Loxodromic Eisenstein series for cofinite Kleinian groups

Abstract

We introduce an Eisenstein series associated to a loxodromic element of cofinite Kleinian groups, namely the loxodromic Eisenstein series, and study its fundamental properties. It is the analogue of the hyperbolic Eisenstein series for Fuchsian groups of the first kind. We prove the convergence and the differential equation associated to the Laplace-Beltrami operator. We also prove the precise spectral expansion associated to the Laplace-Beltrami operator. Furthermore, we derive the analytic continuation with the location of the possible poles and their residues from the spectral expansion.