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The Geometry of Discrete Groups

  • Alan F. Beardon
Textbook

Part of the Graduate Texts in Mathematics book series (GTM, volume 91)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Alan F. Beardon
    Pages 1-8
  3. Alan F. Beardon
    Pages 9-19
  4. Alan F. Beardon
    Pages 20-55
  5. Alan F. Beardon
    Pages 56-82
  6. Alan F. Beardon
    Pages 83-115
  7. Alan F. Beardon
    Pages 116-125
  8. Alan F. Beardon
    Pages 126-187
  9. Alan F. Beardon
    Pages 188-203
  10. Alan F. Beardon
    Pages 204-252
  11. Alan F. Beardon
    Pages 253-286
  12. Alan F. Beardon
    Pages 287-327
  13. Back Matter
    Pages 329-340

About this book

Introduction

This text is intended to serve as an introduction to the geometry of the action of discrete groups of Mobius transformations. The subject matter has now been studied with changing points of emphasis for over a hundred years, the most recent developments being connected with the theory of 3-manifolds: see, for example, the papers of Poincare [77] and Thurston [101]. About 1940, the now well-known (but virtually unobtainable) Fenchel-Nielsen manuscript appeared. Sadly, the manuscript never appeared in print, and this more modest text attempts to display at least some of the beautiful geo­ metrical ideas to be found in that manuscript, as well as some more recent material. The text has been written with the conviction that geometrical explana­ tions are essential for a full understanding of the material and that however simple a matrix proof might seem, a geometric proof is almost certainly more profitable. Further, wherever possible, results should be stated in a form that is invariant under conjugation, thus making the intrinsic nature of the result more apparent. Despite the fact that the subject matter is concerned with groups of isometries of hyperbolic geometry, many publications rely on Euclidean estimates and geometry. However, the recent developments have again emphasized the need for hyperbolic geometry, and I have included a comprehensive chapter on analytical (not axiomatic) hyperbolic geometry. It is hoped that this chapter will serve as a "dictionary" offormulae in plane hyperbolic geometry and as such will be of interest and use in its own right.

Keywords

Finite Geometry Groups Riemann surface complex analysis constraint form hyperbolic geometry matrices transformation

Authors and affiliations

  • Alan F. Beardon
    • 1
  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeEngland

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-1146-4
  • Copyright Information Springer-Verlag Berlin Heidelberg 1983
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-7022-5
  • Online ISBN 978-1-4612-1146-4
  • Series Print ISSN 0072-5285
  • Buy this book on publisher's site