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5. Groups


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      Problem 5.1.

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


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          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.


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                  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]}.


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                      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.


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                              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?


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                                  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.