Discontinuous Groups of Isometries in the Hyperbolic Plane

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Walter de Gruyter, 2003 - Mathematics - 364 pages

This is an introductory textbook on isometry groups of the hyperbolic plane. Interest in such groups dates back more than 120 years. Examples appear in number theory (modular groups and triangle groups), the theory of elliptic functions, and the theory of linear differential equations in the complex domain (giving rise to the alternative name Fuchsian groups).

The current book is based on what became known as the famous Fenchel-Nielsen manuscript. Jakob Nielsen (1890-1959) started this project well before World War II, and his interest arose through his deep investigations on the topology of Riemann surfaces and from the fact that the fundamental group of a surface of genus greater than one is represented by such a discontinuous group. Werner Fenchel (1905-1988) joined the project later and overtook much of the preparation of the manuscript. The present book is special because of its very complete treatment of groups containing reversions and because it avoids the use of matrices to represent Moebius maps.

This text is intended for students and researchers in the many areas of mathematics that involve the use of discontinuous groups.


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Möbius transformations and noneuclidean geometry
7 Isometric transformations
Discontinuous groups of motions and reversions
Surfaces associated with discontinuous groups
Decompositions of groups
Isomorphism and homeomorphism
case of quasicompactness
Symbols and definitions

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Page 356 - L. Bers and I. Kra (Editors), A crash course on Kleinian groups, Lecture Notes in Math., vol. 400, Springer-Verlag, Berlin and New York, 1974.
Page 356 - O. Forster: Lectures on Riemann Surfaces. Grad. Texts in Math. 81, Springer- Verlag, Berlin-Heidelberg-New York, 1991.
Page 355 - RDM Accola: Riemann Surfaces, Theta Functions, and Abelian Automorphisms Groups. Lecture Notes in Math.
Page 355 - Ahlfors: Mobius Transformations in Several Dimensions. School of Mathematics, University of Minnesota, 1 98 1 . [5] LV Ahlfors: Collected Mathematical Papers, 1-2.

About the author (2003)

Asmus L. Schmidt is Associate Professor at the Institute for Mathematical Sciences of the University of Copenhagen, Denmark.

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