We know that every finitely generated metabelian group can be represented by matrices (e.g. see https://link.springer.com/article/10.1007/BF02219822 ). I am particularly interested in how finitely generated abelian-by-$\mathbb Z$ groups (equivalently, finitely generated modules over Laurent polynomial rings) can be represented by matrices.
Is there a nice description of the set of subgroups $S$ and $T$ of the matrix groups such that
- there is a 1 to 1 codependence between $S$ and finitely generated abelian-by-$\mathbb Z$ groups.
- there is a 1 to 1 codependence between $T$ and the finitely generated metabelian group.
I know how to represent some finitely generated abelian-by-$\mathbb Z$ groups by matrices, e.g. the wreath product $\Bbb Z\wr\Bbb Z$ and $BS(1,k)$ groups, I am a bit struggling to use it to find the set of subgroups of matrix group that corresponds to all finitely generated abelian-by-$\mathbb Z$ groups.
Any reference would be really appreciated. Thank you for reading.