Invariant means and the structure of inner amenable groups

Abstract

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 ℓ 2 -Betti number of G with that of the stabilizer subgroups. In addition, 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 fixed price 1 , and we establish 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. As a special case of inner amenability, we consider groups which are stable in the sense of Jones and Schmidt, obtaining a complete characterization of linear groups which are stable. Our analysis of stability leads to many new examples of stable groups; notably, all nontrivial countable subgroups of the group H ( R ) , of piecewise-projective homeomorphisms of the line, are stable.