I'm looking for an example of a finitely presented and finitely generated amenable group, that has a subgroup which is not finitely generated.
The question is easy for finitely generated amenable group and an example is the lamp-lighter group $C_2\wr \mathbb{Z}$.
An Abelian and finitely generated group has no such subgroups. There exists a bigger class of groups with this property?
Best Answer
I don't know much about amenable groups I am afraid, but according to the Wikipedia article, all solvable groups are amenable. So we can take the Baumslag-Solitar group
$B(1,n) = \langle x,y \mid y^{-1}xy = x^n \rangle.$
If we let $N$ be the normal closure of the subgroup generated by $x$, then $N$ is abelian with $G/N$ cyclic, but $N$ is not finitely generated when $n > 1$. Note also that $B(1,n)$ is isomorphic to the subgroup of ${\rm GL}(2, \mathbb{Q})$ generated by
$x = \left(\begin{array}{cc}1&0\\\\1&1\end{array}\right)$ and $y = \left(\begin{array}{cc}n&0\\\\0&1\end{array}\right).$