Torsor Structures on Spanning Trees

We study two actions of the (degree 0) Picard group on the set of the spanning trees of a finite ribbon graph. It is known that these two actions, denoted $\beta_q$ and $\rho_q$ respectively, are independent of the base vertex $q$ if and only if the ribbon graph is planar. Baker and Wang conjectured that in a nonplanar ribbon graph without multiple edges there always exists a vertex $q$ for which $\rho_q\neq\beta_q$. We prove the conjecture and extend it to a class of ribbon graphs with multiple edges. We also give explicit examples exploring the relationship between the two torsor structures in the nonplanar case.