We identify the images of the comparison maps from ordinary homology and Sobolev homology, respectively, to the -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 -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 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.
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Uri Bader, Alex Furman, Roman Sauer, Efficient subdivision in hyperbolic groups and applications. Groups Geom. Dyn. 7 (2013), no. 2, pp. 263–292DOI 10.4171/GGD/182