The critical group of a graph is a finite abelian group whose cardnality is the number of spanning trees. The critical group is an interesting graph invariant in its own right, and it also arises naturally in the theory of a discrete dynamical system with many essentially equivalent formulations - the chip firing game, dollar game, abelian sandpile model, etc. The model describes a certain type of discrete flow along the edges of the graph. We extend the theory of critical groups, associating a family of abelian groups to higher dimensional cell complexes. We show that the groups can be expressed explicitly in terms of the combinatorial Laplacian and that they are finite with orders given by weighted enumerators of their spanning trees. Finally, we describe how the critical groups of a complex represent flow along its faces.

Joint work with Art Duval and Jeremy Martin