Invariant random subgroups of groups acting on rooted trees

TL;DR

This paper studies invariant random subgroups in groups acting on rooted trees, revealing their structure and abundance, especially in branch groups, and extending previous results on ergodic IRS properties.

Contribution

It characterizes ergodic invariant random subgroups in groups acting on rooted trees, showing they contain level stabilizers or derived subgroups, and demonstrates the existence of many atomless ergodic IRS's in weakly branch groups.

Findings

Nontrivial ergodic IRS contains a level stabilizer.

Ergodic IRS in branch groups contains derived subgroups of rigid level stabilizers.

Weakly branch groups have continuum many atomless ergodic IRS's.

Abstract

We investigate invariant random subgroups in groups acting on rooted trees. Let $\mathrm{Alt}_f(T)$ be the group of finitary even automorphisms of the $d$-ary rooted tree $T$. We prove that a nontrivial ergodic IRS of $\mathrm{Alt}_f(T)$ that acts without fixed points on the boundary of $T$ contains a level stabilizer, in particular it is the random conjugate of a finite index subgroup. Applying the technique to branch groups we prove that an ergodic IRS in a finitary regular branch group contains the derived subgroup of a generalized rigid level stabilizer. We also prove that every weakly branch group has continuum many distinct atomless ergodic IRS's. This extends a result of Benli, Grigorchuk and Nagnibeda who exhibit a group of intermediate growth with this property.