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On the Laplacian and the geometry of hyperbolic 3-manifolds. (English) Zbl 0763.53040
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.

##### MSC:
 53C20 Global Riemannian geometry, including pinching 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
##### Keywords:
geometrically finite; injectivity radius; limit set
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