### Growth Rates of Groups associated with Face 2-Coloured Triangulations and Directed Eulerian Digraphs on the Sphere

#### Abstract

Let** **$\mathcal{G}$* *be a properly face $2$-coloured (say black and white) piecewise-linear triangulation of the sphere with vertex set $V$. Consider the abelian group $\mathcal{A}_W$ generated by the set $V$, with relations $r+c+s=0$ for all white triangles with vertices $r$, $c$ and $s$. The group $\mathcal{A}_B$ can be defined similarly, using black triangles. These groups are related in the following manner $\mathcal{A}_W\cong\mathcal{A}_B\cong\mathbb{Z}\oplus\mathbb{Z}\oplus\mathcal{C}$* where** *$\mathcal{C}$ is a finite abelian group.

#### Keywords

Face 2-coloured spherical triangulation; Directed Eulerian spherical embedding; Canonical group; Abelian sand-pile group; Latin bitrade