# Groups, Geometry, and Dynamics

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**Volume 11, Issue 1, 2017, pp. 189–209**

**DOI: 10.4171/GGD/394**

Published online: 2017-04-20

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

Michelle Bucher^{[1]}and Alexey Talambutsa

^{[2]}(1) Université de Genève, Switzerland

(2) Steklov Mathematical Institute of RAS, Moscow, Russia

We prove that for any prime $p \geq 3$ 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 $(1+\sqrt5)/2$. 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.

*Keywords: *Exponential growth rate, actions on trees, lamplighter groups

Bucher Michelle, Talambutsa Alexey: Minimal exponential growth rates of metabelian Baumslag–Solitar groups and lamplighter groups. *Groups Geom. Dyn.* 11 (2017), 189-209. doi: 10.4171/GGD/394