# Groups with finitely generated center

Does every group with a finite classifying space have finitely generated center?

Remarks:

1. 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).

2. Finite classifying space means that the group is the fundamental group of a finite aspherical cell complex.

3. 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.

4. 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.

5. The answer is also yes for elementary amenable groups. (I know a proof, but have no reference).

• @Igor, check Geoff Mess' paper on Seifert conjecture, he had some comments on this question (I forgot what he said though) and I do not have access to his preprint right now. Aug 5 '12 at 4:19
• @Misha, I was unable to find a copy online (and it might not exist given that it was written in 1988). Aug 5 '12 at 11:33
• btw, is there a known $F_\infty$ group with infinitely generated center? Recall that $F_\infty$ means there's a $K(G,1)$ finite in each dimension, so it is weaker the existence of a finite $K(G,1)$. (Abels-Brown found $F_n$ groups with infinitely generated center for all $n$, where $F_n$ means: there exists $K(G,1)$ finite up to dimension $n$.)
– YCor
Aug 5 '12 at 11:52
• @Yves, I don't know such examples. The above results in 4-5 hold for groups of type FP, so assuming type F (like I do) is an overkill, but type F is what I need for an application. Aug 5 '12 at 12:28

The answer to another question implies that there is a finitely presented group which has $\mathbb{Q}$ as its center (or any recursively presentable abelian group). However, you'd have to go through the paper to see if Houcine's proof could produce a group with a finite complex (seems unlikely).