Groups, Geometry, and Dynamics

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Volume 7, Issue 2, 2013, pp. 263–292
DOI: 10.4171/GGD/182

Published online: 2013-05-07

Efficient subdivision in hyperbolic groups and applications

Uri Bader[1], Alex Furman[2] and Roman Sauer[3]

(1) Weizmann Institute of Science, Rehovot, Israel
(2) University of Illinois at Chicago, USA
(3) Karlsruher Institut für Technologie, Germany

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.

Keywords: Hyperbolic groups, measure equivalence, simplicial volume

Bader Uri, Furman Alex, Sauer Roman: Efficient subdivision in hyperbolic groups and applications. Groups Geom. Dyn. 7 (2013), 263-292. doi: 10.4171/GGD/182