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This classic text is at the crossroads of many branches of mathematics. Its main focus is on Thurston's hyperbolization theorem, and it contains numerous open problems and conjectures related to the theorem as well as discussions on related topics.
The main goal of the book is to present a proof of the following. Thurston's Hyperbolization Theorem ("The Big Monster"). 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 "generic case." 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 thefiber bundle case) became available, only a sketch of the proof in the fibered case will be given here.
The main goal of the book is to present a proof of the following. Thurston's Hyperbolization Theorem ("The Big Monster"). 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 "generic case." 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 thefiber bundle case) became available, only a sketch of the proof in the fibered case will be given here.
"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
"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
"We recommend the excellent introduction of the present book for the history of the various contributions, and also for a sketch of the proof itself. . . . This is an important book which had to be written . . . the book contains a lot of material which will be useful for various other directions of research." -- Zentralblatt Math
"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
"We recommend the excellent introduction of the present book for the history of the various contributions, and also for a sketch of the proof itself. . . . This is an important book which had to be written . . . the book contains a lot of material which will be useful for various other directions of research." -- Zentralblatt Math
From the reviews:
"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
"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
"We recommend the excellent introduction of the present book for the history of the various contributions, and also for a sketch of the proof itself. . . . This is an important book which had to be written . . . the book contains a lot of material which will be useful for various other directions of research."
-Zentralblatt Math
"Hyperbolic Manifolds and Discrete Groups is an essential text for anyone working in the topology and geometry of 3-manifolds. It is largely self-contained in that it defines all the needed concepts and machinery and often provides proofs of facts that can be found elsewhere in the literature. This book is most valuable for compiling all the needed concepts in one place. This collection is breath-taking in scope ... . Kapovich's book is an excellent, substantial exposition of the varied aspects of the mathematics present." (Scott Taylor, The Mathematical Association of America, January, 2011)
"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
"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
"We recommend the excellent introduction of the present book for the history of the various contributions, and also for a sketch of the proof itself. . . . This is an important book which had to be written . . . the book contains a lot of material which will be useful for various other directions of research."
-Zentralblatt Math
"Hyperbolic Manifolds and Discrete Groups is an essential text for anyone working in the topology and geometry of 3-manifolds. It is largely self-contained in that it defines all the needed concepts and machinery and often provides proofs of facts that can be found elsewhere in the literature. This book is most valuable for compiling all the needed concepts in one place. This collection is breath-taking in scope ... . Kapovich's book is an excellent, substantial exposition of the varied aspects of the mathematics present." (Scott Taylor, The Mathematical Association of America, January, 2011)