
## 2. Orbit Equivalence of Measure Preserving Actions

1. #### Problem 2.1.

[Miklos Abert] Let $\Gamma_1$ and $\Gamma_2$ be infinite countable groups. Does $\Gamma_1\times\Gamma_2$ have fixed price $1$?
By Abert-Weiss, it is enough to show that Bernoulli action of $\Gamma_1\times\Gamma_2$ have cost equals to $1$.
•     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.

#### Problem 2.2.

[Lewis Bowen] Are the fundamental groups of hyperbolic $3$-manifolds almost treeable?
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.
• #### Problem 2.3.

[Konstantin Medynets] Are $2$- and $3$-generated groups topologically orbit equivalent?
•     Let $\Gamma$ be a non-amenable group with a probability measure preserving action of $\Gamma$ on $(X,\mu)$.

#### Problem 2.4.

[Andreas Thom] Is it true that for any $N$ there exist measurable subsets $A_g\subseteq X$ $(g\in\Gamma)$ such that $\prod\limits_{x\in A_g}(x,xg)$ is a forest with $\sum\limits_{g\in \Gamma} \mu^2(A_g)>N$?
From Gaboriau-Lyons, a measurable group theoretic solution to the von Neumann problem, it follows that $\sum\limits_{g\in \Gamma} \mu(A_g)$ can be arbitrarily large.

Cite this as: AimPL: $L^2$ invariants for finitely generated groups, available at http://aimpl.org/l2invariantsgroups.