The classification of Kleinian surface groups, II: The Ending Lamination Conjecture

Abstract

Thurston’s Ending Lamination Conjecture states that a hyperbolic 3-manifold $N$ with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups; the general case when $N$ has incompressible ends relative to its cusps follows readily. The main ingredient is a uniformly bilipschitz model for the quotient of $\mathbb{H}^3$ by a Kleinian surface group.

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      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {126},
      YEAR = {2004},
      NUMBER = {4},
      PAGES = {713--737},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {57M50 (30F40 57N10)},
      MRNUMBER = {2075479},
      MRREVIEWER = {Darryl McCullough},
      DOI = {10.1353/ajm.2004.0028},
      ZBLNUMBER = {1058.57010},
      }
  • [freedman-hass-scott:area] Go to document M. Freedman, J. Hass, and P. Scott, "Least area incompressible surfaces in $3$-manifolds," Invent. Math., vol. 71, iss. 3, pp. 609-642, 1983.
    @article {freedman-hass-scott:area, MRKEY = {0695910},
      AUTHOR = {Freedman, Michael and Hass, Joel and Scott, Peter},
      TITLE = {Least area incompressible surfaces in {$3$}-manifolds},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {71},
      YEAR = {1983},
      NUMBER = {3},
      PAGES = {609--642},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {57N10 (53A10)},
      MRNUMBER = {0695910},
      MRREVIEWER = {John Hempel},
      DOI = {10.1007/BF02095997},
      ZBLNUMBER = {0482.53045},
      }
  • [hamenstadt:boundary] Go to document U. Hamenstädt, "Train tracks and the Gromov boundary of the complex of curves," in Spaces of Kleinian Groups, Cambridge: Cambridge Univ. Press, 2006, vol. 329, pp. 187-207.
    @incollection {hamenstadt:boundary, MRKEY = {2258749},
      AUTHOR = {Hamenst{ä}dt, Ursula},
      TITLE = {Train tracks and the {G}romov boundary of the complex of curves},
      BOOKTITLE = {Spaces of {K}leinian Groups},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {329},
      PAGES = {187--207},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2006},
      MRCLASS = {30F45 (57M50)},
      MRNUMBER = {2258749},
      MRREVIEWER = {Hugo Parlier},
      ZBLNUMBER = {1117.30036},
      DOI = {10.1017/CBO9781139106993.009},
     }
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    @incollection {harvey:boundary, MRKEY = {0624817},
      AUTHOR = {Harvey, W. J.},
      TITLE = {Boundary structure of the modular group},
      BOOKTITLE = {Riemann Surfaces and Related Topics: {P}roceedings of the 1978 {S}tony {B}rook {C}onference},
      VENUE={{S}tate {U}niv. {N}ew {Y}ork, {S}tony {B}rook, {N}.{Y}., 1978},
      SERIES = {Ann. of Math. Stud.},
      VOLUME = {97},
      PAGES = {245--251},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, N.J.},
      YEAR = {1981},
      MRCLASS = {32G15 (57N05)},
      MRNUMBER = {0624817},
      MRREVIEWER = {B. Maskit},
      ZBLNUMBER = {0461.30036},
      }
  • [harvey:modular] Go to document W. J. Harvey, "Modular groups and representation spaces," in Geometry of Group Representations, Providence, RI: Amer. Math. Soc., 1988, vol. 74, pp. 205-214.
    @incollection {harvey:modular, MRKEY = {0957520},
      AUTHOR = {Harvey, W. J.},
      TITLE = {Modular groups and representation spaces},
      BOOKTITLE = {Geometry of Group Representations},
      VENUE={{B}oulder, {CO},
      1987},
      SERIES = {Contemp. Math.},
      VOLUME = {74},
      PAGES = {205--214},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1988},
      MRCLASS = {14H15 (20F32 20H05 32G15 57S30)},
      MRNUMBER = {0957520},
      MRREVIEWER = {Ruth Charney},
      ZBLNUMBER = {0689.20038},
      DOI = {10.1090/conm/074},
     }
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    @book {hempel, MRKEY = {0415619},
      AUTHOR = {Hempel, John},
      TITLE = {{$3$}-{M}anifolds},
      SERIES = {Ann. of Math. Studies},
      VOLUME={86},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1976},
      PAGES = {xii+195},
      MRCLASS = {57A10},
      MRNUMBER = {0415619},
      MRREVIEWER = {J. S. Birman},
      ZBLNUMBER = {0345.57001},
      }
  • [jaco-rubinstein] Go to document W. Jaco and H. J. Rubinstein, "PL minimal surfaces in $3$-manifolds," J. Differential Geom., vol. 27, iss. 3, pp. 493-524, 1988.
    @article {jaco-rubinstein, MRKEY = {0940116},
      AUTHOR = {Jaco, William and Rubinstein, J. Hyam},
      TITLE = {P{L} minimal surfaces in {$3$}-manifolds},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {27},
      YEAR = {1988},
      NUMBER = {3},
      PAGES = {493--524},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {57N10 (57M99 57Q99)},
      MRNUMBER = {0940116},
      MRREVIEWER = {G. Peter Scott},
      URL = {http://projecteuclid.org/euclid.jdg/1214442006},
      ZBLNUMBER = {0652.57005},
      }
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    @book {jaco, MRKEY = {0565450},
      AUTHOR = {Jaco, William},
      TITLE = {Lectures on Three-Manifold Topology},
      SERIES = {CBMS Reg. Conf. Ser. Math.},
      NUMBER = {43},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1980},
      PAGES = {xii+251},
      ISBN = {0-8218-1693-4},
      MRCLASS = {57N10},
      MRNUMBER = {0565450},
      MRREVIEWER = {G. Peter Scott},
      ZBLNUMBER = {0433.57001},
      }
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    @article {jaco-shalen, MRKEY = {0539411},
      AUTHOR = {Jaco, William H. and Shalen, Peter B.},
      TITLE = {Seifert fibered spaces in {$3$}-manifolds},
      JOURNAL = {Mem. Amer. Math. Soc.},
      FJOURNAL = {Memoirs of the American Mathematical Society},
      VOLUME = {21},
      YEAR = {1979},
      NUMBER = {220},
      PAGES = {viii+192},
      ISSN = {0065-9266},
      CODEN = {MAMCAU},
      MRCLASS = {57N10},
      MRNUMBER = {0539411},
      MRREVIEWER = {Hugh M. Hilden},
      ZBLNUMBER = {0415.57005},
     }
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    @book {johannson:3mfds, MRKEY = {1439249},
      AUTHOR = {Johannson, Klaus},
      TITLE = {Topology and Combinatorics of 3-Manifolds},
      SERIES = {Lect. Notes in Math.},
      VOLUME = {1599},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1995},
      PAGES = {xviii+446},
      ISBN = {3-540-59063-3},
      MRCLASS = {57N10 (57M50)},
      MRNUMBER = {1439249},
      MRREVIEWER = {Colin C. Adams},
      ZBLNUMBER = {0820.57001},
      }
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    @article {jorgensen-marden:convergence, MRKEY = {1060898},
      AUTHOR = {Jørgensen, T. and Marden, A.},
      TITLE = {Algebraic and geometric convergence of {K}leinian groups},
      JOURNAL = {Math. Scand.},
      FJOURNAL = {Mathematica Scandinavica},
      VOLUME = {66},
      YEAR = {1990},
      NUMBER = {1},
      PAGES = {47--72},
      ISSN = {0025-5521},
      CODEN = {MTSCAN},
      MRCLASS = {30F40 (57M99 57S30)},
      MRNUMBER = {1060898},
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      ZBLNUMBER = {0738.30032},
     }
  • [klarreich:boundary] Go to document E. Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space.
    @misc{klarreich:boundary,
      author={Klarreich, E.},
      TITLE={The boundary at infinity of the curve complex and the relative {T}eichmüller space},
      NOTE={preprint},
      URL = {http://www.ericaklarreich.com/curvecomplex.pdf},
     }
  • [kra:spaces] Go to document I. Kra, "On spaces of Kleinian groups," Comment. Math. Helv., vol. 47, pp. 53-69, 1972.
    @article {kra:spaces, MRKEY = {0306485},
      AUTHOR = {Kra, Irwin},
      TITLE = {On spaces of {K}leinian groups},
      JOURNAL = {Comment. Math. Helv.},
      FJOURNAL = {Commentarii Mathematici Helvetici},
      VOLUME = {47},
      YEAR = {1972},
      PAGES = {53--69},
      ISSN = {0010-2571},
      MRCLASS = {30A58},
      MRNUMBER = {0306485},
      MRREVIEWER = {C. Earle},
      DOI = {10.1007/BF02566788},
      ZBLNUMBER = {0239.30020},
      }
  • [luukkainen:isotopy] Go to document J. Luukkainen, "Bi-Lipschitz concordance implies bi-Lipschitz isotopy," Monatsh. Math., vol. 111, iss. 1, pp. 35-46, 1991.
    @article {luukkainen:isotopy, MRKEY = {1089382},
      AUTHOR = {Luukkainen, Jouni},
      TITLE = {Bi-{L}ipschitz concordance implies bi-{L}ipschitz isotopy},
      JOURNAL = {Monatsh. Math.},
      FJOURNAL = {Monatshefte für Mathematik},
      VOLUME = {111},
      YEAR = {1991},
      NUMBER = {1},
      PAGES = {35--46},
      ISSN = {0026-9255},
      CODEN = {MNMTA2},
      MRCLASS = {57N35 (30C99 57N37)},
      MRNUMBER = {1089382},
      MRREVIEWER = {Susan G. Staples},
      DOI = {10.1007/BF01299275},
      ZBLNUMBER = {0719.30013},
      }
  • [marden:geometry] Go to document A. Marden, "The geometry of finitely generated kleinian groups," Ann. of Math., vol. 99, pp. 383-462, 1974.
    @article {marden:geometry, MRKEY = {0349992},
      AUTHOR = {Marden, Albert},
      TITLE = {The geometry of finitely generated kleinian groups},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {99},
      YEAR = {1974},
      PAGES = {383--462},
      ISSN = {0003-486X},
      MRCLASS = {30A58 (22E40)},
      MRNUMBER = {0349992},
      MRREVIEWER = {C. Earle},
      DOI = {10.2307/1971059},
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      }
  • [maskit:self] Go to document B. Maskit, "Self-maps on Kleinian groups," Amer. J. Math., vol. 93, pp. 840-856, 1971.
    @article {maskit:self, MRKEY = {0291453},
      AUTHOR = {Maskit, Bernard},
      TITLE = {Self-maps on {K}leinian groups},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {93},
      YEAR = {1971},
      PAGES = {840--856},
      ISSN = {0002-9327},
      MRCLASS = {30A58},
      MRNUMBER = {0291453},
      MRREVIEWER = {J. Lehner},
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      }
  • [masur-minsky:complex1] 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:complex1, 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-minsky:complex2] Go to document 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-minsky:complex2, 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},
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  • [mcmullen:grafting] Go to document C. T. McMullen, "Complex earthquakes and Teichmüller theory," J. Amer. Math. Soc., vol. 11, iss. 2, pp. 283-320, 1998.
    @article {mcmullen:grafting, MRKEY = {1478844},
      AUTHOR = {McMullen, Curtis T.},
      TITLE = {Complex earthquakes and {T}eichmüller theory},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {11},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {283--320},
      ISSN = {0894-0347},
      MRCLASS = {32G15 (20H10 30F60 57N05)},
      MRNUMBER = {1478844},
      MRREVIEWER = {Darryl McCullough},
      DOI = {10.1090/S0894-0347-98-00259-8},
      ZBLNUMBER = {0890.30031},
     }
  • [minsky:knoxville] Y. N. Minsky, "On Thurston’s ending lamination conjecture," in Low-Dimensional Topology, Int. Press, Cambridge, MA, 1994, vol. III, pp. 109-122.
    @incollection {minsky:knoxville, MRKEY = {1316176},
      AUTHOR = {Minsky, Yair N.},
      TITLE = {On {T}hurston's ending lamination conjecture},
      BOOKTITLE = {Low-Dimensional Topology},
      VENUE={{K}noxville, {TN},
      1992},
      SERIES = {Conf. Proc. Lecture Notes Geom. Topology},
      VOLUME={III},
      PAGES = {109--122},
      PUBLISHER = {Int. Press, Cambridge, MA},
      YEAR = {1994},
      MRCLASS = {57M50 (20H10)},
      MRNUMBER = {1316176},
      MRREVIEWER = {Colin C. Adams},
      ZBLNUMBER = {0846.57010},
      }
  • [minsky:torus] Go to document Y. N. Minsky, "The classification of punctured-torus groups," Ann. of Math., vol. 149, iss. 2, pp. 559-626, 1999.
    @article {minsky:torus, MRKEY = {1689341},
      AUTHOR = {Minsky, Yair N.},
      TITLE = {The classification of punctured-torus groups},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {149},
      YEAR = {1999},
      NUMBER = {2},
      PAGES = {559--626},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {30F40 (57M50)},
      MRNUMBER = {1689341},
      MRREVIEWER = {Darryl McCullough},
      DOI = {10.2307/120976},
      ZBLNUMBER = {0939.30034},
      }
  • [minsky:kgcc] Go to document Y. N. Minsky, "Kleinian groups and the complex of curves," Geom. Topol., vol. 4, pp. 117-148, 2000.
    @article {minsky:kgcc, MRKEY = {2182094},
      AUTHOR = {Minsky, Yair N.},
      TITLE = {Kleinian groups and the complex of curves},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry and Topology},
      VOLUME = {4},
      YEAR = {2000},
      PAGES = {117--148},
      ISSN = {1465-3060},
      MRCLASS = {57M50 (30F40)},
      MRNUMBER = {2182094},
      MRREVIEWER = {Ken-ichi Ohshika},
      DOI = {10.2140/gt.2000.4.117},
      ZBLNUMBER = {0953.30027},
      }
  • [minsky:boundgeom] Go to document Y. N. Minsky, "Bounded geometry for Kleinian groups," Invent. Math., vol. 146, iss. 1, pp. 143-192, 2001.
    @article {minsky:boundgeom, MRKEY = {1859020},
      AUTHOR = {Minsky, Yair N.},
      TITLE = {Bounded geometry for {K}leinian groups},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {146},
      YEAR = {2001},
      NUMBER = {1},
      PAGES = {143--192},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {30F40 (30F60 32G15 37F30 57M50)},
      MRNUMBER = {1859020},
      MRREVIEWER = {Michel Coornaert},
      DOI = {10.1007/s002220100163},
      ZBLNUMBER = {1061.37026},
      }
  • [minsky:warwick] Go to document Y. N. Minsky, "Combinatorial and geometrical aspects of hyperbolic 3-manifolds," in Kleinian Groups and Hyperbolic 3-Manifolds, Komori, Y., Markovic, V., and Series, C., Eds., Cambridge: Cambridge Univ. Press, 2003, vol. 299, p. viii.
    @incollection {minsky:warwick,
      author = {Minsky, Yair N.},
      MRKEY = {2044542},
      TITLE = {Combinatorial and geometrical aspects of hyperbolic 3-manifolds},
      BOOKTITLE={Kleinian Groups and Hyperbolic 3-Manifolds},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {299},
      EDITOR= {Komori, Y. and Markovic, V. and Series, C. },
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2003},
      PAGES = {viii+384},
      ISBN = {0-521-54013-5},
      MRCLASS = {57-06 (30-06 57M50 57N10)},
      MRNUMBER = {2044542},
      DOI = {10.1017/CBO9780511542817},
      ZBLNUMBER = {1062.30053},
     }
  • [minsky:cdm] Y. N. Minsky, "End invariants and the classification of hyperbolic 3-manifolds," in Current Developments in Mathematics, 2002, Int. Press, Somerville, MA, 2003, pp. 181-217.
    @incollection {minsky:cdm, MRKEY = {2062319},
      AUTHOR = {Minsky, Yair N.},
      TITLE = {End invariants and the classification of hyperbolic 3-manifolds},
      BOOKTITLE = {Current Developments in Mathematics, 2002},
      PAGES = {181--217},
      PUBLISHER = {Int. Press, Somerville, MA},
      YEAR = {2003},
      MRCLASS = {57M50 (30F20 30F40 37F30 57N10)},
      MRNUMBER = {2062319},
      MRREVIEWER = {Bruno Zimmermann},
      ZBLNUMBER = {1049.57010},
      }
  • [minsky:ELCI] Go to document Y. N. Minsky, "The classification of Kleinian surface groups. I. Models and bounds," Ann. of Math., vol. 171, iss. 1, pp. 1-107, 2010.
    @article {minsky:ELCI, 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},
     }
  • [mostow:hyperbolic] Go to document G. D. Mostow, "Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms," Inst. Hautes Études Sci. Publ. Math., vol. 34, pp. 53-104, 1968.
    @article {mostow:hyperbolic, MRKEY = {0236383},
      AUTHOR = {Mostow, G. D.},
      TITLE = {Quasi-conformal mappings in {$n$}-space and the rigidity of hyperbolic space forms},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {34},
      YEAR = {1968},
      PAGES = {53--104},
      ISSN = {0073-8301},
      MRCLASS = {30.47},
      MRNUMBER = {0236383},
      MRREVIEWER = {R. J. Cannon},
      DOI = {10.1007/BF02684590},
      ZBLNUMBER = {0189.09402},
      }
  • [namazi:thesis] Go to document H. Namazi, Heegaard Splittings and Hyperbolic Geometry, ProQuest LLC, Ann Arbor, MI, 2005.
    @book {namazi:thesis, MRKEY = {2707466},
      AUTHOR = {Namazi, Hossein},
      TITLE = {Heegaard Splittings and Hyperbolic Geometry},
      NOTE = {Ph.D. thesis, State University of New York at Stony Brook},
      PUBLISHER = {ProQuest LLC, Ann Arbor, MI},
      YEAR = {2005},
      PAGES = {187},
      ISBN = {978-0542-20270-4},
      MRCLASS = {Thesis},
      MRNUMBER = {2707466},
      URL = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3179646},
      }
  • [namazi-souto:uniform-inj] H. Namazi and J. Souto, Revisiting Thurston’s uniform injectivity theorem.
    @misc{namazi-souto:uniform-inj,
      author={Namazi, Hossein and Souto, J.},
      TITLE={Revisiting {T}hurston's uniform injectivity theorem},
      NOTE={in preparation},
      SORTYEAR={2011},
      }
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Authors

Jeffrey F. Brock

Department of Mathematics
Brown University
Box 1917
Providence, RI 02912

Richard D. Canary

Department of Mathematics
University of Michigan
530 Church Street
Ann Arbor, MI 48109-1043

Yair N. Minsky

Mathematics Department
Yale University
PO Box 208283
New Haven, CT 06520-8283