Skip to main content
SpringerLink
Log in

William P. Thurston: “Three-dimensional manifolds, Kleinian groups and hyperbolic geometry”

Bull. Am. Math. Soc., New Ser. 6, 357–379 (1982)

  • Classics Revisited
  • Published:
Jahresbericht der Deutschen Mathematiker-Vereinigung Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Notes

  1. Before 1978 the Bulletin of the AMS did not accept Research announcements longer than 100 lines. For this reason, Thurston could not submit there his preprint about surface diffeomorphisms [96] which dates back to 1976 and which is written in the same informal style as this one. After a couple of refusals, he stopped trying to publish it and waited until 1988 to do so (see the introduction to [96]).

References

  1. Agol, I.: Bounds on exceptional Dehn filling. Geom. Topol. 4, 431–449 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Agol, I.: Tameness of hyperbolic 3-manifolds. math.GT/0405568

  3. Agol, I.: Criteria for virtual fibering. Topology 1, 269–284 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bers, L.: On boundaries of Teichmüller spaces and on Kleinian groups. Ann. Math. 91, 570–600 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bergeron, N., Wise, D.: A boundary criterion for cubulation. Am. J. Math. 134(3), 843–859 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Book  MATH  Google Scholar 

  10. Bishop, C., Jones, P.: Hausdorff dimension and Kleinian groups. Acta Math. 179, 1–39 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Boileau, M., Leeb, B., Porti, J.: Geometrization of 3-dimensional orbifolds. Ann. Math. 162, 195–290 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Boileau, M., Porti, J.: Geometrization of 3-Orbifolds of Cyclic Type. Astérisque, vol. 272. SMF, Paris (2001)

    MATH  Google Scholar 

  13. Bonahon, F.: Bouts des variétés hyperboliques de dimension 3. Ann. Math. 124, 71–158 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  15. Bonahon, F., Otal, J.-P.: Variétés hyperboliques avec géodésiques arbitrairement courtes. Bull. Lond. Math. Soc. 20, 255–261 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  16. Brock, J.: Iteration of mapping classes and limits of hyperbolic 3-manifolds. Invent. Math. 143, 523–570 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bromberg, K.: Projective structures with degenerate holonomy and the Bers density conjecture. Ann. Math. (2) 166, 77–93 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  19. Brock, J., Bromberg, K.: On the density of geometrically finite Kleinian groups. Acta Math. 192, 33–93 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Boyer, S., Zhang, X.: Finite Dehn surgery on knots. J. Am. Math. Soc. 9(4), 1005–1050 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Bowditch, B.: Notes on tameness. Enseign. Math. (2) 56(3–4), 229–285 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  24. Canary, R.: On the Laplacian and the geometry of hyperbolic 3-manifolds. J. Differ. Geom. 36(2), 349–367 (1992)

    MATH  MathSciNet  Google Scholar 

  25. Canary, R.: Ends of hyperbolic 3-manifolds. J. Am. Math. Soc. 6(1), 1–35 (1993)

    MATH  MathSciNet  Google Scholar 

  26. Canary, R.: Algebraic convergence of Schottky groups. Trans. Am. Math. Soc. 337, 235–258 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  27. Canary, R.: A covering theorem for hyperbolic 3-manifolds and its applications. Topology 35(3), 751–778 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  29. Cannon, J., Thurston, W.: Group invariant Peano curves. Geom. Topol. 11, 1315–1356 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. Cooper, D., Hodgson, C., Kerckhoff, S.: Three-dimensional orbifolds and cone-manifolds. MSJ Memoirs, vol. 5. Math. Soc. of Japan, Tokyo (2000)

    MATH  Google Scholar 

  31. Cooper, D., Long, D., Reid, A.: On the virtual Betti numbers of arithmetic hyperbolic 3-manifolds. Geom. Topol. 11, 2265–2276 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Culler, M., Gordon, C., Luecke, J., Shalen, P.: Dehn surgery on knots. Ann. Math. 125, 237–300 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  34. Dunfield, N., Thurston, W.: The virtual Haken conjecture: experiments and examples. Geom. Topol. 7, 399–441 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  35. Floyd, W.: Group completions and limit sets of Kleinian groups. Invent. Math. 57, 205–218 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  37. Gabai, D., Meyerhoff, R., Milley, P.: Mom technology and volumes of hyperbolic 3-manifolds. Comment. Math. Helv. 86, 145–188 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  38. Greenberg, L.: Fundamental polyhedra for kleinian groups. Ann. Math. 84, 433–441 (1966)

    Article  MATH  Google Scholar 

  39. Haglund, F., Wise, D.: Special cube complexes. Geom. Funct. Anal. 17, 1551–1620 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  40. Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)

    MATH  Google Scholar 

  41. Hatcher, A., Thurston, W.: Incompressible surfaces in 2-bridge knot complements. Invent. Math. 79(2), 225–246 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  42. Hempel, J.: 3-Manifolds as viewed from the curve complex. Topology 40(3), 631–657 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  43. Hodgson, C., Kerckhoff, S.: Universal bounds for hyperbolic Dehn surgery. Ann. Math. (2) 162(1), 367–421 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  44. Hodgson, C., Kerckhoff, S.: The shape of hyperbolic Dehn surgery space. Geom. Topol. 12(2), 1033–1090 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  45. Jørgensen, T.: Compact 3-manifolds of constant negative curvature fibering over the circle. Ann. Math. (2) 106(1), 61–72 (1977)

    Article  Google Scholar 

  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)

    Google Scholar 

  47. Kahn, J., Markovic, V.: Immersing almost geodesic surfaces in a closed hyperbolic three-manifold. Ann. Math. 175(3), 1127–1190 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  48. Kapovich, M.: Hyperbolic Manifolds and Discrete Groups. Birkhäuser Classics, Boston (2001)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  52. Kleiner, B., Lott, J.: Notes on Perelman’s papers. Geom. Topol. 12, 2587–2858 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  53. Lackenby, M.: Word hyperbolic Dehn surgery. Invent. Math. 140, 243–282 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  54. Lackenby, M.: Surface subgroups of Kleinian groups with torsion. Invent. Math. 179, 175–190 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  55. Lackenby, M., Meyerhoff, R.: The maximal number of exceptional Dehn surgeries. Invent. Math. 191, 341–382 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  56. Mahan, M.: Cannon-Thurston maps for pared manifolds of bounded geometry. Geom. Topol. 13, 189–245 (2009). arXiv:math/0503581

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  59. McMullen, C.: Iteration on Teichmüller space. Invent. Math. 99, 425–454 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  60. Maclachlan, C., Reid, A.: The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics, vol. 219. Springer, New York (2003)

    Book  MATH  Google Scholar 

  61. Marden, A.: The geometry of finitely generated Kleinian groups. Ann. Math. 99, 383–462 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  62. Meeks, W. III, Scott, P.: Finite group actions on 3-manifolds. Invent. Math. 86, 287–346 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  64. Minsky, Y.: On rigidity, limit sets and end invariants of hyperbolic 3-manifolds. J. Am. Math. Soc. 7, 539–588 (1994)

    MATH  MathSciNet  Google Scholar 

  65. Minsky, Y.: Bounded geometry for Kleinian groups. Invent. Math. 146(1), 143–192 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  66. Minsky, Y.: The classification of Kleinian surface groups. I. Models and bounds. Ann. Math. 171, 1–107 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  67. Morgan, J., Bass, H. (eds.): The Smith Conjecture, New York, 1979. Pure Appl. Math., vol. 112. Academic Press, New York (1984)

    MATH  Google Scholar 

  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)

    Google Scholar 

  69. Morgan, J.: Recent progress on the Poincaré conjecture and the classification of 3-manifolds. Bull. Am. Math. Soc. 42, 57–78 (2005)

    Article  MATH  Google Scholar 

  70. Morgan, J., Tian, G.: Ricci Flow and the Poincaré Conjecture. Clay Mathematics Monographs, vol. 3 (2007)

    MATH  Google Scholar 

  71. Namazi, H., Souto, J.: Heegaard splittings and pseudo-Anosov maps. Geom. Funct. Anal. 19, 1195–1228 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  72. Namazi, H., Souto, J.: Non-realizability and ending laminations: proof of the density conjecture. Acta Math. 209, 323–395 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  75. Neumann, W., Yang, J.: Bloch invariants of hyperbolic 3-manifolds. Duke Math. J. 96(1), 29–59 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  76. Ohshika, K.: Realising end invariants by limits of minimally parabolic, geometrically finite groups. Geom. Topol. 15, 827–890 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  79. Otal, J.-P.: Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3. Astérisque, vol. 235. Soc. Math. France, Paris (1996)

    MATH  Google Scholar 

  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)

    Google Scholar 

  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

    Article  MATH  Google Scholar 

  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)

    Google Scholar 

  86. Riley, R.: Discrete parabolic representations of link groups. Mathematika 22, 141–150 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  87. Riley, R.: A personal account of the discovery of hyperbolic structures on some knot complements. Expo. Math. 31, 104–115 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  88. Scott, P.: Subgroups of surface groups are almost geometric. J. Lond. Math. Soc. 17(3), 555–565 (1978)

    Article  MATH  Google Scholar 

  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)

    Google Scholar 

  90. Soma, T.: Existence of ruled wrappings in hyperbolic 3-manifolds. Geom. Topol. 10, 1173–1184 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  96. Thurston, W.: On the geometry and dynamics of surface diffeomorphisms. Bull. Am. Math. Soc. 19, 417–431 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  100. Wise, D.: The structure of groups with a quasi-convex hierarchy. Elec. Res. Announ. Math. Sci. 16, 44–55 (2009)

    MATH  MathSciNet  Google Scholar 

  101. Yoshida, T.: The η-invariant of hyperbolic 3-manifolds. Invent. Math. 81, 473–514 (1985)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

I thank Brian Bowditch, Jeff Brock, Dick Canary, Alan Reid and Jeff Weeks for their comments on a first version of this text.

Author information

Authors and Affiliations

  1. Institut Mathématique de Toulouse, Université Paul Sabatier, 118, Route de Narbonne, 31062, Toulouse, France

    Jean-Pierre Otal

Authors
  1. Jean-Pierre Otal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Pierre Otal.

Additional information

This is a slightly revised version of a review which appeared first in Zentralblatt für Mathematik (Zbl. 0496.57005).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Otal, JP. William P. Thurston: “Three-dimensional manifolds, Kleinian groups and hyperbolic geometry”. Jahresber. Dtsch. Math. Ver. 116, 3–20 (2014). https://doi.org/10.1365/s13291-014-0079-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1365/s13291-014-0079-5

Keywords

Search

Navigation