Controlled coarse homology and isoperimetric inequalities

TL;DR

This paper introduces a coarse homology theory linked to group amenability, characterizes isoperimetric inequalities via homological vanishing, and explores applications to primitives of volume forms and Poincare inequalities.

Contribution

It develops a new coarse homology framework with growth conditions, relating homological vanishing to isoperimetric inequalities and providing solutions to a homological version of the Burnside problem.

Findings

Homology vanishing characterizes amenability of groups.

Linear control suffices for vanishing of the fundamental class.

Homological obstructions affect weighted Poincare inequalities.

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 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.