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(Z)βZ. We construct new
examples of groups with Shalom's property HFDβ, in particular
among nilpotent-by-cyclic and lacunary hyperbolic groups. Among these examples
we find groups with property HFDβ, which are direct products of
lacunary hyperbolic groups and have arbitrarily large F{\o}lner functions