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CANNON–THURSTON MAPS FOR KLEINIAN GROUPS

Part of: Low-dimensional topology Special aspects of infinite or finite groups Lie groups

Published online by Cambridge University Press:  22 May 2017

MAHAN MJ
MAHAN MJ*
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India; [email protected], [email protected]

Abstract

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We show that Cannon–Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon–Thurston maps for surface groups, we show that Cannon–Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon–Thurston maps for degenerate free groups without parabolics correspond to endpoints of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon–Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.

MSC classification

Primary: 57M50: Geometric structures on low-dimensional manifolds 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 20F65: Geometric group theory 22E40: Discrete subgroups of Lie groups
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 5 , 2017 , e1
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2017

References

Agol, I., ‘Tameness of hyperbolic 3-manifolds’. Preprint, 2004, arXiv:math.GT/0405568.Google Scholar
Brock, J. and Bromberg, K., ‘Density of geometrically finite Kleinian groups’, Acta Math. 192 (2004), 33–93.Google Scholar
Brock, J. and Bromberg, K., ‘Geometric inflexibility and 3-manifolds that fiber over the circle’, J. Topol. 4(1) (2011), 1–38.Google Scholar
Brock, J. and Bromberg, K., ‘Geometric inflexibility of hyperbolic cone-manifolds’. Preprint, 2014, Proceedings of the 2014 MSJ-SI, arXiv:1412.4635, to appear.Google Scholar
Brock, J., Bromberg, K., Evans, R. and Souto, J., ‘Tameness on the boundary and Ahlfors’ measure conjecture’, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 145–166.Google Scholar
Brock, J. F., Canary, R. D. and Minsky, Y. N., ‘The classification of Kleinian surface groups II: the ending lamination conjecture’, Ann. of Math. (2) 176(1) (2012), 1–149.Google Scholar
Brock, J. F., Canary, R. D. and Minsky, Y. N., ‘The classification of finitely generated Kleinian groups’, in preparation, 2014.Google Scholar
Bonahon, F., ‘Bouts de varietes hyperboliques de dimension 3’, Ann. of Math. (2) 124 (1986), 71–158.Google Scholar
Bowditch, B. H., ‘Model geometries for hyperbolic manifolds’. Preprint, Southampton, 2005.Google Scholar
Bowditch, B. H., ‘The ending lamination theorem’. Preprint, Warwick, 2011.Google Scholar
Bowditch, B. H., ‘Relatively hyperbolic groups’, Internat. J. Algebra Comput. 22(3) (2012), 1250016, 66 pp.Google Scholar
Bowditch, B. H., Geometry and Topology Down Under, Contemp. Math., 597 (American Mathematical Society, Providence, RI, 2013), 65–138.Google Scholar
Bromberg, K., ‘Projective structures with degenerate holonomy and the Bers density conjecture’, Ann. of Math. (2) 166(1) (2007), 77–93.Google Scholar
Canary, R. D., ‘Ends of hyperbolic 3 manifolds’, J. Amer. Math. Soc. 6 (1993), 1–35.Google Scholar
Cannon, J. and Thurston, W. P., Group invariant Peano Curves. Preprint, Princeton, 1985.Google Scholar
Cannon, J. and Thurston, W. P., ‘Group invariant Peano curves’, Geom. Topol. 11 (2007), 1315–1356.Google Scholar
Das, S. and Mj, M., ‘Semiconjugacies between relatively hyperbolic boundaries’, Groups Geom. Dyn. 10(2) (2016), 733–752.Google Scholar
Farb, B., ‘Relatively hyperbolic groups’, Geom. Funct. Anal. 8 (1998), 810–840.Google Scholar
Floyd, W. J., ‘Group completions and limit sets of Kleinian groups’, Invent. Math. 57 (1980), 205–218.Google Scholar
Calegari, D. and Gabai, D., ‘Shrink-wrapping and the taming of hyperbolic 3-manifolds’, J. Amer. Math. Soc. 19(2) (2006), 385–446.Google Scholar
Hempel, J., 3 Manifolds, (Princeton University Press, 1976).Google Scholar
Hubbard, J. H., ‘Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz’, inTopological Methods in Modern Mathematics (eds. Goldberg, L. R. and Philips, A. G.) (Publish or Perish Inc., Houston, TX, 1993), 467–511.Google Scholar
Jeon, W., Kim, I., Lecuire, C. and Ohshika, K., ‘Primitive stable representations of free Kleinian groups’, Israel J. Math. 199(2) (2014), 841–866.Google Scholar
Klarreich, E., ‘Semiconjugacies between Kleinian group actions on the Riemann sphere’, Amer. J. Math 121 (1999), 1031–1078.Google Scholar
Kleineidam, G. and Souto, J., ‘Ending laminations in the masur domain’, inKleinian Groups and Hyperbolic 3-Manifolds (Warwick, 2001), (eds. Komori, Y., Markovic, V. and Series, C.) London Math. Soc. Lecture Notes, 299 (Cambridge University Press, Cambridge, 2003), 105–129.Google Scholar
Leininger, C. J., Long, D. D. and Reid, A. W., ‘Commensurators of non-free finitely generated Kleinian groups’, Algebr. Geom. Topol. 11 (2011), 605–624.Google Scholar
Luft, E., ‘Actions of the homeotopy group of an orientable 3-dimensional handlebody’, Math. Ann. 234(3) (1978), 279–292.Google Scholar
McMullen, C. T., Renormalization and 3-manifolds which Fiber over the Circle, (Princeton University Press, Princeton, NJ, 1998).Google Scholar
McMullen, C. T., ‘Local connectivity, Kleinian groups and geodesics on the blow-up of the torus’, Invent. Math. 97 (2001), 95–127.Google Scholar
Milnor, J., ‘Local connectivity of Julia sets: expository lectures’, inThe Mandelbrot Set, Theme and Variations (ed. Lei, T.) (Cambridge University Press, Cambridge, 2000), 67–116.Google Scholar
Minsky, Y. N., ‘The classification of Kleinian surface groups I: models and bounds’, Ann. of Math. (2) 171 (2010), 1–107.Google Scholar
Minsky, Y. N., ‘On dynamics of Out(F n ) on PSL 2(C) characters’, Israel J. Math. 193(1) (2013), 47–70.Google Scholar
Mitra, M., ‘Cannon–Thurston maps for trees of hyperbolic metric spaces’, J. Differential Geom. 48 (1998), 135–164.Google Scholar
Miyachi, H., ‘Semiconjugacies between actions of topologically tame Kleinian groups’. Preprint, 2002.Google Scholar
Mj, M., ‘On discreteness of commensurators’, Geom. and Topol. 15 (2011), 331–350.Google Scholar
Mj, M., ‘Cannon–Thurston maps for surface groups’, Ann. of Math. (2) 179(1) (2014), 1–80.Google Scholar
Mj, M., ‘Ending laminations and Cannon–Thurston Maps, with an appendix by S. Das and M. Mj’, Geom. Funct. Anal. 24 (2014), 297–321.Google Scholar
McCullough, D. and Miller, A., ‘Homeomorphisms of 3-manifolds with compressible boundary’, Mem. Amer. Math. Soc. 61(344) (1986), xii+100 pp.Google Scholar
Mosher, L., ‘Stable Teichmuller quasigeodesics and ending laminations’, Geom. Topol. 7 (2003), 33–90.Google Scholar
McCarthy, J. D. and Papadopoulos, A., ‘Dynamics on Thurston’s sphere of projective measured foliations’, Comment. Math. Helv. 64 (1989), 133–166.Google Scholar
Mj, M. and Pal, A., ‘Relative hyperbolicity, trees of spaces and Cannon–Thurston maps’, Geom. Dedicata 151 (2011), 59–78.Google Scholar
Ohshika, K., ‘Rigidity and topological conjugates of topologically tame Kleinian groups’, Trans. Amer. Math. Soc. 350(10) (1998), 3989–4022.Google Scholar
Otal, J. P., ‘Courants geodesiques et produit libres’, These d’Etat, Universit Paris-Sud, Orsay, 1988.Google Scholar
Otal, J. P., ‘Sur le nouage des geodesiques dans les varietes hyperboliques’, C. R. Acad. Sci. Paris Ser. I Math. 320(7) (1995), 847–852.Google Scholar
Otal, J. P., ‘Les geodesiques fermees d’une variete hyperbolique en tant que noeuds’, inKleinian Groups and Hyperbolic 3-Manifolds (London Math. Soc., Orsay, 2003).Google Scholar
Souto, J., ‘Cannon–Thurston maps for thick free groups’. Preprint, 2006.Google Scholar
Souto, J., ‘Short geodesics in hyperbolic compression bodies are not knotted’. Preprint, 2008.Google Scholar
Sullivan, D., ‘Conformal dynamical systems’, inGeometric Dynamics, Lecture Notes in Mathematics, 1007 (Springer, Berlin, 1983), 725–752.Google Scholar
Sullivan, D., ‘Quasiconformal homeomorphisms and dynamics I: solution of the Fatou–Julia problem on wandering domains’, Ann. of Math. (2) 122 (1985), 401–418.Google Scholar
Suzuki, S., ‘On homeomorphisms of a 3-dimensional handlebody’, Canad. J. Math. 29(1) (1977), 111–124.Google Scholar
Thurston, W. P., The Geometry and Topology of 3-Manifolds (Princeton University Notes, Princeton, NJ, 1980).Google Scholar
Thurston, W. P., ‘Three dimensional manifolds, Kleinian groups and hyperbolic geometry’, Bull. Amer. Math. Soc. 6 (1982), 357–382.Google Scholar