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Co-induction and invariant random subgroups

Kechris, Alexander S. and Quorning, Vibeke (2019) Co-induction and invariant random subgroups. Groups, Geometry, and Dynamics, 13 (4). pp. 1151-1193. ISSN 1661-7207. doi:10.4171/GGD/517.

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In this paper we develop a co-induction operation which transforms an invariant random subgroup of a group into an invariant random subgroup of a larger group. We use this operation to construct new continuum size families of non-atomic, weakly mixing invariant random subgroups of certain classes of wreath products, HNN-extensions and free products with amalgamation. By use of small cancellation theory, we also construct a new continuum size family of non-atomic invariant random subgroups of F₂ which are all invariant and weakly mixing with respect to the action of Aut(F₂). Moreover, for amenable groups Γ ≤ Δ, we obtain that the standard co-induction operation from the space of weak equivalence classes of ΓΓ to the space of weak equivalence classes of Δ is continuous if and only if [Δ:Γ] < ∞ or core Δ(Γ) is trivial. For general groups we obtain that the co-induction operation is not continuous when [Δ:Γ] = ∞. This answers a question raised by Burton and Kechris in [17]. Independently such an answer was also obtained, using a different method, by Bernshteyn in [8].

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Additional Information:© 2019 European Mathematical Society. Received August 10, 2018. Published online: 2019-05-07. ASK was partially supported by NSF Grant DMS-1464475. VQ was partially supported by Lars Hesselholt’s Niels Bohr Professorship. We would like to thank Simon Thomas for a number of useful comments and in particular for bringing up the relevance of small cancellation theory to certain aspects of our work. We also would like to thank an anonymous referee for many useful remarks and corrections.
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University of CopenhagenUNSPECIFIED
Subject Keywords:Co-induction, invariant random subgroups, weak mixing, small cancellation
Issue or Number:4
Classification Code:Mathematics Subject Classification (2010): Primary: 37A15, 28D15, 22F10, 20E06, 20E22; Secondary: 20F06
Record Number:CaltechAUTHORS:20191205-100802061
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Official Citation:Kechris Alexander, Quorning Vibeke: Co-induction and invariant random subgroups. Groups Geom. Dyn. 13 (2019), 1151-1193. doi: 10.4171/GGD/517
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100204
Deposited By: Tony Diaz
Deposited On:05 Dec 2019 18:23
Last Modified:16 Nov 2021 17:52

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