Ferenc Bencs: Invariant random subgroups of groups acting on rooted trees

Speaker: Ferenc Bencs

Title: Invariant random subgroups of groups acting on rooted trees

Abstract: We investigate invariant random subgroups in groups acting on rooted trees. Let Altf(T) be the group of finitary even automorphisms of the d-ary rooted tree T. We prove that a nontrivial ergodic IRS of Altf(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 show that there exists a group of intermediate growth with this property.