Geometry & Topology

Bounds on exceptional Dehn filling II

Ian Agol

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We show that there are at most finitely many one cusped orientable hyperbolic 3–manifolds which have more than eight nonhyperbolic Dehn fillings. Moreover, we show that determining these finitely many manifolds is decidable.

Article information

Geom. Topol., Volume 14, Number 4 (2010), 1921-1940.

Received: 19 March 2008
Revised: 12 June 2010
Accepted: 9 June 2010
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 30F40: Kleinian groups [See also 20H10]

hyperbolic Dehn filling


Agol, Ian. Bounds on exceptional Dehn filling II. Geom. Topol. 14 (2010), no. 4, 1921--1940. doi:10.2140/gt.2010.14.1921.

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  • C C Adams, The noncompact hyperbolic $3$–manifold of minimal volume, Proc. Amer. Math. Soc. 100 (1987) 601–606
  • I Agol, Tameness of hyperbolic $3$–manifolds
  • I Agol, Bounds on exceptional Dehn filling, Geom. Topol. 4 (2000) 431–449
  • I Agol, M Culler, P B Shalen, Dehn surgery, homology and hyperbolic volume, Algebr. Geom. Topol. 6 (2006) 2297–2312
  • J W Anderson, R D Canary, M Culler, P B Shalen, Free Kleinian groups and volumes of hyperbolic $3$–manifolds, J. Differential Geom. 43 (1996) 738–782
  • L Bers, Spaces of Kleinian groups, from: “Several Complex Variables, I (Proc. Conf., Univ. of Maryland, College Park, Md., 1970)”, Springer, Berlin (1970) 9–34
  • S A Bleiler, C D Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996) 809–833
  • F Bonahon, J-P Otal, Variétés hyperboliques à géodésiques arbitrairement courtes, Bull. London Math. Soc. 20 (1988) 255–261
  • B H Bowditch, End invariants of hyperbolic $3$–manifolds, Preprint (2005) Available at \setbox0\makeatletter\@url {\unhbox0
  • B H Bowditch, Notes on tameness, Preprint (2006) Available at \setbox0\makeatletter\@url {\unhbox0
  • J F Brock, K W Bromberg, On the density of geometrically finite Kleinian groups, Acta Math. 192 (2004) 33–93
  • J F Brock, R D Canary, Y N Minsky, Classification of Kleinian surface groups II: the ending lamination conjecture
  • K Bromberg, Projective structures with degenerate holonomy and the Bers density conjecture, Ann. of Math. $(2)$ 166 (2007) 77–93
  • D Calegari, D Gabai, Shrinkwrapping and the taming of hyperbolic $3$–manifolds, J. Amer. Math. Soc. 19 (2006) 385–446
  • C Cao, G R Meyerhoff, The orientable cusped hyperbolic $3$–manifolds of minimum volume, Invent. Math. 146 (2001) 451–478
  • H-D Cao, X-P Zhu, A complete proof of the Poincaré and geometrization conjectures–-application of the Hamilton–Perelman theory of the Ricci flow, Asian J. Math. 10 (2006) 165–492
  • T D Comar, Hyperbolic Dehn surgery and convergence of Kleinian groups, ProQuest LLC, Ann Arbor, MI (1996) PhD Thesis–University of Michigan
  • D Cooper, The volume of a closed hyperbolic $3$–manifold is bounded by $\pi$ times the length of any presentation of its fundamental group, Proc. Amer. Math. Soc. 127 (1999) 941–942
  • M Culler, P B Shalen, Paradoxical decompositions, $2$–generator Kleinian groups, and volumes of hyperbolic $3$–manifolds, J. Amer. Math. Soc. 5 (1992) 231–288
  • D Gabai, R Meyerhoff, P Milley, Minimum volume cusped hyperbolic three-manifolds, J. Amer. Math. Soc. 22 (2009) 1157–1215
  • D Gabai, R Meyerhoff, N Thurston, Homotopy hyperbolic $3$–manifolds are hyperbolic, Ann. of Math. $(2)$ 157 (2003) 335–431
  • C M Gordon, Boundary slopes of punctured tori in $3$–manifolds, Trans. Amer. Math. Soc. 350 (1998) 1713–1790
  • C M Gordon, Dehn filling: a survey, from: “Knot theory (Warsaw, 1995)”, (V F R Jones, J H Kania-Bartoszyńska, J Przytycki, P Traczyk, V G Turaev, editors), Banach Center Publ. 42, Polish Acad. Sci., Warsaw (1998) 129–144
  • M Hildebrand, J Weeks, A computer generated census of cusped hyperbolic $3$–manifolds, from: “Computers and mathematics (Cambridge, MA, 1989)”, (E Kaltofen, S M Watt, editors), Springer, New York (1989) 53–59
  • W H Jaco, P B Shalen, Seifert fibered spaces in $3$–manifolds, Mem. Amer. Math. Soc. 21 (1979) viii+192
  • T Jørgensen, On pairs of once-punctured tori, from: “Kleinian groups and hyperbolic $3$–manifolds (Warwick, 2001)”, (Y Komori, V Markovic, C Series, editors), London Math. Soc. Lecture Note Ser. 299, Cambridge Univ. Press (2003) 183–207
  • R Kirby, editor, Problems in low-dimensional topology, from: “Geometric topology (Athens, GA, 1993)”, (W H Kazez, editor), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 35–473
  • B Kleiner, J Lott, Notes on Perelman's papers, Geom. Topol. 12 (2008) 2587–2855
  • M Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243–282
  • M Lackenby, R Meyerhoff, The maximal number of exceptional Dehn surgeries
  • J Manning, Algorithmic detection and description of hyperbolic structures on closed $3$–manifolds with solvable word problem, Geom. Topol. 6 (2002) 1–25
  • A Marden, The geometry of finitely generated kleinian groups, Ann. of Math. $(2)$ 99 (1974) 383–462
  • R Meyerhoff, A lower bound for the volume of hyperbolic $3$–manifolds, Canad. J. Math. 39 (1987) 1038–1056
  • Y Minsky, The classification of Kleinian surface groups. I. Models and bounds, Ann. of Math. (2) 171 (2010) 1–107
  • J Morgan, G Tian, Ricci flow and the Poincaré conjecture, Clay Math. Monogr. 3, Amer. Math. Soc. (2007)
  • H Namazi, J Souto, Non-realizability and ending laminations: Proof of the Density Conjecture, Preprint (2010) Available at \setbox0\makeatletter\@url {\unhbox0
  • K Ohshika, Realising end invariants by limits of minimally parabolic, geometrically finite groups
  • J-P Otal, Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235, Soc. Math. France (1996)
  • G Perelman, The entropy formula for the Ricci flow and its geometric applications
  • G Perelman, Ricci flow with surgery on three-manifolds
  • S Rafalski, Immersed turnovers in hyperbolic $3$–orbifolds, Groups Geom. Dyn. 4 (2010) 333–376
  • R Riley, Applications of a computer implementation of Poincaré's theorem on fundamental polyhedra, Math. Comp. 40 (1983) 607–632
  • T Soma, Sequences of degree-one maps between geometric $3$–manifolds, Math. Ann. 316 (2000) 733–742
  • T Soma, Existence of ruled wrappings in hyperbolic $3$–manifolds, Geom. Topol. 10 (2006) 1173–1184
  • W P Thurston, Hyperbolic structures on $3$–manifolds, II: Surface groups and $3$–manifolds which fiber over the circle
  • W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url {\unhbox0
  • J R Weeks, SnapPea: A computer program for creating and studying hyperbolic $3$–manifolds Available at \setbox0\makeatletter\@url {\unhbox0
  • J R Weeks, Convex hulls and isometries of cusped hyperbolic $3$–manifolds, Topology Appl. 52 (1993) 127–149