Combinatorial higher dimensional isoperimetry and divergence

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].