Combinatorial higher dimensional isoperimetry and divergence

TL;DR

This paper develops a combinatorial framework for studying isoperimetric problems and divergence in finitely generated groups, simplifying estimates by focusing on specific sphere shapes under certain conditions.

Contribution

It introduces a method to restrict isoperimetric and divergence estimates to 'round' spheres in groups with bounded quasi-geodesic combings, advancing understanding of higher dimensional filling functions.

Findings

Establishes a combinatorial analogy of Federer--Fleming inequality for groups.

Constructs examples of CAT(0)-groups with specific divergence behaviors.

Shows linearity of filling functions above quasi-flat rank in bi-combable groups.

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].