The conjugacy growth of the soluble Baumslag-Solitar groups

TL;DR

This paper analyzes the conjugacy growth in soluble Baumslag-Solitar groups, providing asymptotic formulas, showing the transcendental nature of their conjugacy growth series, and establishing equality between conjugacy and standard growth rates.

Contribution

It offers a complete description of geodesic conjugacy representatives and formulas for the conjugacy growth series in $BS(1,k)$ groups, a novel contribution to understanding their algebraic structure.

Findings

Conjugacy growth series are transcendental.

Conjugacy and standard growth rates are equal in $BS(1,k)$.

Provided asymptotic formulas for conjugacy growth.

Abstract

In this paper we give asymptotics for the conjugacy growth of the soluble Baumslag-Solitar groups $BS(1,k)$, $k\geq 2$, with respect to the standard generating set, by providing a complete description of geodesic conjugacy representatives. We show that the conjugacy growth series for these groups are transcendental, and give formulas for the series. As a result of our computation we also establish that in each $BS(1,k)$ the conjugacy and standard growth rates are equal.