Efficient subdivision in hyperbolic groups and applications

Abstract

We identify the images of the comparison maps from ordinary homology and Sobolev homology, respectively, to the ${\ell }^1$-homology of a word-hyperbolic group with coefficients in complete normed modules. The underlying idea is that there is a subdivision procedure for singular chains in negatively curved spaces that is much more efficient (in terms of the ${\ell }^1$-norm) than barycentric subdivision. The results of this paper are an important ingredient in a forthcoming proof of the authors that hyperbolic lattices in dimension $\ge 3$ are rigid with respect to integrable measure equivalence. Moreover, we prove a new proportionality principle for the simplicial volume of manifolds with word-hyperbolic fundamental groups.