On inner-amenability and boundary actions

Let $\mathrm{\Gamma }$ be a discrete countable group. The first main result of this work is that if $\mathrm{\Gamma }$ is ICC inner-amenable non-amenable then it cannot satisfy the (AO)-property, answering a question posed by C. Anantharaman-Delaroche. It is also proved that if $\mathrm{\Gamma }$ 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.