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Volumes of discrete groups and topological complexity of homology spheres. (English) Zbl 0859.20027

The volume of a discrete group is an invariant, which behaves multiplicatively when passing to a subgroup of finite index. Euler characteristics of Bass-Chiswell are examples. In this paper, we develop a “measure-theoretic” approach to volumes. This enables us to answer a well-known question of Lyndon: what is the relation of deficiency to Euler characteristic and what impact on the structure of the group the value of deficiency has? We also study volumes related to secondary characteristic classes and give applications to topological complexity of homology three-spheres.

MSC:

20F65 Geometric group theory
57M07 Topological methods in group theory
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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References:

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