Abstract
We show that if a hyperbolic 3-manifold M has two toroidal Dehn fillings with distance at least 3, then ∂M consists of at most three tori. As a result, we can obtain an optimal estimate for the number of exceptional slopes on hyperbolic 3-manifolds with boundary a union of at least 4 tori.
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References
Goda H., Teragaito M.: On hyperbolic 3-manifolds realizing the maximal distance between toroidal Dehn fillings. Algebr. Geom. Topol. 5, 463–507 (2005)
Gordon C.McA.: Boundary slopes of punctured tori in 3-manifolds. Trans. Am. Math. Soc. 350, 1713–1790 (1998)
Gordon C.McA.: Small surfaces and Dehn filling. Proc. Kirbyfest Geom. Topol. Monogr. 2, 177–199 (1998)
Gordon C.McA., Litherland R.A.: Incompressible planar surfaces in 3-manifolds. Topol. Appl. 18, 121–144 (1984)
Gordon C.McA., Luecke J.: Dehn surgeries on knots creating essential tori, I. Commun. Anal. Geom. 3, 597–644 (1995)
Gordon C.McA., Luecke J.: Toroidal and boundary-reducing Dehn fillings. Topol. Appl. 93, 77–90 (1999)
Gordon C.McA., Wu Y.-Q.: Toroidal and annular Dehn fillings. Proc. Lond. Math. Soc. 78, 662–700 (1999)
Gordon, C.McA., Wu, Y.-Q.: Toroidal Dehn fillings on hyperbolic 3-manifolds. Mem. Am. Math. Soc. (to appear)
Hayashi C., Motegi K.: Only single twists on unknots can produce composite knots. Trans. Am. Math. Soc. 349, 4465–4479 (1997)
Jaco, W., Shalen, P.: Seifert fibered spaces in 3-manifolds. Mem. Am. Math. Soc. 21(220), viii+192 pp (1979)
Lee S.: Reducing and toroidal Dehn fillings on 3-manifolds bounded by two tori. Math. Res. Lett. 13, 287–306 (2006)
Lee S.: Boundary structure of hyperbolic 3-manifolds admitting annular fillings at large distance. Proc. Am. Math. Soc. 134, 2767–2770 (2006)
Lee S., Oh S., Teragaito M.: Reducing Dehn fillings and small surfaces. Proc. Lond. Math. Soc. 92, 203–223 (2006)
Lee, S., Teragaito, M.: Boundary structure of hyperbolic 3-manifolds admitting annular and toroidal fillings at large distance. Canad. J. Math (to appear)
Oertel U.: Closed incompressible surfaces in complements of star links. Pacific J. Math. 111, 209–230 (1984)
Thurston W.: Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Am. Math. Soc. 6, 357–381 (1982)
Valdez-Śanchez L.: Toroidal and Klein bottle boundary slopes. Topol. Appl. 154, 584–603 (2007)
Wu Y.-Q.: Dehn fillings producing reducible manifolds and toroidal manifolds. Topology 37, 95–108 (1998)
Wu Y.-Q.: Sutured manifold hierarchies, essential laminations, and Dehn surgery. J. Diff. Geom. 48, 407–437 (1998)
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S. Lee was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-314-C00024). M. Teragaito was supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 19540089.
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Lee, S., Teragaito, M. Boundary structure of hyperbolic 3-manifolds admitting toroidal fillings at large distance. Math. Ann. 344, 119–159 (2009). https://doi.org/10.1007/s00208-008-0300-x
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DOI: https://doi.org/10.1007/s00208-008-0300-x