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Ends of hyperbolic $ 3$-manifolds


Author: Richard D. Canary
Journal: J. Amer. Math. Soc. 6 (1993), 1-35
MSC: Primary 57M50; Secondary 30F40
DOI: https://doi.org/10.1090/S0894-0347-1993-1166330-8
MathSciNet review: 1166330
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Abstract: Let $ N = {{\mathbf{H}}^3}/\Gamma $ be a hyperbolic $ 3$-manifold which is homeomorphic to the interior of a compact $ 3$-manifold. We prove that $ N$ is geometrically tame. As a consequence, we prove that $ \Gamma $'s limit set $ {L_\Gamma }$ is either the entire sphere at infinity or has measure zero. We also prove that $ N$'s geodesic flow is ergodic if and only if $ {L_\Gamma }$ is the entire sphere at infinity.


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DOI: https://doi.org/10.1090/S0894-0347-1993-1166330-8
Article copyright: © Copyright 1993 American Mathematical Society

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