Softcover ISBN:  9780821820803 
Product Code:  MMONO/207 
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eBook ISBN:  9781470446321 
Product Code:  MMONO/207.E 
List Price:  $49.00 
MAA Member Price:  $44.10 
AMS Member Price:  $39.20 
Softcover ISBN:  9780821820803 
eBook: ISBN:  9781470446321 
Product Code:  MMONO/207.B 
List Price:  $101.00 $76.50 
MAA Member Price:  $90.90 $68.85 
AMS Member Price:  $80.80 $61.20 
Softcover ISBN:  9780821820803 
Product Code:  MMONO/207 
List Price:  $52.00 
MAA Member Price:  $46.80 
AMS Member Price:  $41.60 
eBook ISBN:  9781470446321 
Product Code:  MMONO/207.E 
List Price:  $49.00 
MAA Member Price:  $44.10 
AMS Member Price:  $39.20 
Softcover ISBN:  9780821820803 
eBook ISBN:  9781470446321 
Product Code:  MMONO/207.B 
List Price:  $101.00 $76.50 
MAA Member Price:  $90.90 $68.85 
AMS Member Price:  $80.80 $61.20 

Book DetailsTranslations of Mathematical MonographsIwanami Series in Modern MathematicsVolume: 207; 2002; 193 ppMSC: Primary 20; 57; 30; Secondary 46
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 graduatelevel course in discrete groups.
ReadershipGraduate students and research mathematicians interested in topology and geometry.

Table of Contents

Chapters

Basic notions for infinite group

Hyperbolic groups

Automatic groups

Kleinian groups

Prospects


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


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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 graduatelevel course in discrete groups.
Graduate students and research mathematicians interested in topology and geometry.

Chapters

Basic notions for infinite group

Hyperbolic groups

Automatic groups

Kleinian groups

Prospects

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