Canonical triangulations of Dehn fillings

Guéritaud, François¹ ; Schleimer, Saul²

¹ Laboratoire Paul–Painlevé, CNRS UMR 8524, Université de Lille 1, 59650 Villeneuve d’Ascq, France
² Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Geometry & topology, Tome 14 (2010) no. 1, pp. 193-242.

Résumé

Every cusped, finite-volume hyperbolic three-manifold has a canonical decomposition into ideal polyhedra. We study the canonical decomposition of the hyperbolic manifold obtained by filling some (but not all) of the cusps with solid tori: in a broad range of cases, generic in an appropriate sense, this decomposition can be predicted from that of the unfilled manifold (a similar result has been independently announced by Akiyoshi [Kokyuroku 1329, RIMS, Kyoto (2003) 121-132]). We also find the canonical decompositions of all hyperbolic Dehn fillings on one cusp of the Whitehead link complement.

Détail

DOI : 10.2140/gt.2010.14.193

Classification : 51H20, 57M50
Keywords: hyperbolic manifold, canonical triangulation, Dehn fillings

Bibliographie

[1] H Akiyoshi, On the hyperbolic manifolds obtained from the Whitehead link, from: "Analysis of discrete groups, II (Kyoto, 1996)", Sūrikaisekikenkyūsho Kōkyūroku 1022 (1997) 213

[2] H Akiyoshi, On the Ford domains of once-punctured torus groups, from: "Hyperbolic spaces and related topics (Japanese) (Kyoto, 1998)", Sūrikaisekikenkyūsho Kōkyūroku 1104 (1999) 109

[3] H Akiyoshi, Finiteness of polyhedral decompositions of cusped hyperbolic manifolds obtained by the Epstein–Penner's method, Proc. Amer. Math. Soc. 129 (2001) 2431

[4] H Akiyoshi, Canonical decompositions of cusped hyperbolic 3-manifolds obtained by Dehn filling, from: "Perspectives of hyperbolic spaces", Kokyuroku 1329, RIMS (2003) 121

[5] H Akiyoshi, M Sakuma, Comparing two convex hull constructions for cusped hyperbolic manifolds, from: "Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001)" (editors Y Komori, V Markovic, C Series), London Math. Soc. Lecture Note Ser. 299, Cambridge Univ. Press (2003) 209

[6] H Akiyoshi, M Sakuma, M Wada, Y Yamashita, J\o rgensen's picture of punctured torus groups and its refinement, from: "Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001)" (editors Y Komori, V Markovic, C Series), London Math. Soc. Lecture Note Ser. 299, Cambridge Univ. Press (2003) 247

[7] H Akiyoshi, M Sakuma, M Wada, Y Yamashita, Punctured torus groups and $2$–bridge knot groups. I, Lecture Notes in Math. 1909, Springer (2007)

[8] P J Callahan, M V Hildebrand, J R Weeks, A census of cusped hyperbolic $3$–manifolds, Math. Comp. 68 (1999) 321

[9] K Chan, Constructing hyperbolic $3$–manifolds, Undergraduate thesis with Craig Hodgson, University of Melbourne (2002)

[10] T A Drumm, J A Poritz, Ford and Dirichlet domains for cyclic subgroups of $\mathrm{PSL}_2(\mathbf{C})$ acting on $\mathbf{H}^3_R$ and $\partial\mathbf{H}^3_R$, Conform. Geom. Dyn. 3 (1999) 116

[11] D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67

[12] D Futer, F Guéritaud, Angled decompositions of arborescent link complements, Proc. Lond. Math. Soc. $(3)$ 98 (2009) 325

[13] F Guéritaud, On an elementary proof of Rivin's characterization of convex ideal hyperbolic polyhedra by their dihedral angles, Geom. Dedicata 108 (2004) 111

[14] F Guéritaud, Géométrie hyperbolique effective et triangulations idéales canoniques en dimension trois, PhD thesis, Université d’Orsay (2006)

[15] F Guéritaud, On canonical triangulations of once-punctured torus bundles and two-bridge link complements, Geom. Topol. 10 (2006) 1239

[16] F Guéritaud, Triangulated cores of punctured-torus groups, J. Differential Geom. 81 (2009) 91

[17] B Jaco, H Rubinstein, Layered-triangulations of $3$–manifolds, Preprint (2006)

[18] T Jørgensen, On cyclic groups of Möbius transformations, Math. Scand. 33 (1973)

[19] T Jørgensen, On pairs of once-punctured tori, from: "Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001)" (editors Y Komori, V Markovic, C Series), London Math. Soc. Lecture Note Ser. 299, Cambridge Univ. Press (2003) 183

[20] M Lackenby, The canonical decomposition of once-punctured torus bundles, Comment. Math. Helv. 78 (2003) 363

[21] B Martelli, C Petronio, Dehn filling of the “magic” $3$–manifold, Comm. Anal. Geom. 14 (2006) 969

[22] W D Neumann, A W Reid, Arithmetic of hyperbolic manifolds, from: "Topology '90 (Columbus, OH, 1990)" (editors B Apanasov, W D Neumann, A W Reid, L Siebenmann), Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 273

[23] I Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. $(2)$ 139 (1994) 553

[24] I Rivin, Combinatorial optimization in geometry, Adv. in Appl. Math. 31 (2003) 242

[25] M Sakuma, J Weeks, Examples of canonical decompositions of hyperbolic link complements, Japan. J. Math. $($N.S.$)$ 21 (1995) 393

[26] W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979)

[27] J R Weeks, SnapPea: a software for the study of hyperbolic manifolds

[28] J R Weeks, Convex hulls and isometries of cusped hyperbolic $3$–manifolds, Topology Appl. 52 (1993) 127