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The Geometry of Discrete Groups

The Geometry of Discrete Groups
Catalogue Information
Field name Details
Dewey Class 512.2
Title The Geometry of Discrete Groups ([EBook] /) / by Alan F. Beardon.
Author Beardon, Alan F.
Other name(s) SpringerLink (Online service)
Publication New York, NY : : Springer New York : : Imprint: Springer, , 1983.
Physical Details XII, 340 p. : online resource.
Series Graduate texts in mathematics 0072-5285 ; ; 91
ISBN 9781461211464
Summary Note 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.:
Contents note 1 Preliminary Material -- 2 Matrices -- 3 Möbius Transformations on ?n -- 4 Complex Möbius Transformations -- 5 Discontinuous Groups -- 6 Riemann Surfaces -- 7 Hyperbolic Geometry -- 8 Fuchsian Groups -- 9 Fundamental Domains -- 10 Finitely Generated Groups -- 11 Universal Constraints on Fuchsian Groups -- References.
System details note Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
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