# Zeta functions of virtually nilpotent groups

@article{Sulca2012ZetaFO, title={Zeta functions of virtually nilpotent groups}, author={Diego Sulca}, journal={Israel Journal of Mathematics}, year={2012}, volume={213}, pages={371-398} }

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 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 Q-Mal’cev completion.

## 3 Citations

Zeta functions of the 3-dimensional almost-Bieberbach groups

- MathematicsJournal of Group Theory
- 2022

Abstract The subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group can be expressed as finite sums of Dirichlet series admitting Euler product…

A remark on the degree of polynomial subgroup growth of nilpotent groups

- Mathematics
- 2021

We show that if two finitely generated nilpotent groups have isomorphic C-Mal’cev completions, then their subgroup (resp. normal) zeta functions have the same abscissa of convergence. A similar…

Statistics of finite degree covers of torus knot complements

- Mathematics
- 2020

In the first part of this paper, we determine the asymptotic subgroup growth of the fundamental group of a torus knot complement. In the second part, we use this to study random finite degree covers…

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