Group actions on spanning trees

Combinatorics Seminar
Tuesday, November 25, 2014 - 13:30
1 hour (actually 50 minutes)
Skiles 005
Georgia Tech
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.