Amenable Invariant Random Subgroups

TL;DR

This paper proves that amenable invariant random subgroups are contained within the amenable radical of a group, and similarly addresses property (T), providing new insights into subgroup structures and their topological properties.

Contribution

It establishes that amenable invariant random subgroups are contained in the amenable radical and proves a related result for property (T), also showing the set of amenable subgroups is Borel measurable.

Findings

Amenable invariant random subgroups lie in the amenable radical.

The set of amenable subgroups is Borel in the Chabauty topology.

A similar statement is proved for property (T).

Abstract

We show that an amenable Invariant Random Subgroup of a locally compact second countable group lives in the amenable radical. This answers a question raised in the introduction of the paper "Kesten's Theorem for Invariant Random Subgroup" by Abert, Glasner and Virag. We also consider, in the opposite direction, property (T), and prove a similar statement for this property. The Appendix by Phillip Wesolek proves that the set of amenable subgroups is a Borel subset in the Chabauty topology.