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


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.


Relative hyperbolicity Cannon-Thurston maps Trees of spaces 

Mathematics Subject Classification (2000)

20F32(Primary) 57M50(Secondary) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Personalised recommendations