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

Millichap, Christian

doi:10.1090/S0002-9939-2015-12395-7

Factorial growth rates for the number of hyperbolic 3-manifolds of a given volume
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by Christian Millichap PDF

Proc. Amer. Math. Soc. 143 (2015), 2201-2214 Request permission

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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