Cannon-Thurston maps for surface groups

Abstract

We prove the existence of Cannon-Thurston maps for simply and doubly degenerate surface Kleinian groups. As a consequence we prove that connected limit sets of finitely generated Kleinian groups are locally connected.

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    @article {klarreich, MRKEY = {1713300},
      AUTHOR = {Klarreich, Erica},
      TITLE = {Semiconjugacies between {K}leinian group actions on the {R}iemann sphere},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {121},
      YEAR = {1999},
      NUMBER = {5},
      PAGES = {1031--1078},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {57M60 (30F40 37F30 57M50 57S30)},
      MRNUMBER = {1713300},
      MRREVIEWER = {Colin C. Adams},
      DOI = {10.1353/ajm.1999.0034},
      ZBLNUMBER = {1011.30035},
      }
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    @article {ctm-locconn, MRKEY = {1859018},
      AUTHOR = {McMullen, Curtis T.},
      TITLE = {Local connectivity, {K}leinian groups and geodesics on the blowup of the torus},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {146},
      YEAR = {2001},
      NUMBER = {1},
      PAGES = {35--91},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {30F40 (30F60 37C85 37F30 57M50)},
      MRNUMBER = {1859018},
      MRREVIEWER = {Petra Bonfert-Taylor},
      DOI = {10.1007/PL00005809},
      ZBLNUMBER = {1061.37025},
      }
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    @article {minsky-jams, MRKEY = {1257060},
      AUTHOR = {Minsky, Yair N.},
      TITLE = {On rigidity, limit sets, and end invariants of hyperbolic {$3$}-manifolds},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {7},
      YEAR = {1994},
      NUMBER = {3},
      PAGES = {539--588},
      ISSN = {0894-0347},
      MRCLASS = {57M50 (30C62 30F40 57M60 57N10)},
      MRNUMBER = {1257060},
      MRREVIEWER = {William Dunbar},
      DOI = {10.2307/2152785},
      ZBLNUMBER = {0808.30027},
      }
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    @article {minsky-elc1, MRKEY = {2630036},
      AUTHOR = {Minsky, Yair N.},
      TITLE = {The classification of {K}leinian surface groups. {I}. {M}odels and bounds},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {171},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {1--107},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {30F40 (20H10 57M50)},
      MRNUMBER = {2630036},
      MRREVIEWER = {Athanase Papadopoulos},
      DOI = {10.4007/annals.2010.171.1},
      ZBLNUMBER = {1193.30063},
     }
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    @article {mitra-ct, MRKEY = {1604882},
      AUTHOR = {Mitra, Mahan},
      TITLE = {Cannon-{T}hurston maps for hyperbolic group extensions},
      JOURNAL = {Topology},
      FJOURNAL = {Topology. An International Journal of Mathematics},
      VOLUME = {37},
      YEAR = {1998},
      NUMBER = {3},
      PAGES = {527--538},
      ISSN = {0040-9383},
      CODEN = {TPLGAF},
      MRCLASS = {57M07 (20F32 57M10)},
      MRNUMBER = {1604882},
      MRREVIEWER = {Sadayoshi Kojima},
      DOI = {10.1016/S0040-9383(97)00036-0},
      ZBLNUMBER = {0907.20038},
      }
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    @article {mitra-trees, MRKEY = {1622603},
      AUTHOR = {Mitra, Mahan},
      TITLE = {Cannon-{T}hurston maps for trees of hyperbolic metric spaces},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {48},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {135--164},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {57M07 (20F32 57M50)},
      MRNUMBER = {1622603},
      MRREVIEWER = {Michah Sageev},
      URL = {http://projecteuclid.org/euclid.jdg/1214460609},
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      }
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      author={Miyachi, H.},
      TITLE = {Semiconjugacies between actions of topologically tame {K}leinian groups},
      NOTE={preprint},
      YEAR={2002},
     }
  • [mahan-amalgeo] M. Mj, Cannon-Thurston maps for surface groups I: amalgamation geometry and split geometry, 2005.
    @misc {mahan-amalgeo,
      author = {Mj, Mahan},
      TITLE = {Cannon-{T}hurston maps for surface groups {I}: amalgamation geometry and split geometry},
      arXiv = {math/0512539},
      YEAR={2005},
      }
  • [mahan-split] M. Mj, Cannon-Thurston maps for surface groups II: Split geometry and the Minsky model, 2006,%%%%.
    @misc {mahan-split,
      author = {Mj, Mahan},
      TITLE = {Cannon-{T}hurston maps for surface groups {II}: {S}plit geometry and the {M}insky model},
      ARXIV={math/0607509v1},
      YEAR={2006},
     }%%%%
  • [mahan-elct] M. Mj, Ending laminations and Cannon-Thurston maps, 2007.
    @misc{mahan-elct,
      author = {Mj, Mahan},
      TITLE = {Ending laminations and {C}annon-{T}hurston maps},
      ARXIV={0701725},
      YEAR={2007},
     }
  • [mahan-pared] Go to document M. Mj, "Cannon-Thurston maps for pared manifolds of bounded geometry," Geom. Topol., vol. 13, iss. 1, pp. 189-245, 2009.
    @article {mahan-pared, MRKEY = {2469517},
      AUTHOR = {Mj, Mahan},
      TITLE = {Cannon-{T}hurston maps for pared manifolds of bounded geometry},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry \& Topology},
      VOLUME = {13},
      YEAR = {2009},
      NUMBER = {1},
      PAGES = {189--245},
      ISSN = {1465-3060},
      MRCLASS = {57M50},
      MRNUMBER = {2469517},
      MRREVIEWER = {Vassilis Metaftsis},
      DOI = {10.2140/gt.2009.13.189},
      ZBLNUMBER = {1166.57009},
      }
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    @incollection {mahan-bddgeo, MRKEY = {2667569},
      AUTHOR = {Mj, Mahan},
      TITLE = {Cannon-{T}hurston maps and bounded geometry},
      BOOKTITLE = {Teichmüller Theory and Moduli Problem},
      SERIES = {Ramanujan Math. Soc. Lect. Notes Ser.},
      VOLUME = {10},
      PAGES = {489--511},
      PUBLISHER = {Ramanujan Math. Soc.},
      ADDRESS = {Mysore},
      YEAR = {2010},
      MRCLASS = {20F65 (57M50)},
      MRNUMBER = {2667569},
      MRREVIEWER = {Thomas Koberda},
      ZBLNUMBER = {1204.57014},
      }
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    @misc {mahan-kl,
      author = {Mj, Mahan},
      TITLE = {Cannon-{T}hurston Maps for {K}leinian groups},
      ARXIV={1002.0996},
      YEAR={2010},
     }
  • [mahan-ibdd] 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.
    @incollection {mahan-ibdd, MRKEY = {2848212},
      AUTHOR = {Mj, Mahan},
      TITLE = {Cannon-{T}hurston maps, $i$-bounded geometry and a theorem of {M}c{M}ullen},
      BOOKTITLE = {Actes du {S}éminaire de {T}héorie {S}pectrale et {G}éometrie. {V}olume 28. {A}nnée 2009--2010},
      SERIES = {Sémin. Théor. Spectr. Géom.},
      VOLUME = {28},
      PAGES = {63--107},
      PUBLISHER = {Univ. Grenoble I},
      ADDRESS = {Saint},
      YEAR = {2011},
      MRCLASS = {57M50 (30F40)},
      MRNUMBER = {2848212},
      MRREVIEWER = {Vassilis Metaftsis},
      ZBLNUMBER = {1237.57018},
      }
  • [masur-minsky] Go to document 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.
    @article {masur-minsky, MRKEY = {1714338},
      AUTHOR = {Masur, Howard A. and Minsky, Yair N.},
      TITLE = {Geometry of the complex of curves. {I}. {H}yperbolicity},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {138},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {103--149},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {57M50 (20F67 30F60 32G15)},
      MRNUMBER = {1714338},
      MRREVIEWER = {Darryl McCullough},
      DOI = {10.1007/s002220050343},
      ZBLNUMBER = {0941.32012},
     }
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    @article {masur-minsky2, MRKEY = {1791145},
      AUTHOR = {Masur, Howard A. and Minsky, Yair N.},
      TITLE = {Geometry of the complex of curves. {II}. {H}ierarchical structure},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {10},
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      NUMBER = {4},
      PAGES = {902--974},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {57M50 (30F60 32G15)},
      MRNUMBER = {1791145},
      MRREVIEWER = {Darryl McCullough},
      DOI = {10.1007/PL00001643},
      ZBLNUMBER = {0972.32011},
     }
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    @article {mahan-pal, MRKEY = {2780738},
      AUTHOR = {Mj, Mahan and Pal, Abhijit},
      TITLE = {Relative hyperbolicity, trees of spaces and {C}annon-{T}hurston maps},
      JOURNAL = {Geom. Dedicata},
      FJOURNAL = {Geometriae Dedicata},
      VOLUME = {151},
      YEAR = {2011},
      PAGES = {59--78},
      ISSN = {0046-5755},
      CODEN = {GEMDAT},
      MRCLASS = {20F65 (57M50)},
      MRNUMBER = {2780738},
      MRREVIEWER = {Vassilis Metaftsis},
      DOI = {10.1007/s10711-010-9519-2},
      ZBLNUMBER = {1222.57013},
      }
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    @article {mahan-reeves, MRKEY = {2421140},
      AUTHOR = {Mj, Mahan and Reeves, Lawrence},
      TITLE = {A combination theorem for strong relative hyperbolicity},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry \& Topology},
      VOLUME = {12},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {1777--1798},
      ISSN = {1465-3060},
      MRCLASS = {20F67 (57M50)},
      MRNUMBER = {2421140},
      DOI = {10.2140/gt.2008.12.1777},
      ZBLNUMBER = {1192.20027},
      }
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    @misc{mahan-series2,
      author={Mj, Mahan and Series, C.},
      TITLE={Limits of limit sets {II}},
      YEAR={2013},
      ARXIV= {1306.3081},
     }
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    @misc{souto-ct,
      author={Souto, J.},
      TITLE={Cannon-{T}hurston maps for thick free groups},
      NOTE={preprint},
      YEAR={2006},
      URL = {http://www.math.ubc.ca/~jsouto/papers/Cannon-Thurston.pdf},
     }
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    @incollection {Thurstonnotes, MRKEY = {0903850},
      AUTHOR = {Canary, Richard D. and Epstein, D. B. A. and Green, P.},
      TITLE = {Notes on notes of {T}hurston},
      BOOKTITLE = {Analytical and Geometric Aspects of Hyperbolic Space},
      VENUE={{C}oventry/{D}urham, 1984},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {111},
      PAGES = {3--92},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1987},
      MRCLASS = {57N10 (32G15 32G99 54A20 57M99 58F17)},
      MRNUMBER = {0903850},
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      }
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    @article {thurstonbams, MRKEY = {0648524},
      AUTHOR = {Thurston, William P.},
      TITLE = {Three-dimensional manifolds, {K}leinian groups and hyperbolic geometry},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {American Mathematical Society. Bulletin. New Series},
      VOLUME = {6},
      YEAR = {1982},
      NUMBER = {3},
      PAGES = {357--381},
      ISSN = {0273-0979},
      CODEN = {BAMOAD},
      MRCLASS = {57N10 (20H15 30F40 57M35 57M40 57S17)},
      MRNUMBER = {0648524},
      MRREVIEWER = {Klaus Johannson},
      DOI = {10.1090/S0273-0979-1982-15003-0},
      ZBLNUMBER = {0496.57005},
      }

Authors

Mahan Mj

Ramakrishna Mission Vivekananda University, West Bengal, India