We study actions of countable discrete groups which are amenable in the sense that there exists a mean on which is invariant under the action of . Assuming that is nonamenable, we obtain structural results for the stabilizer subgroups of amenable actions which allow us to relate the first -Betti number of with that of the stabilizer subgroups. In addition, for any marked finitely generated nonamenable group we establish a uniform isoperimetric threshold for Schreier graphs of , beyond which the group is necessarily weakly normal in .
Even more can be said in the particular case of an atomless mean for the conjugation action—that is, when is inner amenable. We show that inner amenable groups have fixed price , 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 , of piecewise-projective homeomorphisms of the line, are stable.
"Invariant means and the structure of inner amenable groups." Duke Math. J. 169 (13) 2571 - 2628, 15 September 2020. https://doi.org/10.1215/00127094-2019-0070