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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

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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Article copyright: © Copyright 1993 American Mathematical Society