Stabilizers of Actions of Lattices in Products of Groups

TL;DR

This paper proves that ergodic actions of lattices in products of groups are essentially free or transitive under certain conditions, extending previous results and introducing new methods involving contractive maps and invariant random subgroups.

Contribution

It generalizes existing theorems on actions of lattices in semisimple groups by relaxing assumptions and employing novel techniques like relative contractiveness and invariant random subgroups.

Findings

Actions of lattices in higher-rank groups are essentially free.

Actions of product groups with property (T) are either free or transitive.

Introduces new methods for analyzing group actions using contractive maps.

Abstract

We prove that any ergodic nonatomic probability-preserving action of an irreducible lattice in a semisimple group, at least one factor being connected and higher-rank, is essentially free. This generalizes the result of Stuck and Zimmer that the same statement holds when the ambient group is a semisimple real Lie group and every simple factor is higher-rank. We also prove a generalization of a result of Bader and Shalom by showing that any probability-preserving action of a product of simple groups, at least one with property (T), which is ergodic for each simple subgroup is either essentially free or essentially transitive. Our method involves the study of relatively contractive maps and the Howe-Moore property, rather than the relaying on algebraic properties of semisimple groups and Poisson boundaries, and introduces a generalization of the ergodic decomposition to invariant random subgroups of independent interest.