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Algebra and Logic

, Volume 52, Issue 3, pp 222–235 | Cite as

Complementing a subgroup of a hyperbolic group by a free factor

  • F. A. DudkinEmail author
  • K. S. Sviridov
Article
  • 51 Downloads

Let G be a hyperbolic group that is not almost cyclic and H be its quasiconvex subgroup of infinite index. We find necessary and sufficient conditions of there being for H a free subgroup F of rank 2 in G such that F and H generate a free product FHG. It is proved that FH is quasiconvex and that there exists an algorithm for verifying the conditions of the criterion given G and H.

Keywords

hyperbolic group quasiconvex subgroup free product 

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References

  1. 1.
    G. N. Arzhantseva, “On quasiconvex subgroups of word hyperbolic groups,” Geom. Dedic., 87, Nos. 1-3, 191–208 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    M. Gromov, “Hyperbolic groups,” in Essays in Group Theory, Publ., Math. Sci. Res. Inst., 8, S. M. Gersten (ed.) (1987), pp. 75–263.Google Scholar
  3. 3.
    K. S. Sviridov, “Complementing a finite subgroup of a hyperbolic group by a free factor,” Algebra Logika, 49, No. 4, 520–554 (2010).MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren Mathem. Wiss., 319), Berlin, Springer-Verlag (1999).Google Scholar
  5. 5.
    P. S. Novikov and S. I. Adyan, “On infinite periodic groups. I,” Izv. Akad. Nauk SSSR, Ser. Matem., 32, No. 1, 212–244 (1968).Google Scholar
  6. 6.
    P. S. Novikov and S. I. Adyan, “On infinite periodic groups. II,” Izv. Akad. Nauk SSSR, Ser. Matem., 32, No. 2, 251–524 (1968).Google Scholar
  7. 7.
    P. S. Novikov and S. I. Adyan, “On infinite periodic groups. III,” Izv. Akad. Nauk SSSR, Ser. Matem., 32, No. 3, 709–731 (1968).Google Scholar
  8. 8.
    V. Chainikov, Actions of Maximal Growth of Hyperbolic Groups, Preprint; http://arxiv.org/abs/1201.1349.
  9. 9.
    E. Martinez-Pedroza, On Quasiconvexity and Relative Hyperbolic Structures, Preprint; http://arxiv.org/abs/0811.2384.
  10. 10.
    É. Ghys, P. de la Harpe (eds.), Sur les Groupes Hyperboliques d’Aprés Mikhael Gromov, Progr. Math., 83, Boston, MA, Birkhäuser (1990).Google Scholar
  11. 11.
    E. L. Swenson, “Quasi-convex groups of isometries of negatively curved spaces,” Topol. Appl., 110, No. 1, 119–129 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. Minasyan, “Some properties of subsets of hyperbolic groups,” Comm. Alg., 33, No. 3, 909–935 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    O. V. Bogopolskii and V. N. Gerasimov, “Finite subgroups of hyperbolic groups,” Algebra Logika, 34, No. 6, 619–622 (1995).MathSciNetGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.NovosibirskRussia

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