×

Closed quasi-Fuchsian surfaces in hyperbolic knot complements. (English) Zbl 1156.57014

Let \(M\) be an orientable, complete, finite volume hyperbolic manifold with a single cusp. In the paper under review, such an \(M\) is called a hyperbolic knot complement; the knot is meant to lie in an arbitrary closed orientable manifold. It is known that essential surfaces in \(M\) are divided into three types: quasi-Fuchsian, geometrically infinite, and surfaces with accidental parabolics. The following is the main result of the paper.
Theorem. Every hyperbolic knot complement contains a closed quasi-Fuchsian surface.
Since a closed quasi-Fuchsian surface in a hyperbolic knot complement remains essential in all but finitely many Dehn fillings, the theorem has the following consequence.
Corollary. Let \(M\) be a hyperbolic knot complement. Then \(M\) contains a closed essential surface which remains essential in all but finitely many Dehn fillings of \(M\).

MSC:

57M50 General geometric structures on low-dimensional manifolds
57N35 Embeddings and immersions in topological manifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] C C Adams, Toroidally alternating knots and links, Topology 33 (1994) 353 · Zbl 0839.57004 · doi:10.1016/0040-9383(94)90017-5
[2] C C Adams, A W Reid, Quasi-Fuchsian surfaces in hyperbolic knot complements, J. Austral. Math. Soc. Ser. A 55 (1993) 116 · Zbl 0795.57008
[3] M Baker, D Cooper, A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds · Zbl 1151.57014 · doi:10.1112/jtopol/jtn013
[4] F Bonahon, Bouts des variétés hyperboliques de dimension \(3\), Ann. of Math. \((2)\) 124 (1986) 71 · Zbl 0671.57008 · doi:10.2307/1971388
[5] R D Canary, D B A Epstein, P Green, Notes on notes of Thurston, London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 3 · Zbl 0612.57009
[6] D Cooper, D D Long, Virtually Haken Dehn-filling, J. Differential Geom. 52 (1999) 173 · Zbl 1025.57020
[7] D Cooper, D D Long, Some surface subgroups survive surgery, Geom. Topol. 5 (2001) 347 · Zbl 1009.57017 · doi:10.2140/gt.2001.5.347
[8] D Cooper, D D Long, A W Reid, Essential closed surfaces in bounded \(3\)-manifolds, J. Amer. Math. Soc. 10 (1997) 553 · Zbl 0896.57009 · doi:10.1090/S0894-0347-97-00236-1
[9] M Culler, P B Shalen, Bounded, separating, incompressible surfaces in knot manifolds, Invent. Math. 75 (1984) 537 · Zbl 0542.57011 · doi:10.1007/BF01388642
[10] D B A Epstein, A Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 113 · Zbl 0612.57010
[11] I Kapovich, A Myasnikov, Stallings foldings and subgroups of free groups, J. Algebra 248 (2002) 608 · Zbl 1001.20015 · doi:10.1006/jabr.2001.9033
[12] T Li, Immersed essential surfaces in hyperbolic 3-manifolds, Comm. Anal. Geom. 10 (2002) 275 · Zbl 1019.57009
[13] A Marden, The geometry of finitely generated kleinian groups, Ann. of Math. \((2)\) 99 (1974) 383 · Zbl 0282.30014 · doi:10.2307/1971059
[14] K Matsuzaki, M Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press (1998) · Zbl 0892.30035
[15] W Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984) 37 · Zbl 0525.57003 · doi:10.1016/0040-9383(84)90023-5
[16] U Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984) 209 · Zbl 0549.57004 · doi:10.2140/pjm.1984.111.209
[17] J G Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer (1994) · Zbl 0809.51001
[18] P Susskind, Kleinian groups with intersecting limit sets, J. Analyse Math. 52 (1989) 26 · Zbl 0677.30028 · doi:10.1007/BF02820470
[19] W P Thurston, The topology and geomety of \(3\)-manifolds, Lecture notes, Princeton University (1979)
[20] Y Q Wu, Immersed essential surfaces and Dehn surgery, Topology 43 (2004) 319 · Zbl 1047.57009 · doi:10.1016/S0040-9383(03)00046-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.