# Combinatorial higher dimensional isoperimetry and divergence

Behrstock, J
Drutu, C

3 November 2017

## Journal:

Journal of Topology and Analysis

## Last Updated:

2019-08-18T18:01:57.003+01:00

## DOI:

10.1142/S1793525319500225

## abstract:

In this paper we provide a framework for the study of isoperimetric problems
in finitely generated group, through a combinatorial study of universal covers
of compact simplicial complexes. We show that, when estimating filling
functions, one can restrict to simplicial spheres of particular shapes, called
"round" and "unfolded", provided that a bounded quasi-geodesic combing exists.
We prove that the problem of estimating higher dimensional divergence as well
can be restricted to round spheres. Applications of these results include a
combinatorial analogy of the Federer--Fleming inequality for finitely generated
groups, the construction of examples of $CAT(0)$--groups with higher
dimensional divergence equivalent to $x^d$ for every degree d
[arXiv:1305.2994], and a proof of the fact that for bi-combable groups the
filling function above the quasi-flat rank is asymptotically linear
[Behrstock-Drutu].

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