Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



A topological characterisation
of hyperbolic groups

Author: Brian H. Bowditch
Journal: J. Amer. Math. Soc. 11 (1998), 643-667
MSC (1991): Primary 20F32
MathSciNet review: 1602069
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We characterise word hyperbolic groups as those groups which act properly discontinuously and cocompactly on the space of distinct triples of a compact metrisable space. This is, in turn, equivalent to a convergence group for which every point of the space is a conical limit point.

References [Enhancements On Off] (What's this?)

  • [BeM] A.F.Beardon, A.Maskit, Limit points of kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974) 1-12. MR 48:11489
  • [Bo1] B.H.Bowditch, Notes on Gromov's hyperbolicity criterion for path-metric spaces, in ``Group theory from a geometrical viewpoint'' (ed. E.Ghys, A.Haefliger, A.Verjovsky), World Scientific (1991) 64-167. MR 93h:57002
  • [Bo2] B.H.Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J. 77 (1995) 229-274. MR 96b:53056
  • [Bo3] B.H.Bowditch, Cut points and canonical splittings of hyperbolic groups, to appear in Acta. Math.
  • [Bo4] B.H.Bowditch, Convergence groups and configuration spaces, to appear in ``Group Theory Down Under'' (ed. J.Cossey, C.F.Miller, W.D.Neumann, M.Shapiro), de Gruyter.
  • [Bo5] B.H.Bowditch, Relatively hyperbolic groups, preprint, Southampton (1997).
  • [CanS] J.W.Cannon, E.L.Swenson, Recognizing constant curvature discrete groups in dimension 3, to appear in Trans. Amer. Math. Soc. CMP 97:15
  • [CasJ] A.Casson, D.Jungreis, Convergence groups and Seifert fibered $3$-manifolds, Invent. Math. 118 (1994) 441-456. MR 96f:57011
  • [D] M.J.Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985) 449-457. MR 87d:20037
  • [F] E.M.Freden, Negatively curved groups have the convergence property, Ann. Acad. Sci. Fenn. Ser. A Math. 20 (1995) 333-348. MR 96g:20054
  • [Ga] D.Gabai, Convergence groups are fuchsian groups, Ann. of Math. 136 (1992) 447-510. MR 93m:20065
  • [GeM1] F.W.Gehring, G.J.Martin, Discrete quasiconformal groups I, Proc. London Math. Soc. 55 (1987) 331-358. MR 88m:30057
  • [GeM2] F.W.Gehring, G.J.Martin, Discrete quasiconformal groups II, handwritten notes.
  • [GhH] E.Ghys, la Harpe (eds.), Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Maths. 83, Birkhäuser (1990). MR 92f:53050
  • [Gr] M.Gromov, Hyperbolic groups, in ``Essays in Group Theory" (ed. S.M.Gersten) M.S.R.I. Publications No. 8, Springer-Verlag (1987) 75-263. MR 89e:20070
  • [HK] J.Heinonen, P.Koskela, Quasiconformal maps in metric spaces with controlled geometry, to appear in Acta. Math.
  • [K] R.Kirby (ed.), Problems in low-dimensional topology, problem list, Berkeley (1995). CMP 98:01
  • [MT] G.J.Martin, P.Tukia, Convergence groups with an invariant component pair, Amer. J. Math. 114 (1992) 1049-1077. MR 93i:30034
  • [O] J.-P.Otal, Sur la géométrie symplectique de l'espace des géodésiques d'une variété à courbure negative, Rev. Math. Iberoamericana 8 (1992) 441-456. MR 94a:58077
  • [P] F.Paulin, Un groupe hyperbolique est determiné par son bord, J. London Math. Soc. 54 (1996) 50-74. MR 97d:20042
  • [T1] P.Tukia, Homeomorphic conjugates of fuchsian groups, J. reine angew. Math. 391 (1988) 1-54. MR 89m:30047
  • [T2] P.Tukia, Convergence groups and Gromov's metric hyperbolic spaces, New Zealand J. Math. 23 (1994) 157-187. MR 96c:30042
  • [T3] P.Tukia, Conical limit points and uniform convergence groups, preprint, Helsinki (1996).

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 20F32

Retrieve articles in all journals with MSC (1991): 20F32

Additional Information

Brian H. Bowditch
Affiliation: Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, Great Britain

Received by editor(s): March 20, 1997
Received by editor(s) in revised form: February 2, 1998
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society