Thurston's Work on Surfaces

Albert Fathi, François Laudenbach, Valentin Poénaru, Djun Kim, Dan Margalit, Djun Kim, Dan Margalit

• Publisher: Princeton University Press
• Year: 2021
• Print ISBN: 9780691147352
• eBook ISBN: 9781400839032
• Topic: Mathematics
• Subtopic: Algebraic Geometry
• Index: Mathematics

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.

Table of contents

• Cover Page
• Half-title Page
• Title Page
• Copyright Page
• Contents
• Preface
• 1. An Overview of Thurston’s Theorems on Surfaces
• 2. Some Reminders about the Theory of Surface Diffeomorphisms
• 3. Review of Hyperbolic Geometry in Dimension 2
• 4. The Space of Simple Closed Curves in a Surface
• A. Pair of Pants Decompositions of a Surface
• 5. Measured Foliations
• B. Spines of Surfaces
• 6 The Classification of Measured Foliations
• C. Explicit Formulas for Measured Foliations
• 7. Teichm¨uller Space
• 8. The Thurston Compactification of Teichm¨uller Space
• D. Estimates of Hyperbolic Distances
• 9. The Classification of Surface Diffeomorphisms
• 10. Some Dynamics of Pseudo-Anosov Diffeomorphisms
• 11. Thurston’s Theory for Surfaces with Boundary
• 12. Uniqueness Theorems for Pseudo-Anosov Diffeomorphisms
• 13. Constructing Pseudo-Anosov Diffeomorphisms
• 14. Fibrations over S¹ with Pseudo-Anosov Monodromy
• 15. Presentation of the Mapping Class Group
• Bibliography