A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp

Abstract

Let $\hz_{3}$ be three-dimensional hyperbolic space and $\Gamma$ a group of isometries of $\hz_3$ that acts discontinuously on $\hz_{3}$ and that has a fundamental domain of finite hyperbolic volume. The Laplace operator $\MinusDelta$ of $\hz_{3}$ gives rise to a positive, essentially selfadjoint operator on $L^2(\Gamma \backslash \hz_{3})$. The nature of its discrete spectrum $\dspec (\Gamma)$ is still not well understood if $\Gamma$ is not cocompact.

This paper contains a report on a numerical study of $\dspec (\Gamma)$ for various noncocompact groups $\Gamma$. Particularly interesting are the results for some nonarithmetic groups $\Gamma$.