The notion of isoperimetric profiles is a generalization of isoperimetric dimension, which is a large-scale invariant. In the context of discrete groups isoperimetric profiles were introduced by Vershik, but were well-defined only for amenable groups. The purpose of this talk is a definition of an isoperimetric profile of an action of a finitely generated group on a compact Hausdorff space. We show that these profiles share many properties with original invariants for amenable groups/regularly exhaustible open manifolds. We also define the generalized isoperimetric profile of an amenable group via the action of G on its Stone-Cech compactification.

For this last profile we show that it is a quasi-isometry invariant, explore the relation to growth and asymptotic dimension. We also compute the isoperimetric profile for several classes of groups for which the classical profile was not defined, e.g. hyperbolic groups.