Immersing almost geodesic surfaces in a closed hyperbolic three manifold

Abstract

Let $\mathbf{M}^3$ be a closed hyperbolic three manifold. We construct closed surfaces that map by immersions into $\mathbf{M}^3$ so that for each, one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection.

  • [bowen] L. Bowen, Weak Forms of the Ehrenpreis conjecture and the surface subgroup conjecture.
    @misc{bowen,
      author={Bowen, L.},
      TITLE={Weak Forms of the {E}hrenpreis conjecture and the surface subgroup conjecture},
      ARXIV={math/0411662},
     }
  • [brin-gromov] Go to document M. Brin and M. Gromov, "On the ergodicity of frame flows," Invent. Math., vol. 60, iss. 1, pp. 1-7, 1980.
    @article {brin-gromov, MRKEY = {0582702},
      AUTHOR = {Brin, M. and Gromov, M.},
      TITLE = {On the ergodicity of frame flows},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {60},
      YEAR = {1980},
      NUMBER = {1},
      PAGES = {1--7},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {58F17 (58F11)},
      MRNUMBER = {0582702},
      MRREVIEWER = {Gudlaugur Thorbergsson},
      DOI = {10.1007/BF01389897},
      ZBLNUMBER = {0445.58023},
      }
  • [calegari] D. Calegari, SCL, Tokyo: Math. Soc. Japan, 2009, vol. 20.
    @book {calegari, MRKEY = {2527432},
      AUTHOR = {Calegari, Danny},
      TITLE = {{SCL}},
      SERIES = {MSJ Memoirs},
      VOLUME = {20},
      PUBLISHER = {Math. Soc. Japan},
      ADDRESS = {Tokyo},
      YEAR = {2009},
      PAGES = {xii+209},
      ISBN = {978-4-931469-53-2},
      MRCLASS = {57M07 (20F65 20F67 20J05 37E45)},
      MRNUMBER = {2527432},
      MRREVIEWER = {Athanase Papadopoulos},
      ZBLNUMBER = {1187.20035},
      }
  • [epstein] R. D. Canary, D. B. A. Epstein, and P. L. Green, "Notes on notes of Thurston [MR0903850]," in Fundamentals of Hyperbolic Geometry: Selected Expositions, Cambridge: Cambridge Univ. Press, 2006, vol. 328, pp. 1-115.
    @incollection {epstein, MRKEY = {2235710},
      AUTHOR = {Canary, R. D. and Epstein, D. B. A. and Green, P. L.},
      TITLE = {Notes on notes of {T}hurston [{MR}0903850]},
      BOOKTITLE = {Fundamentals of Hyperbolic Geometry: Selected Expositions},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {328},
      PAGES = {1--115},
      NOTE = {With a new foreword by Canary},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2006},
      MRCLASS = {57N10 (32G15 37D40 37F99 54A20 57M99)},
      MRNUMBER = {2235710},
      ZBLNUMBER = {0612.57009},
      }
  • [c-l-r] Go to document D. Cooper, D. D. Long, and A. W. Reid, "Essential closed surfaces in bounded $3$-manifolds," J. Amer. Math. Soc., vol. 10, iss. 3, pp. 553-563, 1997.
    @article {c-l-r, MRKEY = {1431827},
      AUTHOR = {Cooper, D. and Long, D. D. and Reid, A. W.},
      TITLE = {Essential closed surfaces in bounded {$3$}-manifolds},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {10},
      YEAR = {1997},
      NUMBER = {3},
      PAGES = {553--563},
      ISSN = {0894-0347},
      MRCLASS = {57N10 (57M05 57M10 57N35)},
      MRNUMBER = {1431827},
      MRREVIEWER = {Darryl McCullough},
      DOI = {10.1090/S0894-0347-97-00236-1},
      ZBLNUMBER = {0896.57009},
      }
  • [fletcher-markovic] A. Fletcher and V. Markovic, Quasiconformal Maps and Teichmüller Theory, Oxford: Oxford University Press, 2007, vol. 11.
    @book {fletcher-markovic, MRKEY = {2269887},
      AUTHOR = {Fletcher, A. and Markovic, V.},
      TITLE = {Quasiconformal Maps and {T}eichmüller Theory},
      SERIES = {Oxford Grad. Texts Math.},
      VOLUME = {11},
      PUBLISHER = {Oxford University Press},
      ADDRESS = {Oxford},
      YEAR = {2007},
      PAGES = {viii+189},
      ISBN = {978-0-19-856926-8; 0-19-856926-2},
      MRCLASS = {30-01 (30C62 32G15)},
      MRNUMBER = {2269887},
      ZBLNUMBER = {1103.30002},
     }
  • [kour] Go to document C. Kourouniotis, "Complex length coordinates for quasi-Fuchsian groups," Mathematika, vol. 41, iss. 1, pp. 173-188, 1994.
    @article {kour, MRKEY = {1288062},
      AUTHOR = {Kourouniotis, Christos},
      TITLE = {Complex length coordinates for quasi-{F}uchsian groups},
      JOURNAL = {Mathematika},
      FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics},
      VOLUME = {41},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {173--188},
      ISSN = {0025-5793},
      CODEN = {MTKAAB},
      MRCLASS = {30F40 (20H10 30F35 30F60)},
      MRNUMBER = {1288062},
      DOI = {10.1112/S0025579300007270},
      ZBLNUMBER = {0801.30036},
      }
  • [kahn-markovic] J. Kahn and V. Markovic, The Weil-Petersson distance between finite degree covers of punctured Riemann surfaces and random ideal triangulations.
    @misc{kahn-markovic,
      author={Kahn, J. and Markovic, V.},
      TITLE={The {W}eil-{P}etersson distance between finite degree covers of punctured {R}iemann surfaces and random ideal triangulations},
      ARXIV={0806.2304},
     }
  • [lackenby] Go to document M. Lackenby, "Surface subgroups of Kleinian groups with torsion," Invent. Math., vol. 179, iss. 1, pp. 175-190, 2010.
    @article {lackenby, MRKEY = {2563763},
      AUTHOR = {Lackenby, Marc},
      TITLE = {Surface subgroups of {K}leinian groups with torsion},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {179},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {175--190},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {30F40 (20H10)},
      MRNUMBER = {2563763},
      MRREVIEWER = {Jane Gilman},
      DOI = {10.1007/s00222-009-0215-5},
      ZBLNUMBER = {1183.57017},
      }
  • [moore] C. C. Moore, "Exponential decay of correlation coefficients for geodesic flows," in Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics, New York: Springer-Verlag, 1987, vol. 163–181.
    @incollection {moore, MRKEY = {0880376},
      AUTHOR = {Moore, Calvin C.},
      TITLE = {Exponential decay of correlation coefficients for geodesic flows},
      BOOKTITLE = {Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics},
      VENUE={{B}erkeley, {C}alif., 1984},
      SERIES = {Math. Sci. Res. Inst. Publ.},
      NUMBER = {6},
      VOLUME = {163--181},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1987},
      MRCLASS = {58F17 (22E45)},
      MRNUMBER = {0880376},
      ZBLNUMBER = {0625.58023},
      }
  • [pollicott] Go to document M. Pollicott, "Exponential mixing for the geodesic flow on hyperbolic three-manifolds," J. Statist. Phys., vol. 67, iss. 3-4, pp. 667-673, 1992.
    @article {pollicott, MRKEY = {1171148},
      AUTHOR = {Pollicott, Mark},
      TITLE = {Exponential mixing for the geodesic flow on hyperbolic three-manifolds},
      JOURNAL = {J. Statist. Phys.},
      FJOURNAL = {Journal of Statistical Physics},
      VOLUME = {67},
      YEAR = {1992},
      NUMBER = {3-4},
      PAGES = {667--673},
      ISSN = {0022-4715},
      CODEN = {JSTPSB},
      MRCLASS = {58F17 (22E40)},
      MRNUMBER = {1171148},
      MRREVIEWER = {Alexander Starkov},
      DOI = {10.1007/BF01049724},
      ZBLNUMBER = {0892.58060},
      }
  • [ser-tan] Go to document S. P. Tan, "Complex Fenchel-Nielsen coordinates for quasi-Fuchsian structures," Internat. J. Math., vol. 5, iss. 2, pp. 239-251, 1994.
    @article {ser-tan, MRKEY = {1266284},
      AUTHOR = {Tan, Ser Peow},
      TITLE = {Complex {F}enchel-{N}ielsen coordinates for quasi-{F}uchsian structures},
      JOURNAL = {Internat. J. Math.},
      FJOURNAL = {International Journal of Mathematics},
      VOLUME = {5},
      YEAR = {1994},
      NUMBER = {2},
      PAGES = {239--251},
      ISSN = {0129-167X},
      MRCLASS = {32G15 (57M50)},
      MRNUMBER = {1266284},
      MRREVIEWER = {Takayuki Okai},
      DOI = {10.1142/S0129167X94000140},
      ZBLNUMBER = {0816.32017},
      }
  • [series] Go to document C. Series, "An extension of Wolpert’s derivative formula," Pacific J. Math., vol. 197, iss. 1, pp. 223-239, 2001.
    @article {series, MRKEY = {1810217},
      AUTHOR = {Series, Caroline},
      TITLE = {An extension of {W}olpert's derivative formula},
      JOURNAL = {Pacific J. Math.},
      FJOURNAL = {Pacific Journal of Mathematics},
      VOLUME = {197},
      YEAR = {2001},
      NUMBER = {1},
      PAGES = {223--239},
      ISSN = {0030-8730},
      CODEN = {PJMAAI},
      MRCLASS = {30F40 (20H10 57M50)},
      MRNUMBER = {1810217},
      MRREVIEWER = {Fr{é}d{é}ric Paulin},
      ZBLNUMBER = {1065.30044},
      DOI = {10.2140/pjm.2001.197.223},
     }

Authors

Jeremy Kahn

Stony Brook University
Stony Brook, NY

Current address:

Department of Mathematics
Brown University
Box 1917
151 Thayer Street
Providence, RI 02912 Vladimir Markovic

Mathematics 253-37,br/> California Institute of Technology
Pasadena, CA 91125