##
**Cannon-Thurston maps for surface groups.**
*(English)*
Zbl 1301.57013

In a preprint from 1985, Cannon and Thurston raise the following question: Suppose that a surface group \(\pi_1(S)\) acts freely and properly discontinuously on hyperbolic 3-space \(\mathbb H^3\) by isometries such that the quotient manifold has no accidental parabolics. Does the inclusion of the universal covering of \(S\) into \(\mathbb H^3\) extend continuously to the boundary (the compactifications of universal covers)? Such extensions became known as Cannon-Thurston maps (see also the paper of J. W. Cannon and W. P. Thurston [Geom. Topol. 11, 1315–1355 (2007; Zbl 1136.57009)]). Floyd had proved this for geometrically finite Kleinian groups [W. J. Floyd, Invent. Math. 57, 205–218 (1980; Zbl 0428.20022)], Cannon and Thurston proved it for fibers of closed hyperbolic 3-manifolds fibering over the circle (the case of a doubly degenerate Kleinian surface group where the extension gives a sphere-filling or Peano curve; they also pointed out that for a simply degenerate Kleinian surface group this is equivalent, via the Carathéodory extension theorem, to asking if the limit set is locally connected). In the present substantial and involved paper, this is proved in general, for representations of surface groups into \(\text{PSL}_2(\mathbb C)\) without accidental parabolics; in particular, the set of limit points of the image of the representation is locally connected. As a consequence, the author proves in general that connected limit sets of finitely generated Kleinian groups are locally connected (as he notes, it is a classical fact from general topology that a compact, connected, locally connected metric space is homeomorphic to a Peano continuum, i.e. the continuous image of a closed interval; hence, asking if the limit set is locally connected is equivalent to asking if there is some parametrization by a closed interval). The complexity of the proofs is reflected also by the fact that a first version of the paper was submitted already in 2006. An analogous question for general Kleinian groups is stated as problem 14 in Thurston’s famous and most influential problem list on hyperbolic 3-manifolds and Kleinian groups [W. P. Thurston, Bull. Am. Math. Soc., New Ser. 6, 357–379 (1982; Zbl 0496.57005)]; extending the techniques of the present paper, the author offers a positive answer also for this more general case in a preprint [Cannon-Thurston Maps for Kleinian Groups, arXiv1002.0996].

The proofs of the present paper use the Minsky model for surface groups, crucial also for the recent solution of Thurston’s ending lamination conjecture due to Y. Minsky [Ann. Math. (2) 171, No. 1, 1–107 (2010; Zbl 1193.30063)] and Brock-Canary-Minsky [J. F. Brock et al., Ann. Math. (2) 176, No. 1, 1–149 (2012; Zbl 1253.57009)] (roughly stating that the isometry type of a hyperbolic 3-manifold uniformized by a f.g. Kleinian group is determined by its end invariants).

The proofs of the present paper use the Minsky model for surface groups, crucial also for the recent solution of Thurston’s ending lamination conjecture due to Y. Minsky [Ann. Math. (2) 171, No. 1, 1–107 (2010; Zbl 1193.30063)] and Brock-Canary-Minsky [J. F. Brock et al., Ann. Math. (2) 176, No. 1, 1–149 (2012; Zbl 1253.57009)] (roughly stating that the isometry type of a hyperbolic 3-manifold uniformized by a f.g. Kleinian group is determined by its end invariants).

Reviewer: Bruno Zimmermann (Trieste)

### MSC:

57M50 | General geometric structures on low-dimensional manifolds |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

### References:

[1] | W. Abikoff, ”Two theorems on totally degenerate Kleinian groups,” Amer. J. Math., vol. 98, iss. 1, pp. 109-118, 1976. · Zbl 0334.32023 |

[2] | R. C. Alperin, W. Dicks, and J. Porti, ”The boundary of the Gieseking tree in hyperbolic three-space,” Topology Appl., vol. 93, iss. 3, pp. 219-259, 1999. · Zbl 0926.57008 |

[3] | I. Agol, Tameness of hyperbolic 3-manifolds, 2004. |

[4] | J. W. Anderson and B. Maskit, ”On the local connectivity of limit sets of Kleinian groups,” Complex Variables Theory Appl., vol. 31, iss. 2, pp. 177-183, 1996. · Zbl 0869.30034 |

[5] | J. F. Brock and K. W. Bromberg, ”On the density of geometrically finite Kleinian groups,” Acta Math., vol. 192, iss. 1, pp. 33-93, 2004. · Zbl 1055.57020 |

[6] | J. F. Brock, R. D. Canary, and Y. N. Minsky, ”The classification of Kleinian surface groups, II: The ending lamination conjecture,” Ann. of Math., vol. 176, iss. 1, pp. 1-149, 2012. · Zbl 1253.57009 |

[7] | B. Bielefeld, Conformal dynamics problems list. |

[8] | F. Bonahon, ”Bouts des variétés hyperboliques de dimension \(3\),” Ann. of Math., vol. 124, iss. 1, pp. 71-158, 1986. · Zbl 0671.57008 |

[9] | B. H. Bowditch, ”Relatively hyperbolic groups,” Internat. J. Algebra Comput., vol. 22, iss. 3, p. 1250016, 2012. · Zbl 1259.20052 |

[10] | B. H. Bowditch, ”Stacks of Hyperbolic Spaces and Ends of 3 Manifolds,” in Geometry and Topology Down Under, Hodgson, C. D., Jacon, W. H., Scharlemann, M. G., and Tillmann, S., Eds., Providence, RI: Amer. Math., Soc., 2013. · Zbl 1297.57044 |

[11] | B. H. Bowditch, Model geometries for hyperbolic manifolds, 2005. · Zbl 1081.53032 |

[12] | B. H. Bowditch, ”The Cannon-Thurston map for punctured-surface groups,” Math. Z., vol. 255, iss. 1, pp. 35-76, 2007. · Zbl 1138.57020 |

[13] | J. F. Brock and J. Souto, ”Algebraic limits of geometrically finite manifolds are tame,” Geom. Funct. Anal., vol. 16, iss. 1, pp. 1-39, 2006. · Zbl 1095.30034 |

[14] | R. D. Canary, ”Ends of hyperbolic \(3\)-manifolds,” J. Amer. Math. Soc., vol. 6, iss. 1, pp. 1-35, 1993. · Zbl 0810.57006 |

[15] | R. D. Canary, ”A covering theorem for hyperbolic \(3\)-manifolds and its applications,” Topology, vol. 35, iss. 3, pp. 751-778, 1996. · Zbl 0863.57010 |

[16] | J. W. Cannon and W. Dicks, ”On hyperbolic once-punctured-torus bundles,” in Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I, 2002, pp. 141-183. · Zbl 1015.57501 |

[17] | J. W. Cannon and W. Dicks, ”On hyperbolic once-punctured-torus bundles. II. fractal tessellations of the plane,” Geom. Dedicata, vol. 123, pp. 11-63, 2006. · Zbl 1119.57007 |

[18] | M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et Théorie des Groupes, New York: Springer-Verlag, 1990, vol. 1441. · Zbl 0727.20018 |

[19] | R. D. Canary and Y. N. Minsky, ”On limits of tame hyperbolic \(3\)-manifolds,” J. Differential Geom., vol. 43, iss. 1, pp. 1-41, 1996. · Zbl 0856.57011 |

[20] | J. W. Cannon and W. P. Thurston, Group invariant Peano curves, 1985. · Zbl 1136.57009 |

[21] | J. W. Cannon and W. P. Thurston, ”Group invariant Peano curves,” Geom. Topol., vol. 11, pp. 1315-1355, 2007. · Zbl 1136.57009 |

[22] | M. Mj and S. Das, Semiconjugacies between relatively hyperbolic boundaries, 2010. · Zbl 06605290 |

[23] | C. E. Fan, Injectivity radius bounds in hyperbolic convex cores, ProQuest LLC, Ann Arbor, MI, 1997. |

[24] | C. E. Fan, Injectivity radius bounds in hyperbolic I-bundle convex cores, 1999. |

[25] | B. Farb, ”Relatively hyperbolic groups,” Geom. Funct. Anal., vol. 8, iss. 5, pp. 810-840, 1998. · Zbl 0985.20027 |

[26] | W. J. Floyd, ”Group completions and limit sets of Kleinian groups,” Invent. Math., vol. 57, iss. 3, pp. 205-218, 1980. · Zbl 0428.20022 |

[27] | Sur les Groupes Hyperboliques d’après Mikhael Gromov, Ghys, E. and de la Harpe, P., Eds., Boston, MA: Birkhäuser, 1990, vol. 83. · Zbl 0731.20025 |

[28] | M. Gromov, ”Hyperbolic groups,” in Essays in Group Theory, New York: Springer-Verlag, 1987, vol. 8, pp. 75-263. · Zbl 0634.20015 |

[29] | C. D. Hodgson and S. P. Kerckhoff, ”Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery,” J. Differential Geom., vol. 48, iss. 1, pp. 1-59, 1998. · Zbl 0919.57009 |

[30] | C. D. Hodgson and S. P. Kerckhoff, ”Universal bounds for hyperbolic Dehn surgery,” Ann. of Math., vol. 162, iss. 1, pp. 367-421, 2005. · Zbl 1087.57011 |

[31] | J. G. Hocking and G. S. Young, Topology, Reading, MA: Addison-Wesley Publishing Co., 1961. · Zbl 0135.22701 |

[32] | E. Klarreich, ”Semiconjugacies between Kleinian group actions on the Riemann sphere,” Amer. J. Math., vol. 121, iss. 5, pp. 1031-1078, 1999. · Zbl 1011.30035 |

[33] | C. T. McMullen, ”Local connectivity, Kleinian groups and geodesics on the blowup of the torus,” Invent. Math., vol. 146, iss. 1, pp. 35-91, 2001. · Zbl 1061.37025 |

[34] | Y. N. Minsky, ”On rigidity, limit sets, and end invariants of hyperbolic \(3\)-manifolds,” J. Amer. Math. Soc., vol. 7, iss. 3, pp. 539-588, 1994. · Zbl 0808.30027 |

[35] | Y. N. Minsky, ”The classification of Kleinian surface groups. I. Models and bounds,” Ann. of Math., vol. 171, iss. 1, pp. 1-107, 2010. · Zbl 1193.30063 |

[36] | M. Mitra, ”Cannon-Thurston maps for hyperbolic group extensions,” Topology, vol. 37, iss. 3, pp. 527-538, 1998. · Zbl 0907.20038 |

[37] | M. Mitra, ”Cannon-Thurston maps for trees of hyperbolic metric spaces,” J. Differential Geom., vol. 48, iss. 1, pp. 135-164, 1998. · Zbl 0906.20023 |

[38] | H. Miyachi, Semiconjugacies between actions of topologically tame Kleinian groups, 2002. |

[39] | M. Mj, Cannon-Thurston maps for surface groups I: amalgamation geometry and split geometry, 2005. · Zbl 1360.57007 |

[40] | M. Mj, Cannon-Thurston maps for surface groups II: Split geometry and the Minsky model, 2006, · Zbl 1360.57007 |

[41] | M. Mj, Ending laminations and Cannon-Thurston maps, 2007. · Zbl 1297.57040 |

[42] | M. Mj, ”Cannon-Thurston maps for pared manifolds of bounded geometry,” Geom. Topol., vol. 13, iss. 1, pp. 189-245, 2009. · Zbl 1166.57009 |

[43] | M. Mj, ”Cannon-Thurston maps and bounded geometry,” in Teichmüller Theory and Moduli Problem, Mysore: Ramanujan Math. Soc., 2010, vol. 10, pp. 489-511. · Zbl 1204.57014 |

[44] | M. Mj, Cannon-Thurston Maps for Kleinian groups, 2010. · Zbl 1301.57013 |

[45] | M. Mj, ”Cannon-Thurston maps, \(i\)-bounded geometry and a theorem of McMullen,” in Actes du Séminaire de Théorie Spectrale et Géometrie. Volume 28. Année 2009-2010, Saint: Univ. Grenoble I, 2011, vol. 28, pp. 63-107. · Zbl 1237.57018 |

[46] | H. A. Masur and Y. N. Minsky, ”Geometry of the complex of curves. I. Hyperbolicity,” Invent. Math., vol. 138, iss. 1, pp. 103-149, 1999. · Zbl 0941.32012 |

[47] | H. A. Masur and Y. N. Minsky, ”Geometry of the complex of curves. II. Hierarchical structure,” Geom. Funct. Anal., vol. 10, iss. 4, pp. 902-974, 2000. · Zbl 0972.32011 |

[48] | M. Mj and A. Pal, ”Relative hyperbolicity, trees of spaces and Cannon-Thurston maps,” Geom. Dedicata, vol. 151, pp. 59-78, 2011. · Zbl 1222.57013 |

[49] | M. Mj and L. Reeves, ”A combination theorem for strong relative hyperbolicity,” Geom. Topol., vol. 12, iss. 3, pp. 1777-1798, 2008. · Zbl 1192.20027 |

[50] | M. Mj and C. Series, Limits of limit sets II, 2013. · Zbl 1287.30007 |

[51] | J. Souto, Cannon-Thurston maps for thick free groups, 2006. |

[52] | R. D. Canary, D. B. A. Epstein, and P. Green, ”Notes on notes of Thurston,” in Analytical and Geometric Aspects of Hyperbolic Space, Cambridge: Cambridge Univ. Press, 1987, vol. 111, pp. 3-92. · Zbl 0612.57009 |

[53] | W. P. Thurston, ”Three-dimensional manifolds, Kleinian groups and hyperbolic geometry,” Bull. Amer. Math. Soc., vol. 6, iss. 3, pp. 357-381, 1982. · Zbl 0496.57005 |

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