On inner-amenability and boundary actions

Abstract

Let $Γ$ be a discrete countable group. One result in this work is that if $Γ$ is ICC inner-amenable non-amenable then it cannot satisfy the (AO)-property, answering a question posed by C. Anantharaman-Delaroche. A generalization of this phenomenon is also considered. It is also proved that if $Γ$ is a "sufficiently large" discrete subgroup of a product of locally compact second countable bi-exact groups, then it cannot be inner-amenable. Both these results generalize the well-known fact that ICC non-amenable inner-amenable discrete countable groups cannot be bi-exact.