Invariant random subgroups of groups acting on rooted trees

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.

Minor revision, accepted in Transactions of the AMS; 32 pages with Appendix, 4 figures