −1 ˜ ) (N ) = subgroup of the fundamental group .π1 (X, q0 ) and let .ωN : X˜ → X(Is def

X(N ) be the projection defined in Sect. 1.2. Put .(q0 )N = ωN (q˜0 ). For an element .e0 ∈ N ⊂ π1 (X, q0 ) we denote by .(e0 )N the element of .π1 (X(N ), (q0 )N ) that is obtained as follows. Take a curve .γ in X with base point .q0 that represents .e0 ∈ N. def Let .γ˜ be its lift to .X˜ with initial point .q˜0 . Then .γN = ωN (γ˜ ) is a closed curve in q˜0 −1 ˜ .X(N ) = X(Is ) (N ) with base point .(q0 )N . The element of .π1 (X(N ), (q0 )N ) represented by .γN is the required element .(e0 )N . All curves .γN representing .(e0 )N have the form .ωN (γ˜ ) for a curve .γ˜ in .X˜ with initial point .q˜0 and terminal point

380

11 Finiteness Theorems

(Isq˜0 )−1 (e0 )(q˜0 ). Since .ωN ◦ ωN = P, the curve .ωN (γN ) = P(γ˜ ) = γ represents

.e0 for each curve .γ N in .X(N ) that represents .(e0 )N . We obtain .(ωN )∗ ((e0 )N ) = e0 . N2 )∗ ((e0 )N1 ) = (e0 )N2 , For two subgroups .N1 ≤ N2 of .π1 (X, q0 ) we obtain .(ωN 1 .e0 ∈ N1 (see the commutative diagram Fig. 11.1). Let .q˜ be another base point of .X˜ and let .α˜ be a curve in .X˜ with initial point .q˜0 .

def

and terminal point .q. ˜ Let again N be a subgroup of .π1 (X, q0 ). Put .qN = ωN (q). ˜ N ˜ in .X(N ), and the base point .q˜ of .X˜ are compatible, hence, The curve .αN = ω (α) def q˜ −1 q˜0 −1 ˜ ˜ .X(IsαN (N )) = X(Is ) (IsαN (N )) = X(Is ) (N ) = X(N ). N2 also for the projection We will use the previous notation .ωN 1

q˜ −1 q˜ −1 ˜ ˜ ) (IsαN1 (N1 )) → X(Is ) (IsαN2 (N2 )) , X(Is

.

with .N1 ≤ N2 being subgroups of .π1 (X, q0 ) (.N1 may be the identity and .N2 may be .π1 (X, q0 ).) def

def

Put .α = P(α). ˜ For an element .e0 ∈ π1 (X, q0 ) we put .e = Isα (e0 ) ∈ π1 (X, q) and denote by .eN the element of .π1 (X(N ), qN ), that is represented by .ωN (γ˜ ) for a curve .γ˜ in .X˜ with initial point .q˜ and projection .P(γ˜ ) representing e. Again N2 .(ω N1 )∗ (eN1 ) = eN2 for subgroups .N1 ≤ N2 of .π1 (X, q) and .e ∈ N1 , in particular .(ωN )∗ (eN ) = e for a subgroup N of .π1 (X, q) and .e ∈ N. 3.2. The Estimate for a Chosen Pair of Monodromies Since the mapping .f : X → C \ {−1, 1} is irreducible, there exist two elements .e0 , e0

∈ E ⊂ π1 (X, q0 ) such that the monodromies .f∗ (e0 ) and .f∗ (e0

) are not powers of a single conjugate of a power of one of the elements .a1 , .a2 or .a1 a2 . The fundamental group of the Riemann surface .X(e0 , e0

) is a free group in the two generators .(e0 )e0 ,e0

 and





1 .(e )e ,e

 , hence, .X(e , e ) is either a torus with a hole or is equal to .P with three 0 0 0 0 0 holes. Moreover, the system .Ee0 ,e0

 = {(e0 )e0 ,e0

 , (e0

)e0 ,e0

 } of generators of the fundamental group .π1 (X(e0 , e0

), (q0 )e0 ,e0

 ) is associated to a standard bouquet of circles for .X(e0 , e0

). This can be seen as follows. The set of generators .E of .π1 (X, q0 ) is associated to a standard bouquet of circles for X. For each .e0 ∈ E we denote the circle of the bouquet that represents .e0 by .γe0 . For each .e0 ∈ E we ˜ with initial point .q˜0 . Let lift the circle .γe0 with base point .q0 to an arc .γ e0 in .X  D be a small disc in X around .q0 , and .D0 , .De0 , e0 ∈ E, be the preimages of D under the projection .P : X˜ → X, that contain .q˜0 , or the terminal point of .γ e0 , respectively. We assume that D is small enough so that the mentioned preimages



0 ). For .e0 = e , e

the image of D are pairwise disjoint. Put .De0 ,e0

 = ωe0 ,e0  (D 0 0



e ,e  e ,e

  .ω 0 0 (D γe0 ) in .X(e0 , e0

) with 0 ∪ γ e0 ∪ D e0 ) is the union of an arc .ω 0 0 ( two disjoint discs, each containing an endpoint of the arc and one of them equal to



e ,e

 (D 0 ∪ γ  .De ,e

 . For .e0 = e , e the image .ω e0 ∪ D e0 ) is the union of .De0 ,e0

 0 0 0 0 def





with the loop .(γe0 )e0 ,e0

 = ωe0 ,e0  ( γe0 ). For .e0 = e0 , e0

the loop .(γe0 )e0 ,e0

 in

11.3 Holomorphic Mappings into the Twice Punctured Plane

381



X(e0 , e0

) has base point .(q0 )e0 ,e0

 = ωe0 ,e0  (q˜0 ) and represents the generator



.(e0 )e ,e

 of the fundamental group of .π1 (X(e , e ), (q0 )e ,e

 ). 0 0 0 0 0 0 Since the bouquet of circles .∪e0 ∈E γe0 is a standard bouquet of circles for X, the union .(γe0 )e0 ,e0

 ∪ (γe0

)e0 ,e0

 is a standard bouquet of circles in .X(e0 , e0

). This can be seen by looking at the intersections of the loops with a circle that is contained in .De0 ,e0

 and surrounds .(q0 )e0 ,e0

 . By the commutative diagram of coverings the intersection behaviour is the same as for the images of these objects under .ωe0 ,e0

 . Hence, since .(γe0 )e0 ,e0

 and .(γe0

)e0 ,e0

 represent the generators .(e0 )e0 ,e0

 and

.(e )e ,e

 of .Ee ,e

 , the union .(γe )e ,e

 ∪ (γe

)e ,e

 is a standard bouquet of 0 0 0 0 0 0 0 0 0 0 0 circles for .X(e0 , e0

). The set .X(e0 , e0

) is either a torus with a hole or is equal to .P1 with three holes. Apply Proposition 11.4 to the Riemann surface .X(e0 , e0

) with base point .(q0 )e ,e

 , the holomorphic mapping .fe ,e

 = f ◦ ωe ,e

 into .C \ {−1, 1}, and 0 0 0 0 0 0 the set of generators .Ee0 ,e0

 of the fundamental group .π1 (X(e0 , e0

), (q0 )e0 ,e0

 ). We obtain a relatively closed curve .Le0 ,e0

 on which the function .fe0 ,e0

 is real, and a set .(Ee0 ,e0

 ) 2 = {(e 0 )e0 ,e0

 , (e

0 )e0 ,e0

 } which contains one of the elements of .Ee0 ,e0

 . The second element of .(Ee0 ,e0

 ) 2 is either equal to the second element of .Ee0 ,e0

 or to its inverse, or to the product of the two elements (in any order) of





.Ee ,e

 . (We will usually refer to the product .(e )e ,e

 (e )e ,e

 = (e e )e ,e

 , 0 0 0 0 0 0 0 0 0 0 0 0 but we may change the product .(e0 e0

)e0 ,e0

 to the product .(e0

e0 )e0 ,e0

 , without changing the estimate of the .L− of the monodromies of the elements of .E2 .) The free .







homotopy classes .(e 0 )e0 ,e0

 , .(e

0 )e0 ,e0

 , and .(e 0 )e0 ,e0

 (e

0 )e0 ,e0

 = (e 0 e

0 )e0 ,e0

 intersect .Le0 ,e0

 .



Choose a point .qe ,e

 ∈ Le ,e

 and a point .q˜ ∈ X˜ with .ωe0 ,e0  (q) ˜ = qe ,e

 . 0

0

0

0

0

0

Let .α˜ be a curve in .X˜ with initial point .q˜0 and terminal point .q, ˜ and .αe0 ,e0

 =





ωe0 ,e0  (α). ˜ Put .e e ,e

 = Isαe ,e

 ((e 0 )e0 ,e0

 ) and .e

e ,e

 = Isαe ,e

 ((e

0 )e0 ,e0

 ). 0 0 0 0 0 0 0 0 For one out of three Möbius transformations .Ml the mapping .Fe0 ,e0

 = Ml ◦ fe0 ,e0

 = Ml ◦ f ◦ ωe0 ,e0

 takes .Le0 ,e0

 to .(−1, 1), and hence .Fe0 ,e0

 takes a value .q = Fe0 ,e0

 (qe0 ,e0

 ) ∈ (−1, 1) at .qe0 ,e0

 . By Lemma 11.4 each of the .(Fe0 ,e0

 )∗ (e e ,e

 ) and .(Fe0 ,e0

 )∗ (e

e ,e

 ) is the product of at most two 0

0

0

0

elements of .π1 (C \ {−1, 1}, q ) of .L− not exceeding .2π λ3 (X(e0 , e0

)), hence, L− ((Fe0 ,e0

 )∗ (e e ,e

 )) ≤ 4π λ3 (X(e0 , e0

)),

.

0

L− ((F

e0 ,e0



0

)∗ (e

e ,e

 )) 0 0

≤ 4π λ3 (X(e0 , e0

)) .

382

11 Finiteness Theorems



It follows that each of the .(Fe0 ,e0

 )∗ (ee

,e

 ) and .(Fe ,e

 )∗ (ee ,e

 ) is the 0 0 0

0

0

0

product of at most four elements of .π1 (C \ {−1, 1}, q ) of .L− not exceeding



.2π λ3 (X(e , e )), hence, 0 0





L− ((Fe0 ,e0

 )∗ (ee

,e

 )) ≤ 8π λ3 (X(e0 , e0 )),

.

0

0



L− ((Fe0 ,e0

 )∗ (ee



)) 0 ,e0 

≤ 8π λ3 (X(e0 , e0

)) .

It remains to take into account that for a subgroup N of .π1 (X, q0 ) the equation (FN )∗ (eN ) = F∗ (e) holds for each .e ∈ Isα (N ), and .λj (X(N )) ≤ λj (X) for each natural number j .

.

3.3. Other Generators. Intersection of Free Homotopy Classes with a Component of the Zero Set Take any element .e ∈ Isα (E) that is not in .e , e

. Then the monodromy .F∗ (e) is either equal to the identity, or one of the pairs .(F∗ (e), F∗ (e )) or .(F∗ (e), F∗ (e

)) consists of two elements of .π1 (C\{−1, 1}, q ) that are not powers of the same standard generator .aj , .j = 1 or 2. We assume that the second option holds. Interchanging if necessary .e and .e

, we may suppose this option holds for the pair .(F∗ (e), F∗ (e )). Moreover, changing if necessary .e to its inverse .(e )−1 , we may assume that .e is either an element of .Isα (E) or it is a product of two elements of q˜ −1

˜ .Isα (E). The quotient .X(e, e ) = X(Is ) (e, e ) is a Riemann surface whose fundamental group is a free group in two generators. Hence .X(e, e ) is either a torus with a hole or is equal to .P1 with three holes. We consider a diagram of coverings as follows. Let first . X(e ) =

q˜ −1 ˜ X(Is ) (e ) be the annulus with base point .qe  = ωe  (q), ˜ that admits a mapping .ωe  : X(e ) → X that represents .e . By Lemma 11.3 with X e ,e

 −1 ) (L

replaced by .X(e , e

) the connected component .Le  of .(ωe 

ωe  (q) ˜

e ,e

 )

that

.X(e )

contains .qe  = is a relatively closed curve in with limit points on both boundary components. The free homotopy class of the generator .e e  of

.π1 (X(e ), qe  ) intersects .Le  . The mapping .Fe  = Ml ◦ f ◦ ωe  maps .Le  into .(−1, 1), and .Fe  (qe  ) = F ◦ P(q) ˜ = q . q˜ −1 ˜ Next we consider the quotient .X(e , e) = X(Is ) (e , e) whose fundef

damental group is again a free group in two generators. The image .Le ,e = e ,e

ωe  (Le  ) is a connected component of the preimage of .(−1, 1) under .Fe ,e = F ◦ ωe ,e . Indeed, .Le ,e is connected as image of a connected set under a continuous mapping, and e ,e

e ,e

Fe ,e (Le ,e ) = Fe ,e (ωe  (Le  )) = F ◦ ωe ,e ◦ ωe  (Le  )

.

= Fe  (Le  ) ⊂ (−1, 1). e ,e

Moreover, since the mapping .ωe  : X(e ) → X(e , e) is a covering, its restriction .(ImF ◦ωe  )−1 (0) → (ImF ◦ωe ,e )−1 (0) is a covering. Hence the image

11.3 Holomorphic Mappings into the Twice Punctured Plane

383

e ,e

under .ωe  of a connected component of .(ImF ◦ ωe  )−1 (0) is open and closed in e ,e

(ImF ◦ωe ,e )−1 (0). Hence, .Le ,e = ωe  (Le  ) is a connected component of the

.

e ,e

e ,e



˜ = preimage of .(−1, 1) under .Fe ,e . Put .qe ,e = ωe  (qe  ) = ωe  ◦ ωe  (q)

,e  e

ω (q). ˜ Note that .Fe ,e (qe ,e ) = F ◦ ωe ,e (qe ,e ) = F (q) = q . 

We prove now that the free homotopy class .e e ,e in .X(e , e) that is related to .e

intersects .Le ,e . Consider any loop .γ e ,e in .X(e , e) with some base point .q e ,e , 



that represents .e e ,e . There exists a loop .γ e  in .X(e ) which represents .e e  such e ,e

that .ωe  (γ e  ) = γ e ,e . Such a curve .γ e  can be obtained as follows. There is a loop .γ

e ,e in .X(e , e) with base point .qe ,e that represents .(e )e ,e , and a curve .α e ,e in .X(e , e) with initial point .qe ,e and terminal point .q e ,e , such that .γ e ,e is homotopic with fixed endpoints to .(α e ,e )−1 γ

e ,e α e ,e . Consider the lift .γ˜

of .γ

to .X˜ with initial point .q, ˜ and the lift .α˜ of .α with initial e ,e

e ,e

point .q. ˜ The terminal point of .γ˜

e ,e equals .σ (q) ˜ for the covering transformation

σ = (Isq˜ )−1 (e ) = (Isq˜0 )−1 (e 0 ). (See Eq. (1.2).) The terminal point of the curve .(α˜ )−1 γ˜

e ,e σ (α˜ ) is obtained from its initial point by applying the covering

.



transformation .σ . Hence, .ωe  ((α˜ )−1 γ˜

e ,e σ (α˜ )) is a closed curve in .X(e ) that e ,e



represents .e e  and projects to .(α e ,e )−1 γ

e ,e α e ,e under .ωe  . Since .γ e ,e is homotopic to .(α e ,e )−1 γ

e ,e α e ,e with fixed base point, it also has a lift to 

X(e ) which represents .e e  .

.



Since .e e  intersects .Le  , the loop .γ e  has an intersection point .p e  with .Le  . e ,e

The point .p e ,e = ωe  (p e  ) is contained in .γ e ,e and in .Le ,e . We proved that the free homotopy class .e e ,e in .X(e , e) intersects .Le ,e . 

3.4. A System of Generators Associated to a Standard Bouquet of Circles We claim that the system of generators . e e ,e , ee ,e of . π1 (X(e , e), qe ,e ) is associated to a standard bouquet of circles for .X(e , e). If .e ∈ E the claim can be obtained as in paragraph 3.2. Suppose .e = e e

for .e , e

∈ E. Consider the system .E of generators of







.π1 (X, q) that is obtained from .E by replacing .e by .e e . If .e and .e correspond to a handle of X, then .E is associated to a part of a standard bouquet of circles for X, see Fig. 11.2a for the case when .e is represented by an .α-curve and .e

is represented by a .β-curve. The situation when .e is represented by a .β-curve and .e

is represented by an .α-curve is similar. The claim is obtained as in paragraph 3.2. In this case .X(e , e) is a torus with a hole. Suppose one of the pairs .(e, e ) or .(e, e

) corresponds to a handle of X. We assume that e corresponds to an .α-curve and .e corresponds to a .β-curve of a handle of X (see Fig. 11.2b). The remaining cases are treated similarly, maybe, after replacing .e e

by .e

e (see paragraph 3.2). With our assumption .E is not associated

384

11 Finiteness Theorems

Fig. 11.2 Standard bouquets of circles

to a standard bouquet of circles for X. Nevertheless, the pair .(ee ,e , e e ,e ) with





.e = e e is associated to a standard bouquet of circles for .X(e , e). This can be seen as before. Consider a (non-standard) bouquet of circles in X corresponding to

˜ We obtain the union .E and take its union with a disc D around q. Lift this set to .X. ˜ of a collection of arcs in .X with initial point .q, ˜ with a collection of discs in .X˜ around .q ˜ and around the terminal points of the arcs. Take the union of the arcs and the discs.

The image in .X(e , e) of this union under the projection .ωe ,e is the union of the two loops .(γe )e ,e ∪ (γe )e ,e , the disc .De ,e and a set, that is contractible to .De ,e . Looking at the intersection of the two loops with a small circle contained in .De ,e and surrounding .qe ,e , we see as before that .(γe )e ,e ∪ (γe )e ,e is a standard bouquet of circles for .X(e , e). In this case .X(e , e) is a torus with a hole. In the remaining case no pair of generators among e, .e , and .e

corresponds to a handle. In this case again .E does not correspond to a standard bouquet of circles for X. But .{ee ,e , (e e

)e ,e } (maybe, after changing .e e

to .e

e ) corresponds to a standard bouquet of circles for .X(e , e). (See Fig. 11.2c for the case when walking along a small circle around q counterclockwise, we meet the incoming and outgoing

11.3 Holomorphic Mappings into the Twice Punctured Plane

385

rays of representatives of the three elements of .E in the order .e, e , e

. If the order is different the situation is similar, maybe, after replacing .e e

by .e

e .) In this case

.X(e , e) is a planar domain. 3.5. End of the Proof Consider first the case when .X(e , e) is a torus with a hole. Since .e e ,e intersects .Le ,e , we see as in the proof when X itself is a torus with a hole, that the curve .Le ,e cannot be contractible or contractible to the hole, and the intersection number must be different from zero. Then the intersection number



with .Le ,e of the free homotopy class of one of the choices .e±1 e ,e or .(e e)e ,e ,



denoted by .ee ,e , is not zero and has the same sign. By Lemma 11.4 each of the





.(Fe ,e )∗ (e

e ,e ) and .(Fe ,e )∗ (ee ,e ) is the product of at most two elements of

.π1 (C \ {−1, 1}, q ) with .L− not exceeding 

2π λe

.

e ,e

,e

e ,e

≤ 2π λ5 (X),

(11.14)

since .e is the product of at most two elements of .E ∪ E −1 and .e

is the product of at most three elements of .E ∪E −1 . The element e is the product of at most two different elements among the .e and .e

or their inverses. Hence, the monodromy .F∗ (e) = (Fe ,e )∗ (ee ,e ) is the product of at most four elements with .L− not exceeding (11.14). Hence, F∗ (e) ≤ 8π λ5 (X) .

(11.15)

.

Consider now the case when .X(e , e) equals .P1 with three holes. Since .e e ,e and .ee ,e correspond to a standard bouquet of circles for .X(e , e), the curves representing .e e ,e surround counterclockwise one of the holes, denoted by .C , and the curves representing .ee ,e surround counterclockwise another hole, denoted by

.C . After applying a Möbius transformation we may assume that the remaining hole, denoted by .C∞ , contains the point .∞. There are several possibilities for the behaviour of the curve .Le ,e . Since .e e ,e intersects .Le ,e , the curve .Le ,e must have limit points on .C . The first possibility is that .Le ,e has limit points on .∂C

and .∂C

, the second possibility is, .Le ,e has limit points on .C and .C∞ , the third possibility is, .Le ,e has all limit points on .C , and .C

is contained in the bounded connected component of .C \ (Le ,e ∪ C ). 





In the first case the free homotopy classes .e e ,e and .e−1 e ,e have positive intersection number with the suitably oriented curve .Le ,e . In the second case the 



free homotopy classes .e e ,e and .(e e)e ,e have positive intersection number with the suitably oriented curve .Le ,e . In the third case the free homotopy classes of

2



.e

e ,e , .(e e)e ,e and of their product intersect .Le ,e . The first two cases were treated in paragraph 2 of this section. The statement concerning the third case is proved as follows. Any curve that is contained in the complement of .C ∪ Le ,e has either winding number zero around .C (as a curve in the complex plane .C), or its winding number

386

11 Finiteness Theorems

around .C coincides with the winding number around .C

. On the other hand the representatives of the free homotopy class of .e e ,e have winding number 1 around



.C and winding number 0 around .C . The representatives of the free homotopy class 2 of .(e e)e ,e have winding number 2 around .C , and winding number 1 around



2



.C . Hence, .e

e ,e and .(e e)e ,e intersect .Le ,e . By the same argument the free homotopy class of the product of .e e ,e and .(e 2 e)e ,e intersects .Le ,e .

−1



We let .e

e ,e be equal to .ee ,e in the first case, equal to .(e e)e ,e in the second

case, and equal to .(e 2 e)e ,e in the third case. Then e is the product of at most three elements among the .(e )±1 and .(e

)±1 . By Lemma 11.4 each of the .(Fe ,e )∗ (e e ,e ) and .(Fe ,e )∗ (e

e ,e ) is the product of at most two elements of .π1 (C \ {−1, 1}, q ) with .L− not exceeding 2π λe

.

e ,e

,e

e ,e

≤ 2π λ7 (X),

(11.16)

We used that .e is the product of at most two elements of .E ∪ E −1 , .e ∈ E ∪ E −1 and .e

is the product of at most five elements of .E ∪ E −1 . Since e is the product of at most three elements among the .(e )±1 and .(e

)±1 , the monodromy .F∗ (e) = (Fe ,e )∗ (ee ,e ) is the product of at most six elements with .L− not exceeding (11.16). Hence, F∗ (e) ≤ 12π λ7 (X) .

.

(11.17)

The proposition is proved.  

11.4 (g, m)-Bundles over Riemann Surfaces Theorem 11.2 is a consequence of the following theorem on .(0, 3)-bundles with a section. Theorem 11.3 Over a connected Riemann surface of genus g with .m+1 holes there are up to isotopy no more than .(15 exp(6π λ10 (X)))6(2g +m) irreducible holomorphic .(0, 3)-bundles with a holomorphic section. Theorem 11.1 (with a weaker estimate) is a consequence of Theorem 11.3. Indeed, consider holomorphic (smooth, respectively) bundles whose fiber over each point .x ∈ X equals .P1 with set of distinguished points .{−1, 1, f (x), ∞} for a function f which depends holomorphically (smoothly, respectively) on the points .x ∈ X and does not take the values .−1 and 1. Then we are in the situation of Theorem 11.3. The mapping f is reducible, iff the bundle is reducible (see also Lemma 7.3).

11.4 (g, m)-Bundles over Riemann Surfaces

387

The relation between Theorems 11.2 and 11.3 will be obtained by representing families of .(1, 1)-bundles as double branched coverings of special .(0, 4)-bundles (see Definition 8.6) and using Proposition 8.2. Preparation of the Proof of Theorem 11.3 The proof of Theorem 11.3 will go now along the same lines as the proof of Theorem 11.1 with some modifications. Recall that the set H ={{z1 , z2 , z3 } ∈ C3 (C)S3 : the three points z1 , z2 , z3

.

are contained in a real line in the complex plane} is a smooth real hypersurface of .C3 (C)S3 . As before, for each complex affine M  self-mapping   of the complex plane  we consider the diagonal action . M (z1 , z2 , z3 ) = M(z1 ), M(z  2 ), M(z3 ) on points . (z1 , z2 , z3 ) ∈ C3 (C) , and the diagonal action . M {z1 , z2 , z3 } = {M(z1 ), M(z2 ), M(z3 )} on points .{z1 , z2 , z3 } ∈ C3 (C)S3 . The following two lemmas replace Lemma 11.2 in the case of .(0, 3)-bundles with a section. Lemma 11.5 Let A be an annulus with an orientation of simple closed dividing curves and .F : A → C3 (C)S3 be a holomorphic mapping. Suppose .LA is a simple relatively closed curve in A with limit points on both boundary circles of A, and .F (LA ) ⊂ H. Moreover, we assume that for a point .qA ∈ LA the value .F (qA ) is in the totally real subspace .C3 (R)S3 . Let .eA ∈ π1 (A, qA ) be the positively oriented generator of the fundamental group of A with base point .qA . If the braid

def

b = F∗ (eA ) ∈ B3 is different from .σjk Δ2 3 with j equal to 1 or 2, and .k = 0 and

. being integers, then .

L− (ϑ(b)) ≤ 2π λ(A).

(11.18)

.

Notice that the braids .σjk Δ 3 for odd . are exceptional for Theorem 9.4, but not exceptional for Lemma 11.5. The reason is that the braid in Lemma 11.5 is related to a mapping of an annulus, not merely to a mapping of a rectangle. For .t ∈ [0, ∞) t ∈ [1, ∞) def log t we put .log+ t = 0 t ∈ [0, 1) .

Lemma 11.6 If the braid in Lemma 11.5 equals .b = σjk σjk Δ 3 with j and .j equal to 1 or to 2, .j = j , and k and .k being non-zero integers, and . an even integer, then .

log+ (3[

|k | |k| ]) + log+ (3[ ]) ≤ 2π λ(A). 2 2

(11.19)

As before, for a non-negative number x we denote by .[x] the smallest integer not exceeding x.

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11 Finiteness Theorems

Proof of Lemma 11.5 By the same argument as in the proof of Lemma 11.2 we may assume that the annulus A has smooth boundary, the mapping F extends continuously to the closure .A, and the curve .LA is a smooth (connected) curve in .A whose endpoints are on different boundary components of A. By Lemma 9.11 the inequality Λ(btr ) ≤ λ(A)

(11.20)

.

holds. For .b = σjk Δ 3 with j equal to 1 or 2, and .k = 0 and . being integers, the statement of Lemma 11.5 follows from Theorem 9.4 in the same way as Lemma 11.2 follows from Theorem 9.1. For .b = σjk Δ 3 with .k = 0 the statement is trivial since then .ϑ(Id) = Id and .L− (Id) = 0.

To obtain the statement in the remaining case .b = σjk Δ32 +1 with j equal to 1 or 2, and k and . being integers, we use Lemma 11.6. Notice that .σ1 Δ3 = Δ3 σ2

and .σ2 Δ3 = Δ3 σ1 . Hence, .b2 = σjk σjk Δ34 +2 with .σj = σj . Let .ω2 : A2 → A be the twofold unbranched covering of A by an annulus .A2 . The equality .λ(A2 ) = 2λ(A) holds. Let .qA2 be a point in .ω2−1 (qA ), and let .LqA2 be the lift of .LA to .A2 that contains .qA2 . Denote by .γA2 the loop .ω2−1 (γA ) with base point .qA2 . Then .F ◦

ω2 | γA2 represents .b2 and .(b2 )tr . Lemma 11.6 applied to .σjk σjk Δ34 +2 gives the |k| 2 ) = 4π λ(A). Since .ϑ(b) = σ 2[ 2 ]sgn(k) , the estimate .2 log+ (3[ |k| ]) ≤ 2π λ(A j 2   inequality (11.18) follows. The lemma is proved.



Proof of Lemma 11.6 Lemma 3.5 gives .Λtr (σjk σjk Δ 3 ) = Λtr (σjk σjk ). Along the lines of proof of Statement 1 of Lemma 9.4 we see that for .k · k = 0    



Λtr (σjk σjk ) ≥ Λ tr (σjk )pb + Λ pb (σjk )tr .

.

A similar argument as used in the proof of Theorem 9.4 shows  that for any positive integer number . and each of the .σj the inequality .Λ tr (σj± )pb ≥     ±2[ ] ±2[ ] ±2[ ] 1 Λ tr (σj 2 )pb holds. Theorem 9.1 implies .Λ tr (σj 2 )pb ≥ 2π L− (σj 2 ) =

log+ (3[ 2 ]) . Since by (11.20) the inequality .Λtr (σjk σjk Δ 3 ) ≤ λ(A) holds, the lemma is proved.   1 2π

We want to emphasize that periodic 3-braids are not of the form .σjk Δ2 3 for non-zero powers .σjk of a .σj and an integer number . , hence, the lemma is true

also for periodic braids. For each periodic braid b of the form .σ1 σ2 = σ1−1 Δ3 , 2 = σ Δ , .σ σ = σ −1 Δ , .(σ σ )2 = σ Δ , and .Δ the .L (ϑ(b)) .(σ1 σ2 ) 1 3 2 1 3 2 1 2 3 3 − 2 vanishes. However, for instance for the conjugate .σ1−2k Δ3 σ12k = σ1−2k σ22k Δ3 of .Δ3

11.4 (g, m)-Bundles over Riemann Surfaces

389

we have .L− (ϑ(σ1−2k Δ3 σ12k )) = 2 log(3|k|). Another example, for the conjugate −2k .σ σ1 σ2 σ22k of .σ1 σ2 we have 2 σ2−2k σ1 σ2 σ22k = σ2−2k−1 Δ3 σ22k = σ2−2k−1 σ12k Δ3 .

.

and .L− (ϑ(σ2−2k σ1 σ2 σ22k )) equals .2 log(3|k|). Notice that the lemmas and Theorem 9.4 descend to statements on elements of .B3 Z3 rather than on braids. For an element .b of the quotient .B3 Z3 we put .ϑ(b) = ϑ(b) for any representative .b ∈ B3 of .b. Lemma 11.7 below is an analog of Lemma 11.4. It will follow from Lemma 11.5 in the same way as Lemma 11.4 follows from Lemma 11.2. Lemma 11.7 Let X be a connected finite open Riemann surface, and let .F : X → C3 (C)S3 be a non-contractible holomorphic map that is transverse to the hypersurface .H in .C3 (C)S3 . Suppose .L0 is a simple relatively closed curve in X such that .F (L0 ) is contained in .H, and for a point .q ∈ L0 the point .F (q) is contained in the totally real space .C3 (R)S3 . Let .e(1) and .e(2) be primitive elements of .π1 (X, q). Suppose that for .e = e(1) , .e = e(2) , and .e = e(1) e(2) the free homotopy class  .e intersects .L0 . Then either the two monodromies of F modulo the center .F∗ (e(j ) )Z3 , j = 1, 2, are powers of the same element .σj Z3 of .B3 Z3 , or each of them is the product of at most two elements .b1 and .b2 of .B3 Z3 with L− (ϑ(bj )) ≤ 2π λe(1) ,e(2) , j = 1, 2,

.

(11.21)

where 

def





λe(1) ,e(2) = max{λ(A(e(1) )), λ(A(e(2) )), λ(A(e(1) e(2) ))}.

.

Proof Suppose for an element .e ∈ π1 (X, q) the free homotopy class  .e intersects L0 . We claim that all components of .P−1 (L0 ) have limit points on the boundary of ˜ ∼ .X = C+ . Suppose the claim is not true. Then some connected component .L˜ 0 of .P−1 (L0 ) is compact. Lift the holomorphic mapping .F ◦ P : X˜ ∼ = C+ → C3 (C)S3 to a ∼  ˜ holomorphic mapping .F ◦ P : X = C+ → C3 (C). For each .z ∈ C+ we let .Az  be the complex affine mapping that takes the first coordinate of .F ◦ P(z) to .−1, the  third to 1, and  the second to a point .g(z) ∈ C \ {−1, 1}. Then .Az (F ◦ P(z)) = − 1, g(z), 1 , z ∈ C+ . By the conditions of the lemma .g(z) is real on .L˜ 0 . Then, since g is holomorphic on .C+ , .g ≡ c on .C+ for a constant c (see the proof of  Lemma 11.3). It follows that .F ◦ P(z) = A−1 z (c), z ∈ C+ , which means that the image of .F ◦ P is contained in .H. This contradicts the fact that F is not constant and is transverse to .H. The claim is proved. It follows along the lines of the proof of Lemma 11.3 that an analog of Lemma def q˜ −1 ˜ 11.3 holds. More precisely, for any point .q˜ in .P−1 (q), with .A = X(Is ) (e), .

def

def

ωA = ωe,q˜ , and .qA = ωe,q˜ (q) ˜ the connected component .LA ⊂ A of −1 .(ωA ) (L0 ) that contains .qA has limit points on both boundary circles of A. .

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11 Finiteness Theorems

Put .FA = F ◦ ωA . By the conditions of Lemma 11.7 .FA (LA ) = F (L0 ) ⊂ H and FA (qA ) ∈ C3 (R)S3 . Let .eA be the generator of .π1 (A, qA ) for which .ωA (eA ) = e. The mapping .FA : A → C3 (C)S3 , the point .qA and the curve .LA satisfy the conditions of Lemma 11.5. Notice that the equality .(FA )∗ (eA ) = F∗ (e) holds. Hence, if .F∗ (e) is not equal to .σjk Δ2 , .j = 1 or 2, .k = 0, ∈ Z, then inequality (11.18) holds for .F∗ (e). Suppose the two monodromies modulo center .F∗ (e(j ) )Z3 , j = 1, 2, of Lemma 11.7 are not (trivial or non-trivial) powers of the same element .σj Z3 of .B3 Z3 . Then at most two of the elements, . F∗ (e(1) )Z3 , .F∗ (e(2) )Z3 , and (1) e(2) )Z = F (e(1) )Z · F (e(2) )Z , are powers of an element of the .F∗ (e 3 ∗ 3 ∗ 3 form .σj Z3 . If the monodromies modulo center along two elements among .e(1) , .e(2) , and (1) .e e(2) are not (zero or non-zero) powers of a .σj Z3 then by Lemma 11.5 for each of these two monodromies modulo center inequality (11.21) holds, and the third monodromy modulo center is the product of two elements of .B3 Z3 for which inequality (11.21) holds. If the monodromies modulo center along two elements

among .e(1) , .e(2) , and .e(1) e(2) have the form .σjk Z3 and .σjk Z3 , then the .σj and the .σj are different and k and .k are non-zero. The third monodromy modulo

center has the form .σj±k σj±k

Z3 (or the order of the two factors interchanged). .



|k | Lemma 11.6 gives the inequality .log+ (3[ |k| 2 ]) + log+ (3[ 2 ]) ≤ 2π λe(1) ,e(2) . Since

±k ±k k k

.L− (ϑ(σ j )) = log+ (3[ 2 ]) and .L− (ϑ(σj )) = log+ (3[ 2 ]), inequality (11.21) follows for the other two monodromies. The lemma is proved.  

The following lemma is more comprehensive than Lemma 9.10. Lemma 11.8 Let X be a connected finite open Riemann surface, and .F : X → C3 (C)S3 a smooth mapping. Suppose for a base point .q1 of X each element of .π1 (X, q1 ) can be represented by a curve with base point .q1 whose image under F avoids .H. Then all monodromies of F are powers of the same periodic braid of period 3. Proof By Lemma 9.10 the monodromy of F along each element of .π1 (X, q1 ) is the power of a periodic braid with period 3. There is a smooth homotopy .Fs , s ∈ [0, 1], of F , such that .F0 = F , each .Fs is different from F only on a small neighbourhood of .q1 , each .Ft avoids .H on this neighbourhood of .q1 , and .F1 (q1 ) is the set of vertices of an equilateral triangle with barycenter 0. Since F and .F1 are free homotopic, their monodromy homomorphisms are conjugate, and it is enough to prove the statement of the lemma for .F1 . For notational convenience we will keep the notation F for the new mapping and assume that .F (q1 ) is the set of vertices of an equilateral triangle with barycenter 0. Let .e ∈ π1 (X, q1 ) be an arbitrary element. The monodromy .F∗ (e) along the element e is a power of a periodic braid of period 3. Hence, .τ3 (F∗ (e)) is a cyclic permutation. Consider the braid b with base point .F (0) that corresponds to rotation i2π t 3 F (0), t ∈ by the angle . 2π 3 , i.e. it is represented by the geometric braid .t → e k [0, 1], that avoids .H. There exists an integer k such that .F∗ (e) b is a pure braid

11.4 (g, m)-Bundles over Riemann Surfaces

391

that is represented by a mapping that avoids .H. Hence, .F∗ (e) bk represents .Δ2l 3 for some integer l. We proved that for each .e ∈ π1 (X, q1 ) the monodromy .F∗ (e) is represented by rotation of .F (0) around the origin by the angle . 2πj 3 for some integer j . The Lemma is proved.   Let as before X be a finite open connected Riemann surface. The following proposition is the main ingredient of the proof of Theorem 11.3. As before .E ⊂ π1 (X, q0 ) denotes a standard system of generators of the fundamental group with base point .q0 ∈ X that is associated to a standard bouquet of circles for X. Proposition 11.6 Let .(X × P1 , pr1 , E, X) be an irreducible holomorphic special .(0, 4)-bundle over a finite open Riemann surface X, that is not isotopic to a locally holomorphically trivial bundle. Let .F (x), x ∈ X, be the set of finite distinguished points in the fiber over x. Assume that F is transverse to .H. Then there exists a complex affine mapping M and a point .q ∈ X such that .M ◦ F (q) is contained in .C3 (R)S3 , and for an arc .α in X with initial point .q0 and terminal point q and each element .ej ∈ Isα (E) the monodromy modulo center .(M ◦ F )∗ (ej )Z3 can be written as product of at most 6 elements .bj,k , k = 1, 2, 3, 4, 5, 6, of .B3 Z3 with L− (ϑ(bj,k )) ≤ 2π λ10 (X).

.

(11.22)

If X is a torus with a hole the monodromy along each .ej is the product of at most 4 elements with .L− (ϑ(bj,k )) ≤ 2π λ3 (X), and in case of a planar domain the monodromy along each .ej is the product of at most 6 elements with .L− (ϑ(bj,k )) ≤ 2π λ8 (X). If X is the sphere with .m = 3 holes, the monodromy along each .ej is the product of at most 6 elements with .L− (ϑ(bj,k )) ≤ 2π λ5 (X). Proof of Proposition 11.6 Since the bundle is not isotopic to a locally holomorphically trivial bundle, it is not possible that all monodromies are powers of the same periodic braid, and by Lemma 11.8 the set def

L = {z ∈ X : F (z) ∈ H}

.

(11.23)

is not empty. 1. A Torus with a Hole Let X be a torus with a hole and let .E = {e0 , e0

} be a set of generators of .π1 (X, q0 ) that is associated to a standard bouquet of circles for X. There exists a connected component .L0 of L which is not contractible and not contractible to the hole. Indeed, otherwise there would be a base point .q1 and a curve .αq1 that joins .q0 with .q1 , such that for both elements of .Isαq (E) there would be 1 representing loops with base point .q1 which do not meet L, and hence, by Lemma 11.8 the monodromies along both elements would be powers of a single periodic braid of period 3. Hence, as in the proof of Proposition 11.4 there exists a component .L0 of L, which is a simple smooth relatively closed curve in X, such that the free homotopy class of one of the elements of .E, say of .e0 , has positive intersection

392

11 Finiteness Theorems

number with .L0 after orienting .L0 suitably. Put .e 0 = e0 . Moreover, the intersection number with .L0 is positive for the free homotopy class of one of the elements .e0

±1 



or .e0 e0

. Denote this element by .e

0 . (Since .e0 e0

= e0

e0 we may also put .e

0 = e0

e0 if the free homotopy class of .e0 e0

intersects .L0 .) Put .E2 = {e 0 , e

0 }. The free homotopy class of each element of .E2 and of the product of its two elements intersects .L0 . We proved the following claim which we formulate for later use. Claim 11.1 Let X be a torus with a hole and .E = {e0 , e0

} a set of standard generators of .π1 (X, q0 ). There exists a component .L0 of L, which is a simple smooth relatively closed curve in X, and two primitive elements .e 0 and .e

0 in .E such that def

the free homotopy class of each element of .E2

= {e 0 , e

0 } and of the product of its two elements intersects .L0 . Moreover, one of the .e 0 and .e

0 is an element of .E, the other is in .E ∪ E −1 or is the product of two elements of .E. Each element of .E is the product of at most two elements of .E2 ∪ E2 −1 . The proof of the proposition in the case of a torus with a hole is finished as follows. Move the base point .q0 to a point .q ∈ L0 along a curve .α, and consider the respective generators .e = Isα (e 0 ) and .e

= Isα (e

0 ) of the fundamental group .π1 (X, q) with base point q. Since .F (L0 ) ⊂ H there is a complex affine mapping M such that .M ◦ F (q) ∈ C3 (R)S3 . Since F is irreducible, the monodromy maps modulo center .(M ◦ F )∗ (e )Z3 and .(M ◦ F )∗ (e

)Z3 are not powers of a single standard generator .σj Z3 of .B3 Z3 (see Lemma 6.4, or Lemma 7 of [47]). Hence, the second option of Lemma 11.7 occurs. We obtain that each of the .(M ◦ F )∗ (e )Z3 and .(M ◦ F )∗ (e

)Z3 is a product of at most two elements .bj of .B3 S3 with



.L− (ϑ(bj )) ≤ 2π λ3 (X). Hence, .(M ◦ F )∗ (e )Z3 and .(M ◦ F )∗ (e )Z3 are products of at most 4 elements of .B3 Z3 with this property. The proposition is proved for tori with a hole. 2. A Planar Domain Let X be a planar domain. Maybe, after applying a Möbius transformation, we represent X as the Riemann sphere with holes .Cj , .j = 1, . . . , m + 1, such that .Cm+1 contains .∞. Recall, that we have chosen a standard system .E of generators .ej,0 , j = 1, . . . , m, of the fundamental group .π1 (X, q0 ) with base point .q0 that is associated to a standard bouquet of circle for X. Hence, .ej,0 is represented by a loop with base point .q0 that surrounds .Cj counterclockwise and does not surround any other hole. We claim that there is a connected component .L0 of L of one of the following kinds. Either .L0 has limit points on the boundary of two different holes (one of them may contain .∞) (first kind), or a component .L0 has limit points on a single hole .Cj , j ≤ m+1, and .Cj ∪L0 divides the plane .C into two connected components each of which contains a hole (maybe, only the hole containing .∞) (second kind), or there is a compact component .L0 that divides .C into two connected components each of which contains at least two holes (one of them may contain .∞). Indeed, suppose each non-compact component of L has boundary points on the boundary of a single hole and the union of the component with the hole does not separate the remaining holes of X, and for each compact component of L one of the connected

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393

components of its complement in X contains at most one hole. Then there exists a base point .q1 , a curve .αq1 in X with initial point .q0 and terminal point .q1 , and a representative of each element of .Isαq1 (E) ⊂ π1 (X, q1 ) that avoids L. Lemma 11.8 implies that all monodromies modulo center are powers of a single periodic element of .B3 Z3 which is a contradiction. If there is a component .L0 of the first kind we choose the same set of primitive elements .E2 ⊂ E2 ⊂ π1 (X, q0 ) as in the proof of Proposition 11.4 in the planar case. The free homotopy class of each element of .E2 and of the product of two different elements of .E2 intersects .L0 . Moreover, each element of .E is the product of at most two elements of .E2 . Suppose there is no component of the first kind but a component .L0 of the second kind. Assume first that all limit points of .L0 are on the boundary of a hole .Cj that 2 e }. Each element of .E is a does not contain .∞. Put .E3 = {ej,0 } ∪1≤k≤m, k=j {ej,0 k,0 3 primitive element and is the product of at most three generators contained in the set

−1 .E. Further, each element of .E is the product of at most three elements of .E ∪ E 3 3 .

The free homotopy class of each element of .E3 and of each product of two different elements of .E3 intersects .L0 . Indeed, any curve that is contained in   .C \ Cj ∪ L0 has either winding number zero around .Cj (as a curve in the complex plane .C), or its winding number around .Cj coincides with the winding   number around each of the holes in the bounded connected component of .C \ Cj ∪ L0 . On the other hand the representatives of the free homotopy class of .ej,0 have winding number 1 around .Cj and winding number 0 around each other hole that 2 e , does not contain .∞. The representatives of the free homotopy class of .ej,0 k,0 .k ≤ m, k = j , have winding number 2 around .Cj , winding number 1 around .Ck , and winding number zero around each other hole .Cl , l ≤ m. The argument for products of two elements of .E3 is the same. Assume that the limit points of .L0 are on the boundary of the hole .C∞ that contains .∞. Let .Cj0 and .Ck0 be holes that are contained in different components of .C \ (L0 ∪ C∞ ), and let .ej0 ,0 and .ek0 ,0 be the elements of .E whose representatives surround .Cj0 , and .Ck0 respectively. Denote by .E3 the set that consists of the elements .ej0 ,0 ek0 ,0 , . ej20 ,0 ek0 ,0 , and all elements .ej0 ,0 ek0 ,0 e˜0 with .e˜0 running over

.E \ {ej0 ,0 , ek0 ,0 }. Each element of .E is the product of at most 3 elements of .E, and 3 each element of .E is the product of at most 3 elements of .E3 ∪ (E3 )−1 . Each element of .E3 and each product of at most two different elements of .E3

intersects .L0 . Indeed, if a closed curve is contained in one of the components of .C \ (L0 ∪ C∞ ) then its winding number around each hole contained in the other component is zero. But for all mentioned elements there is a hole in each component of .C \ (L0 ∪ C∞ ) such that the winding number of the free homotopy class of the element around the hole does not vanish. Notice that in case of .m + 1 = 3 holes only these two possibilities for the curve .L0 may occur. We proved the following claim which we formulate for later use. Claim 11.2 Suppose X equals .P1 with .m + 1 = 3 holes and .E is a standard system of generators of .π1 (X, q0 ). Then there is a set .E3 = {e 0 , e

0 } ⊂ π1 (X, q0 ), such that

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11 Finiteness Theorems

one of the elements of .E3 is the product of at most two elements of .E ∪ E −1 , and the free homotopy classes of both elements and of their product intersect .L0 . If in addition .L0 has limit points on a hole .Cj with .j ≤ m, then one of the elements of .E3 can be taken to be an element of .E, not merely a product of two such elements. Moreover, e and .e are products of at most three factors, each an element of

−1

.E ∪ E 3 . 3 For the general case of planar domains we also need the following considerations. Suppose there are no components of L of the first or the second kind, but there is a connected component .L0 of L of the third kind. Let .Cj0 be a hole contained in the bounded component of the complement of .L0 in .C, and let .Ck0 , k0 ≤ m, be a hole that is contained in the unbounded component of .C \ L0 . Let .ej0 ,0 and .ek0 ,0 be the elements of .E whose representatives surround .Cj0 , and .Ck0 respectively. Consider the set .E4 consisting of the following elements: .ej0 ,0 ek0 ,0 , .ej20 ,0 ek0 ,0 , and .ej20 ,0 ek0 ,0 e˜0 for each .e˜0 ∈ E different from .ej0 ,0 and .ek0 ,0 . Each element of .E4 is the product of at most 4 elements of .E and each element of .E is the product of at most 3 elements of .E4 ∪ (E4 )−1 . The product of two different elements of .E4 is contained in .E8 . The free homotopy classes of each element of .E4 and of each product of two different elements of .E4 intersect .L0 . Indeed, if a loop is contained in the bounded connected component of .C \ L0 , its winding number around the holes .Cj , j ≤ m, contained in the unbounded component is zero. If a loop is contained in the unbounded connected component of .C \ L0 , its winding numbers around all holes contained in the bounded connected component are equal. But the winding numbers of .ej0 ,0 ek0 ,0 and .ej20 ,0 ek0 ,0 around the hole .Cj0 are positive and the winding numbers around the other holes that are contained in the bounded connected component of .C\L0 vanish, hence the representatives of these two elements cannot be contained in the unbounded component of .C \ L0 . Since the winding numbers of representatives of these elements around .Ck0 are positive, the representatives cannot be contained in the bounded component of .C \ L0 . For representatives of the elements .ej20 ,0 ek0 ,0 e˜0 the winding numbers around .Cj0 equal 2, the winding numbers around any other hole in the bounded component of .C \ L0 are at most 1, and the winding numbers around .Ck0 equal 1. Hence, the free homotopy classes of the mentioned elements must intersect both components of .C \ L0 , hence they intersect .L0 . Representatives of any product of two different elements of .E4 have winding numbers around .Cj0 at least 3, the winding numbers around any other hole in the bounded component of .C \ L0 are at most 1, and the winding numbers around .Ck0 equal 2. Hence, the free homotopy classes of these elements intersect .L0 . We obtained the following Claim 11.3 If X is a planar domain and .E is a standard system of generators of π1 (X, q0 ), then there is a set .E4 ⊂ π1 (X, q0 ), such that each of the elements of .E4 is the product of at most 4 elements of .E ∪ E −1 , and the free homotopy classes of the product of at most two elements of .E4 intersect .L0 . Moreover, e and .e are products of at most three factors, each an element of .E4 ∪ E4 −1 .

.

11.4 (g, m)-Bundles over Riemann Surfaces

395

The proof of the proposition in the planar case is finished as follows. Let .αq be a curve in X with initial point .q0 and terminal point q, and M a complex affine mapping, such that .(M ◦ F )(q) ∈ C3 (R)S3 . Since .M ◦ F is irreducible, the monodromies modulo center of .M ◦ F along the elements of .Isα (E4 ) are not (trivial or non-trivial) powers of a single element .σj Z3 . Hence, for each element of



.Isα (E ) there exists another element of .Isα (E ) so that the second option of Lemma 4 4

11.7 holds for this pair of elements of .Isα (E4 ). Therefore, the monodromy modulo center of .M ◦F along each element of .Isα (E4 ) is the product of at most two elements .bj ∈ B3 Z3 of .L− not exceeding .2π λ8 (X), and the monodromy modulo center of .M ◦ F along each element .Isα (E) is the product of at most 6 elements .bj ∈ B3 Z3 with .L− (ϑ(bj )) not exceeding .2π λ8 (X). Proposition 11.6 is proved in the planar case. 3. The General Case Since not all monodromies are powers of a single element of B3 Z3 that is either periodic or reducible, there exists a pair of generators .e0 , .e0

in .E, such that the monodromies along them are not powers of a single periodic or



reducible element. Consider the projection .ωe0 ,e0  : X˜ → X(e0 , e0

). By the proof for tori with a hole or for .P1 with three holes there exist a relatively closed curve



.Le ,e

 in .X(e , e ) and a Möbius transformation M, such that for .F = M ◦ f 0 0 0 0



the mapping .Fe0 ,e0  = F ◦ ωe0 ,e0

 takes .Le0 ,e0

 into .H, and takes a chosen point .qe ,e

 ∈ Le ,e

 to a point in .C3 (R)S3 . 0 0 0 0 ˜ for which .ωe0 ,e0

 (q) Choose a point .q˜ ∈ X, ˜ = qe ,e

 . Let .α˜ be a curve in .X˜ .

0

0

def





with initial point .q˜0 and terminal point .q. ˜ Then .αe0 ,e0

 = ωe0 ,e0  (α) ˜ is a curve









in .X(e0 , e0 ) with initial point .(q0 )e0 ,e0  and terminal point .qe0 ,e0  , and the curve



.αe ,e

 in .X(e , e ) and the point .q ˜ in the universal covering .X˜ of .X(e0 , e0

) are 0 0 0 0 compatible. Put .α = ωe0 ,e0

 (αe0 ,e0

 ), and for each .e0 ∈ π1 (X, q0 ) we denote as before the element .Isα (e0 ) by e. def

Put .E ∗ = {e0 , e0

}. For an element .e0 ∈ E ∗ we define .(e 0 )e0 ,e0

 and .(e

0 )e0 ,e0

 as in the end of paragraph 3.1 of the proof of Proposition 11.5. The Riemann surface



1 .X(e , e ) is a torus with a hole or .P with three holes. This implies the following 0 0 fact. There are elements .e 0 and .e

0 , one of them contained in .E ∗ or equal to the product of at most two factors among the .e0 and .e0

, the second either contained in .E ∗ ∪ (E ∗ )−1 , or equal to the product of at most three factors among the .e0 and .e0

, such that the free homotopy classes of .(e 0 )e0 ,e0

 , of .(e

0 )e0 ,e0

 , and of their product intersect .Le0 ,e0

 . Moreover, .e0 and .e0

are products of at most three factors, each being either .(e 0 )±1 or .(e

0 )±1 . Put .e e ,e

 = Isαe ,e

 ((e 0 )e0 ,e0

 ), .e

e ,e

 = Isαe ,e

 ((e

0 )e0 ,e0

 ). 0

0

0 0

0

0

0 0

Since the monodromies along .e and .e

are not powers of a single periodic or reducible element, by Lemma 11.7 each monodromy .(Fe0 ,e0

 )∗ (e e ,e

 ) = F∗ (e ) 0

0

and .(Fe0 ,e0

 )∗ (e

e ,e

 ) = F∗ (e

) is the product of at most two elements .bj ∈ 0

0

B3 Z3 with .L− (ϑ(bj )) ≤ 2π λ5 (X). Since .e and .e

are products of at most three

396

11 Finiteness Theorems

elements among .(e )±1 and .(e

)±1 , each of the monodromies .F∗ (e ) and .F∗ (e

) is the product of at most 6 elements .bj ∈ B3 Z3 with .L− (ϑ(bj )) ≤ 2π λ5 (X). Take any element .e0 ∈ E \ {e0 , e0

}. Let .e = Isα (e0 ). Either the pair of monodromies (.F∗ (e ), .F∗ (e)) or the pair of monodromies (.F∗ (e

), .F∗ (e)) does not consist of two powers of the same element of .B3 Z3 that is either periodic or reducible. Suppose this is so for the pair (.F∗ (e ), .F∗ (e)). e ,e



Let .Le 0  be the connected component of .(ωe 0 0 )−1 (Le 0 ,e

0  ) that contains 0  e  .ω 0 (q) ˜ . By the analog of Lemma 11.3, applied to the holomorphic projection q˜0 −1 ˜ .X(Is ) (e 0 ) → X(e 0 , e

0 ) , the free homotopy class .(e 0 )e 0  intersects

.Le  . (For the definition of .(e )e  see the end of the paragraph 3.1 of the 0 0 0 proof of Proposition 11.5.) As in the proof of Proposition 11.5 we consider 

e0 ,e 

the Riemann surface .X(e0 , e 0 ) and the curve .Le0 ,e 0  = ωe  0 (Le 0  ) (see 0 paragraph 3.3. of the proof of Proposition 11.5). As there we see that the free   homotopy class .(e 0 )e0 ,e 0  intersects .Le0 ,e 0  . The system . (e0 )e0 ,e 0  , (e 0 )e0 ,e 0  is associated to a standard bouquet of circles for .X(e0 , e 0 ) (though the system

.(e0 , e ) may not be part of a system of generators that is associated to a standard 0 bouquet of circles for X). This can be seen in the same way as in the proof of Proposition 11.5. We will apply now the arguments, used for .X(e0 , e0

) and the generators .(e0 )e0 ,e0

 , (e0

)e0 ,e0

 of the fundamental group .π1 (X(e0 , e0

), qe0 ,e0

 ), to .X(e0 , e 0 ) and the generators .(e0 )e0 ,e 0  , (e 0 )e0 ,e 0  of the fundamental group

.π1 (X(e0 , e ), qe ,e  ). 0 0 0 In the case when .X(e0 , e 0 ) is a torus with a hole, the intersection number of 



(e 0 )e0 ,e 0  with .Le0 ,e 0  is non-zero. Put .e 0 = e 0 . For one of the choices .e0±1 , or







.e e0 , denoted by .e , the free homotopy classes of .(e )e ,e  , .(e )e ,e  , and of their 0 0 0 0 0 0 0 0 product intersect .Le0 ,e 0  . Since by Claim 11.1 and Claim 11.2 the element .e 0 is the product of at most three elements of .E, the element .e

0 is the product of at most .1 + 3 = 4 elements of .E. Moreover, .e0 is the product of at most two factors, each being .(e 0 )±1 , or .(e

0 )±1 . Hence, .e0 is the product of at most 7 elements of .E. In case .X(e0 , e 0 ) is planar, the curve .Le0 ,e 0  must have limit points on the hole that corresponds to the generator .(e 0 )e0 ,e 0  of the fundamental group







.π1 (X(e0 , e ), qe ,e  ). By Claim 11.2 we find elements .e and .e such that .e = e 0 0 0 0 0 0 0 .

and .e

0 is either equal to .e0−1 , or to the product of at most three factors, one being equal to .e0 and the others equal to .e 0 , and the free homotopy classes of .(e 0 )e0 ,e 0  ,

.(e )e ,e  , and their product intersect .Le ,e  . Moreover, .e0 is the product of at most 0 0 0 0 0 3 factors, each being equal to .(e

0 )±1 or .(e 0 )±1 . Since .e 0 is the product of at most three elements of .E, .e

0 is the product of at most .1 + 2 · 3 = 7 elements of .E. Hence, the elements .e 0 , .e

0 , and .e 0 e

0 are products of at most 10 elements of .E ∪ E −1 . Since for .e = Isα (e0 ), and .e = Isα (e 0 ) the monodromies .F∗ (e) and .F∗ (e 0 ) are not powers of a single periodic or reducible element, for .e = Isα (e 0 ) and .e

= Isα (e

0 ) the monodromies .F∗ (e ) and .F∗ (e

) cannot be powers of a single element

11.4 (g, m)-Bundles over Riemann Surfaces

397

that is either periodic or reducible. Lemma 11.7 implies, that .F∗ (e ) and .F∗ (e

) are products of at most two factors .b with .L− (ϑ(b)) not exceeding .2π λ10 (X). Hence, .F∗ (e) is the product of at most 6 factors .b with .L− (ϑ(b)) not exceeding .2π λ10 (X). We obtain the statement of Proposition 11.6 in the general case. Proposition 11.6 is proved.   Proof of Theorem 11.3 Let X be a connected Riemann surface of genus .g with m + 1 ≥ 1 holes. Since each holomorphic .(0, 3)-bundle with a holomorphic section on X is isotopic to a holomorphic special .(0, 4)-bundle, we need to estimate the number of isotopy classes of irreducible smooth special .(0, 4)-bundles on X, that contain a holomorphic bundle. By Lemma 6.4 the monodromies of an irreducible bundle are not powers of a single element of .B3 Z3 which is conjugate to a .σj Z3 , but they may be powers of a single periodic element of .B3 Z3 (equivalently, the isotopy class may contain a locally holomorphically trivial holomorphic bundle). Consider an irreducible special holomorphic .(0, 4)-bundle on X which is not isotopic to a locally holomorphically trivial bundle. Let .F (x), x ∈ X, be the set of finite distinguished points in the fiber over x. By the Holomorphic Transversality Theorem [51] the mapping .F : X → C3 (C)S3 can be approximated on relatively compact subsets of X by holomorphic mappings that are transverse to .H. Similarly as in the proof of Theorem 11.1 we will therefore assume in the following (after slightly shrinking X to a deformation retract of X and approximating F ) that F is transverse to .H. By Proposition 11.6 there exists a complex affine mapping M and a point .q ∈ X such that .M ◦ F (q) is contained in .C3 (R)S3 , and for an arc .α in X with initial point .q0 and terminal point q and each element .ej ∈ Isα (E) the monodromy .(M ◦ F )∗ (ej )Z3 of the bundle can be written as product of at most 6 elements .bj,k , k = 1, 2, 3, 4, 5, 6, of .B3 Z3 with .

L− (ϑ(bj,k )) ≤ 2π λ10 (X).

.

(11.24)

The mappings F and .M ◦ F from X into the symmetrized configuration space are free homotopic. Consider an isotopy class of special .(0, 4)-bundles that corresponds to a conjugacy class of homomorphisms .π1 (X, q0 ) → B3 Z3 whose image is generated by a single periodic element of .B3 Z3 . Up to conjugacy we may assume that this element is one of the following: .Δ3 Z3 , (σ1 σ2 )Z3 , (σ1 σ2 )−1 Z3 , or .Id. For each of these elements .b the equality .L− (ϑ(b)) = 0 holds. Hence, in this case the isotopy class contains a smooth mapping .F˜ such that for each .ej,0 ∈ E the monodromy .(M ◦ F )∗ (ej,0 )Z3 of the bundle can be written as product of at most 6 elements .bj,k , k = 1, 2, 3, 4, 5, 6, of .B3 Z3 satisfying inequality (11.24). The same argument as in the proof of Theorem 11.1 shows the following fact. Each irreducible free homotopy class of mappings .X → C3 (C)S3 that contains a holomorphic mapping contains a smooth mapping .F˜ such that for each .ej,0 ∈ E

398

11 Finiteness Theorems

the monodromy .F˜∗ (ej,0 )Z3 of the bundle can be written as product of at most 6 elements .bj,k , k = 1, 2, 3, 4, 5, 6, of .B3 Z3 satisfying inequality (11.24). Using Lemma 10.1 (see also Lemma 1 of [45]) the number of elements .b ∈ B3 Z3 (including the identity), for which .L− (ϑ(b)) ≤ 2π λ10 (X), is estimated as def

follows. The element .w = ϑ(b) ∈ PB3 Z3 can be considered as a reduced word in the free group generated by .a1 = σ12 Z3 and .a2 = σ22 Z3 . By Lemma 10.1 there are no more than . 12 exp(6π λ10 (X)) + 1 ≤ 32 exp(6π λ10 (X)) reduced words .w in .a1 and .a2 (including the identity) satisfying the inequality .L− (w) ≤ 2π λ10 (X). For a given element .w ∈ PB3 Z3 (including the identity) we describe now all elements .b of .B3 Z3 with .ϑ(b) = w. If .w = Id these are the following elements. If the first term of .w equals .ajk with .k = 0, then the possibilities are .b = w · (Δ 3 Z3 )

with . = 0 or 1, .b = (σj Z3 )·w·(Δ 3 Z3 ) with . = 0 or 1, or .b = (σj±1

Z3 )· w · (Δ 3 Z3 ) with . = 0 or 1 and .σj = σj . Hence, for .w = Id there are 8 possible choices of elements .b ∈ B3 Z3 with .ϑ(b) = w. If .b = Id then the choices are .Δ Z3 and .(σj±1 Δ )Z3 for .j = 1, 2, and . = 0 or . = 1. These are 10 choices. Hence, there are no more than .15 exp(6π λ10 (X)) different elements .b ∈ B3 Z3 with .L− (ϑ(b)) ≤ 2π λ10 (X). Each monodromy is the product of at most six elements .bj of .B3 Z3 with .L− (ϑ(bj )) ≤ 2π λ10 (X). Hence, for each monodromy there are no more than 6 .(15 exp(6π λ10 (X))) possible choices. We proved that there are up to isotopy no more than .(15 exp(6π λ10 (X)))6(2g +m) irreducible holomorphic .(0, 3)-bundles with a holomorphic section over X. Theorem 11.3 is proved.   sgnk

Notice that we proved a slightly stronger statement, namely, over a Riemann surface of genus g with .m + 1 ≥ 1 holes there are no more than 6(2g +m) isotopy classes of smooth .(0, 3)-bundles with a smooth .(15 exp(6π λ10 (X))) section that contain a holomorphic bundle with a holomorphic section that is either irreducible or isotopic to the trivial bundle. Proof of Theorem 11.2 Proposition 8.2 and Theorem 11.3 imply Theorem 11.2 as follows. Suppose an isotopy class of smooth .(1, 1)-bundles over a finite open Riemann surface X contains a holomorphic bundle. By Proposition 8.2 the class contains a holomorphic bundle which is the double branched covering of a holomorphic special .(0, 4)-bundle. If the .(1, 1)-bundle is irreducible then also the .(0, 4)-bundle  6(2g +m) is irreducible. There are up to isotopy no more than . 15(exp(6π λ10 (X))) holomorphic special .(0, 4)-bundles over X that are either irreducible or isotopic to the trivial bundle. Theorem 8.1 and Theorem 11.3 imply, that there are no more than   6(2g +m) . 15 exp(6π λ10 (X))) conjugacy classes of monodromy homomorphisms that correspond to a special holomorphic .(0, 4)-bundle over X that is either irreducible or isotopic to the trivial bundle. Each monodromy homomorphism of the holomorphic double branched covering is a lift of the respective monodromy homomorphism of the holomorphic special .(0, 4)-bundle. Different lifts of a monodromy mapping class of a special .(0, 4)-bundle differ by involution, and

11.4 (g, m)-Bundles over Riemann Surfaces

399

the fundamental group of X has .2g + m generators. Using Theorem 8.1 for .(1, 1) 6(2g +m) bundles, we see that there are no more than .22g +m 15(exp(6π λ10 (X))) = 2g +m  6 isotopy classes of .(1, 1)-bundles that contain 2 · 15 · exp(36π λ10 (X)) a holomorphic bundle that is either irreducible or isotopic to the trivial bundle. Theorem 11.2 is proved.   For convenience of the reader we give the short reduction of the Corollaries 11.1 and 11.2 to Theorems 11.1 and 11.2, respectively. The arguments of the reduction are known in principle, but the case considered here is especially simple. Proof of Corollary 11.1 We will prove that on a punctured Riemann surface there are no non-constant reducible holomorphic mappings to the twice punctured complex plane and that any non-trivial homotopy class of mappings from a punctured Riemann surface to the twice punctured complex plane contains at most one holomorphic mapping. This implies the corollary. Recall that a holomorphic mapping f from any punctured Riemann surface X to the twice punctured complex plane extends by Picard’s Theorem to a meromorphic function .f c on the closed Riemann surface .Xc . Suppose now that X is a punctured Riemann surface and that the mapping .f : X → C \ {−1, 1} is reducible, i.e. it is homotopic to a mapping into a punctured disc contained in .C \ {−1, 1}. Perhaps after composing f with a Möbius transformation we may suppose that this puncture equals .−1. Then the meromorphic extension .f c omits the value 1. Indeed, if .f c was equal to 1 at some puncture of X, then f would map the boundary of a sufficiently small disc on .Xc that contains the puncture to a loop in .C \ {−1, 1} with nonzero winding number around 1, which contradicts the fact that f is homotopic to a mapping into a disc punctured at .−1 and contained in .C \ {−1, 1}. Hence, .f c is a meromorphic function on a compact Riemann surface that omits a value, and, hence f is constant. Hence, on a punctured Riemann surface there are no non-constant reducible holomorphic mappings to the twice punctured complex plane. Suppose .f1 and .f2 are non-constant homotopic holomorphic mappings from the punctured Riemann surface X to .C\{−1, 1}. Then for their meromorphic extensions c c c c .f 1 and .f2 the functions .f1 − 1 and .f2 − 1 have the same divisor on the closed c Riemann surface .X . Indeed, suppose, for instance, that .f1c − 1 has a zero of order c .k > 0 at a puncture p. Then for the boundary .γ of a small disc in .X around p the curve .(f1 − 1) ◦ γ in .C \ {−2, 0} has index k with respect to the origin. Since .f2 − 1 is homotopic to .f1 − 1 as mapping to .C \ {−2, 0}, the curve .(f2 − 1) ◦ γ is free homotopic to .(f1 −1)◦γ . Hence, .f2 −1 has a zero of order k at p. Applying the same arguments with 0 replaced by .∞, we obtain that .f1c − 1 and .f2c − 1 have the same divisor. Hence, .f1c − 1 and .f2c − 1 differ by a non-zero multiplicative constant. Since the functions are non-constant they must take the value .−2. By the same reasoning as above the functions are equal to .−2 simultaneously. Hence, the multiplicative constant is equal to 1. We proved that non-constant homotopic holomorphic maps from punctured Riemann surfaces to .C \ {−1, 1} are equal.   Proof of Corollary 11.2 We need the following fact. For each special .(0, 4)-bundle F = (X × P1 , pr1 , E, X) there is a finite unramified covering .Pˆ : Xˆ → X of

.

400

11 Finiteness Theorems

ˆ X), for which the X, such that .F lifts to a special .(0, 4)-bundle .(Xˆ × P1 , pr1 , E, ˆ is the union of four disjoint complex curves .E ˆ k , k = 1, 2, 3, 4, complex curve .E 1 each intersecting each fiber .{x} ˆ × P along a single point .(x, ˆ gˆ k (x)). ˆ This can be seen as follows. Let .q0 be the base point of X. The monodromy mapping class along ˆ ∪ ({q0 } × P1 ) each element e of .π1 (X, q0 ) takes the set of distinguished points .E onto itself, permuting them by a permutation .σ (e). Consider the set .N of elements .e ∈ π1 (X, q0 ) for which .σ (e) is the identity. The set N is a normal subgroup of .π1 (X, q0 ). Its index is finite, since two left cosets .e1 N and .e2 N are equal if .σ (e2−1 e1 ) = σ (e2 )−1 σ (e1 ) = Id, and there are only finitely many distinct q˜0 ˜ ˆ ∪ ({q0 } × P1 ). The quotient .Xˆ def = XIs (N ) of permutations of points of .E the universal covering of X by the group of covering transformations corresponding to N and the canonical projection .Xˆ → X define the required covering. To prove the corollary, we have to show first, that any reducible holomorphic .(1, 1)-bundle over a punctured Riemann surface X is locally holomorphically trivial and secondly, that two isotopic (equivalently, smoothly isomorphic) locally holomorphically non-trivial holomorphic .(1, 1)-bundles over X are holomorphically isomorphic. The second fact is obtained as follows. Suppose the locally holomorphically non-trivial holomorphic .(1, 1)-bundles .Fj , j = 1, 2, have conjugate monodromy homomorphisms. By Proposition 8.2 each bundle .Fj , j = 1, 2, is holomorphically isomorphic to a double branched covering of a special holomorphic .(0, 4)-bundle def

(X × P1 , pr1 , E j , X) = Pr(Fj ). The bundles .Pr(Fj ) are isotopic, since they have conjugate monodromy homomorphisms. There is a finite unramified covering .Pˆ : ˆ j , X) Xˆ → X of X, such that the bundles .Pr(Fj ) have isotopic lifts .(Xˆ × P1 , pr1 , E ˆ and for each j the complex curve .E ˆ j is the union of four disjoint complex to .X, k ˆ l , k = 1, 2, 3, 4, each intersecting each fiber .{x} ˆ × P1 along a single curves .E k point .(x, ˆ gˆj (x)). ˆ We may choose the label of the points so that the isotopy moves

.

ˆ k1 to .E ˆ k2 , .k = 1, 2, 3, 4. The lifted bundles are not isotopic the complex curve .E ˆ ∈ P1 are holomorphic. We to the trivial bundle. The mappings .Xˆ  xˆ → gˆjk (x) ˆ = ∞ for each .x. ˆ Define for .j = 1, 2, a holomorphic may assume that .gˆj4 (x) 1 ˆ j , X) by isomorphism of the bundle .(Xˆ × P , pr1 , E

{x} ˆ × P1  (x, ˆ ζ ) → x, ˆ

.

ˆ −ζ gˆj1 (x) gˆj1 (x) ˆ − gˆj2 (x) ˆ

 .

ˆ j of .E ˆ j under the j -th isomorphism intersects the fiber over The image .E each .xˆ ∈ Xˆ along the four points .(x, ˆ 0), (x, ˆ 1), (x, ˆ ∞), and .(x,˚ ˆ gj (x)) ˆ for a 1 ˆ .gj : X → P that avoids .0, 1 and .∞. The mappings . ˚ gj , holomorphic mapping ˚ .j = 1, 2, are homotopic, since the bundles are isotopic. They are not homotopic to a constant mapping since the bundles are not isotopic to the trivial bundle. By g1 and ˚ .g2 coincide. Hence, the the proof of Corollary 11.1 the mappings . ˚

11.4 (g, m)-Bundles over Riemann Surfaces

401

ˆ j , X) are holomorphically isomorphic to the bundle bundles . (Xˆ × P1 , pr1 , E  

1 ˆ ˆ ˆ ∩ {x} .F = (X × P , pr1 , E, X) with . E ˆ × (0, 1,˚ g(x), ˆ ∞) for a ˆ × P1 = {x} .g(x) ˆ : Xˆ → C \ {0, 1}. holomorphic mapping ˚ For each .j = 1, 2, we obtain a bundle that is holomorphically isomorphic to

.Pr(Fj ) by considering the bundle .F and gluing for each .x ∈ X the fibers over ˆ −1 (x) together using complex affine mappings determined by .Pr(Fj ). .x ˆ ∈ (P) The complex affine mappings depend holomorphically on .x. ˆ Each of them takes the unordered triple of finite distinguished points in one fiber to the unordered triple of finite distinguished points in the other fiber over .x. ˆ Assign to each finite k ˆ ˆ distinguished point the number k of the curve .E in .E to which the point belongs. Each of the complex affine mappings is uniquely determined by the permutation of these numbers k, induced by it. Since the bundles .Pr(Fj ) are smoothly isomorphic, the respective permutations for the two bundles coincide. Hence, the bundles .Pr(Fj ) are holomorphically isomorphic to a single bundle that is obtained from .F by a ˆ −1 (x) for .x ∈ X. holomorphic gluing procedure of the fibers over points .xˆ ∈ (P) The bundle .Fj , .j = 1, 2, is a double branched covering of .Pr(Fj ) and the .Pr(Fj ) are holomorphically isomorphic. Moreover, the monodromy homomorphisms of the bundles .Fj are conjugate. Hence, the bundles .Fj , .j = 1, 2, are holomorphically isomorphic. The second fact is proved. The first fact is obtained as follows. After a holomorphic isomorphism we may assume that the reducible holomorphic .(1, 1)-bundle is a double branched covering ˚ ∪ s ∞ , X). After of a reducible special .(0, 4)-bundle .Pr(F) = (X × P1 , pr1 , E a further isomorphism the bundle .Pr(F) lifts to a holomorphic bundle .Pr(F) = ˆ ∪ s ˆ intersects each fiber .{x} × P1 along a set of ∞ , X), ˚ ˚ ˆ such that .E (Xˆ × P1 , pr , E 

1

the form .{x} ˆ × {0, 1,˚ g(x)}. ˆ Since the bundle .F is reducible, the bundle .Pr(F) and also .Pr(F) are reducible. Hence, the mapping ˚ .g is constant by the proof of Corollary 11.1. Hence, the bundle .Pr(F) is holomorphically trivial and, hence, .Pr(F) is locally holomorphically trivial. Since .F is a double branched covering of .Pr(F), the bundle .F is locally holomorphically trivial. The first fact is proved.   



Proof of Proposition 11.1 Denote by .S α a skeleton of .T α,σ ⊂ T α which is the union of two circles each of which lifts under the covering .P : C → T α to a straight line segment which is parallel to an axis in the complex plane. Denote the intersection point of the two circles by .q0 . Note that .S α is a standard bouquet of circles for .T α,σ with base point .q0 , and .P−1 (T α,σ ) is the . σ2 -neighbourhood of −1 (S α ). We may assume that .P−1 (q ) is the lattice .Z + iαZ. .P 0 Denote by e the generator of .π1 (T α,σ , q0 ), that lifts to a vertical line segment and .e the generator of .π1 (T α,σ , q0 ), that lifts to a horizontal line segment. Put

.E = {e, e }. We show first the inequality λ3 (T α,σ ) ≤

.

4(2α + 1) . σ

(11.25)

402

11 Finiteness Theorems

For this purpose we take any primitive element .e

of the fundamental group α,σ , q ) which is the product of at most three factors, each of the factors being .π1 (T 0 an element of .E or the inverse of an element of .E. We represent the element .e

by a piecewise .C 1 mapping .f1 from an interval .[0, l1 ] to the skeleton .S α . We may consider .f1 as a piecewise .C 1 mapping from the circle .R(x ∼ x + l1 ) to the skeleton, and assume that for all points .t of the circle where .f1 is not smooth,

.f1 (t ) = q0 . Let .t0 ∈ [0, l1 ] be a point for which .f1 (t0 ) = q0 . Let .f˜1 be a piecewise smooth mapping from .[t0 , t0 + l1 ] to the universal covering .C of .T α ⊂ T α,σ which projects to .f1 . We may take .f1 so that the equality .|f˜1 | = 1 holds for the derivative

.f˜ of .f˜1 . The mapping may be chosen so that .l1 ≤ 2α + 1. (Recall that .α ≥ 1 and 1 the element e is primitive.) Take any .t for which .f1 is not smooth. We may assume that .f1 is chosen so that the direction of .f˜1 changes by the angle .± π2 at each such point. Hence, there exists a neighbourhood .I (t ) of .t on .(t0 , t0 + 1 ), such that the restriction .f˜1 (t ) covers two sides of a square of side length . σ2 . Denote by .q˜0 the common vertex .f˜1 (t ) of these sides, and by .q˜0

the vertex of the square that is not a vertex of one of the two sides. Replace the union of the two sides of the square that contain .q˜0 by a quarter-circle 2

of radius . σ2 with center at the vertex .q˜0

, and parameterize the latter by .t → σ2 e±i σ t so that the absolute value of the derivative equals 1. Notice that the quarter-circle is shorter than the union of the two sides. Proceed in this way with all such points .t . After a reparameterization we obtain a .C 1 mapping .f˜ of the interval .[0, l] of length l not exceeding .2α + 1 whose image is contained in the union of .P−1 (S α ) with some quarter-circles, such that .|f˜ | = 1. The distance of each point of the image of .f˜ to the boundary of .P−1 (T α,σ ) is not smaller than . σ2 . The mapping .f˜ is piecewise of class .C 2 . The normalization condition .|f˜ | = 1 implies .|f˜

| ≤ σ2 . The projection .f = P ◦ f˜ can be considered as a mapping from the circle α,σ , that represents the free .R(x ∼ x + l) of length l not exceeding .2α + 1 to .T

homotopy class .e of the chosen element of the fundamental group. def Consider the mapping .x + iy → F˜ (x + iy) = f˜(x) + i f˜ (x)y ∈ C, where σ .x + iy runs along the rectangle .Rl = {x + iy ∈ C : x ∈ [0, l], |y| ≤ 4 }. The ∂ image of this mapping is contained in the closure of .P−1 (T α,σ ). Since .2 ∂z F˜ (x + iy) = 2f˜ (x) + i f˜

(x)y and .2 ∂∂z¯ F˜ (x + iy) = i f˜

(x)y, the Beltrami coefficient 

μF˜ (x + iy) =

.

K =

.

1+ 13 1− 13

∂ ˜ z¯ F (x+iy) ∂ ˜ z F (x+iy)

of .F˜ satisfies the inequality .|μF˜ (x + iy)| ≤ 13 . Hence, for

= 2 the mapping .F˜ descends to a K-quasiconformal mapping F from

the annulus .Al to .T α,σ of extremal length .λ(Al ) =

l

σ 2

≤ 2 (2α+1) that represents σ

the free homotopy class of the element .e

of the fundamental group .π1 (T α,σ , q0 ). Realize .Al as an annulus in the complex plane. Let .ϕ be the solution of the Beltrami equation on .C with Beltrami coefficient .μF˜ on .Al and zero else. Then the mapping −1 is a holomorphic mapping of the annulus .ϕ(A ) of extremal length not .g = F ◦ ϕ l

11.4 (g, m)-Bundles over Riemann Surfaces

403

exceeding .Kλ(Al ) ≤ 4(2α+1) into .T α,σ that represents the chosen element of the σ fundamental group .π1 (T α,σ , q0 ). Inequality (11.25) is proved. By Theorem 11.1 for tori with a hole there are up to homotopy no more 5 2 2α+1 α,σ 25 ·32 π 2α+1 σ < 7e2 ·3 π σ non-constant irreducible than .3( 32 e36π λ3 (T ) )2 ≤ 27 4 e holomorphic mappings from .T α,σ to the twice punctured complex plane. 1 We give now the proof of the lower bound. Fix .α ≥ 1 and let .δ ≤ 10 . We consider 5δ the annulus .Aα,5δ = {z ∈ C : |Rez| < 2 }(z ∼ z + αi) of extremal length equal α to . 5δ ≥ 2α. For any natural number j we consider all elements of .π1 (C \ {−1, 1}, 0) of the form a1±2 a2±2 . . . a1±2 a2±2

.

(11.26)

containing 2j terms, each of the form .aj±2 . The choice of the sign in the exponent of each term is arbitrary. There are .22j elements of this kind. α Put .j = [ 60π δ ], where .[x] is the largest integer not exceeding a positive number α x. Then . 5δ ≥ 12j π and Remark 9.1 implies that for each word w of the form ±2 ±2 .a a . . . a1±2 a2±2 with .N = 2j terms there exists a holomorphic mapping .gw :     1 α,5δ2 A , 0(z ∼ z + iα) → C \ {−1, 1}, 0 from .Aα,5δ with base point .0(z ∼ z + iα) to .C \ {−1, 1} with base point 0, that represents w. Moreover, the image of 1 .gw is contained in the domain .{z ∈ C : |z| < C, |z ± 1| > C } for a constant .C > 1. The plan is the following. We will embed .Aα,δ into .T α,δ , restrict .gw to a smaller annulus, and extend the pushforward of the restriction to a smooth mapping ˚ .gw from .T α,δ into .{z ∈ C : |z| < C, |z ± 1| > C1 } such that (with .P being the projection .P : C → T α ) the monodromy along the circle .P({Rez = 0}) with base point .P(0) is equal to (11.26), and the monodromy along .P({Imz = 0} with the same base point equals the identity. We will show the existence of a positive number .ε depending only on C, such that correcting the smooth mapping ˚ .gw on α,σ def ¯ .T = T α,εδ by a solution of a .∂-problem on .T α,σ with .L∞ -norm smaller than C, we obtain a holomorphic mapping .T α,σ → C \ {−1, 1} with monodromies w 2ε log 2 α and .Id. In this way for .σ = εδ we will obtain .22j ≥ 14 e 60π σ non-homotopic holomorphic mappings from .T α,σ to .C \ {−1, 1}. We will now give the detailed proof following the plan. Let .g˜ w be the lift of .gw to a mapping from the strip .{z ∈ C : |Rez| < 5δ 2 } to 1 3δ .{z ∈ C : |z| < C, |z ± 1| > C }. On the thinner strip .{|Rez| < 2 } the derivative of ˜ w satisfies the inequality .|˜g w | ≤ Cδ , since .|˜gw | ≤ C. .g Let .Fα = [− 12 , 12 ) × [− α2 , α2 ) ⊂ C be a fundamental domain for the projection α α,δ = F ∩ P−1 (T α,δ ). Let .χ : [0, 1] → R be a non.P : C → T . Put .Δ α 0 decreasing function of class .C 2 with .χ0 (0) = 0, χ0 (1) = 1, .χ0 (0) = χ0 (1) = 0

404

11 Finiteness Theorems

Fig. 11.3 A fundamental domain for a torus with a hole and the poles of the kernel for the .∂-equation

and .|χ0 (t)| ≤

3 2

+3δ (see also Sect. 9.5). Define .χδ : [ −3δ 2 , 2 ] → [0, 1] by

⎧ 1 3 ⎪ ⎪ ⎨ χ0 ( δ t + 2 ) .χδ (t) = 1 ⎪ ⎪ ⎩χ (− 1 t + 3 ) 0

δ

2

−δ t ∈ [ −3δ 2 , 2 ] +δ t ∈ [ −δ 2 , 2 ]

(11.27)

t ∈ [ 2δ , 3δ 2 ].

Notice that .χδ is a .C 2 -function that vanishes at the endpoints of the interval −3δ +3δ −3δ −δ .[ 2 , 2 ] together with its first derivative, is non-decreasing on .[ 2 , 2 ], and ˚ ˜ w (z) + (1 − χδ (Rez)) g˜ w (0) for non-increasing on .[ 2δ , 3δ 2 ]. Put .gw (z) = χδ (Rez) g ˚ ˜ w (0) for z in the rest z in the intersection of .Δα,δ with .{|Rez| < 3δ }, and .gw (z) = g 2 of .Δα,δ . Put .ϕw (z) = ∂∂z¯ ˚ gw (z) on .Δα,δ . Since . ∂∂z¯ χδ (Rez) = 0 for .|Rez| < 2δ and for def

α,δ \ Q with .Q = ([− 3δ , + 3δ ] × |Rez| > 3δ δ δ 2 , the function .ϕw (z) vanishes on .Δ 2 2 δ δ α,δ [− 2 , 2 ]). On .Qδ ∩ Δ the inequality

.

|ϕw (z)| ≤

.

3 1

3 C 3C |χ (Rez)| |˜gw (z)−˜gw (0)| ≤ · |z| < 2 ·C·2δ = 2 δ 4δ δ 2δ 4δ

(11.28)

−1 (T α,δ ) as continuous holds. Notice that the functions ˚ .gw and .ϕw extend to .P doubly periodic functions. Hence, we may consider them as functions on .T α,δ (see Fig. 11.3).

11.4 (g, m)-Bundles over Riemann Surfaces

405

We want to find now a small positive number .ε that depends only on C, such def

ε there exists a solution .fw of the equation . ∂∂z¯ fw (z) = ϕw (z) that for .σ = εδ ≤ 10 α,σ on .T such that for each z the inequality .|fw (z)| ≤ C1 holds. Then .gw − fw is a holomorphic mapping from .T α,σ to .C \ {−1, 1} whose class has monodromies ε log 2 α equal to (11.26), and to the identity, respectively. This provides the .22j ≥ 14 e 30π σ promised different homotopy classes of mappings from .T α,σ to .C \ {−1, 1}, and, hence proves the lower bound. ¯ To solve the .∂-problem on .T α,εδ = T α,σ , we consider an explicit kernel function which mimics the Weierstraß .℘-function. The author is grateful to Bo Berndtsson who suggested to use this kernel function. Recall that the Weierstraß .℘-function related to the torus .T α is the doubly periodic meromorphic function

℘α (ζ ) =

.

1 + ζ2

 (n,m)∈Z2 \(0,0)

 1 1 − (ζ − n − imα)2 (n + imα)2

on .C. It defines a meromorphic function on .T α with a double pole at the projection of the origin and no other pole. Put .ν = 12 + αi 2 (See Fig. 11.3). Since for .ζ ∈ (Z + iαZ) ∪ (ν + Z + iαZ) the equality .

1 ν 1 − + = (ζ − n − imα) (ζ − n − imα − ν) (ζ − n − imα)2 (ζ

−ν 2 − n − imα − ν)

− n − imα)2 (ζ

holds, and the series with these terms converges uniformly on compact sets not containing poles, the expression def

℘αν (ζ ) =

.

+

1 1 − ζ ζ −ν

 (n,m)∈Z2 \(0,0)

 1 ν 1 − + (ζ − n − imα) (ζ − n − imα − ν) (n + imα)2

defines a doubly periodic meromorphic function on .C with only simple poles. The function descends to a meromorphic function on .T α with two simple poles and no other pole. The support of .ϕw is contained in .Qδ . The set .Qδ is contained in the .2δ-disc in .C (in the Euclidean metric) around the origin. If .ζ is contained in the .2δ-disc around the origin and .z ∈ Δα,δ , then the point .ζ − z is contained in the .2δ-neighbourhood (in .C) of .Δα,δ . By the choice of .δ the distance of any such point .ζ − z to any lattice point .n + iαm except 0 is larger than . 12 − 2δ > 14 . Further, for .z ∈ Δα,δ and .ζ in the

406

11 Finiteness Theorems

2δ-disc around the origin the distance of the point .ζ − z to any point .n + iαm + ν

.

def

1

α,εδ = (including the point .ν) is not smaller than . 12 − 5δ 2 ≥ 4 . Put .Qε,δ = Qδ ∩ Δ 3δ 3δ εδ εδ

εδ εδ δ δ ([− 2 , + 2 ] × [− 2 , + 2 ]) ([− 2 , + 2 ] × [− 2 , + 2 ]). (See Fig. 11.3). Then the function  1 .fw (z) = − ϕw (ζ )℘αν (ζ − z)dm2 (ζ ) , (11.29) π Q ε,δ

for z in .Δα,εδ is holomorphic outside .Q ε,δ and satisfies the equation . ∂∂z¯ fw = ϕw on

. It extends continuously to a doubly periodic function on .P−1 (T α,εδ ) and hence .Q ε,δ descends to a continuous function on .T α,εδ . It remains to estimate the supremum norm of the function .fw on .Δα,σ = Δα,εδ . The following inequality holds for .z ∈ Δα,σ      ν = . ϕ (ζ )℘ (ζ − z)dm (ζ ) w 2 α   Q ε,δ

  

1 1  1  ν  + (℘α (ζ − z) − ) dm2 (ζ ) ≤ ϕw (ζ ) π

ζ −z ζ −z Qε,δ 

 1 1 3C | | + C dm2 (ζ ) . π ζ −z Q ε,δ 2δ

(11.30)

We used the upper bound (11.28) for .ϕw and the fact that for .z ∈ Δα,σ and .ζ 1 in .Q ε,δ the expression .|℘αν (ζ − z) − ζ −z | is bounded by a universal constant .C .

The integral of the second term on the right hand side does not exceed . 3C 2δ · C ·  1 2

4εδ = 6CC εδ. The integral . Q ε,δ | ζ −z |dm2 (ζ ) does not exceed the sum of  1 | dm2 (ζ ) , and . I2 = the two integrals . I1 = (− 3 δ, 3 δ)×(− 1 εδ, 1 εδ) | ζ −z 2 2 2 2  1 (− 12 εδ, 12 εδ)×(− 21 δ, 12 δ) | ζ −z | dm2 (ζ ) . The first integral .I1 is largest when .z = 0. Hence, it does not exceed

 .

√ |ζ | 0 depending only on C so that if .ε < ε0 the supremum norm of .fw is less than . C1 . The proposition is proved.  

Proof of Proposition 11.2 Let . 0 be the length in the Kähler metric of the longest circle in the bouquet. For each natural number k and each positive .σ < σ0 the value .λk (Sσ ) satisfies the inequalities C1

.

0 0 ≤ λk (Sσ ) ≤ C1

σ σ

(11.33)

for constants .C1 and .C1

depending on k, X, S and the Kähler metric. This can be seen by the argument used in the proof of Proposition 11.1. The upper bound in inequalities (11.1) follows from Theorem 11.1. The proof of the lower bound in (11.1) will follow along the same lines as the proof of Proposition 11.1. It will lead to a .∂-problem on an open Riemann surface, for which Hörmander’s .L2 -method can be used. The case of open Riemann surfaces is easier to treat than the general case of pseudo-convex domains. The needed results for Riemann surfaces are explicitly formulated in [68]. To obtain the lower bound we consider the circle of the bouquet S with largest length (in the Kähler metric). After a small deformation of the circle we may assume that it is real analytic outside a small neighbourhood of q. Take a slightly larger neighbourhood of q, so that the part .γ0 of this circle outside this neighbourhood is connected. Consider a conformal mapping .ω of a neighbourhood of .γ0 onto a relatively compact domain V in .C, such that .ω takes .γ0 to an interval on the imaginary axis. We may assume that its length is bigger than .3c 0 for a positive constant c, and the mapping is normalized so that .ω(γ0 ) contains the interval 3c 0 3c 0 .(− 2 , 2 ). Moreover, we may suppose that the orientation of .ω(γ0 ) coincides with the orientation of the imaginary axis. def

For any positive .δ we denote by .Rδ the rectangle .Rδ = {x + iy : x ∈ (−δ, δ), y ∈ (−c 0 − 2δ, c 0 + 2δ)} of extremal length .λ(Rδ ) = c δ0 + 2 in the complex plane. Let .δ0 be a positive number such that .S2δ0 is relatively compact in X and is a deformation retract of X, and for all positive .δ ≤ 2δ0 the rectangle .Rδ is contained in V . We denote by .RδX the curvilinear rectangle .ω−1 (Rδ ) in X. ˚δ def Let .δ ≤ δ0 . Let .R = {x + iy : x ∈ (−δ, δ), y ∈ (−c 0 , c 0 )} ⊂ Rδ be the rectangle in the complex plane with the same center and horizontal side length as .Rδ , ˚δ ) = c 0 . Put .j def = [ c 0 ]. with vertical side length .2c 0 and extremal length .λ(R δ

12δπ

By Remark 9.1 we may represent any word w of the form .a1±2 a2±2 . . . a2±2 in the relative fundamental group .π1 (C\{−1, 1}, (−1, 1)) with 2j terms by a holomorphic ˚δ → {z ∈ C : |z| < C, |z ± 1| > 1 } ⊂ C \ {−1, 1} that vanishes mapping .gw : R C ˚δ at .±ic 0 . The mapping .gw extends by reflection through the horizontal sides of .R ˚δ to a holomorphic function on .Rδ , that we also denote by .gw . Since .|gw | ≤ C on .R for the constant C from Remark 9.1, hence, .|gw | ≤ C also on .Rδ , for any positive C

.α < 1 the inequality .|gw | ≤ δ(1−α) holds for the derivative of the mapping .gw on the smaller rectangle .Rαδ (defined as .Rδ with .δ replaced by .αδ). This fact implies

408

11 Finiteness Theorems

that .|gw | ≤

√ 2Cα 1−α

def

on .Q± αδ = {x + iy : x ∈ (−αδ, αδ), ±y ∈ (c 0 , c 0 + αδ)}. We √

2Cα took into account that .gw (±ic 0 ) = 0. We take .α so that . 1−α < 1 − C1 . With the same function .χ0 as in the proof of Proposition 11.1 we define

⎧ ⎪ ⎪ ⎨1 ˚δ (t) = χ0 ( c 0 +αδ−|t| .χ ) αδ ⎪ ⎪ ⎩0

t ∈ [−c 0 , c 0 ] |t| ∈ (c 0 , c 0 + αδ) ,

(11.34)

t ∈ R \ [−c 0 − αδ, c 0 + αδ] . def

˚δ (Im(z)) and the continuous .(0, 1)Consider the function .g˜w (z) = gw (z) · χ def form .ϕw = ∂¯ g˜w on .Rαδ . The form .ϕw vanishes outside .Q± αδ . Let .ε be a small positive number that does not depend on .δ and will be chosen later. Consider the def

measurable .(0, 1)-form .ϕw,ε on .Rδ that equals .ϕw on .Q± αδ,ε = {x + iy : x ∈ (−αδε, αδε), ±y ∈ (c 0 , c 0 + αδ)} and vanishes outside this set. Extend its pullback under the conformal mapping .ω : RδX → Rδ to a measurable .(0, 1)-form X . on X by putting it equal to zero outside .RδX . Denote the obtained form by .ϕw,ε By Corollary 2.14.2 of [68] there exists a strictly subharmonic exhaustion function .ψ on X. The .L2 -norm of .ϕw,ε √ with respect to the Euclidean metric on the complex plane does not exceed .C2 ε for an absolute constant .C2 . Hence, the X with respect to the Kähler metric and the weight weighted .L2 -norm on X of .ϕw,ε −ψ (see Definition 2.6.1 of [68]) does not exceed .C C √ε for a constant .C that .e 3 2 3 depends on .ψ and on the Kähler metric on the relatively compact subset .V ⊃ RδX X with .∂f X = ϕ X in ¯ w,ε of X. By Corollary 2.12.6 of [68] there exists a function .fw,ε w,ε 2 −ψ the weighted .L -space on X with respect to the Kähler metric and the weight .e√ (see Definition 2.6.1 of [68]), whose norm in this space does not exceed .C4 C3 C2 ε for a constant .C4 depending only on X, .ψ, and the Kähler metric. ± X X Let .(Q± αδ,ε ) be the preimages of .Qαδ,ε under .ω. The function .fw,ε is holo  + X morphic on .X \ (Qαδ,ε )X ∪ (Q− αδ,ε ) . For the above chosen constant .δ0 we put −1 ˜ ± ˜± X ˜ ± def Q δ0 = {x +iy ∈ Rδ0 : ±y ∈ (c 0 −δ0 , c 0 +2δ0 )}, and .(Qδ0 ) = ω (Qδ0 ). Then ± X ˜± X .(Q αδ,ε ) is relatively compact in .(Qδ0 ) . On a relatively compact open subset .X0 of ˜ + )X ∪(Q ˜ − )X ) of X, the supremum norm X, that contains the closed subset .S2δ0 \((Q δ0 δ0 √ X 2 X of .|fw,ε | is estimated by its weighted .L -norm: .|fw,ε | < C5 ε in a neighbourhood ˜ + )X ∪(Q˜ − )X ) for a constant .C5 that depends on the relatively compact of .S2δ0 \((Q δ0 δ0 open subset V of X, on the Kähler metric, on .ψ and on the constants chosen before (see Theorem 2.6.4 of [68]). On the other hand the classical Cauchy-Green formula on the complex plane ˜ − . The ˜+ ∪ Q provides a solution .f˜w,ε of the equation .∂ f˜w,ε = ϕw,ε on the set .Q δ0 δ0 √ supremum norm of the function .f˜w,ε is estimated by .C6 ε for an absolute constant X X − f˜X ˜ + )X ∪ (Q ˜ − )X . The function .fw,ε .C6 . Let .f˜w,ε be the pullback of .f˜w,ε to .(Q w δ0 δ0 + − X X X X | < ˜ ) ∪ (Q ˜ ) and satisfies the inequality .|fw,ε − f˜w,ε is holomorphic on .(Q .

δ0

δ0

11.4 (g, m)-Bundles over Riemann Surfaces

409

√ ˜ + )X ∪ (Q ˜ − )X , that are close to its boundary. (C5 + C6 ) ε at all points of the set .(Q δ0 δ0 ˜ + )X ∪ (Q ˜ − )X . As a consequence, Hence, the inequality is satisfied on .(Q δ0 δ0 √ X ˜ + )X ∪ (Q ˜ − )X . |fw,ε | < (C5 + 2C6 ) ε on (Q δ0 δ0

.

(11.35)

Choose now .ε depending on .C5 and .C6 , so that √ 1 (C5 + 2C6 ) ε < . C

(11.36)

1 on S2δ0 . C

(11.37)

.

Then X |fw,ε |