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)T(d,h) be the complete dd-regular tree of depth hh and let VV be the set of its vertices. Denote the adjacency matrix of T(d,h)T(d,h) by AA and consider the modified Laplacian matrix Δ≔dI−AΔ≔dI−A. Let the rows of ΔΔ span the lattice ΛΛ in ZVZV. The sandpile group G(d,h)G(d,h) of T(d,h)T(d,h) is ZV/ΛZV/Λ. We compute the rank, the exponent, the order, and other structural parameters of the abelian group G(d,h)G(d,h). We find a cyclic Hall-subgroup of order (d−1)h(d−1)h. We prove that the rank of G(d,h)G(d,h) is (d−1)h(d−1)h and that G(d,h)G(d,h) contains a subgroup isomorphic to Zd(d−1)h; therefore, for all primes pp dividing dd, the rank of the Sylow pp-subgroup is maximal (equal to the rank of the entire group). We find that the base -(d−1)(d−1) logarithm of the exponent and of the order are asymptotically 3h2/π23h2/π2 and cd(d−1)hcd(d−1)h, respectively. We conjecture an explicit formula for the ranks of all Sylow subgroups.