Toroidal Manifolds and Dehn Fillings on Links

Abstract

Let M be a hyperbolic 3-manifold such that ∂M has at least two boundary tori ∂0M and ∂1M. Suppose that M contains an essential orientable surface P of genus g with one outer boundary component ∂0P, lying in ∂0M and having slope λ in ∂0M, and p inner boundary components ∂iP, i = 1, · · · , p, each having slope α in ∂1M. Let β be a slope in ∂1M and suppose that M(β) is toroidal. Let ˆ T be a minimal essential torus in M(β), which means that ˆ T is pierced a minimal number of times by the core of the β-Dehnfilling, among all essential tori in M(β). Let T = ˆ T ∩ M and denote by t the number of components of ∂T. In this paper we prove: (i) If t ≥ 3, then
(α, β) ≤ 6 + 10g − 5p, (ii) If t = 2, then
(α, β) ≤ 13 + 24g − 12p, (iii) If t = 1, then
(α, β) ≤ 1.