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Geodesic knots in cusped hyperbolic 3-manifolds. (English) Zbl 1129.53023

Summary: We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Previous results show that at least one geodesic knot always exists [Bull. Lond. Math. Soc. 31, No. 1, 81–86 (1999; Zbl 0955.53025)], and that certain arithmetic manifolds contain infinitely many geodesic knots [J. Differ. Geom. 38, 545–558 (1993; Zbl 0783.53028); Exp. Math. 10, No. 3, 419–436 (2001; Zbl 1014.53025)]. In this paper we show that all cusped orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. Our proof is constructive, and the infinite family of geodesic knots produced, approach a limiting infinite simple geodesic in the manifold.

MSC:

53C22 Geodesics in global differential geometry
57N10 Topology of general \(3\)-manifolds (MSC2010)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57M50 General geometric structures on low-dimensional manifolds

References:

[1] C Adams, J Hass, P Scott, Simple closed geodesics in hyperbolic \(3\)-manifolds, Bull. London Math. Soc. 31 (1999) 81 · Zbl 0955.53025 · doi:10.1112/S0024609398004883
[2] T Chinburg, A W Reid, Closed hyperbolic \(3\)-manifolds whose closed geodesics all are simple, J. Differential Geom. 38 (1993) 545 · Zbl 0783.53028
[3] S M Kuhlmann, Geodesic knots in closed hyperbolic 3-manifolds, in preparation · Zbl 1147.53035 · doi:10.1007/s10711-007-9227-8
[4] S M Kuhlmann, Geodesic knots in hyperbolic 3-manifolds, PhD thesis, University of Melbourne (2005) · Zbl 1147.53035 · doi:10.1007/s10711-007-9227-8
[5] S M Miller, Geodesic knots in the figure-eight knot complement, Experiment. Math. 10 (2001) 419 · Zbl 1014.53025 · doi:10.1080/10586458.2001.10504460
[6] T Sakai, Geodesic knots in a hyperbolic \(3\)-manifold, Kobe J. Math. 8 (1991) 81 · Zbl 0749.57003
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