Unimodularity of invariant random subgroups

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

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2017-06, Vol.369 (6), p.4043-4061
Main Authors: Biringer, Ian, Tamuz, Omer
Format: Article
Language:English
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Research article
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Summary:An invariant random subgroup H≤GH \leq G is a random closed subgroup whose law is invariant to conjugation by all elements of GG. When GG is locally compact and second countable, we show that for every invariant random subgroup H≤GH \leq G there almost surely exists an invariant measure on G/HG/H. Equivalently, the modular function of HH is almost surely equal to the modular function of GG, restricted to HH. 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.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/6755