The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces

The asymptotic distribution of orbits for discrete subgroups of motions in Euclidean and non-Euclidean spaces are found; our principal tool is the wave equation. The results are new for the crystallographic groups in Euclidean space and for those groups in non-Euclidean spaces which have fundamental domains of infinite volume. In the latter case we show that the only point spectrum of the Laplace-Beltrami operator lies in the interval ($\text{?(}\text{(m ? 1)}\text{2}\text{)}{}^2\text{,0}$]; furthermore we show that when the subgroup is nonelementary and the fundamental domain has a cusp, then there is at least one eigenvalue in this interval.