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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?

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    $\begingroup$ Maybe you know about this already, but there is some additional work by Matt Baker and Yao Wang relating rotor-routing of plane graphs to the "Bernardi process" and also work of Baker's student Chi Ho Yuen explaining this relation in terms of "geometric bijections"; see this great blog post by Baker for all the details: mattbaker.blog/2017/09/19/the-combinatorics-of-break-divisors. $\endgroup$ – Sam Hopkins Nov 6 '17 at 0:53
  • $\begingroup$ (This is not directly related to your question, but is more evidence that the planar case is special.) $\endgroup$ – Sam Hopkins Nov 6 '17 at 0:53
  • $\begingroup$ I have read Matt Baker's blog post, but I haven't read his and Yao Wang's paper on the Bernardi process. I'm currently working through ABKS, and will have to check that out afterwards! Hopefully it sheds some light on what happens in the non-planar case. Thanks for the reference! $\endgroup$ – Jonathan Gerhard Nov 8 '17 at 14:34
  • $\begingroup$ This recent paper about what information is contained in the rotor-router torsor seems like it might interest you as well: arxiv.org/abs/1804.07807 $\endgroup$ – Sam Hopkins Jun 15 '18 at 15:48
  • $\begingroup$ @SamHopkins Very interesting, thank you! $\endgroup$ – Jonathan Gerhard Oct 15 '18 at 16:17

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