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

TL;DR

This paper investigates the minimal exponential growth rates of certain metabelian groups, establishing equal rates for Baumslag-Solitar and lamplighter groups for primes p≥3, and providing specific growth rates for p=2, revealing optimal bounds.

Contribution

It proves the equality of growth rates for Baumslag-Solitar and lamplighter groups for primes p≥3 and determines exact growth rates for p=2, refining bounds on group growth.

Findings

Growth rates are equal for p≥3

Exact growth rate for BS(1,2) is the root of x^3 - x^2 - 2

Growth rate for lamplighter group with p=2 is the golden ratio

Abstract

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 [Mann, Journal of Algebra 326, no. 1 (2011) 208--217] for the growth rates of non-semidirect HNN extensions is optimal.