Recognizing Constant Curvature Discrete Groups in Dimension 3

  title={Recognizing Constant Curvature Discrete Groups in Dimension 3},
  author={William P. Thurston and Peter Doyle and Rita Schwarz},
We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3-space H3 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. 
Highly Cited
This paper has 33 citations. REVIEW CITATIONS
26 Citations
9 References
Similar Papers


Publications referenced by this paper.
Showing 1-9 of 9 references

Sur les groupes hyperboliques d’après Mikhael Gromov, available from the authors (The “little green book.”)

  • W. Ballmann, E. Ghys, +4 authors M. Troyanov
  • 1989
Highly Influential
6 Excerpts

Group completions and limit sets of Kleinian groups

  • W. J. Floyd
  • 1992

Notes on word hyperbolic groups, Group theory from a geometric viewpoint (E

  • J. M. Alonso, T. Brady, +5 authors H. Short
  • World Scientific, Singapore,
  • 1991
3 Excerpts

Pinching constants for hyperbolic manifolds

  • M. Gromov, W. Thurston
  • Inv . Math .
  • 1987

A Poisson formula for semisimple Lie groups

  • H. Furstenberg
  • Ann . of Math
  • 1980

The combinatorial structure of cocompact discrete hyperbolic groups

  • B. H. Bowditch, J. W. Cannon
  • Geom . Dedicata
  • 1977

The geometry of discrete groups, Discrete Groups and Automorphic Functions (W

  • A. F. Beardon
  • Academic Press London-New York-San Francisco,
  • 1977
1 Excerpt

A compact Kähler surface of negative curvature not covered by a ball

  • G. D. Mostow, Y.-T. Siu
  • Ann . Math
  • 1973

Discrete quasiconformal groups that are not the quasiconformal conjugates of Möbius groups

  • G. J. Martin
  • Ann . Acad . Sci . Fenn . Ser . A I
  • 1973

Similar Papers

Loading similar papers…