On the Laplacian and the geometry of hyperbolic 3-manifolds (Q1203634)

Let \(N=\mathbb{H}^ 3/\Gamma\) be an infinite volume hyperbolic 3-manifold which is homeomorphic to the interior of a compact manifold. We prove that if \(N\) is not geometrically finite, then \(\lambda_ 0(N)=0\) and if \(N\) is geometrically finite we produce an upper bound for \(\lambda_ 0(N)\) in terms of the volume of the convex core. As a consequence we see that \(\lambda_ 0(N)=0\) if and only if \(N\) 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 \(L_ \Gamma\) has Hausdorff dimension 2.