JournalsggdVol. 13, No. 4pp. 1151–1193

Co-induction and invariant random subgroups

  • Alexander S. Kechris

    California Institute of Technology, Pasadena, USA
  • Vibeke Quorning

    University of Copenhagen, Denmark
Co-induction and invariant random subgroups cover

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Abstract

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\mathbb F_2 which are all invariant and weakly mixing with respect to the action of Aut(F2)(\mathbb F_2).

Moreover, for amenable groups ΓΔ\Gamma\leq \Delta, we obtain that the standard co-induction operation from the space of weak equivalence classes of Γ\Gamma to the space of weak equivalence classes of Δ\Delta is continuous if and only if [Δ:Γ]<[\Delta :\Gamma]<\infty or coreΔ(Γ)_\Delta(\Gamma) is trivial. For general groups we obtain that the co-induction operation is not continuous when [Δ:Γ]=[\Delta:\Gamma]=\infty. 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].

Cite this article

Alexander S. Kechris, Vibeke Quorning, Co-induction and invariant random subgroups. Groups Geom. Dyn. 13 (2019), no. 4, pp. 1151–1193

DOI 10.4171/GGD/517