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Cannon–Thurston maps for hyperbolic free group extensions

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Abstract

This paper gives a detailed analysis of the Cannon–Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free group of finite rank at least 3 and consider a convex cocompact subgroup Γ ≤ Out(F), i.e. one for which the orbit map from Γ into the free factor complex of F is a quasi-isometric embedding. The subgroup Γ determines an extension E Γ of F, and the main theorem of Dowdall–Taylor [DT14] states that in this situation E Γ is hyperbolic if and only if Γ is purely atoroidal.

Here, we give an explicit geometric description of the Cannon–Thurston maps ∂F → ∂E Γ for these hyperbolic free group extensions, the existence of which follows from a general result of Mitra. In particular, we obtain a uniform bound on the multiplicity of the Cannon–Thurston map, showing that this map has multiplicity at most 2 rank(F). This theorem generalizes the main result of Kapovich and Lustig [KL15] which treats the special case where Γ is infinite cyclic. We also answer a question of Mahan Mitra by producing an explicit example of a hyperbolic free group extension for which the natural map from the boundary of Γ to the space of laminations of the free group (with the Chabauty topology) is not continuous.

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References

  1. M. Bestvina and M. Feighn, Outer limits, Preprint 1994; http://andromeda.rutgers.edu/feighn/papers/outer.pdf, 2013.

    Google Scholar 

  2. M. Bestvina and M. Feighn, Hyperbolicity of the complex of free factors, Adv. Math. 256 (2014), 104–155.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Bestvina, M. Feighn and M. Handel, Laminations,trees,and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997), 215–244.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Bestvina and M. Handel, Train tracks and automorphisms of free groups, Ann. of Math. (1992), 1–51.

    Google Scholar 

  5. B. H. Bowditch, The Cannon-Thurston map for punctured-surface groups, Math. Z. 255 (2007), 35–76.

    Article  MathSciNet  MATH  Google Scholar 

  6. B. H. Bowditch, Stacks of hyperbolic spaces and ends of 3-manifolds, in Geometry and topology down under, Contemp. Math., Vol. 597, Amer. Math. Soc., Providence, RI, 2013, pp. 65–138.

    Chapter  Google Scholar 

  7. O. Baker and T. R. Riley, Cannon-Thurston maps do not always exist, Forum Math. Sigma 1 (2013), e3, 11.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Bestvina and P. Reynolds, The boundary of the complex of free factors, Duke Math. J. 164 (2015), 2213–2251.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Brinkmann, Hyperbolic automorphisms of free groups, Geom. Funct. Anal. 10 (2000), 1071–1089.

    Article  MathSciNet  MATH  Google Scholar 

  10. T. Coulbois and A. Hilion, Botany of irreducible automorphisms of free groups, Pacific J. Math. 256 (2012), 291–307.

    Article  MathSciNet  MATH  Google Scholar 

  11. T. Coulbois and A. Hilion, Rips induction: index of the dual lamination of an R-tree, Groups Geom. Dyn. 8 (2014), 97–134.

    Article  MathSciNet  MATH  Google Scholar 

  12. T. Coulbois, A. Hilion and M. Lustig, Which R-trees can be mapped continuously to a current?, Draft preprint, 2006.

    Google Scholar 

  13. T. Coulbois, A. Hilion and M. Lustig, Non-unique ergodicity, observers’ topology and the dual algebraic lamination for R-trees, Illinois J. Math. 51 (2007), 897–911.

    MathSciNet  MATH  Google Scholar 

  14. T. Coulbois, A. Hilion and M. Lustig, R-trees and laminations for free groups. I. Algebraic laminations, J. Lond. Math. Soc. (2) 78 (2008), 723–736.

    Article  MathSciNet  MATH  Google Scholar 

  15. T. Coulbois, A. Hilion and M. Lustig, R-trees and laminations for free groups. II. The dual lamination of an R-tree, J. Lond. Math. Soc. (2) 78 (2008), 737–754.

    Article  MathSciNet  MATH  Google Scholar 

  16. T. Coulbois, A. Hilion and P. Reynolds, Indecomposable FN-trees and minimal laminations, Groups Geom. Dyn. 9 (2015), 567–597.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. M. Cohen and M. Lustig, Very small group actions on R-trees and Dehn twist automorphisms, Topology 34 (1995), 575–617.

    Article  MathSciNet  MATH  Google Scholar 

  18. R. D. Canary, A. Marden and D. Epstein, Fundamentals of hyperbolic manifolds: Selected expositions, Vol. 328, Cambridge University Press, 2006.

    Book  MATH  Google Scholar 

  19. J. W. Cannon and W. P. Thurston, Group invariant Peano curves, Geom. Topol. 11 (2007), 1315–1355.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), 91–119.

    Article  MathSciNet  MATH  Google Scholar 

  21. S. Dowdall and S. J. Taylor, Hyperbolic extensions of free groups, Preprint arXiv:1406.2567, 2014.

    Google Scholar 

  22. S. Dowdall and S. J. Taylor, The co-surface graph and the geometry of hyperbolic free group extensions, Preprint arXiv:1601.00101, 2016.

    Google Scholar 

  23. M. Feighn and M. Handel, The recognition theorem for Out(Fn), Groups Geom. Dyn. 5 (2011), 39–106.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Francaviglia and A. Martino, Metric properties of outer space, Publ. Mat. 55 (2011), 433–473.

    Article  MathSciNet  MATH  Google Scholar 

  25. V. Gerasimov, Floyd maps for relatively hyperbolic groups, Geom. Funct. Anal. 22 (2012), 1361–1399.

    Article  MathSciNet  MATH  Google Scholar 

  26. V. Guirardel, Approximations of stable actions on R-trees, Comment. Math. Helv. 73 (1998), 89–121.

    Article  MathSciNet  MATH  Google Scholar 

  27. V. Guirardel, Actions of finitely generated groups on R-trees, Ann. Inst. Fourier (Grenoble) 58 (2008), 159–211.

    Article  MathSciNet  MATH  Google Scholar 

  28. U. Hamenstädt, The boundary of the free factor graph and the free splitting graph, Preprint arXiv:1211.1630, 2012.

    Google Scholar 

  29. U. Hamenstädt and S. Hensel, Convex cocompact subgroups of Out(Fn), Preprint arXiv:1411.2281, 2014.

    Google Scholar 

  30. M. Handel and L. Mosher, Axes in outer space., Vol. 213, Mem. Amer. Math. Soc., 2011.

    MATH  Google Scholar 

  31. A. Hatcher and K. Vogtmann, The complex of free factors of a free group, Quart. J. Math. 49 (1998), 459–468.

    Article  MathSciNet  MATH  Google Scholar 

  32. W. Jeon, I. Kapovich, C. Leininger and K. Ohshika, Conical limit points and the Cannon-Thurston map, Conform. Geom. Dyn. 20 (2016), 58–80.

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Jäger and M. Lustig, Free group automorphisms with many fixed points at infinity, Geometry & Topology Monographs 14 (2008), 321–333.

    Article  MathSciNet  MATH  Google Scholar 

  34. I. Kapovich and M. Lustig, Geometric intersection number and analogues of the curve complex for free groups, Geom. Topol. 13 (2009), 1805–1833.

    Article  MathSciNet  MATH  Google Scholar 

  35. I. Kapovich and M. Lustig, Ping-pong and Outer space, J. Topol. Anal. 2 (2010), 173–201.

    Article  MathSciNet  MATH  Google Scholar 

  36. I. Kapovich and M. Lustig, Stabilizers of R-trees with free isometric actions of FN, J. Group Theory 14 (2011), 673–694.

    Article  MathSciNet  MATH  Google Scholar 

  37. I. Kapovich and M. Lustig, Invariant laminations for irreducible automorphisms of free groups, Q. J. Math. 65 (2014), 1241–1275.

    Article  MathSciNet  MATH  Google Scholar 

  38. I. Kapovich and M. Lustig, Cannon–Thurston fibers for iwip automorphisms of FN, J. Lond. Math. Soc. (2) 91 (2015), 203–224.

    Article  MathSciNet  MATH  Google Scholar 

  39. E. Klarreich, Semiconjugacies between Kleinian group actions on the Riemann sphere, Amer. J. Math. 121 (1999), 1031–1078.

    Article  MathSciNet  MATH  Google Scholar 

  40. G. Levitt, Automorphisms of hyperbolic groups and graphs of groups, Geom. Dedicata 114 (2005), 49–70.

    Article  MathSciNet  MATH  Google Scholar 

  41. C. Leininger, D. D. Long and A. W. Reid, Commensurators of finitely generated nonfree Kleinian groups, Algebr. Geom. Topol. 11 (2011), 605–624.

    Article  MathSciNet  MATH  Google Scholar 

  42. C. J. Leininger, M. Mj and S. Schleimer, The universal Cannon-Thurston map and the boundary of the curve complex, Comment. Math. Helv. 86 (2011), 769–816.

    Article  MathSciNet  MATH  Google Scholar 

  43. B. Mann, Some hyperbolic Out(FN)-graphs and nonunique ergodicity of very small FN-trees, ProQuest LLC, Ann Arbor, MI, 2014, Thesis (Ph.D.)–The University of Utah.

    Google Scholar 

  44. C. T. McMullen, Local connectivity, Kleinian groups and geodesics on the blowup of the torus, Invent. Math. 146 (2001), 35–91.

    Article  MathSciNet  MATH  Google Scholar 

  45. M. Mitra, Ending laminations for hyperbolic group extensions, Geom. Funct. Anal. 7 (1997), 379–402.

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Mitra, Cannon-Thurston maps for hyperbolic group extensions, Topology 37 (1998), 527–538.

    Article  MathSciNet  MATH  Google Scholar 

  47. M. Mitra, Cannon-Thurston maps for trees of hyperbolic metric spaces, J. Differential Geom. 48 (1998), 135–164.

    MathSciNet  MATH  Google Scholar 

  48. M. Mitra, Coarse extrinsic geometry: a survey, in The Epstein birthday schrift, Geom. Topol. Monogr., Vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 341–364 (electronic).

    Google Scholar 

  49. H. Miyachi, Moduli of continuity of Cannon-Thurston maps, in Spaces of Kleinian groups, London Math. Soc. Lecture Note Ser., Vol. 329, Cambridge Univ. Press, Cambridge, 2006, pp. 121–149.

    Chapter  Google Scholar 

  50. M. Mj, Cannon-Thurston maps for surface groups, Ann. of Math. (2) 179 (2014), 1–80.

    Article  MathSciNet  MATH  Google Scholar 

  51. M. Mj, Ending laminations and Cannon-Thurston maps, Geom. Funct. Anal. 24 (2014), 297–321, With an appendix by Shubhabrata Das and Mj.

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Mj and A. Pal, Relative hyperbolicity, trees of spaces and Cannon-Thurston maps, Geom. Dedicata 151 (2011), 59–78.

    Article  MathSciNet  MATH  Google Scholar 

  53. B. Mann and P. Reynolds, In preparation.

  54. M. Mj and K. Rafi, Algebraic ending laminations and quasiconvexity, Preprint arXiv:1506.08036v2, 2015.

    Google Scholar 

  55. H. Namazi, A. Pettet and P. Reynolds, Ergodic decompositions for folding and unfolding paths in outer space, Preprint arXiv:1410.8870, 2014.

    Google Scholar 

  56. F. Paulin, The Gromov topology on R-trees, Topology Appl. 32 (1989), 197–221.

    Article  MathSciNet  MATH  Google Scholar 

  57. C. Pfaff, Out(F3) Index Realization, Math. Proc. Cambridge Philos. Soc. 159 (2015), 445–458.

    Article  MathSciNet  Google Scholar 

  58. P. Reynolds, Reducing systems for very small trees, Preprint arXiv:1211.3378, 2012.

    Google Scholar 

  59. E. Rips and Z. Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. (2) 146 (1997), 53–109.

    Article  MathSciNet  MATH  Google Scholar 

  60. S. J. Taylor and G. Tiozzo, Random extensions of free groups and surface groups are hyperbolic, Int. Math. Res. Notices (2016), 294–310.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN, 37240, USA

    Spencer Dowdall

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801, USA

    Ilya Kapovich

  3. Department of Mathematics, Yale University, 10 Hillhouse Ave, New Haven, CT, 06520, USA

    Samuel J. Taylor

Authors
  1. Spencer Dowdall
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  2. Ilya Kapovich
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  3. Samuel J. Taylor
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Correspondence to Spencer Dowdall.

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Dowdall, S., Kapovich, I. & Taylor, S.J. Cannon–Thurston maps for hyperbolic free group extensions. Isr. J. Math. 216, 753–797 (2016). https://doi.org/10.1007/s11856-016-1426-2

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  • DOI: https://doi.org/10.1007/s11856-016-1426-2

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