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

Let 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 AW generated by the set V , with relations r+c+s = 0 for all white triangles with vertices r, c and s. The group AB can be defined similarly, using black triangles. These groups are related in the following manner AW ∼= AB ∼= Z⊕ Z⊕ C where C is a finite abelian group. The finite torsion subgroup C is referred to as the canonical group of the triangulation. 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.