William P. Thurston: “Three-dimensional manifolds, Kleinian groups and hyperbolic geometry”: Bull. Am. Math. Soc., New Ser. 6, 357–379 (1982)

Jahresbericht der Deutschen Mathematiker-Vereinigung

Volumn 116, Issue 1, 2014, Pages 3-20

  Otal, Jean Pierre ^a  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

Cube Complex; Fundamental Group; Hyperbolic Manifold; Kleinian Group; Schottky Group

Indexed keywords

EID: 85010375187     PISSN: 00120456     EISSN: 18697135     Source Type: Journal    
DOI: 10.1365/s13291-014-0079-5     Document Type: Article

References (101)

1. Agol, I.: Bounds on exceptional Dehn filling. Geom. Topol. 4, 431–449 (2000) DOI: 10.2140/gt.2000.4.431
2. Agol, I.: Tameness of hyperbolic 3-manifolds. math.GT/0405568
3. Agol, I.: Criteria for virtual fibering. Topology 1, 269–284 (2008) DOI: 10.1112/jtopol/jtn003
4. Agol, I., Groves, D., Manning, J.: The virtual Haken conjecture. arXiv:1204.2810
5. Ahlfors, L.: Fundamental polyhedrons and limit point sets of Kleinian groups. Proc. Natl. Acad. Sci. USA 55, 251–254 (1966) DOI: 10.1073/pnas.55.2.251
6. Anderson, J., Canary, R.: Cores of hyperbolic 3-manifolds and limits of Kleinian groups II. J. Lond. Math. Soc. (2) 61(2), 489–505 (2000) DOI: 10.1112/S0024610799008595
7. Bers, L.: On boundaries of Teichmüller spaces and on Kleinian groups. Ann. Math. 91, 570–600 (1970) DOI: 10.2307/1970638
8. Bergeron, N., Wise, D.: A boundary criterion for cubulation. Am. J. Math. 134(3), 843–859 (2012) DOI: 10.1353/ajm.2012.0020
9. Bessières, L., Besson, G., Boileau, M., Maillot, S., Porti, J.: Geometrisation of 3-manifolds. EMS Tracts in Mathematics, vol. 13. European Mathematical Society, Zurich (2010) DOI: 10.4171/082
10. Bishop, C., Jones, P.: Hausdorff dimension and Kleinian groups. Acta Math. 179, 1–39 (1997) DOI: 10.1007/BF02392718
11. Boileau, M., Leeb, B., Porti, J.: Geometrization of 3-dimensional orbifolds. Ann. Math. 162, 195–290 (2005) DOI: 10.4007/annals.2005.162.195
12. Boileau, M., Porti, J.: Geometrization of 3-Orbifolds of Cyclic Type. Astérisque, vol. 272. SMF, Paris (2001)
13. Bonahon, F.: Bouts des variétés hyperboliques de dimension 3. Ann. Math. 124, 71–158 (1986) DOI: 10.2307/1971388
14. Bonahon, F.: Geometric structures on 3-manifolds. In: Daverman, R.J., Sher, R.B. (eds.) Handbook of Geometric Topology, pp. 93–164. Elsevier, Amsterdam (2001) DOI: 10.1016/B978-044482432-5/50004-4
15. Bonahon, F., Otal, J.-P.: Variétés hyperboliques avec géodésiques arbitrairement courtes. Bull. Lond. Math. Soc. 20, 255–261 (1988) DOI: 10.1112/blms/20.3.255
16. Brock, J.: Iteration of mapping classes and limits of hyperbolic 3-manifolds. Invent. Math. 143, 523–570 (2001) DOI: 10.1007/PL00005799
17. Bromberg, K.: Projective structures with degenerate holonomy and the Bers density conjecture. Ann. Math. (2) 166, 77–93 (2007) DOI: 10.4007/annals.2007.166.77
18. Brock, J., Canary, R., Minsky, Y.: The classification of Kleinian surface groups, II: the ending lamination conjecture. Ann. Math. (2) 176(1), 1–149 (2012) DOI: 10.4007/annals.2012.176.1.1
19. Brock, J., Bromberg, K.: On the density of geometrically finite Kleinian groups. Acta Math. 192, 33–93 (2004) DOI: 10.1007/BF02441085
20. Boyer, S., Zhang, X.: Finite Dehn surgery on knots. J. Am. Math. Soc. 9(4), 1005–1050 (1996) DOI: 10.1090/S0894-0347-96-00201-9
21. Bowditch, B.: Notes on tameness. Enseign. Math. (2) 56(3–4), 229–285 (2010) DOI: 10.4171/LEM/56-3-2
22. Bowditch, B.: The ending lamination theorem. Preprint Warwick, September 2011
23. Calegari, D., Gabai, D.: Shrinkwrapping and the taming of hyperbolic 3-manifolds. J. Am. Math. Soc. 19, 385–446 (2006) DOI: 10.1090/S0894-0347-05-00513-8
24. Canary, R.: On the Laplacian and the geometry of hyperbolic 3-manifolds. J. Differ. Geom. 36(2), 349–367 (1992)
25. Canary, R.: Ends of hyperbolic 3-manifolds. J. Am. Math. Soc. 6(1), 1–35 (1993)
26. Canary, R.: Algebraic convergence of Schottky groups. Trans. Am. Math. Soc. 337, 235–258 (1993) DOI: 10.1090/S0002-9947-1993-1137257-9
27. Canary, R.: A covering theorem for hyperbolic 3-manifolds and its applications. Topology 35(3), 751–778 (1996) DOI: 10.1016/0040-9383(94)00055-7
28. Canary, R.: Marden’s tameness conjecture: history and applications. In: Geometry, Analysis and Topology of Discrete Groups. Adv. Lect. Math., vol. 6, pp. 137–162. International Press, Somerville (2008)
29. Cannon, J., Thurston, W.: Group invariant Peano curves. Geom. Topol. 11, 1315–1356 (2007) DOI: 10.2140/gt.2007.11.1315
30. Cooper, D., Hodgson, C., Kerckhoff, S.: Three-dimensional orbifolds and cone-manifolds. MSJ Memoirs, vol. 5. Math. Soc. of Japan, Tokyo (2000)
31. Cooper, D., Long, D., Reid, A.: On the virtual Betti numbers of arithmetic hyperbolic 3-manifolds. Geom. Topol. 11, 2265–2276 (2007) DOI: 10.2140/gt.2007.11.2265
32. Culler, M., Gordon, C., Luecke, J., Shalen, P.: Dehn surgery on knots. Ann. Math. 125, 237–300 (1987) DOI: 10.2307/1971311
33. Dinkelbach, J., Leeb, B.: Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds. Geom. Topol. 13(2), 1129–1173 (2009) DOI: 10.2140/gt.2009.13.1129
34. Dunfield, N., Thurston, W.: The virtual Haken conjecture: experiments and examples. Geom. Topol. 7, 399–441 (2003) DOI: 10.2140/gt.2003.7.399
35. Floyd, W.: Group completions and limit sets of Kleinian groups. Invent. Math. 57, 205–218 (1980) DOI: 10.1007/BF01418926
36. Gabai, D., Meyerhoff, R., Milley, P.: Mom technology and hyperbolic 3-manifolds. In: Bonk, M., et al. (eds.) the Tradition of Ahlfors-Bers, V. Contemp. Math., vol. 510, pp. 81–108 (2010) DOI: 10.1090/conm/510/2581832
37. Gabai, D., Meyerhoff, R., Milley, P.: Mom technology and volumes of hyperbolic 3-manifolds. Comment. Math. Helv. 86, 145–188 (2011) DOI: 10.4171/CMH/221
38. Greenberg, L.: Fundamental polyhedra for kleinian groups. Ann. Math. 84, 433–441 (1966) DOI: 10.2307/1970456
39. Haglund, F., Wise, D.: Special cube complexes. Geom. Funct. Anal. 17, 1551–1620 (2008) DOI: 10.1007/s00039-007-0629-4
40. Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)
41. Hatcher, A., Thurston, W.: Incompressible surfaces in 2-bridge knot complements. Invent. Math. 79(2), 225–246 (1985) DOI: 10.1007/BF01388971
42. Hempel, J.: 3-Manifolds as viewed from the curve complex. Topology 40(3), 631–657 (2001) DOI: 10.1016/S0040-9383(00)00033-1
43. Hodgson, C., Kerckhoff, S.: Universal bounds for hyperbolic Dehn surgery. Ann. Math. (2) 162(1), 367–421 (2005) DOI: 10.4007/annals.2005.162.367
44. Hodgson, C., Kerckhoff, S.: The shape of hyperbolic Dehn surgery space. Geom. Topol. 12(2), 1033–1090 (2008) DOI: 10.2140/gt.2008.12.1033
45. Jørgensen, T.: Compact 3-manifolds of constant negative curvature fibering over the circle. Ann. Math. (2) 106(1), 61–72 (1977) DOI: 10.2307/1971158
46. Jørgensen, T.: On pairs of once-punctured tori. In: Kleinian Groups and Hyperbolic 3-Manifolds, Warwick, 2001. London Math. Soc., Lecture Notes Ser., vol. 299, pp. 183–207. Cambridge University Press, Cambridge (2003)
47. Kahn, J., Markovic, V.: Immersing almost geodesic surfaces in a closed hyperbolic three-manifold. Ann. Math. 175(3), 1127–1190 (2012) DOI: 10.4007/annals.2012.175.3.4
48. Kapovich, M.: Hyperbolic Manifolds and Discrete Groups. Birkhäuser Classics, Boston (2001)
49. Kerckhoff, S., Thurston, W.: Non-continuity of the action of the modular group at Bers boundary of Teichmüller space. Invent. Math. 100, 25–47 (1990) DOI: 10.1007/BF01231179
50. Kim, I., Lecuire, C., Ohshika, K.: Convergence of freely decomposable Kleinian groups. arXiv:0708.3266
51. Klarreich, E.: Semi-conjugacies between Kleinian group actions on the Riemann sphere. Am. J. Math. 121(5), 1031–1078 (1999) DOI: 10.1353/ajm.1999.0034
52. Kleiner, B., Lott, J.: Notes on Perelman’s papers. Geom. Topol. 12, 2587–2858 (2008) DOI: 10.2140/gt.2008.12.2587
53. Lackenby, M.: Word hyperbolic Dehn surgery. Invent. Math. 140, 243–282 (2000) DOI: 10.1007/s002220000047
54. Lackenby, M.: Surface subgroups of Kleinian groups with torsion. Invent. Math. 179, 175–190 (2010) DOI: 10.1007/s00222-009-0215-5
55. Lackenby, M., Meyerhoff, R.: The maximal number of exceptional Dehn surgeries. Invent. Math. 191, 341–382 (2013) DOI: 10.1007/s00222-012-0395-2
56. Mahan, M.: Cannon-Thurston maps for pared manifolds of bounded geometry. Geom. Topol. 13, 189–245 (2009). arXiv:math/0503581 DOI: 10.2140/gt.2009.13.189
57. Mahan, M.: Cannon-Thurston maps for surface groups. arXiv:math/0607509
58. McMullen, C.: Amenability, Poincaré series and quasiconformal maps. Invent. Math. 97, 95–128 (1989) DOI: 10.1007/BF01850656
59. McMullen, C.: Iteration on Teichmüller space. Invent. Math. 99, 425–454 (1990) DOI: 10.1007/BF01234427
60. Maclachlan, C., Reid, A.: The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics, vol. 219. Springer, New York (2003) DOI: 10.1007/978-1-4757-6720-9
61. Marden, A.: The geometry of finitely generated Kleinian groups. Ann. Math. 99, 383–462 (1974) DOI: 10.2307/1971059
62. Meeks, W. III, Scott, P.: Finite group actions on 3-manifolds. Invent. Math. 86, 287–346 (1986) DOI: 10.1007/BF01389073
63. Meeks, W. III, Yau, S.-T.: Group actions on ${\mathbb{R}}^{3}$. In: Morgan, J., Bass, H. (eds.) The Smith Conjecture, New York, 1979. Pure Appl. Math., vol. 112, pp. 167–179. Academic Press, New York (1984)
64. Minsky, Y.: On rigidity, limit sets and end invariants of hyperbolic 3-manifolds. J. Am. Math. Soc. 7, 539–588 (1994)
65. Minsky, Y.: Bounded geometry for Kleinian groups. Invent. Math. 146(1), 143–192 (2001) DOI: 10.1007/s002220100163
66. Minsky, Y.: The classification of Kleinian surface groups. I. Models and bounds. Ann. Math. 171, 1–107 (2010) DOI: 10.4007/annals.2010.171.1
67. Morgan, J., Bass, H. (eds.): The Smith Conjecture, New York, 1979. Pure Appl. Math., vol. 112. Academic Press, New York (1984)
68. Morgan, J.: On Thurston’s uniformization theorem for three-dimensional manifolds. In: Morgan, J., Bass, H. (eds.) The Smith Conjecture, New York, 1979. Pure Appl. Math., vol. 112, pp. 37–125. Academic Press, New York (1984)
69. Morgan, J.: Recent progress on the Poincaré conjecture and the classification of 3-manifolds. Bull. Am. Math. Soc. 42, 57–78 (2005) DOI: 10.1090/S0273-0979-04-01045-6
70. Morgan, J., Tian, G.: Ricci Flow and the Poincaré Conjecture. Clay Mathematics Monographs, vol. 3 (2007)
71. Namazi, H., Souto, J.: Heegaard splittings and pseudo-Anosov maps. Geom. Funct. Anal. 19, 1195–1228 (2009) DOI: 10.1007/s00039-009-0025-3
72. Namazi, H., Souto, J.: Non-realizability and ending laminations: proof of the density conjecture. Acta Math. 209, 323–395 (2012) DOI: 10.1007/s11511-012-0088-0
73. Neumann, W.: Hilbert’s 3rd problem and invariants of 3-manifolds. In: The Epstein Birthday Schrift. Geometry & Topology Monographs, vol. 1, pp. 383–411 (1998)
74. Neumann, W.: Realizing arithmetic invariants of hyperbolic 3-manifolds. In: Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory. Contemp. Math., vol. 541, pp. 233–246. AMS, Reading (2011) DOI: 10.1090/conm/541/10687
75. Neumann, W., Yang, J.: Bloch invariants of hyperbolic 3-manifolds. Duke Math. J. 96(1), 29–59 (1999) DOI: 10.1215/S0012-7094-99-09602-3
76. Ohshika, K.: Realising end invariants by limits of minimally parabolic, geometrically finite groups. Geom. Topol. 15, 827–890 (2011) DOI: 10.2140/gt.2011.15.827
77. Ohshika, K., Soma, T.: Geometry and topology of geometric limits, I. arXiv:1002.4266v1
78. Otal, J.-P.: Sur la dégénérescence des groupes de Schottky. Duke Math. J. 74, 777–792 (1994) DOI: 10.1215/S0012-7094-94-07429-2
79. Otal, J.-P.: Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3. Astérisque, vol. 235. Soc. Math. France, Paris (1996)
80. Otal, J.-P.: Thurston’s hyperbolization of Haken manifolds. In: Hsiung, C.-C., Yau, S.-T. (eds.) Surveys in Differential Geometry II, pp. 77–194. International Press, Boston (1998)
81. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159
82. Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109
83. Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307245
84. Poincaré, H.: Cinquième complément à l’analysis situs. Rend. Circ. Mat. Palermo 18, 45–110 (1904). See also: Oeuvres, Tome VI, 1953 DOI: 10.1007/BF03014091
85. Reid, A.: The geometry and topology of arithmetic hyperbolic 3-manifolds. In: Proc. Symposium Topology, Complex Analysis and Arithmetic of Hyperbolic Spaces, Kyoto, 2006. RIMS Kokyuroku Series, vol. 1571, pp. 31–58 (2007)
86. Riley, R.: Discrete parabolic representations of link groups. Mathematika 22, 141–150 (1975) DOI: 10.1112/S0025579300005982
87. Riley, R.: A personal account of the discovery of hyperbolic structures on some knot complements. Expo. Math. 31, 104–115 (2013) DOI: 10.1016/j.exmath.2013.01.003
88. Scott, P.: Subgroups of surface groups are almost geometric. J. Lond. Math. Soc. 17(3), 555–565 (1978) DOI: 10.1112/jlms/s2-17.3.555
89. Shalen, P.: The proof in the case of no incompressible surface. In: Morgan, J., Bass, H. (eds.) The Smith Conjecture, New York, 1979. Pure Appl. Math., vol. 112, pp. 21–36. Academic Press, New York (1984)
90. Soma, T.: Existence of ruled wrappings in hyperbolic 3-manifolds. Geom. Topol. 10, 1173–1184 (2006) DOI: 10.2140/gt.2006.10.1173
91. Sullivan, D.: Travaux de Thurston sur les groupes quasi-fuchsiens et sur les variétés hyperboliques de dimension 3 fibrées sur ${\mathbb{S}}^{1}$. In: Séminaire Bourbaki 554. Lecture Notes in Math., vol. 842, pp. 196–214. Springer, Berlin (1981) DOI: 10.1007/BFb0089935
92. Thurston, W.: The geometry and topology of 3-manifolds. Lecture notes, Princeton Math. Dept, 1979
93. Thurston, W.: Hyperbolic geometry and 3-manifolds. In: Low-dimensional topology, Bangor, 1979. London Math. Soc., Lecture Notes Ser., vol. 48, pp. 9–25. Cambridge University Press, Cambridge (1982) DOI: 10.1017/CBO9780511758935.003
94. Thurston, W.: Hyperbolic structures on 3-manifolds: the overall logic. Notes of a workshop at Bowdoin (1980)
95. Thurston, W.: Hyperbolic structures on 3-manifolds, I, deformations of acylindrical manifolds. Ann. Math. 124, 203–246 (1986). arXiv:math/9801019 DOI: 10.2307/1971277
96. Thurston, W.: On the geometry and dynamics of surface diffeomorphisms. Bull. Am. Math. Soc. 19, 417–431 (1988) DOI: 10.1090/S0273-0979-1988-15685-6
97. Thurston, W.: Hyperbolic structures on 3-manifolds, II: surface groups and 3-manifolds which fiber over the circle. arXiv:math/9801058
98. Thurston, W.: Hyperbolic Structures on 3-manifolds, III: deformations of 3-manifolds with incompressible boundary. arXiv:math/9801045
99. Weeks, J.: Computation of hyperbolic structures in knot theory. In: Menasco, B., Thistlethwaite, M. (eds.) Handbook of Knot Theory. Elsevier, Amsterdam (2005)
100. Wise, D.: The structure of groups with a quasi-convex hierarchy. Elec. Res. Announ. Math. Sci. 16, 44–55 (2009)
101. Yoshida, T.: The η-invariant of hyperbolic 3-manifolds. Invent. Math. 81, 473–514 (1985) DOI: 10.1007/BF01388583