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## 5. Groups

1. #### Problem 5.1.

[S. Popa] Characterize all groups which are measure equivalent to free groups, or find new examples of such groups.
• #### Problem 5.2.

[A. Ioana] Find a weird example of a groups $\Gamma$ that is measure equivalent to $\mathbb{F}_2\times \mathbb{F}_2$. Is it true that every such group $\Gamma$ is either a product of groups ME to free groups, or a lattice in a locally compact group which contains $\mathbb{F}_2\times \mathbb{F}_2$ as a lattice?
1. Remark. The group von Neumann algebra version of this states that there are no weird examples. The same holds true in the measure equivalence setting under mild mixing assumptions.
• #### Problem 5.4.

[S. Vaes] Let $G$ be a groups with $\beta_1^{(2)}(G)>0$. Then there is an infinite amenable subgroup, e.g. an element of infinite order, or a finite index subgroup $G_1\leqslant G$ so that $$\beta_1^{(2)}(G)\geq \frac{1}{[G:G_1]}.$$
• #### Problem 5.5.

[S. Vaes] Which groups $\Gamma$ admit a Bernoulli action on $\prod_{g \in \Gamma} (\{0,1\},\mu_g)$ that is nonsingular and of type III?
1. Remark. In https://arxiv.org/abs/1705.00439, it is proved that it is a necessary condition that $\Gamma$ is either amenable or satisfies $\beta_1^{(2)}(\Gamma) > 0$. It is also proved that for many of these groups such a nonsingular type III Bernoulli action exists, e.g. for all the groups satisfying the conclusion of Problem 5.3.
• #### Problem 5.7.

[A. Furman] Is treeability stable under finite index, i.e. if $S\subset R$ is a finite index subequivalence relation of a treaable relation $R$, is $S$ treeable?
• #### Problem 5.8.

[A. Furman] Does every equivalence relation with spectral gap or property (T) measurably contain $\mathbb{F}_2$?

Cite this as: AimPL: Classification of group von Neumann algebras, available at http://aimpl.org/groupvonneumann.