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

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

1998-10-19

arxiv.org/abs/math/9810124v1

J. Geom. Analysis 9 (1999), 18-40

Mathematics

Geometric Topology

31 pages; to appear in J. Geom. Anal

Scientific paper

Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let Lambda(M) be the supremum of the bottom eigenvalue of the Laplacian of N, 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 Lambda(M)=D(M)(2-D(M)) if M is not handlebody or a thickened torus. We characterize exactly when Lambda(M)=1 and D(M)=1 in terms of the characteristic submanifold of the incompressible core of M.

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