  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.
