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Transactions of the American Mathematical Society

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Unimodularity of invariant random subgroups


Authors: Ian Biringer and Omer Tamuz
Journal: Trans. Amer. Math. Soc. 369 (2017), 4043-4061
MSC (2010): Primary 28C10; Secondary 37A20
DOI: https://doi.org/10.1090/tran/6755
Published electronically: October 28, 2016
MathSciNet review: 3624401
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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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Additional Information

Ian Biringer
Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467

Omer Tamuz
Affiliation: Microsoft Research New England, One Memorial Drive, Cambridge, Massachusetts 02142
Address at time of publication: Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena, California 91125

DOI: https://doi.org/10.1090/tran/6755
Keywords: Invariant random subgroups, invariant measures on homogeneous spaces, mass transport principle
Received by editor(s): February 11, 2014
Received by editor(s) in revised form: June 2, 2015
Published electronically: October 28, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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