Zeta functions enumerating normal subgroups of [formula omitted]-groups and their behavior on residue classes

Abstract

Let G be a finitely generated torsion-free nilpotent class-2 group. We investigate how ζG,p◃(s), the local zeta function enumerating normal subgroups of p-power indices in G, varies on residue classes. We first show how the behaviors of local normal subgroup zeta functions ζG,p◃(s) and ζG×Zr,p◃(s) on residue classes are closely related for certain G. Then we show that for G of Hirsch length less than or equal to 7, under certain assumptions the normal subgroup zeta function of G is always a rational function on residue classes. We also show that there are examples of groups with Hirsch length 8 whose normal subgroup zeta function is not a rational function on residue classes.