Coulhon Saloff-Coste isoperimetric inequalities for finitely generated groups

TL;DR

This paper establishes a generalized isoperimetric inequality for finitely generated groups, linking growth, average element length, and F{46}lner function, with implications for amenable groups with exponential growth.

Contribution

It extends the Coulhon Saloff-Coste inequality by incorporating growth and F{46}lner function analysis, providing a more comprehensive understanding of isoperimetric properties.

Findings

Derived a new inequality involving group growth and element length

Reformulated the inequality using the F{46}lner function

Expressed the optimal constant for amenable groups with exponential growth

Abstract

We prove an inequality, valid on any finitely generated group with a fixed finite symmetric generating set, involving the growth of successive balls, and the average length of an element in a ball. It generalizes recent improvements of the Coulhon Saloff-Coste inequality. We reformulate the inequality in terms of the F{\o}lner function; in the case the finitely generated group is amenable with exponential growth, this allows us to express the best possible (outer) constant in the Coulhon Saloff-Coste isoperimetric inequality with the help of a formula involving the growth rate and the asymptotic behavior of the F{\o}lner function.