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 $\text{Aut}(\mathbb{F}_2)$. 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 $\text{core}_\Delta(\Gamma)$ 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. Independently such an answer was also obtained, using a different method, by Bernshteyn.