## Latin trades in groups defined on planar triangulations

For a finite triangulation of the plane with faces properly coloured
white and black, 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. We
show that A has free rank exactly two. Let A* be the
torsion subgroup of A, and B* the corresponding group for
the black triangles. We show that A* and 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*, 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*, so there is an efficient algorithm to
construct the embedding. Contrasting with the spherical case, for
each genus g>0 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 are
bounded. 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.

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