On the Laplacian and the geometry of hyperbolic 3-manifolds

Let N = H 3 / be an infinite volume hyperbolic 3-manifold which is homeomorphic to the interior of a compact manifold. Let Q {N) = inf spec(-) where is the Laplacian acting on functions on TV. We prove that if N is not geometrically finite, then Q (N) = 0, and if N is geometrically finite we produce an upper bound for Q (N) in terms of the volume of the convex core. As a consequence we see that o (N) = 0 if and only if TV is not geometrically finite. We also show that if N has a lower bound for its injectivity radius and is not geometrically finite, then its limit set L has Hausdorff dimension 2.