
## 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.