Hyperbolic Manifolds and Discrete Groups
(Sprache: Englisch)
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,'...
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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
Klappentext zu „Hyperbolic Manifolds and Discrete Groups “
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
Inhaltsverzeichnis zu „Hyperbolic Manifolds and Discrete Groups “
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
Bibliographische Angaben
- Autor: Michael Kapovich
- 2000, XXV, 467 Seiten, 75 Schwarz-Weiß-Abbildungen, Maße: 15,5 x 23,5 cm, Gebunden, Englisch
- Verlag: Springer
- ISBN-10: 0817639047
- ISBN-13: 9780817639044
Sprache:
Englisch
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