Zeta functions of virtually nilpotent groups

TL;DR

This paper investigates the zeta functions of finitely generated virtually nilpotent groups, showing they can be expressed as finite sums of Euler products of cone integrals, and explores invariants related to Mal'cev completions.

Contribution

It establishes the structure of subgroup and normal zeta functions as sums of Euler products and introduces Mal'cev completions as invariants for subgroup growth.

Findings

Zeta functions are finite sums of Euler products of cone integrals over ${Q}$.

Zeta functions have rational abscissa of convergence and meromorphic continuation.

Subgroup growth is an invariant of the ${Q}$-Mal'cev completion.

Abstract

We prove that the subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group are finite sums of Euler products of cone integrals over $\mathbb{Q}$ and we deduce from this that they have rational abscissa of convergence and some meromorphic continuation. We also define Mal'cev completions of a finitely generated virtually nilpotent group and we prove that the subgroup growth and the normal subgroup growth of the latter are invariants of its $\mathbb{Q}$-Mal'cev completion.