Controlled coarse homology and isoperimetric inequalities

Details Controlled coarse homology and isoperimetric inequalities Controlled coarse homology and isoperimetric inequalities

2008-09-19

arxiv.org/abs/0809.3286v4

J. Topol. 3 (2010), no. 2, 443--462

Mathematics

Group Theory

Final version, to appear in the Journal of Topology

Scientific paper

10.1112/jtopol/jtq013

We study a coarse homology theory with prescribed growth conditions. For a finitely generated group G with the word length metric this homology theory turns out to be related to amenability of G. We characterize vanishing of a certain fundamental class in our homology in terms of an isoperimetric inequality on G and show that on any group at most linear control is needed for this class to vanish. The latter is a homological version of the classical Burnside problem for infinite groups, with a positive solution. As applications we characterize existence of primitives of the volume form with prescribed growth and show that coarse homology classes obstruct weighted Poincare inequalities.

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