The critical group of a combinatorial map

Motivated by the appearance of embeddings in the theory of chip-firing and the critical group of a graph, we introduce a version of the critical group (or sandpile group) for combinatorial maps, that is, for graphs embedded in orientable surfaces. We provide several definitions of our critical group, by approaching it through analogues of the cycle-cocycle matrix, the Laplacian matrix, and as the group of critical states of a chip-firing game (or sandpile model) on the edges of a map.

Our group can be regarded as a perturbation of the classical critical group of its underlying graph by topological information, and it agrees with the classical critical group in the plane case. Its cardinality is equal to the number of spanning quasi-trees in a connected map, just as the cardinality of the classical critical group is equal to the number of spanning trees of a connected graph.

Our approach exploits the properties of principally unimodular matrices and the methods of delta-matroid theory.