On diagonal actions of branch groups and the corresponding characters

TL;DR

This paper introduces new concepts of non-free group actions and demonstrates how branch groups exhibit these actions, leading to the construction of diverse ergodic representations and invariant subgroups.

Contribution

It develops the notions of absolutely non-free and perfectly non-free actions and applies them to analyze unitary representations of branch groups.

Findings

Every weakly branch group acts absolutely non-freely on the boundary.

Constructs infinitely many ergodic perfectly non-free actions for each branch group.

Produces infinitely many II$_1$-factor representations and invariant random subgroups.

Abstract

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acts absolutely non-freely on the boundary of the associated rooted tree. Using this result and the symmetrized diagonal actions we construct for every countable branch group infinitely many different ergodic perfectly non-free actions, infinitely many II$_1$-factor representations, and infinitely many continuous ergodic invariant random subgroups.