A Mourre estimate and related bounds for hyperbolic manifolds with cusps of non-maximal rank

Abstract

We consider the Laplace operator on quotients of hyperbolic n-dimensional space by a geometrically finite discrete group of hyperbolic isometries with parabolic subgroups of non-maximal rank. Using methods developed by the first two authors, we prove a “Mourre estimate” and commutator estimates on the Laplacian which imply absolute continuity of the spectrum and quantitative resolvent estimates. These estimates will be used elsewhere to study the scattering matrix and Eisenstein series and their meromorphic continuations, and should be useful in studying trace formulas for these discrete groups.