Hyperbolic Manifolds and Discrete Groups

This classic text is at the crossroads of many branches of mathematics. Its main focus is on Thurston's hyperbolization theorem, and it contains numerous open problems and conjectures related to the theorem as well as discussions on related topics.

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.

Includes beautiful illustrations, a rich set of examples of key concepts, numerous exercises An extensive bibliography and index are complemented by a glossary of terms Presents the first complete proof of the generic case of Thurston's hyperbolization theorem Includes supplementary material: sn.pub/extras

Klappentext

This classic book is at the crossroads of several branches of mathematics: hyperbolic geometry, discrete groups, 3-dimensional topology, geometric group theory, and complex analysis. The main focus throughout the text is on Thurston's hyperbolization theorem, one of the central results of 3-dimensional topology that has completely changed the landscape of the field. The book contains a number of open problems and conjectures related to the hyperbolization theorem as well as rich discussions on related topics including geometric structures on 3-manifolds, higher dimensional negatively curved manifolds, and hyperbolic groups.

Featuring beautiful illustrations, a rich set of examples, numerous exercises, and an extensive bibliography and index, Hyperbolic Manifolds and Discrete Groups continues to serve as an ideal graduate text and comprehensive reference.*

The book is very clearly written and fairly self-contained. It will be useful to researchers and advanced graduate students in the field and can serve as an ideal guide to Thurston's work and its recent developments.

---Mathematical Reviews

Beyond the hyperbolization theorem, this is an important book which had to be written; some parts are still technical and will certainly be streamlined and shortened in the next years, but together with Otal's work a complete published proof of the hyperbolization theorem is finally available. Apart from the proof itself, the book contains a lot of material which will be useful for various other directions of research.*

---Zentralbatt MATH

This book can act as source material for a postgraduate course and as a reference text on the topic as the references are full and extensive. ... The text is self-contained and very well illustrated.

---ASLIB Book Guide

Zusammenfassung
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.

Inhalt
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.