Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


A topological characterisation of hyperbolic groups
HTML articles powered by AMS MathViewer

by Brian H. Bowditch
J. Amer. Math. Soc. 11 (1998), 643-667


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.
  • Alan F. Beardon and Bernard Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974), 1–12. MR 333164, DOI 10.1007/BF02392106
  • B. H. Bowditch, Notes on Gromov’s hyperbolicity criterion for path-metric spaces, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 64–167. MR 1170364
  • B. H. Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J. 77 (1995), no. 1, 229–274. MR 1317633, DOI 10.1215/S0012-7094-95-07709-6
  • B.H.Bowditch, Cut points and canonical splittings of hyperbolic groups, to appear in Acta. Math.
  • 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.
  • B.H.Bowditch, Relatively hyperbolic groups, preprint, Southampton (1997).
  • J.W.Cannon, E.L.Swenson, Recognizing constant curvature discrete groups in dimension 3, to appear in Trans. Amer. Math. Soc.
  • Andrew Casson and Douglas Jungreis, Convergence groups and Seifert fibered $3$-manifolds, Invent. Math. 118 (1994), no. 3, 441–456. MR 1296353, DOI 10.1007/BF01231540
  • M. J. Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985), no. 3, 449–457. MR 807066, DOI 10.1007/BF01388581
  • Eric M. Freden, Negatively curved groups have the convergence property. I, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), no. 2, 333–348. MR 1346817
  • David Gabai, Convergence groups are Fuchsian groups, Ann. of Math. (2) 136 (1992), no. 3, 447–510. MR 1189862, DOI 10.2307/2946597
  • F. W. Gehring and G. J. Martin, Discrete quasiconformal groups. I, Proc. London Math. Soc. (3) 55 (1987), no. 2, 331–358. MR 896224, DOI 10.1093/plms/s3-55_{2}.331
  • F.W.Gehring, G.J.Martin, Discrete quasiconformal groups II, handwritten notes.
  • É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648, DOI 10.1007/978-1-4684-9167-8
  • M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
  • J.Heinonen, P.Koskela, Quasiconformal maps in metric spaces with controlled geometry, to appear in Acta. Math.
  • R.Kirby (ed.), Problems in low-dimensional topology, problem list, Berkeley (1995).
  • Gaven J. Martin and Pekka Tukia, Convergence groups with an invariant component pair, Amer. J. Math. 114 (1992), no. 5, 1049–1077. MR 1183531, DOI 10.2307/2374889
  • Jean-Pierre Otal, Sur la géometrie symplectique de l’espace des géodésiques d’une variété à courbure négative, Rev. Mat. Iberoamericana 8 (1992), no. 3, 441–456 (French). MR 1202417, DOI 10.4171/RMI/130
  • Frédéric Paulin, Un groupe hyperbolique est déterminé par son bord, J. London Math. Soc. (2) 54 (1996), no. 1, 50–74 (French, with French summary). MR 1395067, DOI 10.1112/jlms/54.1.50
  • Pekka Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988), 1–54. MR 961162, DOI 10.1515/crll.1988.391.1
  • Pekka Tukia, Convergence groups and Gromov’s metric hyperbolic spaces, New Zealand J. Math. 23 (1994), no. 2, 157–187. MR 1313451
  • 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
Bibliographic Information
  • Brian H. Bowditch
  • Affiliation: Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, Great Britain
  • Email:
  • Received by editor(s): March 20, 1997
  • Received by editor(s) in revised form: February 2, 1998
  • © Copyright 1998 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 11 (1998), 643-667
  • MSC (1991): Primary 20F32
  • DOI:
  • MathSciNet review: 1602069