Commensurability and quasi-isometry classification for one vertex one loop tubular groups

A tubular group has a graph of groups decomposition with ${\mathbb{Z}}^2$ vertex groups and $\mathbb{Z}$ edge groups. This paper gives a classification of one vertex one loop tubular groups $G_{(k,\mathrm{\ell }),(m,n)}=⟨a,b,t:[a,b]=1,t{a}^m{b}^n{t}^{-1}=a^k{b}^{\mathrm{\ell }}⟩$ up to commensurability and quasi-isometry. We show that tubular groups where the images of the edge maps have nonzero ``intersection number" are all commensurable and so quasi-isometric. When the intersection number is zero, there are two quasi-isometry classes and infinitely many commensurability classes. The nonzero and zero intersection number tubular groups are not quasi-isometric.