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

Abstract

Let G be a properly face 2-coloured (say black and white) piecewiselinear triangulation of the sphere with vertex set V . Consider the abelian group 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 A B can be defined similarly, using black triangles. These groups are related in the following mannerThe finite torsion subgroup C is referred to as the canonical group of the triangulation. Let m t be the maximal order of C over all properly face two-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→∞ (m t ) 1/t .

On which groups can arise as the canonical group of a spherical latin bitrade

“…Note that, if P and Q are two partial latin squares in the same main class, then A P ∼ = A Q . Also, note that two partial latin squares, P and Q, from different main classes may also satisfy A P ∼ = A Q (see Figure 2 in [18]).…”

“…The group C in Theorem 1.3 is referred to as the canonical group of the spherical latin bitrade (see [14,18]).…”

“…Given a face 2-coloured triangulation of the sphere in which the underlying graph is not necessarily simple and leaving the definitions of A W and A B unchanged it is still the case that [1]. In [18] the second author showed that given any finite abelian group Γ there exists a face 2-coloured triangulation of the sphere whose canonical group is isomorphic to Γ. However, unless Γ is a cyclic group the triangulations constructed have underlying graphs that are not simple.…”

“…By [15], the underlying digraph of G has a vertex three-colouring with colour classes R, C and S. Tutte [20] described a construction, from G, of directed Eulerian spherical embeddings D I (G) = D I with vertex set I, where I ∈ {R, C, S}. We give a description of the construction from [18].…”

“…Early motivation [12] for their study arose from considering the differences between the operation tables of a finite group and a latin square of the same order, that is: what is the 'distance' between a group and a latin square? The study of the topological and geometric properties of latin trades has lead to significant progress towards understanding such differences, see for example [8,10,1,18,19], also see [6] for a survey of earlier results.…”