On the sandpile group of regular trees

The sandpile group of a connected graph is the group of recurrent configurations in the abelian sandpile model on this graph. We study the structure of this group for the case of regular trees. A description of this group is the following: Let T (d, h) be the d-regular tree of depth h and let V be the set of its vertices. Denote the adjacency matrix of T (d, h) by A and consider the modified Laplacian matrix ∆ := dI − A. Let the rows of ∆ span the lattice Λ in Z . The sandpile group of T (d, h) is Z /Λ. We compute the rank, the exponent and the order of this abelian group and find a cyclic Hall-subgroup of order (d− 1). We find that the base (d−1)-logarithm of the exponent and of the order are asymptotically 3h/π and cd(d−1) , respectively. We conjecture an explicit formula for the ranks of all Sylow subgroups.