We will begin with a brief introduction to hyperbolic 3-manifolds
and their volumes. Thurston and Jorgensen showed that there is a
finite number N(v) of hyperbolic 3-manifolds with any given volume v.
We will look at the question of how N(v) varies with v.
We show that there is an infinite sequence of closed hyperbolic 3-manifolds
that are uniquely determined by their volumes. The proof uses work of
Neumann-Zagier on the change in volume during hyperbolic Dehn surgery
together with some elementary number theory.
We also describe examples showing that the number of hyperbolic link
complements with volume v can grow at least exponentially fast with v.
(This is joint work with Hidetoshi Masai, Tokyo Institute of Technology)