Matrix
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*serious mathematics, written with the reader in mind*

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We hope that this book will be available by June 2019. 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". Placement of the figures here does not reflect their placement in the book.

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
it possible 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 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 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.

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.
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.

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 an orientable surface with overline *S * compact and * Z* finite. Let * f : S* to *S * be an orientation-preserving
homeomorphism. Denote by *M _{f} *
the mapping torus of

Proving this theorem will take the entire chapter.

Appendix D1 The Nullstellensatz

Appendix D3 The Margulis lemma: another proof

Appendix D3 Amalgamated sums

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

See also:

Volume 1: Teichmüller theory

Volume 2: Surface Homeomorphisms and Rational Functions

Volume 4: Hyperbolization of Haken Manifolds (not yet published)