## Geometry & Topology

### Bounds on exceptional Dehn filling II

Ian Agol

#### Abstract

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

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

Dates
Revised: 12 June 2010
Accepted: 9 June 2010
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883510

Digital Object Identifier
doi:10.2140/gt.2010.14.1921

Mathematical Reviews number (MathSciNet)
MR2680207

Zentralblatt MATH identifier
1201.57011

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds

Keywords
hyperbolic Dehn filling

#### Citation

Agol, Ian. Bounds on exceptional Dehn filling II. Geom. Topol. 14 (2010), no. 4, 1921--1940. doi:10.2140/gt.2010.14.1921. https://projecteuclid.org/euclid.gt/1513883510

#### References

• 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 http://www.warwick.ac.uk/~masgak/preprints.html {\unhbox0
• B H Bowditch, Notes on tameness, Preprint (2006) Available at \setbox0\makeatletter\@url http://www.warwick.ac.uk/~masgak/preprints.html {\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 http://www.ma.utexas.edu/users/hossein/contents/ending6.6.pdf {\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 http://msri.org/publications/books/gt3m/ {\unhbox0
• J R Weeks, SnapPea: A computer program for creating and studying hyperbolic $3$–manifolds Available at \setbox0\makeatletter\@url http://www.geometrygames.org/SnapPea/ {\unhbox0
• J R Weeks, Convex hulls and isometries of cusped hyperbolic $3$–manifolds, Topology Appl. 52 (1993) 127–149