Metric properties of Baumslag-Solitar groups

TL;DR

This paper investigates the metric properties of Baumslag-Solitar groups by estimating their word metrics using Britton's lemma, leading to new lower bounds on their growth rates.

Contribution

It provides the first estimates of the word metric for Baumslag-Solitar groups and derives growth rate bounds based on these estimates.

Findings

Estimated the word metric of Baumslag-Solitar groups.

Derived lower bounds for the groups' growth rates.

Connected metric estimates to algebraic properties of the groups.

Abstract

We compute estimates for the word metric of Baumslag--Solitar groups in terms of the Britton's lemma normal form. As a corollary, we find lower bounds for the growth rate for the groups $BS(p,q)$, with $1<p\le q$.