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

We prove that for any prime $pgeq 3$ the minimal exponential growth rate of the Baumslag-Solitar group $BS(1,p)$ and the lamplighter group $mathcal{L}_p=(mathbb{Z}/pmathbb{Z})wr mathbb{Z}$ 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 $(1+sqrt5)/2$. The latter value also serves to show that the lower bound of A.Mann from [Mann, Journal of Algebra 326, no. 1 (2011) 208--217] for the growth rates of non-semidirect HNN extensions is optimal.