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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-008-0300-x