Recognizing constant curvature discrete groups in dimension 3

Cannon, J.; Swenson, E.

doi:10.1090/S0002-9947-98-02107-2

Recognizing constant curvature discrete groups in dimension 3
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by J. W. Cannon and E. L. Swenson PDF

Trans. Amer. Math. Soc. 350 (1998), 809-849 Request permission

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.

References

É. Ghys, A. Haefliger, and A. Verjovsky (eds.), Group theory from a geometrical viewpoint, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. MR 1170362, DOI 10.1142/1235
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.
A. F. Beardon, The geometry of discrete groups, Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975) Academic Press, London, 1977, pp. 47–72. MR 0474012
Bowditch, B. H., Notes on Gromov’s hyperbolicity criterion, Group theory from a geometric viewpoint, World Scientific, Singapore (to appear).
James W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), no. 2, 123–148. MR 758901, DOI 10.1007/BF00146825
Tim Bedford, Michael Keane, and Caroline Series (eds.), Ergodic theory, symbolic dynamics, and hyperbolic spaces, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991. Papers from the Workshop on Hyperbolic Geometry and Ergodic Theory held in Trieste, April 17–28, 1989. MR 1130170
James W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), no. 2, 155–234. MR 1301392, DOI 10.1007/BF02398434
J. W. Cannon and Daryl Cooper, A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three, Trans. Amer. Math. Soc. 330 (1992), no. 1, 419–431. MR 1036000, DOI 10.1090/S0002-9947-1992-1036000-0
J. W. Cannon, W. J. Floyd, and W. R. Parry, Squaring rectangles: the finite Riemann mapping theorem, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992) Contemp. Math., vol. 169, Amer. Math. Soc., Providence, RI, 1994, pp. 133–212. MR 1292901, DOI 10.1090/conm/169/01656
M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990 (French). Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups]; With an English summary. MR 1075994, DOI 10.1007/BFb0084913
David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694, DOI 10.1201/9781439865699
William J. Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980), no. 3, 205–218. MR 568933, DOI 10.1007/BF01418926
Harry Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335–386. MR 146298, DOI 10.2307/1970220
Harry Furstenberg, Boundaries of Lie groups and discrete subgroups, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 301–306. MR 0430160
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
M. Gromov and W. Thurston, Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), no. 1, 1–12. MR 892185, DOI 10.1007/BF01404671
Tapani Kuusalo, Verallgemeinerter Riemannscher Abbidungssatz und quasikonforme Mannigfaltigkeiten, Ann. Acad. Sci. Fenn. Ser. A I No. 409 (1967), 24 (German). MR 0218560
Joseph Lehner, Discontinuous groups and automorphic functions, Mathematical Surveys, No. VIII, American Mathematical Society, Providence, R.I., 1964. MR 0164033, DOI 10.1090/surv/008
O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463, DOI 10.1007/978-3-642-65513-5
Gaven J. Martin, Discrete quasiconformal groups that are not the quasiconformal conjugates of Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 11 (1986), no. 2, 179–202. MR 853955, DOI 10.5186/aasfm.1986.1113
Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR 0385004
G. D. Mostow and Yum Tong Siu, A compact Kähler surface of negative curvature not covered by the ball, Ann. of Math. (2) 112 (1980), no. 2, 321–360. MR 592294, DOI 10.2307/1971149
Reto A. Rüedy, Deformations of embedded Rieman surfaces, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 385–392. MR 0290419
Siegel, C. L., Topics in Complex Function Theory, vol. II, Wiley-Interscience, New York-London-Sydney-Toronto, 1971.
Dennis Sullivan, 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 (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465–496. MR 624833
Swenson, Eric Lewis, Negatively curved groups and related topics, Ph.D. Dissertation, Brigham Young Univ, 1993.
William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
Pekka Tukia, A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 1, 149–160. MR 639972, DOI 10.5186/aasfm.1981.0625
Pekka Tukia, On quasiconformal groups, J. Analyse Math. 46 (1986), 318–346. MR 861709, DOI 10.1007/BF02796595