Growth rates of groups associated with face 2-coloured triangulations and directed Eulerian digraphs on the sphere

Let $\mathcal{G}$ be a properly face 2-coloured (say black and white) \break 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. The finite torsion subgroup $\mathcal{C}$ is referred to as the canonical group of the triangulation. Let $m_t$ be the maximal order of $\mathcal{C}$ over all properly face two-coloured spherical triangulations with $t$ triangles of each colour. By relating properly face two-coloured spherical triangulations to directed Eulerian spherical embeddings of digraphs whose abelian sand-pile groups are isomorphic to $\mathcal{C}$ we provide improved upper and lower bounds for $\lim {\sup }_{t\to \mathrm{\infty }}(m_t{)}^{1\mathrm{/}t}$.