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Transactions of the American Mathematical Society

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Recognizing constant curvature
discrete groups in dimension 3


Authors: J. W. Cannon and E. L. Swenson
Journal: Trans. Amer. Math. Soc. 350 (1998), 809-849
MSC (1991): Primary 20F32, 30F40, 57N10; Secondary 30C62, 31A15, 30F10
DOI: https://doi.org/10.1090/S0002-9947-98-02107-2
MathSciNet review: 1458317
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Abstract: We characterize those discrete groups $G$ which can act properly discontinuously, isometrically, and cocompactly on hyperbolic $3$-space ${\mathbb H}^3$ in terms of the combinatorics of the action of $G$ on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the Sullivan-Tukia theorem on groups which act uniformly quasiconformally on the $2$-sphere.


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  • 1. Alonso, J. M.; Brady, T.; Cooper, D.; Ferlini, V.; Lustig, M.; Mihalik, M.; Shapiro, M.; and Short, H., Notes on word hyperbolic groups, Group theory from a geometric viewpoint (E. Ghys, A. Häfliger, and A. Verjovsky, eds.), World Scientific, Singapore, 1991. MR 93a:20001
  • 2. Ballmann, W.; Ghys, E.; Haefliger, A.; de la Harpe, P.; Salem, E.; Strebel, R.; and Troyanov, M., Sur les groupes hyperboliques d'après Mikhael Gromov, available from the authors (The ``little green book.''), 1989.
  • 3. Beardon, A. F., The geometry of discrete groups, Discrete Groups and Automorphic Functions (W. J. Harvey, eds.), Academic Press London-New York-San Francisco, 1977, pp. 47-72. MR 57:13670
  • 4. Bowditch, B. H., Notes on Gromov's hyperbolicity criterion, Group theory from a geometric viewpoint, World Scientific, Singapore (to appear).
  • 5. Cannon, J. W., The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), 123-148. MR 86j:20032
  • 6. -, The theory of negatively curved spaces and groups, Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces (Tim Bedford, Michael Keane and Caroline Series, eds.), Oxford Univ. Press, Oxford-New York-Toronto, 1991, pp. 315-369. MR 93e:58002
  • 7. -, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), 155-234. MR 95k:30046
  • 8. Cannon, J. W., and Cooper, Daryl, A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three, Trans. Amer. Math. Soc. 330 (1992), 419-431. MR 92f:22017
  • 9. Cannon, J. W.; Floyd, W. J.; and Parry, Walter, Squaring rectangles: the finite Riemann mapping theorem, The Mathematical Heritage of Wilhelm Magnus - Groups, Geometry & Special Functions, Contemp. Math., vol. 169, Amer. Math. Soc., Providence, RI, 1994, pp. 133-212. MR 95g:20045
  • 10. Coornaert, M.; Delzant, T.; and Papadopoulos, A., Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov, Lecture Notes in Math. 1441, Springer-Verlag, Berlin-Heidelberg-New York, 1990. MR 92f:57003
  • 11. Epstein, D. B. A.; Cannon, J. W.; Holt, D. F.; Levy, S. V. F.; Paterson, M. S.; and Thurston, W. P., Word Processing in Groups, Jones and Bartlett, Boston-London, 1992. MR 93i:20036
  • 12. Floyd, W. J., Group completions and limit sets of Kleinian groups, Inv. Math. 57 (1980), 205-218. MR 81e:57002
  • 13. Furstenberg, H., A Poisson formula for semi-simple Lie groups, Ann. of Math. 77 (1963), 335-386. MR 26:3820
  • 14. -, Boundaries of Lie groups and discrete subgroups, Actes, Congrès Intern. Math. (Nice, 1970), vol. 2, pp. 301-306. MR 55:3167
  • 15. Gromov, M., Hyperbolic groups, Essays in group theory (S. M. Gersten, ed.), MSRI Publ. 8, Springer-Verlag, Berlin-Heidelberg-New York, 1987, pp. 75-263. MR 89e:20070
  • 16. Gromov, M., and Thurston, W., Pinching constants for hyperbolic manifolds, Inv. Math. 89 (1987), 1-12. MR 88e:53058
  • 17. Kuusalo, Tapani, Verallgemeinerter Riemannscher Abbildungssatz und quasikonforme Mannigfaltigkeiten, Ann. Acad. Sci. Fenn. Ser. A I 409 (1967), 24 pages. MR 36:1645
  • 18. Lehner, Joseph, Discontinuous Groups and Automorphic Functions, Amer. Math. Soc., Providence, R. I. (1964). MR 29:1332
  • 19. Lehto, O., and Virtanen, K. I., Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin-Heidelberg-New York, 1973. MR 49:9202
  • 20. Martin, G. J., Discrete quasiconformal groups that are not the quasiconformal conjugates of Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I 11 (1986), 179-202. MR 89d:30025
  • 21. Maskit, B., Kleinian Groups, Springer-Verlag, Berlin-Heidelberg-New York, 1988. MR 90a:30132
  • 22. Mostow, G. D., Strong Rigidity of Locally Symmetric Spaces, Princeton Univ. Press, Princeton, N. J., 1973. MR 52:5874
  • 23. Mostow, G. D., and Siu, Y. -T., A compact Kähler surface of negative curvature not covered by a ball, Ann. Math 112 (1980), 321-360. MR 82f:53075
  • 24. Rüedy, Reto A., Deformations of embedded Riemann surfaces, Advances in the Theory of Riemann Surfaces (Ahlfors, Bers, et al., eds.), Princeton Univ. Press, Princeton, N. J., 1971, pp. 385-392. MR 44:7600
  • 25. Siegel, C. L., Topics in Complex Function Theory, vol. II, Wiley-Interscience, New York-London-Sydney-Toronto, 1971.
  • 26. Sullivan, Dennis, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (Irwin Kra and Bernard Maskit, eds.), Princeton University Press, Princeton, N.J., 1981, pp. 465-496. MR 83f:58052
  • 27. Swenson, Eric Lewis, Negatively curved groups and related topics, Ph.D. Dissertation, Brigham Young Univ, 1993.
  • 28. Thurston, W. P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982), 357-381. MR 83h:57019
  • 29. Tukia, P., A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 149-160. MR 83b:30019
  • 30. Tukia, Pekka, On quasiconformal groups, J. Analyse Math. 46 (1986), 318-346. MR 87m:30043

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Additional Information

J. W. Cannon
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: cannon@math.byu.edu

E. L. Swenson
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: eric@math.byu.edu

DOI: https://doi.org/10.1090/S0002-9947-98-02107-2
Received by editor(s): July 13, 1994
Received by editor(s) in revised form: November 14, 1996
Additional Notes: This research was supported in part by The Geometry Center at the University of Minnesota, a Science and Technology Center funded by NSF, DOE, and Minnesota Technology, Inc.; and by NSF Research Grant No. DM-8902071.
Article copyright: © Copyright 1998 American Mathematical Society

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