The canonical groups of properly face 2-coloured triangulations of the sphere: What groups can you get?

Let G be a properly face 2-coloured spherical triangulation, whose underlying graph is simple and finite, with face colour classes W and B derived from a separated and connected latin bitrade (W,B); i.e. a spherical latin bitrade. Let A be the abelian group constructed by labelling the vertices with commuting indeterminates and adding relations which say that the labels around each white triangle add to the identity. Further let A* be the finite torsion subgroup of A, the canonical group of the triangulation. This talk investigates a question asked by Cavenagh and Wanless, namely: Which abelian groups arise as the canonical group of some spherical latin bitrade?.