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 complete 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 Δ ≔ d I − A . Let the rows of Δ span the lattice Λ in Z V . The sandpile group G ( d , h ) of T ( d , h ) is Z V / Λ . We compute the rank, the exponent, the order, and other structural parameters of the abelian group G ( d , h ) . We find a cyclic Hall-subgroup of order ( d − 1 ) h . We prove that the rank of G ( d , h ) is ( d − 1 ) h and that G ( d , h ) contains a subgroup isomorphic to Z d ( d − 1 ) h ; therefore, for all primes p dividing d , the rank of the Sylow p -subgroup is maximal (equal to the rank of the entire group). We find that the base - ( d − 1 ) logarithm of the exponent and of the order are asymptotically 3 h 2 / π 2 and c d ( d − 1 ) h , respectively. We conjecture an explicit formula for the ranks of all Sylow subgroups.