Growth in higher Baumslag-Solitar groups

TL;DR

This paper investigates the growth properties of certain Baumslag-Solitar groups, demonstrating that these groups exhibit rational growth and providing explicit subgroup growth series calculations for small cases.

Contribution

It introduces a method to compute subgroup growth series in Baumslag-Solitar groups and proves rational growth for all dimensions m.

Findings

Groups have rational growth for all m

Method for computing subgroup growth series

Explicit calculations for m ≤ 10

Abstract

We study the HNN extension of $\mathbb{Z}^m$ given by the cubing endomorphism $g\mapsto g^3$, and prove that such groups have rational growth. To do so, we describe a method of computing the subgroup growth series of the horocyclic subgroup $\mathbb{Z}^m$ in this family of examples, prove that for all $m$ the group has rational growth. In the appendix, the subgroup growth series has been computed for all $m \leq 10$.