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.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 2-coloured spherical triangulations with $t$ triangles of each colour. By relating such a triangulation to certain directed Eulerian spherical embeddings of digraphs whose abelian sand-pile groups are isomorphic to the triangulation's canonical group we provide improved upper and lower bounds for $\lim \sup_{t\rightarrow\infty}(m_t)^{1/t}$.
Highlights
Let mt be the maximal order of C over all properly face 2-coloured spherical triangulations with t triangles of each colour. By relating such a triangulation to certain directed Eulerian spherical embeddings of digraphs whose abelian sand-pile groups are isomorphic to the triangulation’s canonical group we provide improved upper and lower bounds for lim supt→∞(mt)1/t
In the same paper the question of the growth rate of the maximal order of C, in the terminology established in [15] the canonical group of the face 2-coloured spherical triangulation, was raised
Let mt be the maximal order of the canonical group over all properly face 2-coloured spherical triangulations whose underlying graphs are simple and have t faces of each colour
Summary
Suppose that there exists a face 2-coloured, black and white say, triangulation of the sphere, i.e. a spherical triangulation, G which has G as its underlying graph. In the same paper the question of the growth rate of the maximal order of C, in the terminology established in [15] the canonical group of the face 2-coloured spherical triangulation, was raised. Let mt be the maximal order of the canonical group over all properly face 2-coloured spherical triangulations whose underlying graphs are simple and have t faces of each colour. In order to establish these new bounds we will make use of a connection between canonical groups of face 2-coloured spherical triangulations and abelian sand-pile groups of the digraphs underlying directed Eulerian spherical embeddings.
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