Isoperimetric inequalities, shapes of Følner sets and groups with Shalom's property $H_{\mathrm{FD}}$

Abstract

We prove an isoperimetric inequality for groups. As an application, we obtain lower bound on Følner functions in various nilpotent-by-cyclic groups. Under a regularity assumption, we obtain a characterization of Følner functions of these groups. As another application, we evaluate the asymptotics of the Følner function of $Sym(\mathbb{Z})⋊\mathbb{Z}$. We construct new examples of groups with Shalom's property $H_{\mathrm{FD}}$, in particular among nilpotent-by-cyclic and lacunary hyperbolic groups. Among these examples we find groups with property $H_{\mathrm{FD}}$, which are direct products of lacunary hyperbolic groups and have arbitrarily large Følner functions.

arXiv commentA symmetry assumption on the set T is added in Theorem 1.1