Quantitative Amenability for Actions of Finitely Generated Groups

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. The decay of such isoperimetric profile is asymptotically invariant if the Radon-Nikodym derivatives is L∞ bounded, and invariant up to a polynomial bound in most cases. 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. We proved that these two isoperimetric profiles are asymptotic same if the group has a sequence of Følner multi-tilings. Moreover, we talk about example of the isoperimetric profile for the actions of free groups on its Poisson boundary in the case of simple random walks, and gives its lower boundary of the decay.