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We prove an isoperimetric inequality for groups. As an application, we obtainlower bound on F{\o}lner functions in various nilpotent-by-cyclic groups. Undera regularity assumption, we obtain a characterization of F{\o}lner functions ofthese groups. As another application, we evaluate the asymptotics of theF{\o}lner function of $Sym(\mathbb{Z})\rtimes {\mathbb{Z}}$. We construct newexamples of groups with Shalom's property $H_{\mathrm{FD}}$, in particularamong nilpotent-by-cyclic and lacunary hyperbolic groups. Among these exampleswe find groups with property $H_{\mathrm{FD}}$, which are direct products oflacunary hyperbolic groups and have arbitrarily large F{\o}lner functions.

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