Ordering BS$(1,3)$ using the Magnus transformation

Abstract

Following a similar treatment of the Baumslag–Solitar group $BS\left(1,2\right)$ by Bahls, we modify a transformation developed by Magnus to linearly order the group $BS\left(1,3\right)$ given by the presentation $\langle a,b\mid ab=b{a}^3\rangle$. We demonstrate how this same method will fail to admit such a treatment of the groups $BS\left(1,n\right)$, $n\ge 4$.