Short geodesics in hyperbolic 3-manifolds.

*(English)*Zbl 1221.57027For 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.

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.

Reviewer: Ken’ichi Ohshika (Osaka)

##### MSC:

57M50 | General geometric structures on low-dimensional manifolds |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

53C22 | Geodesics in global differential geometry |

##### References:

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