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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
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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Additional Information

J. W. Cannon
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

E. L. Swenson
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

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