Efficient subdivision in hyperbolic groups and applications

We identify the images of the comparision maps from ordinary homology andSobolev homology, respectively, to the $l^1$-homology of a word-hyperbolicgroup with coefficients in complete normed modules. The underlying idea is thatthere is a subdivision procedure for singular chains in negatively curvedspaces that is much more efficient (in terms of the $l^1$-norm) thanbarycentric subdivision. The results of this paper are an important ingredientin a forthcoming proof of the authors that hyperbolic lattices in dimension atleast 3 are rigid with respect to integrable measure equivalence. Moreover, weprove a proportionality principle for the simplicial volume of negativelycurved manifolds with regard to integrable measure equivalence.