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.
Readership
Graduate students and research mathematicians interested in
topology and geometry.
