Ends of hyperbolic 3-manifolds

Canary, Richard D.

doi:10.1090/S0894-0347-1993-1166330-8

Ends of hyperbolic $3$-manifolds
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by Richard D. Canary

J. Amer. Math. Soc. 6 (1993), 1-35

DOI: https://doi.org/10.1090/S0894-0347-1993-1166330-8

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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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