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CONTROLLED COARSE HOMOLOGY AND ISOPERIMETRIC INEQUALITIES

By Piotr W. Nowak

Abstract

Abstract. 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 can be viewed as a positive solution to a homological version of the Burnside problem. We also give homological estimates on invariants like isodiametric profiles and type of asymptotic dimension. As applications we characterize existence of primitives of the volume form with prescribed growth and show that coarse homology classes obstruct weighted Poincaré inequalities. 1

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