Grundlehren der mathematischen Wissenschaften

Kleinian Groups

Authors: Maskit, Bernard

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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.

Table of contents (10 chapters)

Table of contents (10 chapters)

Buy this book

eBook $89.00
price for USA in USD (gross)
  • ISBN 978-3-642-61590-0
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $149.99
price for USA in USD
  • ISBN 978-3-540-17746-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $119.99
price for USA in USD
  • ISBN 978-3-642-64878-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Kleinian Groups
Authors
Series Title
Grundlehren der mathematischen Wissenschaften
Series Volume
287
Copyright
1988
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-61590-0
DOI
10.1007/978-3-642-61590-0
Hardcover ISBN
978-3-540-17746-3
Softcover ISBN
978-3-642-64878-6
Series ISSN
0072-7830
Edition Number
1
Number of Pages
XIII, 328
Topics