Geometric invariants and modules of type FP∞ over constructible nilpotent-by-abelian groups

We construct examples of modules of type FP∞ over constructible nilpotent-by-abelian groups. In particular abelian normal subgroups of such groups considered as modules via conjugation have this property. The proof relies on some of Cartan‘s results about homology groups and the theory of geometric invariants (of discrete groups) due to Bieri, Strebel, Neumann and Renz.