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Vol3 and other exceptional hyperbolic 3-manifolds

Authors: K. N. Jones and A. W. Reid
Journal: Proc. Amer. Math. Soc. 129 (2001), 2175-2185
MSC (2000): Primary 57M50
Published electronically: December 4, 2000
MathSciNet review: 1825931
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Abstract: D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional hyperbolic manifolds in their proof that a manifold which is homotopy equivalent to a hyperbolic manifold is hyperbolic. These families are each conjectured to consist of a single manifold. In fact, an important point in their argument depends on this conjecture holding for one particular exceptional family. In this paper, we prove the conjecture for that particular family, showing that the manifold known as $\mathrm{Vol}3$ in the literature covers no other manifold. We also indicate techniques likely to prove this conjecture for five of the other six families.

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Additional Information

K. N. Jones
Affiliation: Department of Mathematical Sciences, Ball State University, Muncie, Indiana 47304

A. W. Reid
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712

Keywords: Hyperbolic 3-manifold, arithmetic manifold, homotopy hyperbolic 3-manifold
Received by editor(s): April 19, 1999
Received by editor(s) in revised form: October 13, 1999, and November 8, 1999
Published electronically: December 4, 2000
Additional Notes: The first author was partially supported by Ball State University.
The second author was partially supported by the Royal Society, NSF, the A. P. Sloan Foundation and a grant from the Texas Advanced Research Program.
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2000 American Mathematical Society

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