Isoperimetric inequalities, shapes of Følner sets and groups with Shalom’s property H FD

We prove an isoperimetric inequality for groups. As an application we show that any Grigrochuk group of intermediate growth has at least exponential Følner function. As another application, we obtain lower bounds on Følner functions in various nilpotent-by-cyclic groups. Under a regularity assumption, we obtain a characterization of Følner functions of these groups. As a further application, we evaluate the asymptotics of the Følner function of Sym ( ℤ ) ⋊ ℤ . We study examples of groups with Shalom’s property H FD among nilpotent-by-cyclic groups. We show that there exist lacunary hyperbolic groups with property H FD . We find groups with property H FD , which are direct products of lacunary hyperbolic groups and have arbitrarily large Følner functions.