Georgia Institute of Technology College of Sciences

The Jacobian group Jac(G) of a finite graph G is a finite abelian group whose cardinality is the number of spanning trees of G. It is natural to wonder whether there is a canonical simply transitive action of Jac(G) on the set of spanning trees which "explains" this numerical coincidence. Surprisingly, this turns out to be related to topological embeddings: we will explain a certain precise sense in which the answer is yes if and only if G is planar. We will also explain how tropical geometry sheds an interesting new light on this picture.