Quantitative Amenability for Actions of Finitely Generated Groups

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 the essentially-free action is equivalent to amenability of the action in the sense of Zimmer. For measure-preserving actions, we find the bounds between the original and generalized isoperimetric profiles for measure-preserving actions.Comment: 12 page