I've copied over this question from what I asked on StackExchange, in the hope that an expert here can readily answer the question.

Is there an example of a group $G=K\rtimes \mathbb{Z}$ satisfying the following three conditions?

- $G$ is finitely generated;
- $K$ is
**not**finitely generated; - the fixed points of $\phi(1)$, which is the automorphism on $K$ corresponding to $1_\mathbb{Z}$, are
**not**a finitely generated group.

I suspect there is an example, but I don't have enough experience with infinite groups to come up with one right away.

$\\\\$

One idea, which may or may not work:

finding matrices $M_1,\ldots,M_k\in\text{GL}_n(\mathbb{C})$ and $g\in\text{GL}_n(\mathbb{C})$

such that $S=\langle g^{-n}\,M_j\,g^n\rangle_{1\leq j\leq k,\;n\in\mathbb{Z}} $ is not finitely generated

but such that $g$ commutes with surprisingly many matrices in $S$.