Invariant Means and the Structure of Inner Amenable Groups

We study actions of countable discrete groups which are amenable in the sense that there exists a mean on X which is invariant under the action of G. Assuming that G is nonamenable, we obtain structural results for the stabilizer subgroups of amenable actions which allow us to relate the first `-Betti number of G with that of the stabilizer subgroups. An analogous relationship is also shown to hold for cost. This relationship becomes even more pronounced for transitive amenable actions, leading to a simple criterion for vanishing of the first `-Betti number and triviality of cost. Moreover, for any marked finitely generated nonamenable group G we establish a uniform isoperimetric threshold for Schreier graphs G/H of G, beyond which the group H is necessarily weakly normal in G. Even more can be said in the particular case of an atomless mean for the conjugation action – that is, when G is inner amenable. We show that inner amenable groups have cost 1 and moreover they have fixed price. We establish Ufin-cocycle superrigidity for the Bernoulli shift of any nonamenable inner amenable group. In addition, we provide a concrete structure theorem for inner amenable linear groups over an arbitrary field. We also completely characterize linear groups which are stable in the sense of Jones and Schmidt. Our analysis of stability leads to many new examples of stable groups; notably, all nontrivial countable subgroups of the group H(R), recently studied by Monod, are stable. This includes nonamenable groups constructed by Monod and by Lodha and Moore, as well as Thompson’s group F .