Center of a finitely generated group

Here is an example I encountered recently (which unfortunately completely wrecked a proof I was working on at the time).

Start with a covering group $N$ of an infinite elementary abelian $p$-group for a prime $p$. This is not unique, but if we choose $p$ odd, then we can make it have exponent $p$. Then $N$ has a presentation

$\langle\ y_i, z_{jk}\ (i,j,k \in \mathbb{Z}, j<k) \mid [y_j,y_k] = z_{jk}\ (j<k),\ z_{jk} {\rm\ central},\ y_i^p=z_{jk}^p=1\ \rangle.$

This group has an automorphism of infinite order that maps $y_i \mapsto y_{i+1}$, $z_{jk} \mapsto z_{j+1,k+1}$.

Take the semidirect product of $N$ with an infinite cyclic group $\langle x \rangle$ inducing this automorphism, and factor out the normal closure of the elements $z_{j,j+t}^{-1} z_{j+1,j+t+1}$ for all $t>0$. This yields a 2-generator group with presentation

$\langle\ y_1, x \mid y_1^p=1, [y_j,y_k] {\rm\ central\ for\ all\ } j<k\ \rangle,$

where $y_j$ is an abbreviation for $y_1^{x^j}$. Its centre is elementary abelian and generated by the infinite set $[y_1,y_{1+t}]$ for $t>0$.

V. N. Remeslennikov found a finitely presented group whose center is not finitely generated. But I didn't check his construction.

Remeslennikov, V. N. A finitely presented group whose center is not finitely generated. (Russian) Algebra i Logika 13 (1974), no. 4, 450–459, 488.

English translation: Algebra and Logic 13 (1974), no. 4, 258–264 (1975)