Latin trades in groups defined on planar triangulations

For a finite triangulation of the plane with faces properly coloured white and black, let A_W 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. We show that A _W has free rank exactly two. Let A_W^* be the torsion subgroup of A_W, and A_B^* the corresponding group for the black triangles. We show that A_W ^* and A_B^* have the same order, and conjecture that they are isomorphic. For each spherical latin trade W, we show there is a unique disjoint mate B such that (W,B) is a connected and separated bitrade. The bitrade (W,B) is associated with a two-colourable planar triangulation and we show that W can be embedded in A_W ^*, thereby proving a conjecture due to Cavenagh and Drápal. The proof involves constructing a (0,1) presentation matrix whose permanent and determinant agree up to sign. The Smith normal form of this matrix determines A_W^*, so there is an efficient algorithm to construct the embedding. Contrasting with the spherical case, for each genus g≥1 we construct a latin trade which is not embeddable in any group and another that is embeddable in a cyclic group. We construct a sequence of spherical latin trades which cannot be embedded in any family of abelian groups whose torsion ranks arbounded. Also, we show that any trade that can be embedded in a finitely generated abelian group can be embedded in a finite abelian group. As a corollary, no trade can be embedded in a free abelian group.