Hyperbolic Manifolds and Discrete Groups

✍ Scribed by Michael Kapovich (auth.)

• Publisher: Birkhäuser Basel
• Year: 2010
• Tongue: English
• Leaves: 485
• Series: Modern Birkhäuser Classics
• Edition: 1
• Category: Library

✦ Synopsis

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 Thurston’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

✦ Table of Contents

Front Matter....Pages i-xxvii
Three-Dimensional Topology....Pages 1-21
Thurston Norm....Pages 23-30
Geometry of Hyperbolic Space....Pages 31-56
Kleinian Groups....Pages 57-118
Teichmüller Theory of Riemann Surfaces....Pages 119-133
Introduction to Orbifold Theory....Pages 135-159
Complex Projective Structures....Pages 161-167
Sociology of Kleinian Groups....Pages 169-218
Ultralimits of Metric Spaces....Pages 219-225
Introduction to Group Actions on Trees....Pages 227-241
Laminations, Foliations, and Trees....Pages 243-278
Rips Theory....Pages 279-332
Brooks’ Theorem and Circle Packings....Pages 333-350
Pleated Surfaces and Ends of Hyperbolic Manifolds....Pages 351-368
Outline of the Proof of the Hyperbolization Theorem....Pages 369-376
Reduction to the Bounded Image Theorem....Pages 377-381
The Bounded Image Theorem....Pages 383-395
Hyperbolization of Fibrations....Pages 397-401
The Orbifold Trick....Pages 403-416
Beyond the Hyperbolization Theorem....Pages 417-431
Back Matter....Pages 433-467

✦ Subjects

Topology; Group Theory and Generalizations; Geometry; Manifolds and Cell Complexes (incl. Diff.Topology)