Document Type

Article

Publication Date

3-2015

Department

Mathematics

Abstract

Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it has a canonical rotor-routing action associated to it; i.e., the rotor-routing action is actually independent of the choice of root vertex. It is well known that the spanning trees of a planar graph G are in canonical bijection with those of its planar dual G∗, and furthermore that the sandpile groups of G and G∗ are isomorphic. Thus, one can ask: are the two rotor-routing actions, of the sandpile group of G on its spanning trees, and of the sandpile group of G∗ on its spanning trees, compatible under plane duality? In this paper, we give an affirmative answer to this question, which had been conjectured by Baker.

DOI

10.1137/140982015

Required Publisher's Statement

Original version is available from the publisher at: http://epubs.siam.org/doi/abs/10.1137/140982015

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