Invariant Random Subgroups of Groups Acting on Rooted Trees

Bencs, Ferenc and Tóth, László Márton (2021) Invariant Random Subgroups of Groups Acting on Rooted Trees. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. pp. 1-32. ISSN 0002-9947 (print); 1088-6850 (online)

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Abstract

We investigate invariant random subgroups in groups acting on rooted trees. Let Altf (T) be the group of finitary even automorphisms of the dary 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 exhibit a group of intermediate growth with this property.

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