Hyperbolic Manifolds and Discrete Groups

The main goal of the book is to present a proof of the following. Thurston's Hyperbolization Theorem ("The Big Monster"). Suppose that M is a compact atoroidal Haken 3-manifold that has zero Euler characteristic. Then the interior of M admits a complete hyperbolic metric of finite volume. This theorem establishes a strong link between the geometry and topology 3 of 3-manifolds and the algebra of discrete subgroups of Isom(JH[ ). It completely changed the landscape of 3-dimensional topology and theory of Kleinian groups. Further, it allowed one to prove things that were beyond the reach of the standard 3-manifold technique as, for example, Smith's conjecture, residual finiteness of the fundamental groups of Haken manifolds, etc. In this book we present a complete proof of the Hyperbolization Theorem in the "generic case." Initially we planned 1 including a detailed proof in the remaining case of manifolds fibered over § as well. However, since Otal's book [Ota96] (which treats the fiber bundle case) became available, only a sketch of the proof in the fibered case will be given here.

Three-Dimensional Topology.- Thurston Norm.- Geometry of Hyperbolic Space.- Kleinian Groups.- Teichmüller Theory of Riemann Surfaces.- to Orbifold Theory.- Complex Projective Structures.- Sociology of Kleinian Groups.- Ultralimits of Metric Spaces.- to Group Actions on Trees.- Laminations, Foliations, and Trees.- Rips Theory.- Brooks’ Theorem and Circle Packings.- Pleated Surfaces and Ends of Hyperbolic Manifolds.- Outline of the Proof of the Hyperbolization Theorem.- Reduction to the Bounded Image Theorem.- The Bounded Image Theorem.- Hyperbolization of Fibrations.- The Orbifold Trick.- Beyond the Hyperbolization Theorem.