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 [BD2], and a proof of the fact that for bi-combable groups the filling function above the quasi-flat rank is asymptotically linear [BD3].

Higher dimensional divergence for mapping class groups

“…Since a finite (n + 1)-presentation of a group composed only of simplices can always be found, it suffices to restrict to simplicial complexes when discussing filling problems. Throughout the paper, we make use of the terminology related to the n-dimensional filling functions in the simplicial setting as described in full detail in [BD2]. We briefly recall a number of relevant notions and results.…”

“…In [BD2] we proved that under certain conditions, which are satisfied in the presence of a bounded quasi-geodesic combing, an arbitrary sphere has a partition into round spheres, such that the sum of the volumes of the spheres in the partition is bounded by a multiple of the volume of the initial sphere. Here a partition consists of the following: finitely many spheres (more generally, hypersurfaces) h 1 , .…”

“…. , h n compose a partition of a sphere h if by filling all of them one obtains a ball filling h (see [BD2]). The hypersurfaces h 1 , .…”

“…For the inductive construction of CAT (0)-spaces with pathological divergence behaviour described in the above theorem, the sharp estimate from above of the divergence is obtained by using results from [BD2] (see Theorem 2.8), which allow us to restrict our study to round spheres (i.e., k-dimensional spheres with diameter controlled by the k-th root of the volume), and a careful study of the structure of round spheres in the CAT (0)-spaces that we construct.…”

“…In Section 2 we recall some basic notions and establish notation which we will use in the paper, in particular some combinatorial terminology and results on mapping class groups. Also, we recall background from [BD2] on combinatorial formulations of isoperimetric and divergence functions and their behavior in the presence of a combing. Section 3 contains the theorem describing the behavior of divergence functions in mapping class groups.…”