Unimodularity of invariant random subgroups

Biringer, Ian; Tamuz, Omer

doi:10.1090/tran/6755

Unimodularity of invariant random subgroups
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by Ian Biringer and Omer Tamuz PDF

Trans. Amer. Math. Soc. 369 (2017), 4043-4061 Request permission

Abstract:

An invariant random subgroup $H \leq G$ 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 $H \leq G$ 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.

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