Recognizing constant curvature discrete groups in dimension 3

We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3-space \Ht 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 curv...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 1998-02, Vol.350 (2), p.809-849
Main Authors: Cannon, J. W., Swenson, E. L.
Format: Article
Language:English
Subjects:
Conformity
Curvature
Homeomorphism
Infinity
Integers
Mathematical constants
Mathematical theorems
Topological theorems
Vertices
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Summary:We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3-space \Ht 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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-98-02107-2