  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.
