Orbit Equivalence of Measure Preserving Actions · Question 2.2

Workshop introduction

This problem list concerns the study of the asymptotic behavior of some natural invariants of finitely generated groups.

Question context

{\bf Definition:} $\Gamma$ is almost treable if it admits a free action on probability space such that the equivalence relation associated to this action is almost treeable, i.e., it is an increasing union of treeable subequivalence relations.

The property is stable under measure equivalence and taking subgroups. From the property it follows that $\Gamma$ has the Haagerup property and is sofic. Moreover, $\lambda_{cb}(\Gamma)=1$ if $\Gamma$ is almost treeable.

Examples include: $\mathbb{F}_2\times H$, where $H$ is amenable.

Historical workshop record. Current open/solved status has not been independently reassessed.

Status note in the source

This would be true if we knew that surface groups semidirect product with $\mathbb{Z}$ are strongly almost treeable. By strongly almost treeable we mean that every free probability measure preserving action is almost treeable.