Hyperbolic Manifolds and Discrete Groups (eBook)
XXVI, 470 Seiten
Birkhäuser Boston (Verlag)
978-0-8176-4913-5 (ISBN)
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 the text is on the 'Big Monster,' i.e., on Thurston's hyperbolization theorem, which has not only completely changes the landscape of 3-dimensinal topology and Kleinian group theory but is one of the central results of 3-dimensional topology.
The book is fairly self-contained, replete with beautiful illustrations, a rich set of examples of key concepts, numerous exercises, and an extensive bibliography and index. It should serve as an ideal graduate course/seminar text or as a comprehensive reference.
The main goal of the book is to present a proof of the following. Thurston's Hyperbolization Theorem ("e;The Big Monster"e;). 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 "e;generic case."e; 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.
Preface 7
Contents 22
1. Three-Dimensional Topology 27
2. Thurston Norm 48
3. Geometry of Hyperbolic Space 56
4. Kleinian Groups 82
5. Teichmüller Theory of Riemann Surfaces 144
6. Introduction of Orbifold Theory 159
7. Complex Projective Structures 184
8. Sociology of Kleinian Groups 191
9. Ultralimits of Metric Spaces 241
10. Introduction to Group Actions on Treens 248
11. Laminations, Foliations, and Trees 263
12. Rips Theory 299
13. Brooks' Theorem and Circle Packings 353
14. Pleated Surfaces and Ends of Hyperbolic Manifolds 371
15. Outline of the Proof of the Hyperbolization Theorem 389
16. Reduction of the Bounded Image Theorem 397
17. The Bounded Image Theorem 402
18. Hyperbolization of Fibrations 415
19. The Orbifold Trick 420
20. Beyond the Hyperbolization Theorem 434
References 449
Index 477
| Erscheint lt. Verlag | 4.8.2009 |
|---|---|
| Reihe/Serie | Modern Birkhäuser Classics | Modern Birkhäuser Classics |
| Zusatzinfo | XXVI, 470 p. 78 illus. |
| Verlagsort | Boston |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| Technik | |
| Schlagworte | 3-dimensional topology • compactification • Complex Analysis • foliation • geometric structures on 3-manifolds • group theory • Homeomorphism • Hyperbolic Geometry • hyperbolic manifolds • Kleinian groups • manifold • Otal's proof • Rips theory • Thurston's hyperbolization theorem • Topology |
| ISBN-10 | 0-8176-4913-1 / 0817649131 |
| ISBN-13 | 978-0-8176-4913-5 / 9780817649135 |
| Haben Sie eine Frage zum Produkt? |
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