Invariant random subgroups and action versus representation maximality

TL;DR

This paper demonstrates that for free ergodic measure-preserving actions of the free group, weak containment of actions does not imply weak containment of their Koopman representations, using a specially constructed invariant random subgroup.

Contribution

It introduces a new invariant random subgroup supported on maximal actions, showing a distinction between action and representation maximality.

Findings

Weak containment of actions is not equivalent to weak containment of Koopman representations.

Constructed an invariant random subgroup supported on maximal actions.

Highlights differences between action and representation maximality in free groups.

Abstract

We show that weak containment of free ergodic measure-preserving actions of $\mathbf{F}_\infty$ is not equivalent to weak containment of the corresponding Koopman representations. This result is based on the construction of an invariant random subgroup of $\mathbf{F}_\infty$ which is supported on the maximal actions.