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Teichmüller Theory and Applications to Geometry, Topology, and Dynamics

Volume 3: 3-Manifolds that fiber over the circle

We hope that this book will be available by the end of 2016. Here we give the opening paragraphs of each chapter, as they now stand, and some figures. In the book, labels are included in the figures, using "puts". For a preview of the final volume, please go to Teichmuller Theory Volume 4

Chapter 11 Geometry of hyperbolic space

In this chapter we give a short introduction to the geometry of hyperbolic space and to Kleinian groups.  We  begin in Sections 11.1-11.3 with hyperbolic space and its group of automorphisms.  In Section 11.4 we discuss elementary and non-elementary Kleinian groups.  Section 11.5 defines the limit set of a Kleinian group and gives some of its basic properties.

 We go on in Section 11.6 to the Jørgensen inequality, a beautiful and rather mysterious inequality that exploits the discreteness of a Kleinian group in an essential way.  This makes itpossible to give in Section 11.7 a fairly easy proof of the  Margulis lemma. This is a 3-dimensional analogue of the collaring theorem (Theorem 3.8.3 in Volume 1), giving a complete description of the thin parts of a hyperbolic 3-manifold; it provides an analogue of the plumbing picture of hyperbolic surfaces given in Section 3.8.

We find the proof of J
ørgensen's inequality  hard to motivate,   so  we also give in Appendix D2 a different proof of the Margulis lemma, which is  cruder, but much easier to understand.
Thurston's hyperbolization theorems require thinking  not just about individual Kleinian groups,
but also about their limits.   There are two notions of ``limit of a sequence of Kleinian groups'', with surprisingly different properties:  algebraic limits, which are relatively easy to understand, and geometric limits, which can be amazingly complicated.   Algebraic limits are described in Section 11.8, geometric limits in Section 11.9.

Thurston's second hyperbolization theorem requires the Klein-Maskit combination theorems, which will eventually allow us to glue together hyperbolic manifolds. These theorems,  proved in Section 11.10,  also illustrate the power of 3-dimensional thinking about Kleinian groups; it is quite hard to see why the combined groups as defined are discrete if one sticks to the action on the Riemann sphere.

 Finally, in Section 11.11,  we   prove the 3-dimensional analogue of the Poincaré polygon theorem.

Chapter 12 Rigidity theorems

In the first half of this chapter we  prove  three great theorems, due to Ahlfors, McMullen, and Mostow. Although the statements look quite different, they are technically closely related: they are all concerned with the construction of  Beltrami forms on \partial \overline H3 invariant under a Kleinian group G.  Since these theorems show that such Beltrami forms are very restricted, they are called  rigidity theorems.   

In the second half (Sections 12.5-12.8) we prove that the central hypothesis of McMullen's rigidity theorem  holds for quasi-Fuchsian groups. We need  this to prove the hyperbolization theorem for 3-manifolds  that fiber over the circle. Along the way we develop several  topics, including laminations and pleated surfaces,  of great interest in their own right.

Chapter 13 Hyperbolization of 3-manifolds that fiber over the circle

One of the great accomplishments of nineteenth century mathematics was  showing that all surfaces of genus  g >= 2 admit hyperbolic structures. We  now present a 3-dimensional version of this theorem: Theorem 13.0.1. This is one of the great accomplishments of twentieth century mathematics.
Let S=\overline S-Z be an orientable surface with 
S=\overline S compact and  Z  finite.  Let  f : S \to be an orientation-preserving  homeomorphism. Denote by  M f  the mapping torus of  f , i.e.,  the quotient of
[0,1] x  S  by the equivalence
relation that identifies  (0, x)  to  (1, f (x)). Every 3-manifold that fibers over the circle is of this form; the corresponding map f is called the holonomy map. It is  defined only up to isotopy and so is really an element of the mapping class group MCG(S). 

Theorem 13.0.1  (Hyperbolization of 3-manifolds that fiber over the circle) The 3-manifold
M f  admits a complete hyperbolic structure if and only if the holonomy map f is homotopic to a pseudo-Anosov

 Proving this theorem will take the entire chapter.  One direction - that if
M f  admits a hyperbolic structure,  then f is homotopic to a pseudo-Anosov  homeomorphism - is easy; at least it follows easily from Theorem 8.1.4 on  the classification of homeomorphisms of surfaces. This is the content of Section 13.1. 

The proof of the other direction - that if f is homotopic to a pseudo-Anosov map, then
M f  carries a hyperbolic structure - is long and hard. We discuss the main idea in   Section 13.2. In Section 13.3 we prove the compactness of Bers slices;  Sections 13.4--13.7 are devoted to the double limit theorem. We complete the proof of the hyperbolization theorem in Section 13.8.

We begin with Example 13.0.3, which  illustrates the theorem  for the complement of the figure-eight knot.  We will need the notion of  splitting  an n-dimensional manifold-with-boundary  X along a properly embedded  (n-1) -dimensional submanifold  S ....

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


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