  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.
