Growth in higher Baumslag-Solitar groups

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 $m\le 10$.

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