Minimal exponential growth rates of metabelian Baumslag–Solitar groups and lamplighter groups

Abstract

We prove that for any prime $p\ge 3$ the minimal exponential growth rate of the Baumslag–Solitar group BS$(1,p)$ and the lamplighter group ${\mathcal{L}}_p=(\mathbb{Z}/p\mathbb{Z})\wr \mathbb{Z}$ are equal. We also show that for $p=2$ this claim is not true and the growth rate of BS$(1,2)$ is equal to the positive root of $x^3-x^2-2$, whilst the one of the lamplighter group ${\mathcal{L}}_2$ is equal to the golden ratio $(1+\sqrt{5})/2$. The latter value also serves to show that the lower bound of A.Mann from[9] for the growth rates of non-semidirect HNN extensions is optimal.