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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 $Pic^0(G)$...

Let G be a finite undirected connected graph. A divisor on G is an element of the free abelian group Div(G) on the vertices of G (or an integer-valued function on the vertices.) Summing over all ...