Efficient subdivision in hyperbolic groups and applications

TL;DR

This paper introduces an efficient subdivision method for singular chains in negatively curved spaces, linking homology theories and contributing to rigidity results for hyperbolic lattices and a proportionality principle for simplicial volume.

Contribution

It develops a subdivision procedure that improves the analysis of homology in hyperbolic groups, aiding in rigidity proofs and volume comparisons.

Findings

Identifies images of comparison maps in homology theories.

Establishes an efficient subdivision procedure for negatively curved spaces.

Proves a proportionality principle for simplicial volume.

Abstract

We identify the images of the comparision maps from ordinary homology and Sobolev homology, respectively, to the $l^1$-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 $l^1$-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 at least 3 are rigid with respect to integrable measure equivalence. Moreover, we prove a proportionality principle for the simplicial volume of negatively curved manifolds with regard to integrable measure equivalence.