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Invariant random subgroups of groups acting on hyperbolic spaces

Author: D. Osin
Journal: Proc. Amer. Math. Soc. 145 (2017), 3279-3288
MSC (2010): Primary 20F65
Published electronically: February 24, 2017
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Abstract: Suppose that a group $ G$ acts non-elementarily on a hyperbolic space $ S$ and does not fix any point of $ \partial S$. A subgroup $ H\le G$ is geometrically dense in $ G$ if the limit sets of $ H$ and $ G$ on $ \partial S$ coincide and $ H$ does not fix any point of $ \partial S$. We prove that every invariant random subgroup of $ G$ is either geometrically dense or contained in the elliptic radical (i.e., the maximal normal elliptic subgroup of $ G$). In particular, every ergodic measure preserving action of an acylindrically hyperbolic group on a Borel probability space $ (X,\mu )$ either has finite stabilizers $ \mu $-almost surely or otherwise the stabilizers are very large (in particular, acylindrically hyperbolic) $ \mu $-almost surely.

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Additional Information

D. Osin
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

Received by editor(s): January 18, 2016
Received by editor(s) in revised form: September 9, 2016
Published electronically: February 24, 2017
Additional Notes: This work was supported by the NSF grant DMS-1308961
Communicated by: Kevin Whyte
Article copyright: © Copyright 2017 American Mathematical Society

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