On inner-amenability and boundary actions

Let $\Gamma$ be a discrete countable group.

One result in this work is that if $\Gamma$ 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 $\Gamma$ 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.