A new proof of the growth rate of the solvable Baumslag-Solitar groups

TL;DR

This paper presents a new, straightforward method to compute the growth rate of solvable Baumslag-Solitar groups by constructing a regular language of geodesics, simplifying previous calculations.

Contribution

It introduces a regular language of geodesics for $BS(1,n)$ that directly yields the group's growth rate, providing a simpler proof than earlier methods.

Findings

Growth rate of $BS(1,n)$ matches the language's growth rate

Established a regular language of geodesics for a large set of elements

Connected growth rate calculation with conjugation curvature properties

Abstract

We exhibit a regular language of geodesics for a large set of elements of $BS(1,n)$ and show that the growth rate of this language is the growth rate of the group. This provides a straightforward calculation of the growth rate of $BS(1,n)$, which was initially computed by Collins, Edjvet and Gill in [5]. Our methods are based on those we develop in [8] to show that $BS(1,n)$ has a positive density of elements of positive, negative and zero conjugation curvature, as introduced by Bar-Natan, Duchin and Kropholler in [1].