Groups acting on hyperbolic space

by Jürgen Elstrodt

This book deals with a broad range of topics from the theory of automorphic functions on three-dimensional hyperbolic space and its arithmetic, group-theoretic, and geometric ramifications. Starting off with several models of hyperbolic space and its group of motions the authors discuss the spectral theory of the Laplacian and Selberg's theory for cofinite groups. This culminates in explicit versions of the Selberg trace formula and the Selberg zeta-function. The interplay with arithmetic is demonstrated by means of the groups PSL (2) over rings of quadratic integers, their Eisenstein series and their associated Hermitian forms. A comprehensive chapter on concrete examples of arithmetic and non-arithmetic cofinite groups enhances the usefulness of this work for a wide audience of mathematicians.