On a co-induction question of Kechris

TL;DR

This paper addresses a question by Kechris about property MD in groups, proving that under certain conditions, property MD and weak containment of actions are preserved when extending from a subgroup to the whole group.

Contribution

It provides new results on property MD and weak containment of actions for groups with amenable and residually finite quotients, extending previous understanding.

Findings

If $H<G$ is normal with property MD and $G/H$ is amenable and residually finite, then $G$ has property MD.

Under the same conditions, induced actions combined with free actions of $G/H$ weakly contain original actions.

For any subgroup $H<G$ with an amenable action on $G/H$, the induced action weakly contains the original action for Gaussian actions.

Abstract

This note answers a question of Kechris: if $H<G$ is a normal subgroup of a countable group $G$, $H$ has property MD and $G/H$ is amenable and residually finite then $G$ also has property MD. Under the same hypothesis we prove that for any action $a$ of $G$, if $b$ is a free action of $G/H$, and $b_G$ is the induced action of $G$ then $\CInd_H^G(a|H) \times b_G$ weakly contains $a$. Moreover, if $H<G$ is any subgroup of a countable group $G$, and the action of $G$ on $G/H$ is amenable, then $\CInd_H^G(a|H)$ weakly contains $a$ whenever $a$ is a Gaussian action.