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Factorial growth rates for the number of hyperbolic 3-manifolds of a given volume

Author: Christian Millichap
Journal: Proc. Amer. Math. Soc. 143 (2015), 2201-2214
MSC (2010): Primary 52A22; Secondary 46B09
Published electronically: January 16, 2015
MathSciNet review: 3314126
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Abstract: The work of Jørgensen and Thurston shows that there is a finite number $ N(v)$ of orientable hyperbolic $ 3$-manifolds with any given volume $ v$. In this paper, we construct examples showing that the number of hyperbolic knot complements with a given volume $ v$ can grow at least factorially fast with $ v$. A similar statement holds for closed hyperbolic $ 3$-manifolds, obtained via Dehn surgery. Furthermore, we give explicit estimates for lower bounds of $ N(v)$ in terms of $ v$ for these examples. These results improve upon the work of Hodgson and Masai, which describes examples that grow exponentially fast with $ v$. Our constructions rely on performing volume preserving mutations along Conway spheres and on the classification of Montesinos knots.

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Christian Millichap
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

Received by editor(s): September 3, 2012
Received by editor(s) in revised form: November 4, 2013
Published electronically: January 16, 2015
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2015 American Mathematical Society