# Co-induction and invariant random subgroups

### Alexander S. Kechris

California Institute of Technology, Pasadena, USA### Vibeke Quorning

University of Copenhagen, Denmark

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

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