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Short geodesics in hyperbolic 3-manifolds. (English) Zbl 1221.57027
For a closed 3-manifold which is either fibred over \(S^1\) or admits a Heegaard decomposition, a simple closed curve in \(M\) is said to be unknotted if it is ambient isotopic into a fibre or a Heegaard surface. More generally, a disjoint collection of simple closed curves in \(M\) is said to be unlinked if the curves are ambient isotopic into disjoint fibres or disjoint Heegaard surfaces, in such a way that one curve lies on one surface.
J.-P. Otal [in Kleinian groups and hyperbolic 3-manifolds. Proceedings of the Warwick workshop, Warwick, UK, September 11–14, 2001. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 299, 95–104 (2003; Zbl 1049.57007)] showed that for a closed hyperbolic \(3\)-manifold \(M\) fibred over \(S^1\), there is a constant depending only on the genus of fibres such that any closed geodesic with length less than \(\epsilon\) is unknotted. J. Souto [Short geodesics in hyperbolic compression bodies are not knotted, preprint] showed that if a closed hyperbolic \(3\)-manifold \(M\) has a strongly irreducible Heegaard decomposition, there is a constant \(\epsilon\) depending only on the genus of the Heegaard decomposition such that any disjoint collection of closed geodesics is unlinked.
In the present paper, the author shows that in the same setting as Souto’s, there is a computable function \(\epsilon\) of the genus such that any closed geodesic with length less than \(\epsilon\) is unknotted. The main technique used in this paper is a sweep-out by surfaces with bounded area.
MSC:
57M50 General geometric structures on low-dimensional manifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
53C22 Geodesics in global differential geometry
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References:
[1] J Cerf, Sur les difféomorphismes de la sphère de dimension trois \((\Gamma _4=0)\), Lecture Notes in Math. 53, Springer (1968) · Zbl 0164.24502
[2] J Johnson, Locally unknotted spines of Heegaard splittings, Algebr. Geom. Topol. 5 (2005) 1573 · Zbl 1120.57007
[3] J Johnson, Notes on Heegaard splittings, Preprint (2006)
[4] R Meyerhoff, A lower bound for the volume of hyperbolic \(3\)-manifolds, Canad. J. Math. 39 (1987) 1038 · Zbl 0694.57005
[5] J P Otal, Les géodésiques fermées d’une variété hyperbolique en tant que nœuds (editors Y Komori, V Markovic, C Series), London Math. Soc. Lecture Note Ser. 299, Cambridge Univ. Press (2003) 95 · Zbl 1049.57007
[6] J T Pitts, J H Rubinstein, Applications of minimax to minimal surfaces and the topology of \(3\)-manifolds (editors J E Hutchinson, L E Simon), Proc. Centre Math. Anal. Austral. Nat. Univ. 12, Austral. Nat. Univ. (1987) 137 · Zbl 0639.49030
[7] J H Rubinstein, M Scharlemann, Comparing Heegaard splittings of non-Haken \(3\)-manifolds, Topology 35 (1996) 1005 · Zbl 0858.57020
[8] M Scharlemann, Local detection of strongly irreducible Heegaard splittings, Topology Appl. 90 (1998) 135 · Zbl 0926.57018
[9] J Schultens, Heegaard splittings of Seifert fibered spaces with boundary, Trans. Amer. Math. Soc. 347 (1995) 2533 · Zbl 0851.57017
[10] J Souto, Short geodesics in hyperbolic compression bodies are not knotted, Preprint (2008)
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