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

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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 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 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 C we provide improved upper and lower bounds for ₜ→∞(mₜ)¹/ᵗ.