Synopses & Reviews
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 Thurstona (TM)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.
Synopsis
Hyperbolic Manifolds and Discrete Groups 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 is on the "Big Monster," i.e., on Thurston's hyperbolization theorem, which has not only completely changed the landscape of 3-dimensional topology and Kleinian group theory but is one of the central results of 3-dimensional topology. The book presents the first complete proof of Thurston's hyperbolization theorem in the "generic case" and an outline of Otal's proof of the hyperbolization theorem for manifolds fibered over the circle.
This important work contains an extended treatment of the theory of Kleinian groups and group actions on trees, including such key topics as:
* the Kazhdan--Margulis--Zassenhaus theorem
* the Klein and Maskit combination theorems
* the Mostow rigidity theorem
* the Douady--Earle extension theorem for homeomorphisms of the circle
* the smoothness theorem for representation varieties of Kleinian groups
* the Ahlfors finiteness theorem
* the Brooks deformation theorem
* characterization of pseudo-Anosov homeomorphisms
* compactification of character varieties via group actions on trees
Table of Contents
UPDATED, 6/29/2000
[see attached for complete TOC]
Introduction * 1. Three-dimensional Topology * 2. Thurston Norm * 3.
Geometry of the Hyperbolic Space * 4. Kleinian Groups * 5.
Teichm\:uller Theory of Riemann Surfaces * 6. Introduction to the
Orbifold Theory * 7. Complex Projective Structures * 8. Sociology of
Kleinian Groups * 9. Ultralimits of Metric Spaces * 10. Introduction
to Group Actions on Trees * 11. Laminations, Foliations and Trees *
12. Rips' Theory * 13. Brooks' Theorem and Circle Packings * 14.
Pleated Surfaces and Ends of Hyperbolic Manifolds * 15. Outline of the
Proof of the Hyperbolization Theorem * 16. Reduction to The Bounded
Image Theorem * 17. The Bounded Image Theorem * 18. Hyperbolization of
Fibrations * 19. The Orbifold Trick * 20. Beyond the Hyperbolization
Theorem * Bibliography * Index