Minimum volume cusped hyperbolic three-manifolds

Gabai, David; Meyerhoff, Robert; Milley, Peter

doi:10.1090/S0894-0347-09-00639-0

Minimum volume cusped hyperbolic three-manifolds
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by David Gabai, Robert Meyerhoff and Peter Milley

J. Amer. Math. Soc. 22 (2009), 1157-1215

DOI: https://doi.org/10.1090/S0894-0347-09-00639-0

Published electronically: May 1, 2009

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Abstract:

This paper is the second in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic $3$-manifolds. Using Mom technology, we prove that any one-cusped hyperbolic $3$-manifold with volume $\le 2.848$ can be obtained by a Dehn filling on one of $21$ cusped hyperbolic $3$-manifolds. We also show how this result can be used to construct a complete list of all one-cusped hyperbolic $3$-manifolds with volume $\le 2.848$ and all closed hyperbolic $3$-manifolds with volume $\le 0.943$. In particular, the Weeks manifold is the unique smallest volume closed orientable hyperbolic $3$-manifold.

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