Unimodularity of invariant random subgroups
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- by Ian Biringer and Omer Tamuz PDF
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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
- MR Author ID: 898902
- Received by editor(s): February 11, 2014
- Received by editor(s) in revised form: June 2, 2015
- Published electronically: October 28, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 4043-4061
- MSC (2010): Primary 28C10; Secondary 37A20
- DOI: https://doi.org/10.1090/tran/6755
- MathSciNet review: 3624401