Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds

Richard D. Canary

Yair N. Minsky

Edward C. Taylor


Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let L(M) be the supremum of the bottom l0(N) of the spectrum of the Laplacian where N varies over all hyperbolic 3-manifolds homeomorphic to the interior of M. Similarly, we let D(M) be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of M. We observe that L(M) = D(M)(2-D(M)) if M is not handlebody or a thickened torus. We characterize exactly when L(M) = 1 and D(M) = 1 in terms of the characteristic submanifold of the incompressible core of M.

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On 23 Nov 1999, 17:42.