The geometry and spectra of hyperbolic manifolds

This paper is a self-contained discussion of the relationship between spectral and geometric properties of a class of hyperbolic manifolds. After a review of the fundamentals of hyperbolic manifolds, aspects of the theory for the compact case and the finite-volume case are discussed. The main emphasis of this work is on a class of infinite-volume hyperbolic manifolds ℳ which arise as quotients of hyperbolic space H ⁿ by discrete subgroups G{cyrillic}, i.e. ℳ =H ⁿ/G{cyrillic}. This paper describes joint work with R G Froese and P A Perry. For these infinite-volume hyperbolic manifolds, there are very few eigenvalues, so most of the spectral information in carried by the generalized eigenfunctions of the Laplacian. These eigenfunctions can be constructed from the asymptotics of the Green's function. It is shown how the asymptotic geometry of the manifold determines the asymptotic behavior of the Green's function, and hence the eigenfunctions, near infinity. This information is used to construct an S-matrix for the manifold which is a pseudo-differential operator acting on sections of a fibre bundle over the boundary of the manifold at infinity. The meromorphic properties of this operator and its inverse, as a function of the spectral parameter, are described. A functional relation between the S-matrix and the generalized eigenfunctions is derived. An important consequence of this relation and the meromorphicity of the S-matrix and its inverse is the existence of the meromorphic continuation of the Eisenstein series associated with the discrete group G{cyrillic}. Finally, an overview of recent progress and some open problems are presented, including a discussion of the asymptotic behavior of the counting function for the scattering poles.