Originally published as volume 183 in the series: Progress in Mathematics
2010, XXVII, 467p. 78 illus..
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Includes beautiful illustrations, a rich set of examples of key concepts, and numerous exercises
Contains an extensive bibliography and index that are complemented by a glossary of terms
Presents the first complete proof of the generic case of Thurston’s hyperbolization theorem
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.
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.
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
Content Level »Research
Keywords »3-dimensional topology - Compactification - Group theory - Homeomorphism - Kleinian groups - Otal's proof - Rips theory - Thurston's hyperbolization theorem - complex analysis - foliation - geometric structures on 3-manifolds - hyperbolic geometry - hyperbolic manifolds - manifold - topology