Abstract
This talk is a survey of work done jointly, mainly with Peter Lax [3,4,5] but also with Bettina Wilkott and Alex Woo [10], over the past ten years. It deals with the spectrum of the perturbed (and unperturbed) Laplace-Beltrami operator acting on automorphic functions on an n-dimensional hyperbolic space IHn. The associated discrete subgroup Γ of motions is assumed to have the finite geometric property, but is otherwise unrestricted. This means that the fundamental domain, when derived by the polygonal method, has a finite number of sides; its volume may be finite or infinite and it can have cusps of arbitrary rank. With the obvious identifications the fundamental domain can be treated as a manifold, M.
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