Abstract
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 ( −( (m − 1) 2 ) 2,0 ]; 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.
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