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Iwanami Series in Modern Mathematics
Discrete Groups
Ken'ichi Ohshika, Osaka University, Japan
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Translations of Mathematical Monographs
Iwanami Series in Modern Mathematics
2002; 193 pp; softcover
Volume: 207
ISBN-10: 0-8218-2080-X
ISBN-13: 978-0-8218-2080-3
List Price: US$43 Member Price: US$34.40
Order Code: MMONO/207

This book deals with geometric and topological aspects of discrete groups. The main topics are hyperbolic groups due to Gromov, automatic group theory, invented and developed by Epstein, whose subjects are groups that can be manipulated by computers, and Kleinian group theory, which enjoys the longest tradition and the richest contents within the theory of discrete subgroups of Lie groups.

What is common among these three classes of groups is that when seen as geometric objects, they have the properties of a negatively curved space rather than a positively curved space. As Kleinian groups are groups acting on a hyperbolic space of constant negative curvature, the technique employed to study them is that of hyperbolic manifolds, typical examples of negatively curved manifolds. Although hyperbolic groups in the sense of Gromov are much more general objects than Kleinian groups, one can apply for them arguments and techniques that are quite similar to those used for Kleinian groups. Automatic groups are further general objects, including groups having properties of spaces of curvature 0. Still, relationships between automatic groups and hyperbolic groups are examined here using ideas inspired by the study of hyperbolic manifolds. In all of these three topics, there is a "soul" of negative curvature upholding the theory. The volume would make a fine textbook for a graduate-level course in discrete groups.

Graduate students and research mathematicians interested in topology and geometry.

Reviews

"Discrete Groups gives a straightforward and very readable introduction to three related topics. It manages to be both thorough and precise at the same time ... clear and complete proofs are given of the results presented ... a handy reference ... would happily recommend it to a graduate student."

-- Bulletin of the LMS