Unimodularity of Invariant Random Subgroups

Abstract

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

23 pages, one figure

References in corpus (1)

• Amenable Invariant Random Subgroups

Cited by in corpus (4)

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