Closed quasi-Fuchsian surfaces in hyperbolic knot complements

Masters, Joseph D¹ ; Zhang, Xingru¹

¹ Mathematics Department, SUNY at Buffalo, Buffalo, NY 14290, USA

Geometry & topology, Tome 12 (2008) no. 4, pp. 2095-2171.

Résumé

We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.

Détail

DOI : 10.2140/gt.2008.12.2095

Keywords: hyperbolic manifold, quasi-Fuchsian surface, $\pi_1$–injective surface

Bibliographie

[1] C C Adams, Toroidally alternating knots and links, Topology 33 (1994) 353

[2] C C Adams, A W Reid, Quasi-Fuchsian surfaces in hyperbolic knot complements, J. Austral. Math. Soc. Ser. A 55 (1993) 116

[3] M Baker, D Cooper, A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds

[4] F Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. $(2)$ 124 (1986) 71

[5] R D Canary, D B A Epstein, P Green, Notes on notes of Thurston, from: "Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984)", London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 3

[6] D Cooper, D D Long, Virtually Haken Dehn-filling, J. Differential Geom. 52 (1999) 173

[7] D Cooper, D D Long, Some surface subgroups survive surgery, Geom. Topol. 5 (2001) 347

[8] D Cooper, D D Long, A W Reid, Essential closed surfaces in bounded $3$-manifolds, J. Amer. Math. Soc. 10 (1997) 553

[9] M Culler, P B Shalen, Bounded, separating, incompressible surfaces in knot manifolds, Invent. Math. 75 (1984) 537

[10] D B A Epstein, A Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, from: "Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984)", London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 113

[11] I Kapovich, A Myasnikov, Stallings foldings and subgroups of free groups, J. Algebra 248 (2002) 608

[12] T Li, Immersed essential surfaces in hyperbolic 3-manifolds, Comm. Anal. Geom. 10 (2002) 275

[13] A Marden, The geometry of finitely generated kleinian groups, Ann. of Math. $(2)$ 99 (1974) 383

[14] K Matsuzaki, M Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press (1998)

[15] W Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984) 37

[16] U Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984) 209

[17] J G Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer (1994)

[18] P Susskind, Kleinian groups with intersecting limit sets, J. Analyse Math. 52 (1989) 26

[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