Kleinian Groups, Laplacian on Forms and Currents at Infinity

Abstract

In this note we consider the spectrum of the Laplacian acting on the space of (co-closed) differential forms on the quotient of $n$-dimensional hyperbolic space by a co-compact Kleinian group. Using a result of P.-Y. Gaillard we relate these to currents on the sphere at infinity of hyperbolic space with distinctive transformation properties under the action of the group. We analyse these currents using zeta-functions and Ruelle’s Transfer operator. This represents a partial extension of earlier work of the author related to Fuchsian groups. In an appendix we propose an alternative approach to related questions.