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  • © 2010

Hyperbolic Manifolds and Discrete Groups

Birkhäuser

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  • Includes beautiful illustrations, a rich set of examples of key concepts, numerous exercises

  • An extensive bibliography and index are complemented by a glossary of terms

  • Presents the first complete proof of the generic case of Thurston’s hyperbolization theorem

  • Includes supplementary material: sn.pub/extras

Part of the book series: Modern Birkhäuser Classics (MBC)

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  • ISBN: 978-0-8176-4913-5
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Table of contents (20 chapters)

  1. Front Matter

    Pages i-xxvii
  2. Three-Dimensional Topology

    • Michael Kapovich
    Pages 1-21
  3. Thurston Norm

    • Michael Kapovich
    Pages 23-30
  4. Geometry of Hyperbolic Space

    • Michael Kapovich
    Pages 31-56
  5. Kleinian Groups

    • Michael Kapovich
    Pages 57-118
  6. Teichmüller Theory of Riemann Surfaces

    • Michael Kapovich
    Pages 119-133
  7. Introduction to Orbifold Theory

    • Michael Kapovich
    Pages 135-159
  8. Complex Projective Structures

    • Michael Kapovich
    Pages 161-167
  9. Sociology of Kleinian Groups

    • Michael Kapovich
    Pages 169-218
  10. Ultralimits of Metric Spaces

    • Michael Kapovich
    Pages 219-225
  11. Introduction to Group Actions on Trees

    • Michael Kapovich
    Pages 227-241
  12. Laminations, Foliations, and Trees

    • Michael Kapovich
    Pages 243-278
  13. Rips Theory

    • Michael Kapovich
    Pages 279-332
  14. Brooks’ Theorem and Circle Packings

    • Michael Kapovich
    Pages 333-350
  15. Pleated Surfaces and Ends of Hyperbolic Manifolds

    • Michael Kapovich
    Pages 351-368
  16. Reduction to the Bounded Image Theorem

    • Michael Kapovich
    Pages 377-381
  17. The Bounded Image Theorem

    • Michael Kapovich
    Pages 383-395
  18. Hyperbolization of Fibrations

    • Michael Kapovich
    Pages 397-401
  19. The Orbifold Trick

    • Michael Kapovich
    Pages 403-416

About this book

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

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

Reviews

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)

Authors and Affiliations

  • Dept. Mathematics, University of California, Davis, Davis, U.S.A.

    Michael Kapovich

Bibliographic Information

Buying options

eBook
USD 109.00
Price excludes VAT (USA)
  • ISBN: 978-0-8176-4913-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 139.99
Price excludes VAT (USA)