Unimodularity of invariant random subgroups

Abstract

An invariant random subgroup H ≤ G H \leq G is a random closed subgroup whose law is invariant to conjugation by all elements of G G . When G G is locally compact and second countable, we show that for every invariant random subgroup H ≤ G H \leq G there almost surely exists an invariant measure on G / H G/H . Equivalently, the modular function of H H is almost surely equal to the modular function of G G , restricted to H H .

We use this result to construct invariant measures on orbit equivalence relations of measure preserving actions. Additionally, we prove a mass transport principle for discrete or compact invariant random subgroups.