Hyperbolic Manifolds and Discrete Groups /
Kapovich, Michael, 1963-
Hyperbolic Manifolds and Discrete Groups / Michael Kapovich. - Boston : Birkhauser, 2001. - xxv, 467 p. ; 24 cm. - Progress in mathematics ; v. 183 .
Includes bibliographical references (pp433-460) and index.
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
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; and, compactification of character varieties via group actions on trees.
0817639047 3764339047
Hyperbolic spaces.
Discrete groups.
QA685 / .K36 2000
Hyperbolic Manifolds and Discrete Groups / Michael Kapovich. - Boston : Birkhauser, 2001. - xxv, 467 p. ; 24 cm. - Progress in mathematics ; v. 183 .
Includes bibliographical references (pp433-460) and index.
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
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; and, compactification of character varieties via group actions on trees.
0817639047 3764339047
Hyperbolic spaces.
Discrete groups.
QA685 / .K36 2000