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

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Craig Hodgson (Melbourne University,
visiting Boston College)

Abstract:
The work of Thurston and Jorgensen shows that there is a finite number N(v)
of orientable hyperbolic 3-manifolds with any given volume v.
We will look at the question of how N(v) varies with v.

The work of Gabai-Meyerhoff-Milley shows that the Weeks manifold is the unique
orientable hyperbolic 3-manifold of lowest volume v_1 = 0.9427... , so N(v_1) = 1.We show that there is an infinite sequence of closed hyperbolic 3-manifolds
that are uniquely determined by their volume. 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 describe examples showing that the number of hyperbolic link complements
with volume v can grow at least exponentially fast with v.

(Joint work with Hidetoshi Masai, Tokyo Institute of Technology)