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  • © 1988

Kleinian Groups

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

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  • ISBN: 978-3-642-61590-0
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Table of contents (10 chapters)

  1. Front Matter

    Pages I-XIII
  2. Fractional Linear Transformations

    • Bernard Maskit
    Pages 1-14
  3. Discontinuous Groups in the Plane

    • Bernard Maskit
    Pages 15-40
  4. Covering Spaces

    • Bernard Maskit
    Pages 41-52
  5. Groups of Isometries

    • Bernard Maskit
    Pages 53-83
  6. The Geometric Basic Groups

    • Bernard Maskit
    Pages 84-114
  7. Geometrically Finite Groups

    • Bernard Maskit
    Pages 115-134
  8. Combination Theorems

    • Bernard Maskit
    Pages 135-170
  9. A Trip to the Zoo

    • Bernard Maskit
    Pages 171-213
  10. B-Groups

    • Bernard Maskit
    Pages 214-248
  11. Function Groups

    • Bernard Maskit
    Pages 249-318
  12. Back Matter

    Pages 319-328

About this book

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 geomeĀ­ 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

  • Dept. of Mathematics, SUNY at Stony Brook, Stony Brook, USA

    Bernard Maskit

Bibliographic Information

Buying options

eBook USD 109.00
Price excludes VAT (USA)
  • ISBN: 978-3-642-61590-0
  • 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 159.99
Price excludes VAT (USA)
Hardcover Book USD 149.99
Price excludes VAT (USA)