Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces; the case of finite volume

Abstract:

Let $\Gamma$ be a discrete subgroup of automorphisms of ${{\mathbf {H}}^n}$, with fundamental polyhedron of finite volume, finite number of sides, and $N$ cusps. Denote by ${\Delta _\Gamma }$ the Laplace-Beltrami operator acting on functions automorphic with respect to $\Gamma$. We give a new short proof of the fact that ${\Delta _\Gamma }$ has absolutely continuous spectrum of uniform multiplicity $N$ on $( - \infty ,{((n - 1)/2)^2})$, plus a standard discrete spectrum. We show that this property of the spectrum is unchanged under arbitrary perturbation of the metric on a compact set. Our method avoids Eisenstein series entirely and proceeds instead by constructing explicitly a translation representation for the associated wave equation.