We prove a vanishing theorem for a convex cocompact hyperbolic manifold which relates its L2-cohomology and the Hausdorff dimension of its limit set. The borderline case is shown to characterize the manifold completely.
Xiaodong Wang. "On the L2-cohomology of a convex cocompact hyperbolic manifold." Duke Math. J. 115 (2) 311 - 327, 1 November 2002. https://doi.org/10.1215/S0012-7094-02-11523-3