Stabilizers of actions of lattices in products of groups

Abstract

We prove that any ergodic non-atomic probability-preserving action of an irreducible lattice in a semisimple group, with at least one factor being connected and of higher-rank, is essentially free. This generalizes the result of Stuck and Zimmer [Stabilizers for ergodic actions of higher rank semisimple groups. Ann. of Math. (2)139(3) (1994), 723–747], who found that the same statement holds when the ambient group is a semisimple real Lie group and every simple factor is of higher-rank. We also prove a generalization of a result of Bader and Shalom [Factor and normal subgroup theorems for lattices in products of groups. Invent. Math.163(2) (2006), 415–454] by showing that any probability-preserving action of a product of simple groups, with at least one having 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 relying on algebraic properties of semisimple groups and Poisson boundaries, and introduces a generalization of the ergodic decomposition to invariant random subgroups, which is of independent interest.