Transactions of the American Mathematical Society

Author(s): J. W. Cannon; E. L. Swenson
Journal: Trans. Amer. Math. Soc. 350 (1998), 809-849.
MSC (1991): Primary 20F32, 30F40, 57N10; Secondary 30C62, 31A15, 30F10
Retrieve article in: PDF
This article is available free of charge

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.

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

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 20F32, 30F40, 57N10, 30C62, 31A15, 30F10

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: 10.1090/S0002-9947-98-02107-2
PII: S 0002-9947(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.
Copyright of article: Copyright 1998, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS