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Thurston's Work on Surfaces (MN-48)


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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 Poenaru 1.1 Introduction 1 1.2 The Space of Simple Closed Curves 2 1.3 Measured Foliations 3 1.4 Teichmuller 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 Poenaru 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 Poenaru 3.1 A Little Hyperbolic Geometry 25 3.2 The Teichmuller 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 Poenaru 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 Francois Laudenbach 5.1 Measured Foliations and the Euler-Poincare Formula 56 5.2 Poincare 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 Poenaru 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 Teichmuller Space 107 Adrien Douady; notes by Francois Laudenbach Chapter 8 The Thurston Compactification of Teichmuller Space 118 Albert Fathi and Francois 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 Poenaru 9.1 Preliminaries 135 9.2 Rational Foliations (the Reducible Case) 136 9.3 Arational Measured Foliations 137 9.4 Arational Foliations with lambda = 1 (the Finite Order Case) 140 9.5 Arational Foliations with lambda 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 Francois Laudenbach 11.1 The Spaces of Curves and Measured Foliations 177 11.2 Teichmuller 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 Poenaru 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 Francois 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 Francois 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

About the Author

Albert Fathi is professor at the Ecole Normale Superieure de Lyon. Francois Laudenbach is professor emeritus at the University of Nantes. Valentin Poenaru is professor emeritus at the Universite 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).


"[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

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