Isoperimetry in Finitely Generated Groups

TL;DR

This paper explores isoperimetric inequalities in finitely generated groups, establishing a lower bound via the $$-transform of the growth function and deriving asymptotic estimates for the F{}lner function.

Contribution

It introduces a novel lower bound on isoperimetric quotients using the $$-transform, linking geometric group theory with transform techniques.

Findings

Lower bound on isoperimetric quotient given by the $$-transform.

Asymptotic estimates for the F{}lner function.

Discussion of properties and computational methods for the $$-transform.

Abstract

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 $\U-$transform of its growth function, which is a variant of the Legendre transform. From this lower bound, we obtain some asymptotic estimates for the F{\o}lner function of the group. The paper also includes a discussion of some basic definitions from Geometric Group Theory and some basic properties of the $\U$-transform, including some computational techniques and its relation with the Legendre transform.