  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 $\lim \sup_{t\rightarrow\infty}(m_t)^{1/t}$.
