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 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].