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Kleinian Groups

  • Bernard Maskit

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 287)

Table of contents

  1. Front Matter
    Pages I-XIII
  2. Bernard Maskit
    Pages 1-14
  3. Bernard Maskit
    Pages 15-40
  4. Bernard Maskit
    Pages 41-52
  5. Bernard Maskit
    Pages 53-83
  6. Bernard Maskit
    Pages 84-114
  7. Bernard Maskit
    Pages 115-134
  8. Bernard Maskit
    Pages 135-170
  9. Bernard Maskit
    Pages 171-213
  10. Bernard Maskit
    Pages 214-248
  11. Bernard Maskit
    Pages 249-318
  12. Back Matter
    Pages 319-328

About this book

Introduction

The modern theory of Kleinian groups starts with the work of Lars Ahlfors and Lipman Bers; specifically with Ahlfors' finiteness theorem, and Bers' observation that their joint work on the Beltrami equation has deep implications for the theory of Kleinian groups and their deformations. From the point of view of uniformizations of Riemann surfaces, Bers' observation has the consequence that the question of understanding the different uniformizations of a finite Riemann surface poses a purely topological problem; it is independent of the conformal structure on the surface. The last two chapters here give a topological description of the set of all (geometrically finite) uniformizations of finite Riemann surfaces. We carefully skirt Ahlfors' finiteness theorem. For groups which uniformize a finite Riemann surface; that is, groups with an invariant component, one can either start with the assumption that the group is finitely generated, and then use the finiteness theorem to conclude that the group represents only finitely many finite Riemann surfaces, or, as we do here, one can start with the assumption that, in the invariant component, the group represents a finite Riemann surface, and then, using essentially topological techniques, reach the same conclusion. More recently, Bill Thurston wrought a revolution in the field by showing that one could analyze Kleinian groups using 3-dimensional hyperbolic geome­ try, and there is now an active school of research using these methods.

Keywords

Area Dimension Finite Group theory Invariant Riemann surface approximation convergence field finite group function hyperbolic geometry mapping theorem uniformization

Authors and affiliations

  • Bernard Maskit
    • 1
  1. 1.Dept. of MathematicsSUNY at Stony BrookStony BrookUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-61590-0
  • Copyright Information Springer-Verlag Berlin Heidelberg 1988
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-64878-6
  • Online ISBN 978-3-642-61590-0
  • Series Print ISSN 0072-7830
  • Buy this book on publisher's site