On inner-amenability and boundary actions

TL;DR

This paper explores the properties of inner-amenability in discrete groups, showing that certain non-amenable groups cannot have the (AO)-property or be inner-amenable if they are ICC, and extends these results to large subgroups of product groups.

Contribution

It establishes new restrictions on inner-amenability related to the (AO)-property and bi-exactness in ICC non-amenable groups and their subgroups.

Findings

ICC non-amenable inner-amenable groups cannot satisfy (AO)

Large subgroups of product bi-exact groups are not inner-amenable

Generalizes the fact that ICC non-amenable inner-amenable groups are not bi-exact

Abstract

Let $\Gamma$ be a discrete countable group. The first main result of 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. 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.