Isoperimetry in Finitely Generated Groups

We revisit the isoperimetric inequalities for finitely generated groups introduced and studied by N. Varopoulos, T. Coulhon and L. Saloff-Coste. Namely we show that a lower bound on the isoperimetric quotient of finite subsets in a finitely generated group is given by the \(\mathcal {U}-\) transform of its growth function, which is a variant of the Legendre transform. From this lower bound, we obtain some asymptotic estimates for the Følner function of the group. The paper also includes a discussion of some basic definitions from Geometric Group Theory and some basic properties of the \(\mathcal {U}\) -transform, including some computational techniques and its relation with the Legendre transform.