Thurston's Work on Surfaces

by Albert Fathi, François Laudenbach, Valentin Poénaru, Djun Kim (Translator), Dan Margalit (Translator)

Overview

This book provides a detailed exposition of William Thurston's work on surface homeomorphisms, available here for the first time in English. Based on material of Thurston presented at a seminar in Orsay from 1976 to 1977, it covers topics such as the space of measured foliations on a surface, the Thurston compactification of Teichmüller space, the Nielsen-Thurston classification of surface homeomorphisms, and dynamical properties of pseudo-Anosov diffeomorphisms. Thurston never published the complete proofs, so this text is the only resource for many aspects of the theory.

Thurston was awarded the prestigious Fields Medal in 1982 as well as many other prizes and honors, and is widely regarded to be one of the major mathematical figures of our time. Today, his important and influential work on surface homeomorphisms is enjoying continued interest in areas ranging from the Poincaré conjecture to topological dynamics and low-dimensional topology.

Conveying the extraordinary richness of Thurston's mathematical insight, this elegant and faithful translation from the original French will be an invaluable resource for the next generation of researchers and students.

Product Details

ISBN-13:	9780691147352
Publisher:	Princeton University Press
Publication date:	04/01/2012
Series:	Mathematical Notes , #48
Pages:	272
Product dimensions:	6.10(w) x 9.20(h) x 0.80(d)

About the Author

Albert Fathi is professor at the École Normale Supérieure de Lyon. François Laudenbach is professor emeritus at the University of Nantes. Valentin Poénaru is professor emeritus at the Université Paris-Sud, Orsay. Djun Kim is a Skylight research associate in mathematics at the University of British Columbia. Dan Margalit is assistant professor of mathematics at Georgia Institute of Technology. He is the coauthor of A Primer on Mapping Class Groups (Princeton).

Table of Contents

Preface ix
Foreword to the First Edition ix
Foreword to the Second Edition x
Translators’ Notes xi
Acknowledgments xii
Abstract xiii

Chapter 1 An Overview of Thurston’s Theorems on Surfaces 1
Valentin Poénaru
1.1 Introduction 1
1.2 The Space of Simple Closed Curves 2
1.3 Measured Foliations 3
1.4 Teichmüller Space 5
1.5 Pseudo-Anosov Diffeomorphisms 6
1.6 The Case of the Torus 8

Chapter 2 Some Reminders about the Theory of Surface Diffeomorphisms 14
Valentin Poénaru
2.1 The Space of Homotopy Equivalences of a Surface 14
2.2 The Braid Groups 15
2.3 Diffeomorphisms of the Pair of Pants 19

Chapter 3 Review of Hyperbolic Geometry in Dimension 2 25
Valentin Poénaru
3.1 A Little Hyperbolic Geometry 25
3.2 The Teichmüller Space of the Pair of Pants 27
3.3 Generalities on the Geometric Intersection of Simple Closed Curves 35
3.4 Systems of Simple Closed Curves and Hyperbolic Isometries 42
V4 The Space of Simple Closed Curves in a Surface 44
Valentin Poénaru
4.1 The Weak Topology on the Space of Simple Closed Curves 44
4.2 The Space of Multicurves 46
4.3 An Explicit Parametrization of the Space of Multicurves 47
A Pair of Pants Decompositions of a Surface 53
Albert Fathi

Chapter 5 Measured Foliations 56
Albert Fathi and François Laudenbach
5.1 Measured Foliations and the Euler-Poincaré Formula 56
5.2 Poincaré Recurrence and the Stability Lemma 59
5.3 Measured Foliations and Simple Closed Curves 62
5.4 Curves as Measured Foliations 71
B Spines of Surfaces 74
Valentin Poénaru

Chapter 6 The Classification of Measured Foliations 77
Albert Fathi
6.1 Foliations of the Annulus 78
6.2 Foliations of the Pair of Pants 79
6.3 The Pants Seam 84
6.4 The Normal Form of a Foliation 87
6.5 Classification of Measured Foliations 92
6.6 Enlarged Curves as Functionals 97
6.7 Minimality of the Action of the Mapping Class Group on PMF 98
6.8 Complementary Measured Foliations 100
C Explicit Formulas for Measured Foliations 101
Albert Fathi

Chapter 7 Teichmüller Space 107
Adrien Douady; notes by François Laudenbach

Chapter 8 The Thurston Compactification of Teichmüller Space 118
Albert Fathi and François Laudenbach
8.1 Preliminaries 118
8.2 The Fundamental Lemma 121
8.3 The Manifold T 125
D Estimates of Hyperbolic Distances 128
Albert Fathi
D.1 The Hyperbolic Distance from i to a Point _z0 128
D.2 Relations between the Sides of a Right Hyperbolic Hexagon 129
D.3 Bounding Distances in Pairs of Pants 131

Chapter 9 The Classification of Surface Diffeomorphisms 135
Valentin Poénaru
9.1 Preliminaries 135
9.2 Rational Foliations (the Reducible Case) 136
9.3 Arational Measured Foliations 137
9.4 Arational Foliations with λ = 1 (the Finite Order Case) 140
9.5 Arational Foliations with λ 6= 1 (the Pseudo-Anosov Case) 141
9.6 Some Properties of Pseudo-Anosov Diffeomorphisms 150

Chapter 10 Some Dynamics of Pseudo-Anosov Diffeomorphisms 154
Albert Fathi and Michael Shub
10.1 Topological Entropy 154
10.2 The Fundamental Group and Entropy 157
10.3 Subshifts of Finite Type 162
10.4 The Entropy of Pseudo-Anosov Diffeomorphisms 165
10.5 Constructing Markov Partitions for Pseudo-Anosov Diffeomorphisms
171
10.6 Pseudo-Anosov Diffeomorphisms are Bernoulli 173

Chapter 11 Thurston’s Theory for Surfaces with Boundary 177
François Laudenbach
11.1 The Spaces of Curves and Measured Foliations 177
11.2 Teichmüller Space and Its Compactification 179
11.3 A Sketch of the Classification of Diffeomorphisms 180
11.4 Thurston’s Classification and Nielsen’s Theorem 184
11.5 The Spectral Theorem 188

Chapter 12 Uniqueness Theorems for Pseudo-Anosov Diffeomorphisms 191
Albert Fathi and Valentin Poénaru
12.1 Statement of Results 191
12.2 The Perron-Frobenius Theorem and Markov Partitions 192
12.3 Unique Ergodicity 194
12.4 The Action of Pseudo-Anosovs on PMF 196
12.5 Uniqueness of Pseudo-Anosov Maps 204

Chapter 13 Constructing Pseudo-Anosov Diffeomorphisms 208
François Laudenbach
13.1 Generalized Pseudo-Anosov Diffeomorphisms 208
13.2 A Construction by Ramified Covers 209
13.3 A Construction by Dehn Twists 210

Chapter 14 Fibrations over S¹ with Pseudo-Anosov Monodromy 215
David Fried
14.1 The Thurston Norm 216
14.2 The Cone C of Nonsingular Classes 218
14.3 Cross Sections to Flows 224

Chapter 15 Presentation of the Mapping Class Group 231
François Laudenbach and Alexis Marin
15.1 Preliminaries 231
15.2 A Method for Presenting the Mapping Class Group 232
15.3 The Cell Complex of Marked Functions 234
15.4 The Marking Complex 238
15.5 The Case of the Torus 241

Bibliography 243
Index 251

Editorial Reviews

"[T]he translation is a most welcome addition for those who use FLP as a reference. . . . It is the reviewer's hope that this new version will also introduce Thurston's brilliant insights and imagination to even wider audiences and help inspire the present and future generations to pick up where he left off."—-Dan Margalit, Bulletin of the American Mathematical Society

From the Publisher