Boundary Structure of Hyperbolic 3-Manifolds Admitting Annular and Toroidal Fillings at Large Distance

Teragaito

2008

Math. Ann.

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“…Proof Using the proof of [14,Lemma 2.5(1)], one can show that G β does not contain two S-cycles on disjoint label pairs. This implies that G α has at most two S-vertices.…”

Section: Lemma 411mentioning

confidence: 99%

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

Teragaito

2008

Math. Ann.

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“…Put X = M (γ i ). Then X is irreducible by [8,10], and atoroidal by [7]. Since ∂M is a union of at least four tori, X has at least three torus boundary components…”

Section: Lemma 2 M (γ I ) Is Homeomorphic To P × Smentioning

confidence: 99%

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

2006

Proc. Amer. Math. Soc.

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Abstract. We show that if a hyperbolic 3-manifold M with ∂M a union of tori admits two annular Dehn fillings at distance ∆ ≥ 3, then M is bounded by at most three tori.An annulus or torus embedded in a 3-manifold is essential if it is incompressible, boundary-incompressible and is not boundary-parallel. A 3-manifold is annular (resp. toroidal) if it contains an essential annulus (resp. torus). Otherwise, it is anannular (resp. atoroidal). Also, a 3-manifold is irreducible if any embedded 2-sphere bounds an embedded 3-ball, and is boundary-irreducible if its boundary is incompressible. Thurston [9] has shown that a compact, orientable 3-manifold M with non-empty boundary is irreducible, boundary-irreducible, atoroidal and anannular if and only if it is hyperbolic, in the sense that M with its boundary tori removed admits complete finite volume hyperbolic structure with totally geodesic boundary.Let M be a compact, connected, orientable 3-manifold with torus boundary component T 0 . The Dehn filling of M with slope γ is the manifold M (γ) obtained by attaching a solid torus V γ to M along their boundary so that a meridian of V γ is identified with a curve of slope γ on T 0 . For two slopes γ 1 , γ 2 on T 0 , ∆(γ 1 , γ 2 ) denotes their minimal geometric intersection number and is called a distance between the slopes.Suppose that given a hyperbolic 3-manifold M , there are two slopes γ 1 , γ 2 on T 0 such that both M (γ 1 ) and M (γ 2 ) are annular. Gordon [2] showed that ∆(γ 1 , γ 2 ) ≤ 5, and furthermore, together with Wu [5], showed that there are only three specific manifolds M realizing ∆(γ 1 , γ 2 ) = 4 and 5. These manifolds are the exteriors of the Whitehead link, the Whitehead sister link and the 2-bridge link corresponding to 3/10 in the 3-sphere S 3 . Also, he [3, Theorem 5.3] constructed a hyperbolic kcomponent link in S 2 × S 1 for any k ≥ 4 whose exterior realizes ∆(γ 1 , γ 2 ) = 2, and he asked [3, Question 5.3] what is the maximal value for ∆(γ 1 , γ 2 ) if M is bounded by at least four tori. In this paper, we give an answer to this question. Theorem 1. Let M be a hyperbolic 3-manifold with ∂M a union of tori. Suppose that there are two slopes γ 1 and γ 2 on a specified boundary torus of M such that

Lens spaces and toroidal Dehn fillings

2009

Math. Z.

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