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Some surface subgroups survive surgery. (English) Zbl 1009.57017
It is proved that if $$M$$ is a hyperbolic 3-manifold with a single torus boundary component, then all but finitely many surgeries on $$M$$ contain the fundamental group of a closed orientable surface of genus at least two.
Reviewer: Daryl Cooper

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
Dehn surgery; surface subgroups; hyperbolic 3-manifold
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##### References:
 [1] F Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. $$(2)$$ 124 (1986) 71 · Zbl 0671.57008 [2] , The Smith conjecture, Pure and Applied Mathematics 112, Academic Press (1984) · Zbl 0599.57001 [3] S A Bleiler, C D Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996) 809 · Zbl 0863.57009 [4] 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 [5] D Cooper, D D Long, Virtually Haken Dehn-filling, J. Differential Geom. 52 (1999) 173 · Zbl 1025.57020 [6] 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 [7] M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. $$(2)$$ 125 (1987) 237 · Zbl 0633.57006 [8] M P do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser (1992) · Zbl 0752.53001 [9] S R Fenley, Quasi-Fuchsian Seifert surfaces, Math. Z. 228 (1998) 221 · Zbl 0902.57003 [10] J Hempel, 3-Manifolds, Ann. of Math. Studies 86, Princeton University Press (1976) · Zbl 0345.57001 [11] W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, American Mathematical Society (1980) · Zbl 0433.57001 [12] T Li, Immersed essential surfaces in hyperbolic 3-manifolds, Comm. Anal. Geom. 10 (2002) 275 · Zbl 1019.57009 [13] W P Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series 35, Princeton University Press (1997) · Zbl 0873.57001
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