Normal zeta functions of small $\mathfrak{T}_2$-groups and their behaviour on residue classes

TL;DR

This paper investigates the behavior of normal subgroup zeta functions of small nilpotent class-2 torsion-free groups, revealing rationality in low Hirsch lengths and non-rationality at length 8, with connections to Higman's PORC conjecture.

Contribution

It demonstrates that for small groups of Hirsch length ≤7, the normal zeta functions are rational on residue classes, and provides counterexamples at length 8, linking to Higman's PORC conjecture.

Findings

Normal zeta functions are rational on residue classes for groups with Hirsch length ≤7.

Counterexamples exist at Hirsch length 8 where zeta functions are not rational.

The work connects the behavior of zeta functions to Higman's PORC conjecture.

Abstract

Let $G$ be a finitely generate nilpotent class-2 torsion-free group. We study how the zeta function enumerating normal subgroups of G varies on residue classes. In particular, we show that for small such $G$ of Hirsch length less than or equal to 7, the normal zeta functions are generically always rational functions on residue classes. We then show that there are examples of groups with Hirsch length 8 whose normal zeta function is not a rational function on residue classes. We observe the connection to Higman's PORC conjecture.