Characteristic varieties of nilpotent groups and applications

We compute the characteristic varieties and the Alexander polynomial of a finitely generated nilpotent group. We show that the first characteristic variety may be used to detect nilpotence. We use the Alexander polynomial to deduce that the only torsion-free, finitely generated nilpotent groups with positive deficiency are $\mathbb{Z}$ and ${\mathbb{Z}}^2$, extending a classical result on nilpotent link groups.