Zeta functions of the 3-dimensional almost-Bieberbach groups

TL;DR

This paper computes the subgroup and normal zeta functions of 3-dimensional almost-Bieberbach groups, showing their meromorphic continuation and functional equations, advancing understanding of zeta functions in virtually nilpotent groups.

Contribution

It provides explicit formulas and analytic properties of zeta functions for 3-dimensional almost-Bieberbach groups, including complete calculations for torsion-free cases.

Findings

Zeta functions admit Euler product factorization.

They can be meromorphically continued to the entire complex plane.

They satisfy local functional equations.

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 factorization. We compute these series except for a finite number of local factors when the group is virtually nilpotent of Hirsch length 3. We deduce that they can be meromorphically continued to the whole complex plane and that they satisfy local functional equations. The complete computation (with no exception of local factors) is presented for those groups that are also torsion-free, that is, for the 3-dimensional almost-Bieberbach groups.