A formula for the normal subgroup growth of Baumslag-Solitar groups

TL;DR

This paper derives an exact formula for counting normal subgroups of finite index in Baumslag-Solitar groups BS(p,q) when p and q are coprime, revealing new structural insights and examples in profinite group theory.

Contribution

It provides the first explicit formula for normal subgroup growth in BS(p,q) for coprime p and q, distinguishing these groups from general finite index subgroup counts.

Findings

Exact formula for normal subgroup counts in BS(p,q)

Identification of a profinite group with an Euler product zeta function

Demonstration that normal subgroup growth differs from all subgroup growth

Abstract

We give an exact formula for the number of normal subgroups of each finite index in the Baumslag-Solitar group BS(p,q) when p and q are coprime. Unlike the formula for all finite index subgroups, this one distinguishes different Baumslag-Solitar groups and is not multiplicative. This allows us to give an example of a finitely generated profinite group which is not virtually pronilpotent but whose zeta function has an Euler product.