×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, J., Intersections of analytically and geometrically finite subgroups of Kleinian groups, Trans. amer. math. soc., 343, 87-98, (1994) · Zbl 0802.30036
[2] Alonso, J.; Brady, T.; Cooper, D.; Delzant, T.; Ferlini, V.; Lustig, M.; Mihalik, M.; Shapiro, M.; Short, H., Notes on word hyperbolic groups, () · Zbl 0849.20023
[3] Bowditch, B., Discrete parabolic groups, J. differential geom., 38, 559-583, (1993) · Zbl 0793.53029
[4] Bowditch, B., Geometric finiteness for hyperbolic groups, J. funct. anal., 113, 245-317, (1993) · Zbl 0789.57007
[5] Bowditch, B., Geometric finiteness with variable negative curvature, Duke math. J., 77, 229-274, (1995) · Zbl 0877.57018
[6] Cannon, J., The theory of negatively curved spaces and groups, (), 315-369 · Zbl 0764.57002
[7] Cannon, J.; Swenson, E., Recognizing constant curvature discrete groups in dimension 3, Trans. amer. math. soc., 350, 2, 809-849, (1998) · Zbl 0910.20024
[8] Coornaert, M., Sur le domaine de discontinuité pour LES groupes d’isométrie d’un espace métrique hyperbolique, Rend. sem. mate. univ. Cagliari, 59, 185-195, (1989) · Zbl 0799.53054
[9] Coornaert, M., Mesures de patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Pacific J. math., 159, 241-270, (1993) · Zbl 0797.20029
[10] Coornaert, M.; Delzant, T.; Papadopoulos, A., Géométrie et théorie des groupes, Lecture notes in mathematics, 1441, (1991), Springer Berlin
[11] Gerston, S.M.; Short, H., Rational subgroups of biautomatic groups, Ann. of math., 134, 125-158, (1991) · Zbl 0744.20035
[12] Ghy, E.; de la Harpe, P., Sur LES groupes hyperboliques d’apres mikael Gromov, Progress in mathematics, 83, (1990), Birkhäuser Zürich
[13] Gitik, R.; Mitra, M.; Rips, E.; Sageev, M., Widths of subgroups, Trans. amer. math. soc., 350, 1, 321-329, (1998) · Zbl 0897.20030
[14] Gromov, M., Hyperbolic groups, () · Zbl 0634.20015
[15] Kapovich, I.; Short, H., Greenberg’s theorem for quasi-convex subgroups of word hyperbolic groups, Canad. J. math., 48, 6, 1224-1244, (1996) · Zbl 0873.20025
[16] Nicholls, P., The ergodic theory of discrete groups, London math. soc. lecture note ser., 143, (1989), Cambridge University Press Cambridge · Zbl 0674.58001
[17] Susskind, P.; Swarup, G., Limit sets of geometrically finite hyperbolic groups, Amer. J. math., 114, 233-250, (1992) · Zbl 0791.30039
[18] Swarup, G., Geometric finiteness and rationality, J. pure appl. algebra, 88, 327-333, (1993) · Zbl 0806.57003
[19] Swenson, E., Boundary dimension in negatively curved spaces, Geom. dedicata, 57, 297-303, (1995) · Zbl 0837.53037
[20] Swenson, E., Limit sets in the boundary of negatively curved groups, (1994), Preprint
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.