A Numerical Study of Eigenvalues of the Hyperbolic Laplacian for Polyhedra with One Cusp

Abstract

Let H₃ be three-dimensional hyperbolic space and Γ a group of isometries of H₃ that acts discontinuously on H₃ and that has a fundamental domain of finite hyperbolic volume. The laplace operator –δ of H₃ gives rise to a positive, essentiallv selfadjoint operator on L ² (Γ\H₃). The nature of its discrete spectrum dspec Γ is still not well understood if Γ is not cocompact.

This paper contains a report on a numerical study of dspec Γ for various noncocompact groups Γ. Particularly interesting are the results for some nonarithmetic groups Γ.