Stabilizers of Stationary Actions of Lattices in Semisimple Groups

TL;DR

This paper proves that stationary actions of certain lattices in semisimple groups almost surely have either finite or finite index stabilizers, revealing a dichotomy in their stabilizer structure.

Contribution

It establishes a new dichotomy result for stationary actions of lattices in semisimple groups with higher-rank factors, characterizing their stabilizers.

Findings

Stationary actions have finite stabilizers almost surely.

Stationary actions have finite index stabilizers almost surely.

A dichotomy in stabilizer types for these actions.

Abstract

Every stationary action of a strongly irreducible lattice or commensurator of such a latiice in a general semisimple group, with at least one higher-rank connected factor, either has finite stabilizers almost surely or finite index stabilizers almost surely.