Hodge theory on hyperbolic manifolds of infinite volume

Abstract

Let $Y=\Gamma\backslash H^n$ be a quotient of the hyperbolic space by the action of a discrete convex-cocompact group of isometries. We describe certain spaces of $\Gamma$-invariant currents on the sphere at infinity of $H^n$ with support on the limit set of $\Gamma$. These spaces are finite-dimensional. The main result identifies the cohomology of $Y$ with a quotient of such spaces. We explain in which sense this result generalizes the classical Hodge theorem for compact quotients. We obtain analogous results for the cohomology groups $H^p(\Gamma,F)$, where $F$ is a finite-dimensional representation of the full group of orientation preserving isometries of $H^n$.