Teichmüller Theory and Applications to Geometry, Topology, and Dynamics
Volume 3: 3Manifolds 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 11.111.3 with hyperbolic space and its group of automorphisms. In Section
11.4 we discuss elementary and nonelementary 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 3dimensional analogue of the collaring theorem (Theorem 3.8.3 in Volume 1), giving a complete description of the thin parts of a hyperbolic 3manifold; it provides an analogue of the plumbing picture of hyperbolic surfaces given in Section 3.8.
We find the proof of Jørgensen'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 KleinMaskit combination theorems, which will eventually allow us to glue together hyperbolic manifolds. These theorems, proved in Section 11.10, also illustrate the power of 3dimensional 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 3dimensional 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 H^{3} 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 12.512.8) we prove that the central
hypothesis of McMullen's rigidity theorem holds for
quasiFuchsian groups. We need this to prove the hyperbolization
theorem for 3manifolds 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 3manifolds 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 3dimensional version of
this theorem: Theorem 13.0.1. This is one of the great accomplishments
of twentieth century mathematics.
Let S=\overline SZ be an orientable surface with S=\overline S compact and Z finite. Let f : S \to S be an orientationpreserving 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 3manifold 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 3manifolds that fiber over the circle) The 3manifold M_{ }_{f}_{ }admits a complete hyperbolic structure if and only if the holonomy map f is homotopic to a pseudoAnosov
homeomorphism.
Proving this theorem will take the entire chapter. One direction  that if M_{ }_{f}_{ } admits a hyperbolic structure, then f is homotopic to a pseudoAnosov 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 pseudoAnosov 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 13.413.7 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 figureeight knot. We will need the
notion of splitting an ndimensional manifoldwithboundary X along a properly embedded (n1) 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

