Toroidal Manifolds and Dehn Fillings on Links

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-Dehn filling, 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 − 5 p , (ii) If t = 2, then �(α, β) ≤ 13 + 24g − 12 p , (iii) If t = 1, then �(α, β) ≤ 1. Let M be a compact, connected, orientable and irreducible 3-manifold with a toral boundary component T. The unoriented isotopy class of a non-trivial simple closed curve on T is called its slope, and if α and β are slopes on T then �(α, β) will denote their minimal geometric intersection number. Let γ be a slope on T, and let M(γ) be the 3-manifold obtained by γ-Dehn filling. Thus M(γ) = M ∪ J, where J is a solid torus, glued to M along T in such a way that γ bounds a disk in J. A surface in 3-manifold is called essential if it is properly embedded and either (i) incompressible, not parallel to a subsurface of the boundary of the 3-manifold, and not a 2-sphere, or (ii) a 2-sphere that does not bound a 3-ball. A 3-manifold is said to be toroidal if it contains an essential torus. Recall that Thurston has shown that if M is hyperbolic then M(γ) is hyperbolic for all but finitely many slopes γ. Furthermore if M(γ) is not hyperbolic, then it is either reducible, toroidal or a Seifert fiber space. 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. We assume that the genus g of P is minimal (over all essential

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