Vol3 and other exceptional hyperbolic 3-manifolds

D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional hyperbolic manifolds in their proof that a manifold which is homotopy equivalent to a hyperbolic manifold is hyperbolic. These families are each conjectured to consist of a single manifold. In fact, an important point in their argument depends on this conjecture holding for one particular exceptional family. In this paper, we prove the conjecture for that particular family, showing that the manifold known as V o l 3 Vol3 in the literature covers no other manifold. We also indicate techniques likely to prove this conjecture for five of the other six families.