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 ${\ell }^2$-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 $H\left(\mathbb{R}\right)$, of piecewise-projective homeomorphisms of the line, are stable.