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CANNON–THURSTON MAPS DO NOT ALWAYS EXIST

Part of:Special aspects of infinite or finite groups

Published online by Cambridge University Press:  11 September 2013

O. BAKER and
T. R. RILEY
O. BAKER
Affiliation:
Department of Mathematics & Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, L8S 4K1, Canada
T. R. RILEY
Affiliation:
Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, NY 14853, USA tim.riley@math.cornell.edu

Abstract

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We construct a hyperbolic group with a hyperbolic subgroup for which inclusion does not induce a continuous map of the boundaries.

MSC classification

Secondary:20F67: Hyperbolic groups and nonpositively curved groups

Information

Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 1 , 2013, e3
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2013.

References

Baker, O. and Riley, T. R., Cannon–Thurston maps for hyperbolic hydra. arXiv:1209.0815, (2012).Google Scholar
Barnard, J., Brady, N. and Dani, P., ‘Super-exponential distortion of subgroups of $\mathrm{CAT} (- 1)$ groups’, Algebr. Geom. Topol. 7 (2007), 301–308.Google Scholar
Bestvina, M., Questions in geometric group theory. http://www.math.utah.edu/~bestvina/.Google Scholar
Bonahon, F., ‘Bouts des variétés hyperboliques de dimension 3’, Ann. of Math. (2) 124(1) (1986), 71–158.Google Scholar
Bonahon, F., ‘Geodesic currents on negatively curved groups’, in Arboreal Group Theory (Berkeley, CA, 1988), Math. Sci. Res. Inst. Publ., 19 (Springer, New York, 1991), 143–168.CrossRefGoogle Scholar
Bridson, M. R. and Haefliger, A., Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften, 319 (Springer, 1999).Google Scholar
Cannon, J. W. and Thurston, W. P., ‘Group invariant Peano curves’, Geom. Topol. 11 (2007), 1315–1355.Google Scholar
Floyd, W. J., ‘Group completions and limit sets of Kleinian groups’, Invent. Math. 57(3) (1980), 205–218.Google Scholar
Gerasimov, V. and Potyagailo, L., Similar relatively hyperbolic actions of a group. arXiv:1305.6649, (2013).Google Scholar
Gersten, S. M., ‘Introduction to hyperbolic and automatic groups’, in Summer School in Group Theory, Banff, 1996, CRM Proc. Lecture Notes, 17 (American Mathematical Society, Providence, RI, 1999), 45–70.Google Scholar
Ghys, E. and de la Harpe, P.  (eds.) Sur les Groups Hyperbolic d’après Mikhael Gromov, Progress in Mathematics, 83 (Birkhäuser, 1990).Google Scholar
Gromov, M., ‘Hyperbolic groups’, in Essays in Group Theory, MSRI Publications, 8 (ed. Gersten, S. M.) (Springer, 1987), 75–263.Google Scholar
Kapovich, I., ‘The combination theorem and quasiconvexity’, Internat. J. Algebra Comput. 11(2) (2001), 185–216.CrossRefGoogle Scholar
Kapovich, I. and Benakli, N., ‘Boundaries of hyperbolic groups’, in Combinatorial and Geometric Group Theory (New York, 2000/Hoboken, NJ, 2001), Contemporary Mathematics, 296 (American Mathematical Society, Providence, RI, 2002), 39–93.Google Scholar
Kapovich, M., Problems on boundaries of groups and Kleinian groups, collected at a 2005 American Institute of Mathematics workshop. http://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.Google Scholar
Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory, Classics in Mathematics (Springer, Berlin, 2001), Reprint of the 1977 edition.Google Scholar
Matsuda, Y. and Oguni, S., On Cannon–Thurston maps for relatively hyperbolic groups. arXiv:1206.5868, (2012).Google Scholar
McMullen, C. T., ‘Local connectivity, Kleinian groups and geodesics on the blowup of the torus’, Invent. Math. 146(1) (2001), 35–91.Google Scholar
Minsky, Y. N. N., ‘On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds’, J. Amer. Math. Soc. 7(3) (1994), 539–588.Google Scholar
Mitra, M., ‘Ending laminations for hyperbolic group extensions’, Geom. Funct. Anal. 7(2) (1997), 379–402.Google Scholar
Mitra, M., ‘Cannon–Thurston maps for hyperbolic group extensions’, Topology 37(3) (1998), 527–538.Google Scholar
Mitra, M., ‘Cannon–Thurston maps for trees of hyperbolic metric spaces’, J. Differential Geom. 48 (1998), 135–164.Google Scholar
Mitra, M., ‘Coarse extrinsic geometry: a survey’, in The Epstein Birthday Schrift, Geom. Topol. Monogr., 1 (Geometry & Topology Publications, Coventry, 1998), 341–364.Google Scholar
Mj, M., Cannon–Thurston maps for Kleinian groups. arXiv:1002.0996, (2010).Google Scholar
Mj, M., ‘Cannon–Thurston maps for surface groups’, Annals of Math. 179 (2014), 1–80.Google Scholar
Rips, E., ‘Subgroups of small cancellation groups’, Bull. Lond. Math. Soc. 14(1) (1982), 45–47.Google Scholar
Stillwell, J., Classical Topology and Combinatorial Group Theory, 2nd edn. Graduate Texts in Mathematics (Springer, 1993).Google Scholar
Thurston, W. P., ‘Three-dimensional manifolds, Kleinian groups and hyperbolic geometry’, Bull. Amer. Math. Soc. (N.S.) 6(3) (1982), 357–381.Google Scholar
Wise, D. T., ‘Incoherent negatively curved groups’, Proc. Amer. Math. Soc. 126(4) (1998), 957–964.CrossRefGoogle Scholar