On the number of hyperbolic 3-manifolds of a given volume

TL;DR

This paper demonstrates that certain hyperbolic 3-manifolds are uniquely determined by their volumes and explores the growth rate of hyperbolic link complements with fixed volume, revealing infinite sequences and exponential growth patterns.

Contribution

It provides explicit examples of hyperbolic 3-manifolds uniquely determined by volume and analyzes the growth rate of such manifolds with respect to volume.

Findings

Infinite sequences of hyperbolic 3-manifolds uniquely determined by volume.

Volumes of these manifolds converge to the volume of the figure eight knot complement.

Number of hyperbolic link complements with fixed volume can grow exponentially.

Abstract

The work of Jorgensen and Thurston shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. We show that there is an infinite sequence of closed orientable hyperbolic 3-manifolds, obtained by Dehn filling on the figure eight knot complement, that are uniquely determined by their volumes. This gives a sequence of distinct volumes x_i converging to the volume of the figure eight knot complement with N(x_i) = 1 for each i. We also give an infinite sequence of 1-cusped hyperbolic 3-manifolds, obtained by Dehn filling one cusp of the (-2,3,8)-pretzel link complement, that are uniquely determined by their volumes amongst orientable cusped hyperbolic 3-manifolds. Finally, we describe examples showing that the number of hyperbolic link complements with a given volume v can grow at least exponentially fast with v.