P2-reducing and toroidal Dehn fillings

Jin, Gyo Taek; Lee, Sangyop; Oh, Seungsang; Teragaito, Masakazu

doi:10.1017/s0305004102006412

Cited by 10 publications

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References 17 publications

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“…(See also [17] for a short proof.) In [16], we have shown the conclusions for the two cases (P, T ) and (P, K). In [16], we have shown the conclusions for the two cases (P, T ) and (P, K).…”

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confidence: 77%

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Reducing Dehn Fillings and Small Surfaces

Lee

Teragaito

2005

Proc. Lond. Math. Soc.

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Let M be a compact, connected, orientable 3-manifold with a torus boundary component ∂ 0 M . Let γ be a slope on ∂ 0 M , that is, the isotopy class of an essential simple closed curve on ∂ 0 M . The 3-manifold obtained from M by γ-Dehn filling is defined to be M (γ) = M ∪ V γ , where V γ is a solid torus glued to M along ∂ 0 M in such a way that γ bounds a meridian disk in V γ .By a small surface we mean one with non-negative Euler characteristic including non-orientable surfaces. Such surfaces play a special role in the theory of 3dimensional manifolds. We shall say that a 3-manifold M is hyperbolic if M with its boundary tori removed has a complete hyperbolic structure of finite volume with totally geodesic boundary. Thurston's geometrization theorem for Haken manifolds [22] asserts that a hyperbolic 3-manifold M with non-empty boundary contains no essential small surfaces. Furthermore, if M is hyperbolic, then the Dehn filling M (γ) is also hyperbolic for all but finitely many slopes [22], and a good deal of attention has been directed towards obtaining a more precise quantification of this statement.Let us say that a 3-manifold is of type S, D, A or T if it contains an essential orientable small surface which is an essential sphere, disk, annulus or torus, and of type P, B or K if it contains a non-orientable small surface which is a projective plane, Möbius band or Klein bottle, respectively. In particular, 3-manifolds of type S, D, A and T are called reducible, ∂-reducible, annular and toroidal, respectively. The distance ∆(γ 1 , γ 2 ) between two slopes on a torus is their minimal geometric intersection number. The bound ∆(X 1 , X 2 ) is the least non-negative number m such that if M is a hyperbolic manifold which admits two Dehn fillings M (γ 1 ) and M (γ 2 ) of types X 1 and X 2 , respectively, then ∆(γ 1 , γ 2 ) m. Surveys of the known bounds for various choices (X 1 , X 2 ) and the maximal values realized by known examples are given in [5,7,25].In this paper we consider the case when M (γ 1 ) is of type S or P, and M (γ 2 ) is of type S, P, A, B, T or K, and recover the known bounds for ∆(γ 1 , γ 2 ) in all eleven cases by a unified argument. Suppose that M (γ i ) contains such a small surface F i . Then we may assume that F i meets the attached solid torus V γi in a finite collection of meridian disks, and is chosen so that the number of disks n i is minimal among all such surfaces in M (γ i ). The new results are the restrictions on n 2 in the cases (S/P, A) and (S/P, B) when ∆(γ 1 , γ 2 ) = 2 and in the cases (S, T ) and (S, K) when ∆(γ 1 , γ 2 ) = 3. The main results of this paper are the following.sangyop lee, seungsang oh and masakazu teragaito Theorem 1.1. Suppose that M is hyperbolic. If M (γ 1 ) and M (γ 2 ) are of type S or P, then ∆(γ 1 , γ 2 ) 1.The original proof of the case (S, S) is very complicated [10]. Our proof is remarkably short, although, like [10], it is based on the analysis of intersections of two surfaces. Recently, Hoffman and Matignon [15] also gave a short proof, along almo...

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“…Assume that n 2 = 2 when G 2 is of type K. This case is done by Theorem 5.2(2) and the argument of [JLOT,Section6], which can be carried over without change.…”

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confidence: 99%

Reducing Dehn Fillings and Small Surfaces

Lee

Teragaito

2005

Proc. Lond. Math. Soc.

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“…(2) If G K contains an S-cycle, then M(π) contains a projective plane [3]. But this contradicts [5,Theorem 1.2].…”

Section: Seungsang Ohmentioning

confidence: 98%

Reducing Spheres and Klein Bottles after Dehn Fillings

2003

Can. math. bull.

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“…Since P is non-orientable, we cannot give a sign to a vertex of G P . However, there is a way to give a sign to an edge of G P (see [10]). Then the parity rule survives without any change.…”

Section: Klein Bottlementioning

confidence: 99%

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

Lee¹,

Teragaito²

2008

Can. j. math.

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Abstract. For a hyperbolic 3-manifold M with a torus boundary component, all but finitely many Dehn fillings yield hyperbolic 3-manifolds. In this paper, we will focus on the situation where M has two exceptional Dehn fillings: an annular filling and a toroidal filling. For such a situation, Gordon gave an upper bound of 5 for the distance between such slopes. Furthermore, the distance 4 is realized only by two specific manifolds, and 5 is realized by a single manifold. These manifolds all have a union of two tori as their boundaries. Also, there is a manifold with three tori as its boundary which realizes the distance 3. We show that if the distance is 3 then the boundary of the manifold consists of at most three tori.

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P2-reducing and toroidal Dehn fillings

Cited by 10 publications

References 17 publications

Reducing Dehn Fillings and Small Surfaces

Reducing Dehn Fillings and Small Surfaces

Reducing Spheres and Klein Bottles after Dehn Fillings

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

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