# The sandpile group of a trinity and a canonical definition for the planar Bernardi action

## TL;DR

This paper provides a canonical definition for the planar Bernardi and rotor-routing actions on sandpile groups, establishing their compatibility with planar duality and revealing hidden symmetries through a generalized setting involving trinities.

## Contribution

It introduces a canonical framework for the planar Bernardi/rotor-routing actions and a canonical isomorphism between sandpile groups of dual graphs, clarifying previous confusions.

## Key findings

- Canonical definition for planar Bernardi/rotor-routing actions
- Compatibility with planar duality proved with a short argument
- Extension of results to balanced plane digraphs and trinities

## Abstract

Baker and Wang define the so-called Bernardi action of the sandpile group of a ribbon graph on the set of its spanning trees. This potentially depends on a fixed vertex of the graph but it is independent of the base vertex if and only if the ribbon structure is planar, moreover, in this case the Bernardi action is compatible with planar duality. Earlier, Chan, Church and Grochow and Chan, Glass, Macauley, Perkinson, Werner and Yang proved analogous results about the rotor-routing action. Baker and Wang moreover showed that the Bernardi and rotor-routing actions coincide for plane graphs. We clarify this still confounding picture by giving a canonical definition for the planar Bernardi/rotor-routing action, and also a canonical isomorphism between sandpile groups of planar dual graphs. Our canonical definition implies the compatibility with planar duality via an extremely short argument. We also show hidden symmetries of the problem by proving our results in the slightly more general setting of balanced plane digraphs. Any balanced plane digraph gives rise to a trinity, i.e., a triangulation of the sphere with a three-coloring of the $0$-simplices. Our most important tool is a group associated to trinities, introduced by Cavenagh and Wanless, and a result of a subset of the authors characterizing the Bernardi bijection in terms of a dissection of a root polytope.