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 {abikoff-ct, MRKEY = {0396937},
AUTHOR = {Abikoff, William},
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} -
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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},
} -
[mitra-ct]
M. Mitra, "Cannon-Thurston maps for hyperbolic group extensions," Topology, vol. 37, iss. 3, pp. 527-538, 1998.
@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},
} -
[mitra-trees]
M. Mitra, "Cannon-Thurston maps for trees of hyperbolic metric spaces," J. Differential Geom., vol. 48, iss. 1, pp. 135-164, 1998.
@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},
ZBLNUMBER = {0906.20023},
} -
[miyachi-ct] H. Miyachi, Semiconjugacies between actions of topologically tame Kleinian groups, 2002.
@misc{miyachi-ct,
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]
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},
} -
[mahan-bddgeo] M. Mj, "Cannon-Thurston maps and bounded geometry," in Teichmüller Theory and Moduli Problem, Mysore: Ramanujan Math. Soc., 2010, vol. 10, pp. 489-511.
@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},
} -
[mahan-kl] M. Mj, Cannon-Thurston Maps for Kleinian groups, 2010.
@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]
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},
} -
[masur-minsky2]
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.
@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},
YEAR = {2000},
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},
} -
[mahan-pal]
M. Mj and A. Pal, "Relative hyperbolicity, trees of spaces and Cannon-Thurston maps," Geom. Dedicata, vol. 151, pp. 59-78, 2011.
@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},
} -
[mahan-reeves]
M. Mj and L. Reeves, "A combination theorem for strong relative hyperbolicity," Geom. Topol., vol. 12, iss. 3, pp. 1777-1798, 2008.
@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},
} -
[mahan-series2] M. Mj and C. Series, Limits of limit sets II, 2013.
@misc{mahan-series2,
author={Mj, Mahan and Series, C.},
TITLE={Limits of limit sets {II}},
YEAR={2013},
ARXIV= {1306.3081},
} -
@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},
} -
[Thurstonnotes] 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.
@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},
MRREVIEWER = {William Dunbar},
ZBLNUMBER = {0612.57009},
} -
[thurstonbams]
W. P. Thurston, "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry," Bull. Amer. Math. Soc., vol. 6, iss. 3, pp. 357-381, 1982.
@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},
}