Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T07:24:54.247Z Has data issue: false hasContentIssue false
  • English
  • Français

Greenberg's Theorem for Quasiconvex Subgroups of Word Hyperbolic Groups

Published online by Cambridge University Press:  20 November 2018

Ilya Kapovich  and
Hamish Short
Ilya Kapovich
Affiliation:
City College, 138th Street and Convent Avenue, New York, NY, USA 10031 e-mail: ilya@groups.sci.ccny.cuny.edu
Hamish Short
Affiliation:
C.M.I., Université de Provence, 39 Rue Joliot-Curie, 13452 Marseille Cedex 13, France e-mail: hamish@claudia.univ-mrs.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Analogues of a theorem of Greenberg about finitely generated subgroups of free groups are proved for quasiconvex subgroups of word hyperbolic groups. It is shown that a quasiconvex subgroup of a word hyperbolic group is a finite index subgroup of only finitely many other subgroups.

Keywords

20F3257Mword hyperbolic groupboundaryquasiconvex
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 6 , 01 December 1996 , pp. 1224 - 1244
Copyright
Copyright © Canadian Mathematical Society 1996

References

[ABC] Alonso, J., Brady, T., Cooper, D., Ferlini, V., Lustig, M., Mihalik, M., Shapiro, M. and Short, H., Notes on hyperbolic groups, In: Group theory from a geometric viewpoint, Proc. ICTR Trieste, World Scientific, Singapore, 1991, 3–63.Google Scholar
[Ba] Bass, H., Covering theory for graphs of groups, J. Pure Appl. Algebra 89(1993), 3–47.Google Scholar
[Bo1] Bowditch, B., Notes on Gromov's hyperbolicity criterion for path-metric spaces, In: Group theory from a geometric viewpoint, (eds. Ghys, E., Haefliger, A. and Verkovsky, A.), 1991, 64–167.Google Scholar
[Bo2] Bowdich, B., Geometrical finiteness for hyperbolic groups, J. Funct. Anal. (2)113(1993), 245–317.Google Scholar
[Br] Brady, N., Branched coverings of cubical complexes and subgroups of hyperbolic groups, University of Utah, 1995, preprint.Google Scholar
[BMS] Baumslag, G., Miller, C. F. III and Short, H., Unsolvable problems about small cancellation and word hyperbolic groups, Bull. London Math. Soc. (1)26(1994), 97–101.Google Scholar
[BS] Bridson, M. and Swamp, G., On Hausdorff-Gromov convergence and a theorem of Paulin, Enseign. Math. (2)40(1994), no. 3-4, 267–289.Google Scholar
[CDP] Coornaert, M., Delzant, T. and Papadopoulos, A., Notes sur les groupes hyerboliques de Gromov, no. 1441, Springer-Verlag, 1990.Google Scholar
[De] Dehn, M., Papers on Group Theory and Topology, translated and introduced by Stillwell, J., Springer-Verlag, 1987.Google Scholar
[ECHLPT] Epstein, D. B. A., Cannon, J. W., Holt, D. F., Levy, S. V. F., Paterson, M. S. and Thurston, W. P., Word Processing in Groups, Jones and Bartlett, MA, 1992.Google Scholar
[F] Floyd, W., Group completions and limit sets of Kleinian groups, Invent. Math. 57(1980), 205–218.Google Scholar
[G] Greenberg, L., Discrete Groups of Motions, Canad. J. Math. 12(1960), 414–425.Google Scholar
[GH] Ghys, E. and Harpe, P. de la (editors), Sur les groupes hyperboliques d'aprés Mikhael Gromov, In: Papers from the Swiss seminar on hyperbolic groups held in Bern, 1988, (eds. Ghys, E. and Harpe, P. de la), Progress in Mathematics 83, 1990.Google Scholar
[Gr] Gromov, M., Hyperbolic Groups, In: Essays in group theory, (ed. Gersten, S. M.), MSRI Publ. 8, Springer, 1987, 75–263.Google Scholar
[GMRS] Gitik, R., Mitra, M., Rips, E. and Sageev, M., Widths of subgroups, Trans. Amer. Math. Soc, to appear.Google Scholar
[GS] Gersten, S. M. and Short, H., Rational subgroups of biautomatic groups, Ann. of Math. (2)134(1991), no. 1, 125–158.Google Scholar
[LS] Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory, Springer-Verlag, Berlin-Heidleberg-New York, 1977.Google Scholar
[Mo] Morgan, J., On Thurston's uniformization theorem for three-dimensional manifolds, In: The Smith Conjecture, (eds. Bass, H. and Morgan, J.), Academic Press, 1984, 37—126.Google Scholar
[MT] Mihalik, M. and Towle, W., Quasiconvex subgroups of negatively curved groups, J. Pure Appl. Algebra 3(1995), 297–301.Google Scholar
[N] Neumann, W., The fixed group of an automorphism of a word hyperbolic group is rational, Invent. Math. 110(1992), 147–150.Google Scholar
[Pi] Pittet, Ch., Surface groups and quasiconvexity, In: Geometric Group Theory 1, Sussex, 1991, London Math. Soc. Lecture Notes Series 181, Cambridge University Press, Cambridge, 1993, 169–175.Google Scholar
[R] Rips, E., Subgroups of small cancellation groups, Bull. London Math Soc. 14(1982), 45—47.Google Scholar
[Sho] Short, H., Quasiconvexity and a Theorem of Howson ‘s, In: Group theory from a geometric viewpoint, Proc. ICTP. Trieste, World Scientific, Singapore, 1991.Google Scholar
[Sta] Stallings, J. R., Topology of finite graphs, Invent. Math. 71(1983), 551–565.Google Scholar
[Swa] Swarup, G. A., Geometric finiteness and rationality, J. Pure Appl. Algebra 86(1993), 327–333.Google Scholar
[Swe] Swensen, E., Limit sets in the boundary of word hyperbolic groups, 1994, preprint.Google Scholar