Orbit equivalence and total weak mixing of free group actions

Abstract

We prove that the orbit equivalence class of every free ergodic probability-measure-preserving (pmp) action of a free group contains a totally weak mixing action. Equivalently, every ergodic treeable pmp equivalence relation of cost $n\in \mathbf{N}\cup \{\infty \}$ is generated by a free totally weak mixing action of ${\mathbf{F}}_n$. This answers a question of Miller and Tserunyan. The proof goes by considering a Polish space of edge slidings along a fixed mixing transformation and proving that for every $w\ne e\in {\mathbf{F}}_n$ the set of edge slidings that produce an action with $w$ weakly mixing forms a comeager set.

6 pages