2. Orbit Equivalence of Measure Preserving Actions

Problem 2.1.
[Miklos Abert] Let $\Gamma_1$ and $\Gamma_2$ be infinite countable groups. Does $\Gamma_1\times\Gamma_2$ have fixed price $1$? 
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? 
Problem 2.3.
[Konstantin Medynets] Are $2$ and $3$generated groups topologically orbit equivalent? 
Let $\Gamma$ be a nonamenable 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 $\sum\limits_{g\in \Gamma} \mu^2(A_g)>N$?
Cite this as: AimPL: $L^2$ invariants for finitely generated groups, available at http://aimpl.org/l2invariantsgroups.