Co-induction and Invariant Random Subgroups

TL;DR

This paper introduces a co-induction operation for invariant random subgroups, constructing large families of such subgroups in various groups and analyzing the continuity of this operation, answering a question by Burton and Kechris.

Contribution

It develops a new co-induction method for invariant random subgroups and explores its properties across different group classes, providing new constructions and continuity results.

Findings

Constructed continuum families of invariant random subgroups in various groups.

Established conditions for the continuity of co-induction in amenable groups.

Answered a question on co-induction continuity posed by Burton and Kechris.

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.