Stabilizer Rigidity in Irreducible Group Actions

TL;DR

This paper investigates the stabilizer subgroups in irreducible actions of certain groups, demonstrating their co-amenability and deriving new rigidity results that extend previous theorems in the field.

Contribution

It introduces new rigidity results for irreducible group actions, generalizing prior theorems by employing intermediate factor theorems and co-amenability concepts.

Findings

Stabilizers are co-amenable in their normal closure.

Irreducible invariant random subgroups are co-amenable.

Derived rigidity results strengthen previous theorems.

Abstract

We consider irreducible actions of locally compact product groups, and of higher rank semi-simple Lie groups. Using the intermediate factor theorems of Bader-Shalom and Nevo-Zimmer, we show that the action stabilizers, and all irreducible invariant random subgroups, are co-amenable in their normal closure. As a consequence, we derive rigidity results on irreducible actions that generalize and strengthen the results of Bader-Shalom and Stuck-Zimmer.