Geometry & Topology

Characteristic subsurfaces, character varieties and Dehn fillings

Steve Boyer, Marc Culler, Peter B Shalen, and Xingru Zhang

Full-text: Open access


Let M be a one-cusped hyperbolic 3–manifold. A slope on the boundary of the compact core of M is called exceptional if the corresponding Dehn filling produces a non-hyperbolic manifold. We give new upper bounds for the distance between two exceptional slopes α and β in several situations. These include cases where M(β) is reducible and where M(α) has finite π1, or M(α) is very small, or M(α) admits a π1–injective immersed torus.

Article information

Geom. Topol., Volume 12, Number 1 (2008), 233-297.

Received: 23 November 2006
Accepted: 31 October 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M50: Geometric structures on low-dimensional manifolds 57M99: None of the above, but in this section

characteristic subsurfaces character varieties Dehn filling


Boyer, Steve; Culler, Marc; Shalen, Peter B; Zhang, Xingru. Characteristic subsurfaces, character varieties and Dehn fillings. Geom. Topol. 12 (2008), no. 1, 233--297. doi:10.2140/gt.2008.12.233.

Export citation


  • L Ben Abdelghani, S Boyer, A calculation of the Culler–Shalen seminorms associated to small Seifert Dehn fillings, Proc. London Math. Soc. $(3)$ 83 (2001) 235–256
  • S Boyer, On the local structure of ${\rm SL}(2,{\mathbb C})$–character varieties at reducible characters, Topology Appl. 121 (2002) 383–413
  • S Boyer, M Culler, P B Shalen, X Zhang, Characteristic subsurfaces and Dehn filling, Trans. Amer. Math. Soc. 357 (2005) 2389–2444
  • S Boyer, C M Gordon, X Zhang, Dehn fillings of large hyperbolic $3$–manifolds, J. Differential Geom. 58 (2001) 263–308
  • S Boyer, X Zhang, Finite Dehn surgery on knots, J. Amer. Math. Soc. 9 (1996) 1005–1050
  • S Boyer, X Zhang, On Culler–Shalen seminorms and Dehn filling, Ann. of Math. $(2)$ 148 (1998) 737–801
  • S Boyer, X Zhang, On simple points of character varieties of $3$–manifolds, from: “Knots in Hellas '98 (Delphi)”, Ser. Knots Everything 24, World Sci. Publ., River Edge, NJ (2000) 27–35
  • S Boyer, X Zhang, Virtual Haken $3$–manifolds and Dehn filling, Topology 39 (2000) 103–114
  • S Boyer, X Zhang, A proof of the finite filling conjecture, J. Differential Geom. 59 (2001) 87–176
  • A Casson, D Jungreis, Convergence groups and Seifert fibered $3$–manifolds, Invent. Math. 118 (1994) 441–456
  • M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. $(2)$ 125 (1987) 237–300
  • M Culler, P B Shalen, Varieties of group representations and splittings of $3$–manifolds, Ann. of Math. $(2)$ 117 (1983) 109–146
  • N M Dunfield, Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. 136 (1999) 623–657
  • D Gabai, Foliations and the topology of $3$–manifolds. II, J. Differential Geom. 26 (1987) 461–478
  • D Gabai, Convergence groups are Fuchsian groups, Ann. of Math. $(2)$ 136 (1992) 447–510
  • D Gabai, Homotopy hyperbolic $3$–manifolds are virtually hyperbolic, J. Amer. Math. Soc. 7 (1994) 193–198
  • D Gabai, G R Meyerhoff, N Thurston, Homotopy hyperbolic $3$–manifolds are hyperbolic, Ann. of Math. $(2)$ 157 (2003) 335–431
  • C M Gordon, Dehn filling: a survey, from: “Knot theory (Warsaw, 1995)”, Banach Center Publ. 42, Polish Acad. Sci., Warsaw (1998) 129–144
  • C M Gordon, J Luecke, Reducible manifolds and Dehn surgery, Topology 35 (1996) 385–409
  • W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, American Mathematical Society, Providence, R.I. (1980)
  • S Lee, Reducing and toroidal Dehn fillings on $3$–manifolds bounded by two tori, Math. Res. Lett. 13 (2006) 287–306
  • P E Newstead, Introduction to moduli problems and orbit spaces, volume 51 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Tata Institute of Fundamental Research, Bombay (1978)
  • S Oh, Reducible and toroidal $3$–manifolds obtained by Dehn fillings, Topology Appl. 75 (1997) 93–104
  • P Scott, A new proof of the annulus and torus theorems, Amer. J. Math. 102 (1980) 241–277
  • J-P Serre, Représentations linéaires des groupes finis, revised edition, Hermann, Paris (1978)
  • F Waldhausen, On irreducible $3$–manifolds which are sufficiently large, Ann. of Math. $(2)$ 87 (1968) 56–88
  • Y Q Wu, Incompressibility of surfaces in surgered $3$–manifolds, Topology 31 (1992) 271–279
  • Y-Q Wu, Dehn fillings producing reducible manifolds and toroidal manifolds, Topology 37 (1998) 95–108
  • Y-Q Wu, Standard graphs in lens spaces, Pacific J. Math. 220 (2005) 389–397