Stabilizers of Ergodic Actions of Lattices and Commensurators

TL;DR

This paper establishes a dichotomy for ergodic measure-preserving actions of certain lattices and their commensurators, showing they either have finite orbits or finite stabilizers, based on properties of the acting groups.

Contribution

It extends the understanding of stabilizer structures in ergodic actions to lattices and commensurators in semisimple groups with specific properties, using the Howe-Moore property and property (T).

Findings

Dichotomy for ergodic actions: finite orbits or finite stabilizers.

Results apply to lattices and commensurators in semisimple groups with rank ≥ 2.

Generalization to actions in products of groups with at least one totally disconnected factor.

Abstract

We prove that any ergodic measure-preserving action of an irreducible lattice in a semisimple group, with finite center and each simple factor having rank at least two, either has finite orbits or has finite stabilizers. The same dichotomy holds for many commensurators of such lattices. The above are derived from more general results on groups with the Howe-Moore property and property $(T)$. We prove similar results for commensurators in such groups and for irreducible lattices (and commensurators) in products of at least two such groups, at least one of which is totally disconnected.