Does every group with a finite classifying space have finitely generated center?
If $G$ is a finitely generated group with infinitely generated center $Z(G)$, then the quotient $G/Z(G)$ is not finitely presented (as follows from a result of B.H Newmann).
Finite classifying space means that the group is the fundamental group of a finite aspherical cell complex.
I suspect the above question is a well-known open problem, but cannot find it stated in the literature, so a reference would be appreciated.
Alperin-Shalen (Inventiones, 1982) showed that the answer is yes for every subgroup of $GL_n(K)$ where $n>0$ and $K$ is a field of characteristic zero.
The answer is also yes for elementary amenable groups. (I know a proof, but have no reference).