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3. More problems

    1. Problem 3.1.

      [Roman Sauer] Do closed, aspherical manifolds with OE fundamental groups have the same simplicial volume?

      If, in addition, the simplicial volumes are positive, are the dimensions of the manifolds equal?
        • Problem 3.2.

          [Miklos Abert] Let $\Gamma/\mathbb{Z}_2$ be a sofic group. Is $\Gamma$ sofic?
            •     It is known that for every finitely generated group we have $$\beta_1^{(2)}(\Gamma)\leq d(\Gamma)-1.$$ where $d$ stands for the minimal number of generators of $\Gamma$.

              A group $\Gamma$ is normally generated by $S\subseteq \Gamma$ if the only normal subgroup of $\Gamma$ containing $S$ is $\Gamma$ itself. Let $nrk(\Gamma)$ be the normal rank of $\Gamma$, i.e., $nrk(\Gamma)$ is the minimal number of normal generators.

              Problem 3.3.

              [Andreas Thom] Do we have $\beta_1^{(2)}(\Gamma)\leq nrk(\Gamma)-1$ for a torsion free group $\Gamma$?
                1. Remark. The question is true for groups that are limits of left-orderable amenable groups. Also for groups where every non-trivial finitely generated subgroup surjects on $\mathbb{Z}$).

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