A Primer on Mapping Class Groups (PMS-49) (Princeton Mathematical)
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A Primer on Mapping Class Groups Princeton Mathematical Series EDITORS:PHILLIP A. GRIFFITHS,JOHN N. MATHER, AND ELIAS M. STEIN 1. The Classical Groups by Hermann Weyl 8. Theory of Lie Groups: I by C. Chevalley 9. Mathematical Methods of Statistics by Harald Cramer´ 14. The Topology of Fibre Bundles by Norman Steenrod 17. Introduction to Mathematical Logic, Vol. I by Alonzo Church 19. Homological Algebra by H. Cartan and S. Eilenberg 28. Convex Analysis by R. T. Rockafellar 30. Singular Integrals and Differentiability Properties of Functions by E. M. Stein 32. Introduction to Fourier Analysis on Euclidean Spaces by E. M. Stein and G. Weiss 33. Etale´ Cohomology by J. S. Milne 35. Three-Dimensional Geometry and Topology, Volume 1 by William P. Thurston. Edited by Silvio Levy 36. Representation Theory of Semisimple Groups: An Overview Based on Examples by Anthony W. Knapp 38. Spin Geometry by H. Blaine Lawson, Jr., and Marie-Louise Michel- sohn 43. Harmonic Analysis: Real Variable Methods, Orthogonality, and Os- cillatory Integrals by Elias M. Stein 44. Topics in Ergodic Theory by Ya. G. Sinai 45. Cohomological Induction and Unitary Representations by Anthony W. Knapp and David A. Vogan, Jr. 46. Abelian Varieties with Complex Multiplication and Modular Func- tions by Goro Shimura 47. Real Submanifolds in Complex Space and Their Mappings by M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild 48. Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane by Kari Astala, Tadeusz Iwaniec, and Gaven Martin 49. A Primer on Mapping Class Groups by Benson Farb and Dan Mar- galit A Primer on Mapping Class Groups Benson Farb and Dan Margalit PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Copyright c 2012 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved ISBN 978-0-691-14794-9 Library of Congress Cataloging-in-Publication Data Farb, Benson. A primer on mapping class groups / Benson Farb, Dan Margalit. p. cm. – (Princeton mathematical series) Includes bibliographical references and index. ISBN 978-0-691-14794-9 (hardback) 1. Mappings (Mathematics) 2. Class groups (Mathematics) I. Margalit, Dan, 1976– II. Title. III. Series QA360.F37 2011 512.7’4–dc22 2011008491 British Library Cataloging-in-Publication Data is available This book has been composed in Times and Helvetica. The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper. ∞ Printed in the United States of America 10987654321 To Amie and Kathleen This page intentionally left blank Contents Preface xi Acknowledgments xiii Overview 1 PART 1. MAPPING CLASS GROUPS 15 1. Curves, Surfaces, and Hyperbolic Geometry 17 1.1 Surfaces and Hyperbolic Geometry 17 1.2 Simple Closed Curves 22 1.3 The Change of Coordinates Principle 36 1.4 Three Facts about Homeomorphisms 41 2. Mapping Class Group Basics 44 2.1 Definition and First Examples 44 2.2 Computations of the Simplest Mapping Class Groups 47 2.3 The Alexander Method 58 3. Dehn Twists 64 3.1 Definition and Nontriviality 64 3.2 Dehn Twists and Intersection Numbers 69 3.3 Basic Facts about Dehn Twists 73 3.4 The Center of the Mapping Class Group 75 3.5 Relations between Two Dehn Twists 77 3.6 Cutting, Capping, and Including 82 4. Generating the Mapping Class Group 89 4.1 The Complex of Curves 92 4.2 The Birman Exact Sequence 96 4.3 Proof of Finite Generation 104 4.4 Explicit Sets of Generators 107 5. Presentations and Low-dimensional Homology 116 5.1 The Lantern Relation and H1(Mod(S); Z) 116 5.2 Presentations for the Mapping Class Group 124 5.3 Proof of Finite Presentability 134 viii CONTENTS 5.4 Hopf’s Formula and H2(Mod(S); Z) 140 5.5 The Euler Class 146 5.6 Surface Bundles and the Meyer Signature Cocycle 153 6. The Symplectic Representation and the Torelli Group 162 6.1 Algebraic Intersection Number as a Symplectic Form 162 6.2 The Euclidean Algorithm for Simple Closed Curves 166 6.3 Mapping Classes as Symplectic Automorphisms 168 6.4 Congruence Subgroups, Torsion-free Subgroups, and Residual Finiteness 176 6.5 The Torelli Group 181 6.6 The Johnson Homomorphism 190 7. Torsion 200 7.1 Finite-order Mapping Classes versus Finite-order Homeomorphisms 200 7.2 Orbifolds, the 84(g − 1) Theorem, and the 4g +2Theorem 203 7.3 Realizing Finite Groups as Isometry Groups 213 7.4 Conjugacy Classes of Finite Subgroups 215 7.5 Generating the Mapping Class Group with Torsion 216 8. The Dehn–Nielsen–Baer Theorem 219 8.1 Statement of the Theorem 219 8.2 The Quasi-isometry Proof 222 8.3 Two Other Viewpoints 236 9. Braid Groups 239 9.1 The Braid Group: Three Perspectives 239 9.2 Basic Algebraic Structure of the Braid Group 246 9.3 The Pure Braid Group 248 9.4 Braid Groups and Symmetric Mapping Class Groups 253 PART 2. TEICHMULLER¨ SPACE AND MODULI SPACE 261 10. Teichm¨ullerSpace 263 10.1 Definition of Teichm¨ullerSpace 263 10.2 Teichm¨uller Space of the Torus 265 10.3 The Algebraic Topology 269 10.4 Two Dimension Counts 272 10.5 The Teichm¨uller Space of a Pair of Pants 275 10.6 Fenchel–Nielsen Coordinates 278 10.7 The 9g − 9 Theorem 286 11. Teichm¨uller Geometry 294 11.1 Quasiconformal Maps and an Extremal Problem 294 11.2 Measured Foliations 300 11.3 Holomorphic Quadratic Differentials 308 11.4 Teichm¨uller Maps and Teichm¨uller’s Theorems 320 11.5 Gr¨otzsch’s Problem 325 CONTENTS ix 11.6 Proof of Teichm¨uller’s Uniqueness Theorem 327 11.7 Proof of Teichm¨uller’s Existence Theorem 330 11.8 The Teichm¨uller Metric 337 12. Moduli Space 342 12.1 Moduli Space as the Quotient of Teichm¨uller Space 342 12.2 Moduli Space of the Torus 345 12.3 Proper Discontinuity 349 12.4 Mumford’s Compactness Criterion 353 12.5 The Topology at Infinity of Moduli Space 359 12.6 Moduli Space as a Classifying Space 362 PART 3. THE CLASSIFICATION AND PSEUDO-ANOSOV THEORY 365 13. The Nielsen–Thurston Classification 367 13.1 The Classification for the Torus 367 13.2 The Three Types of Mapping Classes 370 13.3 Statement of the Nielsen–Thurston Classification 376 13.4 Thurston’s Geometric Classification of Mapping Tori 379 13.5 The Collar Lemma 380 13.6 Proof of the Classification Theorem 382 14. Pseudo-Anosov Theory 390 14.1 Five Constructions 391 14.2 Pseudo-Anosov Stretch Factors 403 14.3 Properties of the Stable and Unstable Foliations 408 14.4 The Orbits of a Pseudo-Anosov Homeomorphism 414 14.5 Lengths and Intersection Numbers under Iteration 419 15. Thurston’s Proof 424 15.1 A Fundamental Example 424 15.2 A Sketch of the General Theory 434 15.3 Markov Partitions 442 15.4 Other Points of View 445 Bibliography 447 Index 465 This page intentionally left blank Preface Our goal in this book is to explain as many important theorems, examples, and techniques as possible, as quickly and directly as possible, while at the same time giving (nearly) full details and keeping the text (nearly) self- contained. This book contains some simplifications of known approaches and proofs, the exposition of some results that are not readily available, and some new material as well. We have tried to incorporate many of the “greatest hits” of the subject, as well as its small quirks and gems. There are a number of other references that cover various of the topics we cover here (and more). We would especially like to mention the books by Abikoff [1], Birman [24], Casson–Bleiler [44], Fathi–Laudenbach–Po´enaru [61], and Hubbard [97], as well as the survey papers by Harer [84] and Ivanov [107]. The works of Bers [14, 15] on Teichm¨uller’s theorems and on the Nielsen–Thurston classification theorem have had a particularly strong influence on this book. The first author learned much of what he knows about these topics from his advisor Bill Thurston, his teacher Curt McMullen, and his collabora- tors Lee Mosher and Howard Masur. The second author’s perspective on this subject was greatly influenced by his advisor Benson Farb, his mentors Mladen Bestvina and Joan Birman, and his collaborator Chris Leininger. This book in particular owes a debt to notes the first author took from a course given by McMullen at Berkeley in 1991. Benson Farb and Dan Margalit Chicago and Atlanta, January 2011 This page intentionally left blank Acknowledgments We would like to thank Mohammed Abouzaid, Yael Algom-Kfir, Javier Aramayona, Jessica Banks, Mark Bell, Jeff Brock, Xuanting Cai, Mered- ith Casey, Pallavi Dani, Florian Deloup, John Franks, Siddhartha Gadgil, Bill Goldman, Carlos Segovia Gonz´alez, Michael Handel, John Harer, Pe- ter Horn, Nikolai Ivanov, Jamie Jorgensen, Krishna Kaipa, Keiko Kawa- muro, Richard Kent, Sarah Kitchen, Mustafa Korkmaz, Maria Luisa, Feng Luo, Joseph Maher, Kathryn Mann, Vladimir Markovic, Howard Masur, Jon McCammond, Curt McMullen, Ben McReynolds, Guido Mislin, Ma- han Mj, Jamil Mortada, Lee Mosher, Zbigniew Nitecki, Jeremy Pecharich, Rita Jimenez Rolland, Simon Rose, Keefe San Agustin, Travis Schedler, Paul Seidel, Ignat Soroko, Steven Spallone, Harold Sultan, Macky Suzuki, Genevieve Walsh, Richard Webb, Alexander Wickens, Bert Wiest, Jenny Wilson, Rebecca Winarski, Stefan Witzel, Scott Wolpert, Kevin Wortman, Alex Wright, and Bruno Zimmermann for comments and questions. We would also like to thank Robert Bell, Tara Brendle, Jeff Carlson, Tom Church, Spencer Dowdall, Moon Duchin, David Dumas, Jeff Frazier, Daniel Groves, Asaf Hadari, Allen Hatcher, Thomas Koberda, Justin Malestein, Johanna Mangahas, Erika Meucci, Catherine Pfaff, Kasra Rafi, and Saul Schleimer for thorough comments on large portions of the book.