Read more
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
List 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
