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Quasi-convex groups of isometries of negatively curved spaces. (English) Zbl 0973.20037
B. H. Bowditch has given the most careful analyses to date of the various possible definitions of geometrically finite groups [see J. Funct. Anal. 113, No. 2, 245-317 (1993; Zbl 0789.57007) and Duke Math. J. 77, No. 2, 229-274 (1995; Zbl 0877.57018)]. The author extends most of these results to Gromov hyperbolic metric spaces. Because of problems with finite generation, the author restricts himself to the case where there are no parabolic elements. He proves the equivalence of five conditions: the group is quasiconvex; all limit points are conical; all limit point are horospherical; the action is cocompact on the weak convex hull of the limit set; the action is cocompact on the union of the space and the domain of discontinuity at infinity.
Reviewer: J.W.Cannon (Provo)

MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M07 Topological methods in group theory 57M60 Group actions on manifolds and cell complexes in low dimensions 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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