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Lectures on Hyperbolic Geometry

  • Riccardo Benedetti
  • Carlo Petronio

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Riccardo Benedetti, Carlo Petronio
    Pages 1-43
  3. Riccardo Benedetti, Carlo Petronio
    Pages 45-82
  4. Riccardo Benedetti, Carlo Petronio
    Pages 83-131
  5. Riccardo Benedetti, Carlo Petronio
    Pages 133-157
  6. Riccardo Benedetti, Carlo Petronio
    Pages 159-272
  7. Riccardo Benedetti, Carlo Petronio
    Pages 273-320
  8. Back Matter
    Pages 321-333

About this book

Introduction

In recent years hyperbolic geometry has been the object and the preparation for extensive study that has produced important and often amazing results and also opened up new questions. The book concerns the geometry of manifolds and in particular hyperbolic manifolds; its aim is to provide an exposition of some fundamental results, and to be as far as possible self-contained, complete, detailed and unified. Since it starts from the basics and it reaches recent developments of the theory, the book is mainly addressed to graduate-level students approaching research, but it will also be a helpful and ready-to-use tool to the mature researcher. After collecting some classical material about the geometry of the hyperbolic space and the Teichmüller space, the book centers on the two fundamental results: Mostow's rigidity theorem (of which a complete proof is given following Gromov and Thurston) and Margulis' lemma. These results form the basis for the study of the space of the hyperbolic manifolds in all dimensions (Chabauty and geometric topology); a unified exposition is given of Wang's theorem and the Jorgensen-Thurston theory. A large part is devoted to the three-dimensional case: a complete and elementary proof of the hyperbolic surgery theorem is given based on the possibility of representing three manifolds as glued ideal tetrahedra. The last chapter deals with some related ideas and generalizations (bounded cohomology, flat fiber bundles, amenable groups). This is the first book to collect this material together from numerous scattered sources to give a detailed presentation at a unified level accessible to novice readers.

Keywords

Cohomology Flat Fiber Bundles Geometry of Manifolds Hyperbolic Geometry manifold

Authors and affiliations

  • Riccardo Benedetti
    • 1
  • Carlo Petronio
    • 1
  1. 1.Dipartimento di MatematicaUniversità degli Studi di PisaPisaItaly

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-58158-8
  • Copyright Information Springer-Verlag Berlin Heidelberg 1992
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-540-55534-6
  • Online ISBN 978-3-642-58158-8
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • Buy this book on publisher's site
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