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Transactions of the American Mathematical Society

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Characteristic random subgroups of geometric groups and free abelian groups of infinite rank

Authors: Lewis Bowen, Rostislav Grigorchuk and Rostyslav Kravchenko
Journal: Trans. Amer. Math. Soc. 369 (2017), 755-781
MSC (2010): Primary 20K20, 20K27, 20P05, 20E07
Published electronically: May 6, 2016
MathSciNet review: 3572253
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Abstract: We show that if $ G$ is a non-elementary word hyperbolic group, mapping class group of a hyperbolic surface or the outer automorphism group of a non-abelian free group, then $ G$ has $ 2^{\aleph _0}$ many non-atomic ergodic invariant random subgroups. If $ G$ is a non-abelian free group, then $ G$ has $ 2^{\aleph _0}$ many non-atomic $ G$-ergodic characteristic random subgroups. We also provide a complete classification of characteristic random subgroups of free abelian groups of countably infinite rank and elementary $ p$-groups of countably infinite rank.

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Additional Information

Lewis Bowen
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712

Rostislav Grigorchuk
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Rostyslav Kravchenko
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Address at time of publication: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208

Keywords: Invariant random subgroup, free abelian group
Received by editor(s): May 19, 2014
Received by editor(s) in revised form: January 20, 2015
Published electronically: May 6, 2016
Additional Notes: The first author was supported by NSF grant DMS-0968762 and NSF CAREER Award DMS-0954606
The second author was supported by NSF grant DMS-1207699
Article copyright: © Copyright 2016 American Mathematical Society

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