Thurston’s work on surfaces. Transl. from the French by Djun Kim and Dan Margalit. (English) Zbl 1244.57005

Fathi, Albert; Laudenbach, François; Poénaru, Valentin

Mathematical Notes (Princeton) 48. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14735-2/pbk; 978-1-4008-3903-2/ebook). xiii, 255 p. (2012).

The present book provides the first English translation of the authors’ classic text “Travaux de Thurston sur les surfaces”, the French original of which was first published in 1979 as a volume of “Seminaire Orsay” in [Astérisque, No. 66–67, Paris: Société Mathématique de France (1979; Zbl 0406.00016)]. A corrected reprint of this original 1979 edition appeared in (1991; Zbl 0731.57001), but it was to take more than another twenty years to finally present the well-deserved English version of this unique source book.
Based on the pioneering work of William Thurston on surfaces, including measured foliations, the compactification of Teichmüller space, the classification of surface diffeomorphisms, and the dynamical properties of pseudo-Anosov diffeomorphisms of surfaces, the book under review provides a detailed exposition of these topics in a number of individual exposés, which originally were presented at a seminar in Paris-Orsay from 1976 to 1977.
All the exposés forming this book have been reviewed back in 1979, and therefore we may refer to those original reviews as for the details. As the translators have pointed out, they have attempted to stay as faithful to the original texts as possible. At the same time, they have made many small modifications in order to elucidate the structure of the theory, modernize the language (after more than 30 years), and clarify the details of some definitions and proofs. Recall that the contents and their reviews can be listed as follows.

1.

V. Poénaru: An overview of Thurston’s theorems on surfaces (Zbl 0446.57005);

2.

V. Poénaru: Some reminders about the theory of surface diffeomorphisms (Zbl 0446.57006);

3.

V. Poénaru: Review of hyperbolic geometry in dimension 2 (Zbl 0446.57007);

4.

V. Poénaru: The space of simple closed curves in a surface (Zbl 0446.57008);

A.

A. Fathi: Pair of pants decompositions of a surface (Zbl 0446.57009);

5.

A. Fathi and F. Laudenbach: Measured foliations (Zbl 0446.57010);

B.

V. Poénaru: Spines of surfaces (Zbl 0446.57011);

6.

A. Fathi: The classification of measured foliations (Zbl 0446.57012);

C.

A. Fathi: Explicit formulas for measured foliations (Zbl 0446.57013);

7.

A. Douady: Teichmüller space (Zbl 0446.57014);

8.

A. Fathi and F. Laudenbach: The Thurston compactification of Teichmüller space (Zbl 0446.57015);

D.

A. Fathi: Estimates of hyperbolic distances (Zbl 0446.57016);

9.

V. Poenaru: The classification of surface diffeomorphisms (Zbl 0446.57017);

10.

A. Fathi and M. Shub: Some dynamics of pseudo-Anosov diffeomorphisms (Zbl 0446.57022);

11.

F. Laudenbach: Thurston’s theory for surfaces with boundary (Zbl 0446.57018);

12.

A. Fathi and V. Poénaru: Uniqueness theorems, for pseudo-Anosov diffeomorphisms (Zbl 0446.57019);

13.

F. Laudenbach: Constructing pseudo-Anosov diffeomorphisms (Zbl 0446.57020);

14.

D. Fried: Fibrations over \(S^1\) with pseudo-Anosov monodromy (Zbl 0446.57023);

15.

F. Laudenbach and A. Marin: Presentation of the mapping class group (Zbl 0446.57021).

William Thurston was awarded the prestigious Fields Medal in 1982 for his epoch-making work on surfaces, which is still enjoying continued interest in various areas of contemporary geometry, analysis, theory of dynamical systems, and mathematical physics. However, as W. Thurston barely published complete proofs of his fundamental constructions and theorems, the French original of the present text was the only source for many details of Thurston’s work in the past three decades. No doubt, this masterly English translation of the original French classic will make it accessible for a much wider audience of readers worldwide, thereby strengthening its role as an invaluable resource in the relevant literature, and that for the next generation of researchers and students likewise.

Reviewer: Werner Kleinert (Berlin)

MSC:

57-03	History of manifolds and cell complexes
01A75	Collected or selected works; reprintings or translations of classics
57M50	General geometric structures on low-dimensional manifolds
57N05	Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
30F60	Teichmüller theory for Riemann surfaces
00B15	Collections of articles of miscellaneous specific interest

Keywords:

translations of classics; topology of 2-manifolds; measured foliations; pseudo-Anosov diffeomorphisms; Teichmüller spaces

Citations:

Zbl 0406.00016; Zbl 0731.57001; Zbl 0446.57005; Zbl 0446.57006; Zbl 0446.57007; Zbl 0446.57008; Zbl 0446.57009; Zbl 0446.57010; Zbl 0446.57012; Zbl 0446.57013; Zbl 0446.57014; Zbl 0446.57015; Zbl 0446.57016; Zbl 0446.57017; Zbl 0446.57022; Zbl 0446.57018; Zbl 0446.57019; Zbl 0446.57020; Zbl 0446.57021; Zbl 0446.57023; Zbl 0446.57011