Abelian Sandpiles

Sandpiles models have nice physical properties. However, they can also be studied from an algebraic point of view. In this thesis we consider the abelian sandpile model. We begin by examining the group structure of this model. The elements of this group, configurations of the sandpile, can be added by placing them on top of each other and subsequently letting them topple to a stable configuration. We show this operation is well defined. It turns out that the sandpile groups are in fact finitely generated. The structure of the group can be determined using the minor method. We define four equivalence relations on the elements of the sandpile group. Each of these is generated by a different interpretation of the idea that configurations can be toppled into each other and are thus ‘equivalent’. We compare these equivalence classes and show that some of them are in fact equivalent.