Unimodularity of Invariant Random Subgroups

Creators: Biringer, Ian; Tamuz, Omer

Abstract

An invariant random subgroup H≤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≤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.

Additional Information

© 2016 American Mathematical Society. Received by the editors February 11, 2014 and, in revised form, June 2, 2015. Article electronically published on October 28, 2016. The authors are indebted to Lewis Bowen, who first posed the question that led to Theorem 1.1, suggested the inclusion of the statement of Theorem 1.3, and inspired the discussion in Remark 3.3. The first author was partially supported by NSF grant DMS-1308678, and he would like to thank Miklos Abért for numerous conversations, in particular those relating to the mass transport principle, as without his input the statement given here might not have been considered or solved. The second author would like to thank Yair Hartman for enlightening discussions. Both authors would also like to thank the referee for greatly improving the readability and accuracy of the paper.

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