Boundary structure of hyperbolic 3-manifolds admitting toroidal fillings at large distance

Abstract

We show that if a hyperbolic 3-manifold M has two toroidal Dehn fillings with distance at least 3, then ∂ M consists of at most three tori. As a result, we can obtain an optimal estimate for the number of exceptional slopes on hyperbolic 3-manifolds with boundary a union of at least 4 tori.