In this question from 2012, Jordan Ellenberg asks if the set of spanning trees of a graph $G$ is naturally a torsor for the critical group (also called the sandpile group or the picard group $Pic^0(G)$) of $G$. In other words, is the action described by the rotor-routing process in "Chip-Firing and Rotor-Routing on Directed Graphs" by Alexander E. Holroyd, Lionel Levine, Karola Meszaros, Yuval Peres, James Propp and David B. Wilson independent of the basepoint vertex chosen?
A lovely answer is given in "Rotor-routing and spanning trees on planar graphs" by Melody Chan, Thomas Church, Joshua A. Grochow:
The $Pic^0(G)$-torsor structure is independent of the basepoint vertex $v$ if and only if $G$ is a planar ribbon graph.
I highly recommend reading Jordan's question and the resulting discussion for a better summary of the required background. Here's what this question is about.
We can rephrase this result as follows: Let $\mathcal{T}$ be the set of spanning trees of $G$, and $D \in Pic^0(G)$. Define the map $\phi_D: V(G) \rightarrow Aut(\mathcal{T})$ by $\phi_D(v) = D_v$. That is, we send a vertex to the action of $D$ on $\mathcal{T}$ with basepoint vertex $v$. Then the theorem of Chan, Church, and Grochow says
The map $\phi_D: V(G) \rightarrow Aut(\mathcal{T})$ is a point map for all divisors $D \in Pic^0(G)$ if and only if $G$ is planar.
Vague Question) What does the map $\phi_D$ look like for non-planar ribbon graphs $G$?
This question is fairly vague, and I'd be happy reading anything in the neighborhood of an answer to it. One way to make this more precise is as follows:
Question) What are the integers $|\phi_D(V(G))|$ for $D \in Pic^0(G)$? For a planar graph, these are all $1.$ For a non-planar graph, is there some way (involving either the combinatorial properties of the divisors or the graph-theoretic properties of the spanning trees) to deduce these values?
