Invariant random subgroups of the free group

TL;DR

This paper explores the diversity of invariant random subgroups in nonabelian free groups, revealing a rich variety of IRS's, contrasting with the rigidity observed in higher rank semisimple Lie groups with property (T).

Contribution

It demonstrates that nonabelian free groups possess a large and diverse collection of invariant random subgroups, highlighting fundamental differences from higher rank semisimple Lie groups.

Findings

Nonabelian free groups have a large 'zoo' of IRS's.

Contrasts with the rigidity of IRS's in higher rank semisimple Lie groups.

Shows fundamental differences in subgroup structures between free groups and higher rank groups.

Abstract

Let $G$ be a locally compact group. A random closed subgroup with conjugation-invariant law is called an {\em invariant random subgroup} or IRS for short. We show that each nonabelian free group has a large "zoo" of IRS's. This contrasts with results of Stuck-Zimmer which show that there are no non-obvious IRS's of higher rank semisimple Lie groups with property (T).