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Geometriae Dedicata

, Volume 151, Issue 1, pp 59–78 | Cite as

Relative hyperbolicity, trees of spaces and Cannon-Thurston maps

  • Mahan MjEmail author
  • Abhijit Pal
Original Paper

Abstract

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalizes a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.

Keywords

Relative hyperbolicity Cannon-Thurston maps Trees of spaces 

Mathematics Subject Classification (2000)

20F32(Primary) 57M50(Secondary) 

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References

  1. 1.
    Bestvina M., Feighn M.: A combination theorem for negatively curved groups. J. Differ. Geom. 35, 85–101 (1992)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bowditch, B.H.: Relatively hyperbolic groups. Southampton (1997) (preprint)Google Scholar
  3. 3.
    Bowditch, B.H.: The Cannon-Thurston map for punctured surface groups. Southampton (2002) (preprint)Google Scholar
  4. 4.
    Bowditch, B.H.: Stacks of hyperbolic spaces and ends of 3 manifolds. Southampton (2002) (preprint)Google Scholar
  5. 5.
    Cannon, J., Thurston, W.P.: Group invariant Peano curves. Princeton (1989) (preprint)Google Scholar
  6. 6.
    Cannon J., Thurston W.P.: Group invariant Peano curves. Geom. Topol. 11, 1315–1356 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Farb B.: Relatively hyperbolic groups. Geom. Funct. Anal. 8, 810–840 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ghys, E., de la Harpe P. (eds.): Sur les groupes hyperboliques d’apres Mikhael Gromov. Progress in Mathematics, vol. 83. Birkhauser, Boston Ma (1990)Google Scholar
  9. 9.
    Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (eds.) Essays in Group Theory, MSRI Publications, vol. 8, pp. 75–263. Springer (1985)Google Scholar
  10. 10.
    Minsky Y.N.: On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds. J.A.M.S. 7, 539–588 (1994)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Mitra M.: Cannon-Thurston maps for hyperbolic group extensions. Topology 37, 527–538 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Mitra M.: Cannon-Thurston maps for trees of hyperbolic metric spaces. J. Differ. Geom. 48, 135–164 (1998)zbMATHGoogle Scholar
  13. 13.
    Mj, M.: Cannon-Thurston maps and bounded geometry. Teichmuller theory and moduli problems no. 10. In: Proceedings of Workshop on Teichmuller Theory at HRI, Allahabad, published by Ramanujan Mathematical Society, arXiv:math.GT/0603729, pp. 489–511 (2009)Google Scholar
  14. 14.
    Mj M.: Cannon-Thurston maps for pared manifolds of bounded geometry. Geom. Topol. 13, 189–245 (2009) arXiv:math.GT/0503581MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Mj M., Reeves L.: A combination theorem for strong relative hyperbolicity. Geom. Topol. 12, 1777–1798 (2008) arXiv:math.GT/0611601MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Mosher L.: A hyperbolic-by-hyperbolic hyperbolic group. Proc. AMS 125, 3447–3455 (1997)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.RKM Vivekananda UniversityBelur MathIndia
  2. 2.Stat-Math Unit, Indian Statistical InstituteKolkataIndia

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