Co-induction and invariant random subgroups

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 $\Gamma \le \Delta$, 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].