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Matrix Editions future books

 

 We have decided to break up volume 2 of the Teichmüller theory book (Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 2: Four Theorems by William Thurston) into three shorter volumes:

Volume 2: Surface Homeomorphisms and Rational Functions

Volume 3: 3-Manifolds that fiber over the circle

Volume 4: Hyberbolization of Haken Manifolds

Our hope is that Volume 2 will be published by summer 2016, and Volume 2 by the end of the year.  Volume 3 presents greater mathematical difficulties, and we hesitate to hazard a date.

We plan to use color in the illustrations.

Table of Contents for Volume 2:  Surface Homeomorphisms and Rational Functions

Chapter 8 The classification of homeomorphisms of surfaces
    8.1 The classification theorem
    8.2 Periodic and reducible homeomorphisms
    8.3 Pseudo-Anosov homeomorphisms
    8.4 Proof of the classification theorem
    8.5 The structure in the reducible case

Chapter 9  Dynamics of polynomials
    9.1 Julia sets
    9.2 Fixed points
    9.3 Green's functions, Böttcher coordinates
    9.4 Extending
f_0 to S^1
    9.5 External rays at rational angles land

Chapter 10 Rational functions
    10.1 Introduction
    10.1 Thurston mappings
    10.2 Thurston mapps associated to spiders
    10.3 Thurston obstructions for spider maps and Levy cycles
    10.4 Julia sets of quadratic polynomials with superattracting cycles
    10.5 Parameter spaces for quadratic polynomials
    10.6 The Thurston pullback mapping s_f
    10.7 The derivative and coderivative of
s_f
    10.8 The necessity of the eigenvalue criterion
    10.9 Convergence in moduli spaces implies convergence in Teichmüller space
    10.10 Asymptotic geometry of Riemann surfaces
    10.11 Sufficiency of the eigenvalue criterion

Appendix C1  The Perron-Frobenius theorem
Appendix C2  The Alexander trick
Appendix C3  Homotopy implies isotopy
Appendix C4  The mapping class group and outer automorphisms
Appendix C5  Totally real stretch factors
Appendix C6  Irrationally indifferent fixed points
Appendix C7  Examples of Thurston pullback maps
Appendix C8  Branched maps with nonhyperbolic orbifolds
Appendix C9  The Sullivan dictionary

Table of Contents for Volume 3:  3-Manifolds that fiber over the circle

Chapter 11 Geometry of hyperbolic space
Chapter 12 Rigidity theorems
Chapter 13 Hyperbolization of 3-manifolds that fiber over the circle

Appendix D1  The Nullstellensatz and Selberg's  lemma
Appendix D2  Fundamental groups, amalgamated sums,  and HNN extensions
Appendix D3  The Margulis lemma: another proof
Appendix D4  Arithmetic Kleinian groups
Appendix D5  Ergodic flow and Mostow rigidity
Appendix D6  Sullivan's rigidity theorem
Appendix D7  Positivity of the Riemann matrix

Table of Contents for Volume 4:  Hyberbolization of Haken Manifolds

 Chapter 14  Hierarchies in 3-manifolds
 Chapter 15  Andreev's theorem
 Chapter 16  Hyberbolization of Haken manifolds

 Appendix E1  Triangulations
 Appendix E2 Connected sums

 

From the preface to volume 1:

Not only are the theorems of extraordinary beauty in themselves, but the methods of proof that Thurston introduced were so novel and displayed such amazing geometric insight that to this day they have barely entered the accepted methods of mathematicians in the field.

The results sound more or less unrelated, but they are linked by a common thread: each one goes from topology to geometry. Each says that either a topological problem has a natural geometry, or there is an understandable obstruction.

The proofs are closely related: you use the topology to set up an analytic mapping from a Teichmüller space to itself; the geometry arises from a fixed point of this mapping. Thurston proceeds to show that if there is no fixed point, then some system of simple closed curves on the surface is an obstruction to finding a solution.

Thus the proofs of the theorems are somehow similar, although the details and difficulty are very different....

 

 

 

 

 

Contents:

Chapter 8 The Classification of Homeomorphisms of Surfaces

Chapter 9 Rational Functions

Chapter 10 Geometry of Hyperbolic Space

Chapter 11 The Ahlfors Finiteness Theorem and Rigidity Theorems

Chapter 12 A Hyperbolic Structure for Mapping Tori of Pseudo-Anosov Homeomorphisms

Chapter 13 Some Topology of 3-manifolds: Hierarchies

Chapter 14 Andreev's Theorem on Hyperbolic Polyhedra

Chapter 15 The Skinning Lemma



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