Geometriae Dedicata

, Volume 167, Issue 1, pp 35–67 | Cite as

Limits of limit sets I

Original Paper


We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon–Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometric limit is geometrically finite, the maps on limit sets converge pointwise but not uniformly.


Kleinian group Limit set Geometrically finite Cannon–Thurston map 

Mathematics Subject Classification (2010)

30F40 57M50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agol, I.: Tameness of hyperbolic 3-manifolds. arXiv:math/0405568v1, (2004)Google Scholar
  2. 2.
    Bridson, M., Haefliger, A.: Metric spaces of non-positive curvature. In: Springer Grundlehren 319, Springer, Berlin, (1999)Google Scholar
  3. 3.
    Brock J.: Iteration of mapping classes and limits of hyperbolic 3-manifolds. Inventiones Math. 1043, 523–570 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Calegari D., Gabai D.: Shrinkwrapping and the taming of hyperbolic 3-manifolds. J. Am. Math. Soc. 19, 385–446 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Canary R.: Ends of hyperbolic 3-manifolds. J. Am. Math. Soc. 6, 1–35 (1993)MathSciNetMATHGoogle Scholar
  6. 6.
    Cannon J., Thurston W.P.: Group invariant Peano curves. Geom. Topol. 11, 1315–1355 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Evans, R.: Deformation spaces of hyperbolic 3-manifolds: strong convergence and tameness. Ph.D. Thesis, University of Michigan, (2000)Google Scholar
  8. 8.
    Evans R.: Weakly type-preserving sequences and strong convergence. Geometriae Dedicata 108, 71–92 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fenchel, W., Nielsen, J.: Discontinuous groups of isometries in the hyperbolic plane. In: de Gruyter Studies in Mathematics 319, Berlin (2003)Google Scholar
  10. 10.
    Floyd W.: Group completions and limit sets of Kleinian groups. Inventiones Math. 57, 205–218 (1980)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Francaviglia S.: Constructing equivariant maps for representations. Ann. Inst. Fourier 59, 393–428 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ghys, E., de la Harpe, P., (eds): Sur les groupes hyperboliques d’après Mikhael Gromov. In: Progress in Mathematics, vol. 83. Birkhauser, Boston, (1990)Google Scholar
  13. 13.
    Kerckhoff S., Thurston W.: Non-continuity of the action of the modular group at the Bers’ boundary of Teichmüller Space. Inventiones Math. 100, 25–48 (1990)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kuusalo T.: Boundary mappings of geometric isomorphisms of Fuchsian groups. Ann. Acad. Sci. Fennicae Ser. A Math. 545, 1–7 (1973)Google Scholar
  15. 15.
    Jørgensen T., Marden A.: Algebraic and geometric convergence of Kleinian groups. Math. Scand. 66, 47–72 (1990)MathSciNetGoogle Scholar
  16. 16.
    Mañé R., Sad P., Sullivan D.: On the dynamics of rational maps. Ann. Sci. École. Norm. Sup. 16, 193–217 (1983)MATHGoogle Scholar
  17. 17.
    Marden A.: Outer Circles: An Introduction to Hyperbolic 3-Manifolds. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  18. 18.
    McMullen C.T.: Hausdorff dimension and conformal dynamics I: strong convergence of Kleinian groups. J. Differ. Geom. 51, 471–515 (1999)MathSciNetMATHGoogle Scholar
  19. 19.
    McMullen C.T.: Local connectivity, Kleinian groups and geodesics on the blow-up of the torus. Inventiones Math. 97, 95–127 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Minsky Y.N.: The classification of punctured-torus groups. Ann. Math. 149, 559–626 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Minsky Y.N.: The classification of Kleinian surface groups I: models and bounds. Ann. Math. 171, 1–107 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mitra M.: Cannon–Thurston maps for hyperbolic group extensions. Topology 37, 527–538 (1998)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Miyachi, H.: Moduli of continuity of Cannon–Thurston maps. In: Spaces of Kleinian groups; London Mathematical Society Lecture Notes 329, pp. 121–150. Cambridge University Press, (2006)Google Scholar
  24. 24.
    Mj, M.: Cannon–Thurston Maps for Kleinian Groups. preprint, arXiv:1002.0996, (2010)Google Scholar
  25. 25.
    Mj, M., Series, C.: Limits of limit sets II. In preparationGoogle Scholar
  26. 26.
    Nielsen J.: Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. Acta Math. 50, 189–358 (1927)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Swarup G.A.: Two finiteness properties in 3-manifolds. Bull. Lond. Math. Soc. 12, 296–302 (1980)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Thurston W.: Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Am. Math. Soc. 50, 357–382 (1982)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tukia P.: On isomorphisms of geometrically finite Möbius groups. IHES Publ. 61, 127–140 (1985)Google Scholar
  30. 30.
    Tukia, P.: A remark on a paper by Floyd. In: Holomorphic Functions and Moduli, vol. II; MSRI Publ. 11, pp. 165–172. Springer, New York (1988)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.School of Mathematical SciencesRKM Vivekananda UniversityDt. HowrahIndia
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

Personalised recommendations