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

We prove an isoperimetric inequality for groups. As an application, we obtain lower bound on F{\o}lner functions in various nilpotent-by-cyclic groups. Under a regularity assumption, we obtain a characterization of F{\o}lner functions of these groups. As another application, we evaluate the asymptotics of the F{\o}lner function of $Sym(\mathbb{Z})\rtimes {\mathbb{Z}}$. 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 F{\o}lner functions.
Comment: A symmetry assumption on the set T is added in Theorem 1.1