Results 1 
5 of
5
Recognizing constant curvature discrete groups in dimension 3
 TRANS. AMER. MATH. SOC.
, 1998
"... We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3space 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 cu ..."
Abstract

Cited by 20 (8 self)
 Add to MetaCart
We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3space 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 SullivanTukia theorem on groups which act uniformly quasiconformally on the 2sphere.
Negatively Curved Groups Have The Convergence Property I
 ANNALES ACADEMIAE SCIENTIARUM FENNICAE
, 1995
"... It is known that the Cayley graph \Gamma of a negatively curved (Gromovhyperbolic) group G has a welldefined boundary at infinity @ \Gamma . Furthermore, @ \Gamma is compact and metrizable. In this paper I show that G acts on @ \Gamma as a convergence group. This implies that if @ \Gamma ' S 1 ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
It is known that the Cayley graph \Gamma of a negatively curved (Gromovhyperbolic) group G has a welldefined boundary at infinity @ \Gamma . Furthermore, @ \Gamma is compact and metrizable. In this paper I show that G acts on @ \Gamma as a convergence group. This implies that if @ \Gamma ' S 1 , then G is topologically conjugate to a cocompact Fuchsian group.
Hyperbolic geometry
 In Flavors of geometry
, 1997
"... 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65 ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65
The boundary of a Busemann space
 Proc. Amer. Math. Soc
, 1997
"... Abstract. Let X be a proper Busemann space. Then there is a well defined boundary, ∂X, for X. Moreover, if X is (Gromov) hyperbolic (resp. nonpositively curved), then this boundary is homeomorphic to the hyperbolic (resp. nonpositively curved) boundary. The boundary of a (Gromov) hyperbolic space ( ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Abstract. Let X be a proper Busemann space. Then there is a well defined boundary, ∂X, for X. Moreover, if X is (Gromov) hyperbolic (resp. nonpositively curved), then this boundary is homeomorphic to the hyperbolic (resp. nonpositively curved) boundary. The boundary of a (Gromov) hyperbolic space (and hence of a (Gromov) hyperbolic group) was introduced in Gromov’s now famous article on hyperbolic groups [G1]. Since then, this notion has received much attention and provided many interesting results (see [F], [G1], [GH], [Sw]). This notion of boundary has been generalized
Negatively Curved Groups and the Convergence Property II: Transitivity in Negatively Curved Groups
 ANNALES ACADEMIAE SCIENTIARUM FENNICAE
, 1996
"... Let G be a negatively curved group. This paper continues the classification of limit points of G that began in part I. A probability measure is constructed on the space at infinity and with respect to this measure almost every point at infinity is shown to be line transitive. ..."
Abstract
 Add to MetaCart
Let G be a negatively curved group. This paper continues the classification of limit points of G that began in part I. A probability measure is constructed on the space at infinity and with respect to this measure almost every point at infinity is shown to be line transitive.