Quantitative Amenability for Actions of Finitely Generated Groups

TL;DR

This paper introduces a generalized concept of isoperimetric profiles for group actions, linking their decay to amenability, and explores relationships between these profiles and group properties in measure-preserving contexts.

Contribution

It extends classical isoperimetric profiles to actions of finitely generated groups and establishes their equivalence to amenability in the sense of Zimmer.

Findings

Decay of isoperimetric profiles characterizes amenability of actions

Relation between action profiles and group profiles in measure-preserving cases

Generalization of isoperimetric concepts to orbit graphings

Abstract

We generalize the notion of isoperimetric profiles of finitely generated groups to their actions by measuring the boundary of finite subgraphings of the orbit graphing. We prove that like the classical isoperimetric profiles for groups, decay of the isoperimetric profile for an essentially-free action is equivalent to amenability of the action in the sense of Zimmer.For measure-preserving actions, we relate the isoperimetric profiles of the actions and the group.