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

Abstract

The work of Jørgensen and Thurston shows that there is a finite number N ( v ) N(v) of orientable hyperbolic 3 3 -manifolds with any given volume v v . In this paper, we construct examples showing that the number of hyperbolic knot complements with a given volume v v can grow at least factorially fast with v v . A similar statement holds for closed hyperbolic 3 3 -manifolds, obtained via Dehn surgery. Furthermore, we give explicit estimates for lower bounds of N ( v ) N(v) in terms of v v for these examples. These results improve upon the work of Hodgson and Masai, which describes examples that grow exponentially fast with v v . Our constructions rely on performing volume preserving mutations along Conway spheres and on the classification of Montesinos knots.