LOXODROMIC EISENSTEIN SERIES THROUGH DEGENERATION

Abstract

The loxodromic Eisenstein series is defined for a loxodromic element of cofinite Kleinian groups. It is the analogue of the ordinary Eisenstein series associated to cusps. We study the asymptotic behavior of the loxodromic Eisenstein series for degenerating sequences of three-dimensional hyperbolic manifolds of finite volume. In particular, we prove that if the loxodromic element corresponds to the degenerating geodesic, then the associated loxodromic Eisenstein series converges to the ordinary Eisenstein series associated to the newly developing cusp on the limit manifold.