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

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.