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 F 2 which are all invariant and weakly mixing with respect to the action of Aut(F 2 ).Moreover, for amenable groups Γ ≤ ∆, we obtain that the standard co-induction operation from the space of weak equivalence classes of Γ to the space of weak equivalence classes of ∆ is continuous if and only if [∆ : Γ] < ∞ or core ∆ (Γ) is trivial. For general groups we obtain that the co-induction operation is not continuous when [∆ : Γ] = ∞. This answers a question raised by Burton and Kechris in [BK18]. Independently such an answer was also obtained, using a different method, by Bernshteyn in [B18].

Faithful invariant random subgroups in acylindrically hyperbolic groups

“…The following proposition will be very useful for us. It is due to Kechris‐Quorning [19, Proposition 7.1] following Ioana [17, Lemma 2.1]. For the convenience of the reader, we repeat the proof.…”

“…Example Let $$\Gamma _0 \leqslant \Gamma$$ be countable groups. Kechris and Quorning defined a co‐induction operation $$\operatorname{CIND}_{\Gamma ^{0}}^{\Gamma }:\operatorname{IRS}(\Gamma _0)\rightarrow \operatorname{IRS}(\Gamma )$$ in [19]. For any $$\theta \in \operatorname{IRS}(\Gamma _0)$$, the IRS $$\operatorname{CIND}_{\Gamma _0}^\Gamma (\theta )$$ is $$\bigcap _{g\in \Gamma /\Gamma _0} g_*\theta$$.…”

“…Instead of the intersection, one can also use the group generated by subgroups (see [19, Remark 5.2.(3)]). Definition The group generated by subgroups defines a map $$\langle \rangle :{\rm Sub}(G)\times {\rm Sub}(G)\rightarrow {\rm Sub}(G), (\Delta _1,\Delta _2)\mapsto \langle \Delta _1,\Delta _2\rangle$$.…”

“…In this section, we prove our theorems. The main ingredients are a construction of Kechris and Quorning (in [19]) and a theorem of Sun (in [26]). We first reproduce the needed results of Sun.…”

“…and Quorning defined a co-induction operation CIND Γ Γ 0 ∶ IRS(Γ 0 ) → IRS(Γ) in[19]. For any 𝜃 ∈ IRS(Γ 0 ), the IRS CIND Γ Γ 0 (𝜃) is ⋂ g∈Γ∕Γ 0 g * 𝜃.…”