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

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Abstract

Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let Λ(M) be the supremum of λ0(N) where N varies over all hyperbolic 3-manifolds homeomorphic to the interior of N. 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 A(M) = D(M)(2 − D(M)) if M is not handlebody or a thickened torus. We characterize exactly when A(M) = 1 and D(M) = 1 in terms of the characteristic submanifold of the incompressible core of M.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 48109, Ann Arbor, MI

    Richard D. Canary

  2. Department of Mathematics, SUNY-Stony Brook, 11794, Stony Brook, NY

    Yair N. Minsky

  3. Department of Mathematics, University of Michigan, 48109-1189, Ann Arbor, MI

    Edward C. Taylor

Authors
  1. Richard D. Canary
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  2. Yair N. Minsky
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  3. Edward C. Taylor
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Correspondence to Richard D. Canary.

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Communicated by Linda Keen

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Canary, R.D., Minsky, Y.N. & Taylor, E.C. Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds. J Geom Anal 9, 17–40 (1999). https://doi.org/10.1007/BF02923086

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  • DOI: https://doi.org/10.1007/BF02923086

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