Abstract:
We will begin with a brief introduction to hyperbolic 3manifolds
and their volumes. Thurston and Jorgensen showed that there is a
finite number N(v) of hyperbolic 3manifolds 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 3manifolds
that are uniquely determined by their volumes. The proof uses work of
NeumannZagier 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)
