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Geometric and Functional Analysis

, Volume 24, Issue 1, pp 297–321 | Cite as

Ending Laminations and Cannon–Thurston Maps

  • Mahan Mj
Article

Abstract

In earlier work, we had shown that Cannon–Thurston maps exist for Kleinian surface groups without accidental parabolics. In this paper we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon–Thurston map is not one-to-one.

Mathematics Subject Classification (1991)

57M50 20F67 (Primary) 20F65 22E40 (Secondary) 

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References

  1. Ago04.
    I. Agol. Tameness of hyperbolic 3-manifolds. preprint, arXiv:math.GT/0405568, 2004.
  2. BCM12.
    J.F. Brock, R.D. Canary and Y.N. Minsky. The Classification of Kleinian surface groups II: The Ending Lamination Conjecture. Ann. of Math., (1)176, (2012), 1–149 (arXiv:math/0412006).
  3. Bon86.
    F. Bonahon. Bouts de varietes hyperboliques de dimension 3. Ann. of Math., (2)124, (1986), 71–158.Google Scholar
  4. Bow07.
    Bowditch B.H.: The Cannon–Thurston map for punctured surface groups. Math. Z., 255, 35–76 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. Can93.
    R.D. Canary. Ends of hyperbolic 3 manifolds. J. Amer. Math. Soc., (1993), 1–35.Google Scholar
  6. CB87.
    A. Casson and S. Bleiler. Automorphisms of Surfaces after Nielsen and Thurston. London Math. Soc. Student Texts, Cambridge (1987).Google Scholar
  7. CEG87.
    R.D. Canary, D.B.A. Epstein and P. Green. Notes on Notes of Thurston. in Analytical and Geometric Aspects of Hyperbolic Spaces, (1987), 3–92.Google Scholar
  8. CG06.
    D. Calegari and D. Gabai. Shrink-wrapping and the Taming of Hyperbolic 3-manifolds. J. Amer. Math. Soc., (2)19 (2006) 385–446.Google Scholar
  9. CT85.
    J. Cannon and W. P. Thurston. Group Invariant Peano Curves. preprint, Princeton (1985).Google Scholar
  10. CT07.
    J. Cannon and W.P. Thurston. Group Invariant Peano Curves. Geom. Topol, 11, (2007), 1315–1355.Google Scholar
  11. Far98.
    Farb B: Relatively hyperbolic groups. Geom. Funct. Anal., 8, 810–840 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  12. LLR11.
    C.J. Leininger, D.D. Long and A.W. Reid. Commensurators of non-free finitely generated Kleinian groups. Algebr. Geom. Topol., 11 arXiv:math/0908.2272, (2011), 605–624.
  13. McM01.
    McMullen C.T.: Local connectivity, Kleinian groups and geodesics on the blow-up of the torus. Invent. math., 97, 95–127 (2001)CrossRefMathSciNetGoogle Scholar
  14. Min94.
    Y.N. Minsky. On Rigidity, Limit Sets, and End Invariants of Hyperbolic 3-Manifolds. J. Amer. Math. Soc., 7, (1994),539–588.Google Scholar
  15. Min10.
    Y.N. Minsky. The Classification of Kleinian surface groups I: Models and bounds. Ann. of Math., (1)171, (2010), 1–107.Google Scholar
  16. Mit97.
    M. Mitra. Ending Laminations for Hyperbolic Group Extensions. Geom. Funct. Anal., 7, (1997), 379–402.Google Scholar
  17. Mit98.
    M.Mitra. Cannon–Thurston Maps for Hyperbolic Group Extensions. Topology, 37, (1998), 527–538.Google Scholar
  18. Mj11.
    M. Mj. Cannon–Thurston Maps for Surface Groups: An Exposition of Amalgamation Geometry and Split Geometry. preprint, arXiv:math.GT/0512539, (2005).
  19. Mj10.
    M. Mj. Cannon–Thurston Maps for Kleinian Groups. preprint, arXiv:math/1002.0996, (2010).
  20. Mj11.
    M.Mj. On Discreteness of Commensurators. Geom. Topol., 15, arXiv:math.GT/0607509, (2011), 331–350.
  21. Mj14.
    M. Mj. Cannon–Thurston Maps for Surface Groups. Ann. of Math., (1)179, (2014), 1–80.Google Scholar
  22. MP11.
    M. Mj and A. Pal. Relative Hyperbolicity, Trees of Spaces and Cannon–Thurston Maps. Geom. Dedicata, 151, arXiv:0708.3578, (2011), 59–78.
  23. PH92.
    R. Penner and J. Harer. Combinatorics of train tracks. Ann. Math. Studies, 125, Princeton University Press (1992).Google Scholar
  24. Sha91.
    P. B. Shalen. Dendrology and its applications. In: Group Theory from a Geometrical Viewpoint, (E. Ghys, A. Haefliger, A. Verjovsky eds.) (1991) pp. 543–616.Google Scholar
  25. Thu80.
    W. P. Thurston. The Geometry and Topology of 3-Manifolds. Princeton University Notes, (1980).Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.RKM Vivekananda UniversityBelur MathIndia

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