Characteristic varieties of nilpotent groups and applications

Abstract

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.

14 pages

References in corpus (4)

• Non-finiteness properties of fundamental groups of smooth projective varieties
• Formality, Alexander invariants, and a question of Serre
• A two component link with Alexander polynomial one is concordant to the Hopf link
• Quasi-Kähler Bestvina-Brady groups

Cited by in corpus (6)

• Topology and geometry of cohomology jump loci
• Fundamental groups, Alexander invariants, and cohomology jumping loci
• Characteristic varieties and Betti numbers of free abelian covers
• Cohomology rings and formality properties of nilpotent groups
• Flat bundles and Hyper-Hodge decomposition on solvmanifolds
• Morse-Novikov cohomology of almost nonnegatively curved manifolds