Sandpile group of infinite graphs

Abstract

For a finite connected graph $G$ and a non-empty subset $S$ of its vertices thought of sinks, the so-called critical group (or sandpile group) $C(G,S)$ has been studied for a long time. We present a class of graphs where such an extension can be made in a unified way. Similar extension was made by Maes, C. and Redig, F. and Saada, E., but we propose a more algebraic point of view. Namely, consider a $C$-net $S\subset {\mathbb{Z}}^2$. We define a sandpile dynamics on ${\mathbb{Z}}^2$ with the set $S$ of sinks. For such a choice of sinks, a relaxation of any bounded state is well defined. This allows us to define a group $C({\mathbb{Z}}^2,S)$ of recurrent states of this model. We show that $C({\mathbb{Z}}^2,S)$ is isomorphic to a group of $S^1$-valued discrete harmonic functions on ${\mathbb{Z}}^2\setminus S$. Examples of $S$, for which $C({\mathbb{Z}}^2,S)$ has no torsion or has all torsions, are provided. Pontryagin dual point of view is investigated. A discussion about perspectives of a sandpile group for ${\mathbb{Z}}^2$ as a projective limit concludes this work.

added a discussion, example with a hure pile, proofs are simplified