Teichmüller curves, triangle groups, and Lyapunov exponents

Abstract

We construct a Teichmüller curve uniformized by a Fuchsian triangle group commensurable to $\Delta(m,n,\infty)$ for every $m,n\leq \infty$. In most cases, for example when $m\neq n$ and $m$ or $n$ is odd, the uniformizing group is equal to the triangle group $\Delta(m,n,\infty)$. Our construction includes the Teichmüller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. For small $m$, we find billiard tables that generate these Teichmüller curves. We interpret some of the so-called Lyapunov exponents of the Kontsevich-Zorich cocycle as normalized degrees of a natural line bundle on a Teichmüller curve. We determine the Lyapunov exponents for the Teichmüller curves we construct.

  • [BGGS80] G. Benettin, I. Galgini, A. Giorgilli, and J. -M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them, Part I (Meccanica, 1980).
    @misc{BGGS80,
      author={Benettin, G. and Galgini, I. and Giorgilli, A. and Strelcyn, J.-M.},
      TITLE={Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them, Part I (Meccanica, 1980)},
      PAGES={9--20},
     }
  • [Bo01] Go to document I. I. Bouw, "The $p$-rank of ramified covers of curves," Compositio Math., vol. 126, iss. 3, pp. 295-322, 2001.
    @article {Bo01, MRKEY = {1834740},
      AUTHOR = {Bouw, Irene I.},
      TITLE = {The {$p$}-rank of ramified covers of curves},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {126},
      YEAR = {2001},
      NUMBER = {3},
      PAGES = {295--322},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14H30},
      MRNUMBER = {2002e:14045},
      MRREVIEWER = {Helmut V{ö}lklein},
      DOI = {10.1023/A:1017513122376},
      ZBLNUMBER = {1056.14036},
      }
  • [Bo04] Go to document I. I. Bouw, "Reduction of the Hurwitz space of metacyclic covers," Duke Math. J., vol. 121, iss. 1, pp. 75-111, 2004.
    @article {Bo04, MRKEY = {2031166},
      AUTHOR = {Bouw, Irene I.},
      TITLE = {Reduction of the {H}urwitz space of metacyclic covers},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {121},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {75--111},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {14H30 (14G32)},
      MRNUMBER = {2005d:14042},
      MRREVIEWER = {Rachel J. Pries},
      DOI = {10.1215/S0012-7094-04-12113-X},
      ZBLNUMBER = {1056.14037},
      }
  • [Bo05] I. I. Bouw, "Pseudo-elliptic bundles, deformation data and the reduction of Galois covers, Habilitation thesis," PhD Thesis , Essen, 2005.
    @phdthesis{Bo05,
      author={Bouw, Irene I.},
      TITLE = {Pseudo-elliptic bundles, deformation data and the reduction of Galois covers, Habilitation thesis},
      SCHOOL={Essen},
      YEAR={2005},
      }
  • [An] Go to document S. Wewers and I. I. Bouw, "Alternating groups as monodromy groups in positive characteristic," Pacific J. Math., vol. 222, iss. 1, pp. 185-200, 2005.
    @article {An, MRKEY = {2200250},
      AUTHOR = {Wewers, Stefan and Bouw, Irene I.},
      TITLE = {Alternating groups as monodromy groups in positive characteristic},
      JOURNAL = {Pacific J. Math.},
      FJOURNAL = {Pacific Journal of Mathematics},
      VOLUME = {222},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {185--200},
      ISSN = {0030-8730},
      CODEN = {PJMAAI},
      MRCLASS = {14H30 (14H10)},
      MRNUMBER = {2006k:14050},
      MRREVIEWER = {Rachel J. Pries},
      DOI = {10.2140/pjm.2005.222.185},
      ZBLNUMBER = {1111.14021},
      }
  • [BoWe05] I. I. Bouw and S. Wewers, "Indigenous bundles with nilpotent $p$-curvature," Internat. Math. Res. Not., vol. 2006, p. I, 2006.
    @article {BoWe05, MRKEY = {2219211},
      AUTHOR = {Bouw, Irene I. and Wewers, Stefan},
      TITLE = {Indigenous bundles with nilpotent {$p$}-curvature},
      JOURNAL = {Internat. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2006},
      PAGES = {Art. ID 89254, 37},
      ISSN = {1073-7928},
      MRCLASS = {14H60 (14H30)},
      MRNUMBER = {2007i:14032},
      MRREVIEWER = {Sabin Cautis},
      VOLUME = {2006},
      ZBLNUMBER={1121.14026},
      }
  • [CoWo90] P. Cohen and J. Wolfart, "Modular embeddings for some nonarithmetic Fuchsian groups," Acta Arith., vol. 56, iss. 2, pp. 93-110, 1990.
    @article {CoWo90, MRKEY = {1075639},
      AUTHOR = {Cohen, Paula and Wolfart, J{ü}rgen},
      TITLE = {Modular embeddings for some nonarithmetic {F}uchsian groups},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
      VOLUME = {56},
      YEAR = {1990},
      NUMBER = {2},
      PAGES = {93--110},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11F06 (11G18 14G35)},
      MRNUMBER = {92d:11039},
      MRREVIEWER = {C. Maclachlan},
      ZBLNUMBER={0717.14014},
      }
  • [CFS82] I. P. Cornfeld, S. V. Fomin, and Y. G. Sinaui, Ergodic Theory, New York: Springer-Verlag, 1982.
    @book {CFS82, MRKEY = {832433},
      AUTHOR = {Cornfeld, I. P. and Fomin, S. V. and Sina{\u\i},
      Ya. G.},
      TITLE = {Ergodic Theory},
      SERIES = {Grundl. Math. Wissen.},
      NUMBER = {245},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1982},
      PAGES = {x+486},
      ISBN = {0-387-90580-4},
      MRCLASS = {28D05 (54H20 58F11)},
      MRNUMBER = {87f:28019},
      MRREVIEWER = {D. Newton},
      ZBLNUMBER = {0493.28007},
      }
  • [De70] P. Deligne, Équations Différentielles à Points Singuliers Réguliers, New York: Springer-Verlag, 1970, vol. 163.
    @book {De70, MRKEY = {0417174},
      AUTHOR = {Deligne, Pierre},
      TITLE = {\'{E}quations Différentielles à Points Singuliers Réguliers},
      SERIES = {Lecture Notes in Math.},
      VOLUME={163},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1970},
      PAGES = {iii+133},
      MRCLASS = {14D05 (14C30)},
      MRNUMBER = {54 \#5232},
      MRREVIEWER = {Helmut Hamm},
      ZBLNUMBER = {0244.14004},
      }
  • [De87] P. Deligne, "Un théorème de finitude pour la monodromie," in Discrete Groups in Geometry and Analysis, Boston, MA: Birkhäuser, 1987, pp. 1-19.
    @incollection {De87, MRKEY = {900821},
      AUTHOR = {Deligne, P.},
      TITLE = {Un théorème de finitude pour la monodromie},
      BOOKTITLE = {Discrete Groups in Geometry and Analysis},
      VENUE={{N}ew {H}aven, {C}onn., 1984},
      SERIES = {Progr. Math.},
      NUMBER = {67},
      PAGES = {1--19},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1987},
      MRCLASS = {14D05 (14D20 32G13)},
      MRNUMBER = {88h:14013},
      MRREVIEWER = {L. B{\u{a}}descu},
      ZBLNUMBER = {0656.14010},
      }
  • [Fo02] Go to document G. Forni, "Deviation of ergodic averages for area-preserving flows on surfaces of higher genus," Ann. of Math., vol. 155, iss. 1, pp. 1-103, 2002.
    @article {Fo02, MRKEY = {1888794},
      AUTHOR = {Forni, Giovanni},
      TITLE = {Deviation of ergodic averages for area-preserving flows on surfaces of higher genus},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {155},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {1--103},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {37A25 (32G15 37C40 37D25 37D40 37E35 53D25)},
      MRNUMBER = {2003g:37009},
      MRREVIEWER = {Howard Masur},
      DOI = {10.2307/3062150},
      ZBLNUMBER = {1034.37003},
      }
  • [Fo05] Go to document G. Forni, "On the Lyapunov exponents of the Kontsevich-Zorich cocycle," in Handbook of Dynamical Systems, Elsevier B. V., Amsterdam, 2006, vol. 1B, pp. 549-580.
    @incollection {Fo05, MRKEY = {2186248},
      AUTHOR = {Forni, Giovanni},
      TITLE = {On the {L}yapunov exponents of the {K}ontsevich-{Z}orich cocycle},
      BOOKTITLE = {Handbook of Dynamical Systems},
      VOLUME={1{B}},
      PAGES = {549--580},
      PUBLISHER = {Elsevier B. V., Amsterdam},
      YEAR = {2006},
      MRCLASS = {37A25 (32G15 37A20 37C40 37E20 37F30)},
      MRNUMBER = {2010c:37004},
      MRREVIEWER = {Serge L. Tabachnikov},
      DOI = {10.1016/S1874-575X(06)80033-0},
      ZBLNUMBER = {1130.37302},
      }
  • [HaSt] Go to document D. Harbater and K. F. Stevenson, "Patching and thickening problems," J. Algebra, vol. 212, iss. 1, pp. 272-304, 1999.
    @article {HaSt, MRKEY = {1670658},
      AUTHOR = {Harbater, David and Stevenson, Katherine F.},
      TITLE = {Patching and thickening problems},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {212},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {272--304},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {14B10 (12F12 14B12 14H30)},
      MRNUMBER = {2000b:14004},
      MRREVIEWER = {Hiroaki Nakamura},
      DOI = {10.1006/jabr.1998.7574},
      ZBLNUMBER = {0944.14013},
      }
  • [Ho09] P. Hooper, Grid graphs and lattice surfaces, preprint, 2009.
    @misc{Ho09,
      author={Hooper, P.},
      TITLE={Grid graphs and lattice surfaces, preprint},
      YEAR={2009},
      }
  • [HuSc01] Go to document P. Hubert and T. A. Schmidt, "Invariants of translation surfaces," Ann. Inst. Fourier $($Grenoble$)$, vol. 51, iss. 2, pp. 461-495, 2001.
    @article {HuSc01, MRKEY = {1824961},
      AUTHOR = {Hubert, P. and Schmidt, T. A.},
      TITLE = {Invariants of translation surfaces},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {51},
      YEAR = {2001},
      NUMBER = {2},
      PAGES = {461--495},
      ISSN = {0373-0956},
      CODEN = {AIFUA7},
      MRCLASS = {32G15 (37D50 57N16)},
      MRNUMBER = {2003e:32023},
      MRREVIEWER = {Athanase Papadopoulos},
      URL = {http://aif.cedram.org/item?id=AIF_2001__51_2_461_0},
      ZBLNUMBER = {0985.32008},
      }
  • [Ka70] Go to document N. M. Katz, "Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin," Inst. Hautes Études Sci. Publ. Math., iss. 39, pp. 175-232, 1970.
    @article {Ka70, MRKEY = {0291177},
      AUTHOR = {Katz, Nicholas M.},
      TITLE = {Nilpotent connections and the monodromy theorem: {A}pplications of a result of {T}urrittin},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {39},
      YEAR = {1970},
      PAGES = {175--232},
      ISSN = {0073-8301},
      MRCLASS = {14A99 (32C15)},
      MRNUMBER = {45 \#271},
      MRREVIEWER = {S. S. Shatz},
      URL = {http://www.numdam.org/item?id=PMIHES_1970__39__175_0},
      ZBLNUMBER = {0221.14007},
      }
  • [KeSm00] Go to document R. Kenyon and J. Smillie, "Billiards on rational-angled triangles," Comment. Math. Helv., vol. 75, iss. 1, pp. 65-108, 2000.
    @article {KeSm00, MRKEY = {1760496},
      AUTHOR = {Kenyon, Richard and Smillie, John},
      TITLE = {Billiards on rational-angled triangles},
      JOURNAL = {Comment. Math. Helv.},
      FJOURNAL = {Commentarii Mathematici Helvetici},
      VOLUME = {75},
      YEAR = {2000},
      NUMBER = {1},
      PAGES = {65--108},
      ISSN = {0010-2571},
      CODEN = {COMHAX},
      MRCLASS = {37D50 (37J99)},
      MRNUMBER = {2001e:37046},
      MRREVIEWER = {Serge L. Tabachnikov},
      DOI = {10.1007/s000140050113},
      ZBLNUMBER = {0967.37019},
      }
  • [KMS86] Go to document S. Kerckhoff, H. Masur, and J. Smillie, "Ergodicity of billiard flows and quadratic differentials," Ann. of Math., vol. 124, iss. 2, pp. 293-311, 1986.
    @article {KMS86, MRKEY = {855297},
      AUTHOR = {Kerckhoff, Steven and Masur, Howard and Smillie, John},
      TITLE = {Ergodicity of billiard flows and quadratic differentials},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {124},
      YEAR = {1986},
      NUMBER = {2},
      PAGES = {293--311},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {58F17 (30F30 58F11)},
      MRNUMBER = {88f:58122},
      MRREVIEWER = {Eugene Gutkin},
      DOI = {10.2307/1971280},
      ZBLNUMBER = {0637.58010},
      }
  • [Ko97] M. Kontsevich, "Lyapunov exponents and Hodge theory," in The Mathematical Beauty of Physics, World Sci. Publ., River Edge, NJ, 1997, pp. 318-332.
    @incollection {Ko97, MRKEY = {1490861},
      AUTHOR = {Kontsevich, M.},
      TITLE = {Lyapunov exponents and {H}odge theory},
      BOOKTITLE = {The Mathematical Beauty of Physics },
      VENUE={{S}aclay, 1996},
      SERIES = {Adv. Ser. Math. Phys.},
      NUMBER = {24},
      PAGES = {318--332},
      PUBLISHER = {World Sci. Publ., River Edge, NJ},
      YEAR = {1997},
      MRCLASS = {58F11 (82C28)},
      MRNUMBER = {99b:58147},
      MRREVIEWER = {Boris Hasselblatt},
      ZBLNUMBER = {1058.37508},
      }
  • [Kr03] R. Krikorian, "Déviations de moyennes ergodiques, flots de Teichmüller et cocycle de Kontsevich-Zorich," in Séminaire Bourbaki 927 (2003), Paris: Soc. Math. France, 2005, vol. 299, pp. 59-93.
    @incollection{Kr03, MRKEY = {2167202},
      AUTHOR = {Krikorian, Rapha{ë}l},
      TITLE = {Déviations de moyennes ergodiques, flots de {T}eichmüller et cocycle de {K}ontsevich-{Z}orich},
      BOOKTITLE = {S{é}minaire Bourbaki {\bf 927} {\rm (2003)}},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      PUBLISHER={Soc. Math. France},
      ADDRESS={Paris},
      VOLUME = {299},
      YEAR = {2005},
      PAGES = {59--93},
      ISSN = {0303-1179},
      MRCLASS = {37E35 (32G15 37A25 37C40 37D40)},
      MRNUMBER = {2006k:37109},
      MRREVIEWER = {Boris Hasselblatt},
      ZBLNUMBER={1094.37021},
      }
  • [Lo05] Go to document P. Lochak, "On arithmetic curves in the moduli spaces of curves," J. Inst. Math. Jussieu, vol. 4, iss. 3, pp. 443-508, 2005.
    @article {Lo05, MRKEY = {2197065},
      AUTHOR = {Lochak, Pierre},
      TITLE = {On arithmetic curves in the moduli spaces of curves},
      JOURNAL = {J. Inst. Math. Jussieu},
      FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu},
      VOLUME = {4},
      YEAR = {2005},
      NUMBER = {3},
      PAGES = {443--508},
      ISSN = {1474-7480},
      MRCLASS = {32G15 (14H10 37F99)},
      MRNUMBER = {2006m:32010},
      MRREVIEWER = {Martin M{ö}ller},
      DOI = {10.1017/S1474748005000101},
      ZBLNUMBER = {1094.14018},
      }
  • [Ma91] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, New York: Springer-Verlag, 1991.
    @book {Ma91, MRKEY = {1090825},
      AUTHOR = {Margulis, G. A.},
      TITLE = {Discrete Subgroups of Semisimple {L}ie Groups},
      SERIES = {Ergeb. Math. Grenzgeb.},
      NUMBER = {17},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1991},
      PAGES = {x+388},
      ISBN = {3-540-12179-X},
      MRCLASS = {22E40 (20Hxx 22-02 22D40)},
      MRNUMBER = {92h:22021},
      MRREVIEWER = {Gopal Prasad},
      ZBLNUMBER = {0732.22008},
      }
  • [Mr75] Go to document H. Masur, "On a class of geodesics in Teichmüller space," Ann. of Math., vol. 102, iss. 2, pp. 205-221, 1975.
    @article {Mr75, MRKEY = {0385173},
      AUTHOR = {Masur, Howard},
      TITLE = {On a class of geodesics in {T}eichmüller space},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {102},
      YEAR = {1975},
      NUMBER = {2},
      PAGES = {205--221},
      ISSN = {0003-486X},
      MRCLASS = {32G15 (30A52)},
      MRNUMBER = {52 \#6038},
      MRREVIEWER = {Jane Gilman},
      DOI = {10.2307/1971031},
      ZBLNUMBER={0322.32010},
      }
  • [MaTa02] Go to document H. Masur and S. Tabachnikov, "Rational billiards and flat structures," in Handbook of Dynamical Systems, Amsterdam: North-Holland, 2002, vol. 1A, pp. 1015-1089.
    @incollection {MaTa02, MRKEY = {1928530},
      AUTHOR = {Masur, Howard and Tabachnikov, Serge},
      TITLE = {Rational billiards and flat structures},
      BOOKTITLE = {Handbook of Dynamical Systems},
      VOLUME={1{A}},
      PAGES = {1015--1089},
      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
      YEAR = {2002},
      MRCLASS = {37-02 (28D05 30F60 37A25 37D50 37F99)},
      MRNUMBER = {2003j:37002},
      MRREVIEWER = {Richard Kenyon},
      DOI = {10.1016/S1874-575X(02)80015-7},
      ZBLNUMBER = {1057.37034},
      }
  • [McM03] Go to document C. T. McMullen, "Billiards and Teichmüller curves on Hilbert modular surfaces," J. Amer. Math. Soc., vol. 16, iss. 4, pp. 857-885, 2003.
    @article {McM03, MRKEY = {1992827},
      AUTHOR = {McMullen, Curtis T.},
      TITLE = {Billiards and {T}eichmüller curves on {H}ilbert modular surfaces},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {16},
      YEAR = {2003},
      NUMBER = {4},
      PAGES = {857--885},
      ISSN = {0894-0347},
      MRCLASS = {32G15 (37D50)},
      MRNUMBER = {2004f:32015},
      MRREVIEWER = {Richard Kenyon},
      DOI = {10.1090/S0894-0347-03-00432-6},
      ZBLNUMBER={1030.32012},
      }
  • [McM04c] Go to document C. T. McMullen, "Teichmüller curves in genus two: torsion divisors and ratios of sines," Invent. Math., vol. 165, iss. 3, pp. 651-672, 2006.
    @article {McM04c, MRKEY = {2242630},
      AUTHOR = {McMullen, Curtis T.},
      TITLE = {Teichmüller curves in genus two: torsion divisors and ratios of sines},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {165},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {651--672},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14H15 (30F10 32G15)},
      MRNUMBER = {2007f:14023},
      MRREVIEWER = {Thomas A. Schmidt},
      DOI = {10.1007/s00222-006-0511-2},
      ZBLNUMBER={1103.14014},
      }
  • [McM05] Go to document C. T. McMullen, "Prym varieties and Teichmüller curves," Duke Math. J., vol. 133, iss. 3, pp. 569-590, 2006.
    @article {McM05, MRKEY = {2228463},
      AUTHOR = {McMullen, Curtis T.},
      TITLE = {Prym varieties and {T}eichmüller curves},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {133},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {569--590},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {32G15 (14H40 37D50 57M50)},
      MRNUMBER = {2007a:32018},
      MRREVIEWER = {Thomas A. Schmidt},
      DOI = {10.1215/S0012-7094-06-13335-5},
      ZBLNUMBER={1099.14018},
      }
  • [Mo99] S. Mochizuki, Foundations of $p$-adic Teichmüller Theory, Providence, RI: Amer. Math. Soc., 1999.
    @book {Mo99, MRKEY = {1700772},
      AUTHOR = {Mochizuki, Shinichi},
      TITLE = {Foundations of {$p$}-adic {T}eichmüller Theory},
      SERIES = {{\rm AMS/IP} Studies in Advanced Mathematics},
      NUMBER = {11},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1999},
      PAGES = {xii+529},
      ISBN = {0-8218-1190-8},
      MRCLASS = {14F30 (11G20 14D22 14G20)},
      MRNUMBER = {2000i:14025},
      MRREVIEWER = {Dan Abramovich},
      ZBLNUMBER={0969.14013},
      }
  • [Mo04b] Go to document M. Möller, "Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve," Invent. Math., vol. 165, iss. 3, pp. 633-649, 2006.
    @article {Mo04b, MRKEY = {2242629},
      AUTHOR = {M{ö}ller, Martin},
      TITLE = {Periodic points on {V}eech surfaces and the {M}ordell-{W}eil group over a {T}eichmüller curve},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {165},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {633--649},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14D07 (32G15)},
      MRNUMBER = {2007e:14012},
      MRREVIEWER = {Thomas A. Schmidt},
      DOI = {10.1007/s00222-006-0510-3},
      ZBLNUMBER={1111.14019},
      }
  • [Mo04a] Go to document M. Möller, "Variations of Hodge structures of a Teichmüller curve," J. Amer. Math. Soc., vol. 19, iss. 2, pp. 327-344, 2006.
    @article {Mo04a, MRKEY = {2188128},
      AUTHOR = {M{ö}ller, Martin},
      TITLE = {Variations of {H}odge structures of a {T}eichmüller curve},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {19},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {327--344},
      ISSN = {0894-0347},
      MRCLASS = {32G15 (14D07 32G20)},
      MRNUMBER = {2007b:32026},
      MRREVIEWER = {Thomas A. Schmidt},
      DOI = {10.1090/S0894-0347-05-00512-6},
      ZBLNUMBER={1090.32004},
      }
  • [Os68] V. I. Oseledec, "A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems," Trans. Moscow Math. Soc., vol. 19, pp. 197-231, 1968.
    @article {Os68, MRKEY = {0240280},
      AUTHOR = {Oseledec, V. I.},
      TITLE = {A multiplicative ergodic theorem. {C}haracteristic {L}japunov, exponents of dynamical systems},
      JOURNAL = {Trans. Moscow Math. Soc.},
      VOLUME = {19},
      YEAR = {1968},
      PAGES = {197--231},
      ISSN = {0134-8663},
      MRCLASS = {28.70 (34.00)},
      MRNUMBER = {39 \#1629},
      MRREVIEWER = {J{ó}zsef Sz{ü}cs},
      ZBLNUMBER={0236.93034},
      }
  • [Pe84] Go to document C. A. M. Peters, "A criterion for flatness of Hodge bundles over curves and geometric applications," Math. Ann., vol. 268, iss. 1, pp. 1-19, 1984.
    @article {Pe84, MRKEY = {744325},
      AUTHOR = {Peters, C. A. M.},
      TITLE = {A criterion for flatness of {H}odge bundles over curves and geometric applications},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {268},
      YEAR = {1984},
      NUMBER = {1},
      PAGES = {1--19},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {14D05 (14C30 14J15 32J25)},
      MRNUMBER = {85m:14014},
      MRREVIEWER = {Jerome William Hoffman},
      DOI = {10.1007/BF01463870},
      ZBLNUMBER = {0548.14004},
      }
  • [Pu01] Go to document J. Puchta, "On triangular billiards," Comment. Math. Helv., vol. 76, iss. 3, pp. 501-505, 2001.
    @article {Pu01, MRKEY = {1854695},
      AUTHOR = {Puchta, Jan-Christoph},
      TITLE = {On triangular billiards},
      JOURNAL = {Comment. Math. Helv.},
      FJOURNAL = {Commentarii Mathematici Helvetici},
      VOLUME = {76},
      YEAR = {2001},
      NUMBER = {3},
      PAGES = {501--505},
      ISSN = {0010-2571},
      CODEN = {COMHAX},
      MRCLASS = {37D50 (11N36)},
      MRNUMBER = {2002f:37060},
      MRREVIEWER = {Richard Kenyon},
      DOI = {10.1007/PL00013215},
      ZBLNUMBER = {01701563},
      }
  • [Si72] Go to document D. Singerman, "Finitely maximal Fuchsian groups," J. London Math. Soc., vol. 6, pp. 29-38, 1972.
    @article {Si72, MRKEY = {0322165},
      AUTHOR = {Singerman, David},
      TITLE = {Finitely maximal {F}uchsian groups},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {6},
      YEAR = {1972},
      PAGES = {29--38},
      ISSN = {0024-6107},
      MRCLASS = {30A58},
      MRNUMBER = {48 \#529},
      MRREVIEWER = {A. M. Macbeath},
      DOI = {10.1112/jlms/s2-6.1.29},
      ZBLNUMBER = {0251.20052},
      }
  • [Ta77] Go to document K. Takeuchi, "Arithmetic triangle groups," J. Math. Soc. Japan, vol. 29, iss. 1, pp. 91-106, 1977.
    @article {Ta77, MRKEY = {0429744},
      AUTHOR = {Takeuchi, Kisao},
      TITLE = {Arithmetic triangle groups},
      JOURNAL = {J. Math. Soc. Japan},
      FJOURNAL = {Journal of the Mathematical Society of Japan},
      VOLUME = {29},
      YEAR = {1977},
      NUMBER = {1},
      PAGES = {91--106},
      ISSN = {0025-5645},
      MRCLASS = {10D05},
      MRNUMBER = {55 \#2754},
      MRREVIEWER = {Roy Fuller},
      DOI = {10.2969/jmsj/02910091},
      ZBLNUMBER={0365.20055},
      }
  • [Ve89] Go to document W. A. Veech, "Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards," Invent. Math., vol. 97, iss. 3, pp. 553-583, 1989.
    @article {Ve89, MRKEY = {1005006},
      AUTHOR = {Veech, W. A.},
      TITLE = {Teichmüller curves in moduli space, {E}isenstein series and an application to triangular billiards},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {97},
      YEAR = {1989},
      NUMBER = {3},
      PAGES = {553--583},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {58F17 (11F11 32G15)},
      MRNUMBER = {91h:58083a},
      DOI = {10.1007/BF01388890},
      ZBLNUMBER={0676.32006},
      }
  • [ViZu04] Go to document E. Viehweg and K. Zuo, "A characterization of certain Shimura curves in the moduli stack of abelian varieties," J. Differential Geom., vol. 66, iss. 2, pp. 233-287, 2004.
    @article {ViZu04, MRKEY = {2106125},
      AUTHOR = {Viehweg, Eckart and Zuo, Kang},
      TITLE = {A characterization of certain {S}himura curves in the moduli stack of abelian varieties},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {66},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {233--287},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {14D07 (14G35 14K10)},
      MRNUMBER = {2006a:14015},
      MRREVIEWER = {Conjeeveram S. Rajan},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1102538611},
      ZBLNUMBER = {1078.11043},
      }
  • [Vo96] Go to document Y. B. Vorobets, "Plane structures and billiards in rational polygons: the Veech alternative," Russian Math. Surveys, vol. 51, iss. 5(311), pp. 779-817, 1996.
    @article {Vo96, MRKEY = {1436653},
      AUTHOR = {Vorobets, Ya. B.},
      TITLE = {Plane structures and billiards in rational polygons: the {V}eech alternative},
      JOURNAL = {Russian Math. Surveys},
      VOLUME = {51},
      YEAR = {1996},
      NUMBER = {5(311)},
      PAGES = {779--817},
      ISSN = {0042-1316},
      MRCLASS = {58F11 (58F05 58F17)},
      MRNUMBER = {97j:58092},
      MRREVIEWER = {Serge E. Troubetzkoy},
      DOI = {10.1070/RM1996v051n05ABEH002993},
      }
  • [Wa98] Go to document C. C. Ward, "Calculation of Fuchsian groups associated to billiards in a rational triangle," Ergodic Theory Dynam. Systems, vol. 18, iss. 4, pp. 1019-1042, 1998.
    @article {Wa98, MRKEY = {1645350},
      AUTHOR = {Ward, Clayton C.},
      TITLE = {Calculation of {F}uchsian groups associated to billiards in a rational triangle},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {18},
      YEAR = {1998},
      NUMBER = {4},
      PAGES = {1019--1042},
      ISSN = {0143-3857},
      MRCLASS = {30F35 (20H10)},
      MRNUMBER = {2000b:30065},
      MRREVIEWER = {Bruno Zimmermann},
      DOI = {10.1017/S0143385798117479},
      ZBLNUMBER = {0915.58059},
      }
  • [We99] S. Wewers, "Deformation of tame admissible covers of curves," in Aspects of Galois Theory, Cambridge: Cambridge Univ. Press, 1999, pp. 239-282.
    @incollection {We99, MRKEY = {1708609},
      AUTHOR = {Wewers, Stefan},
      TITLE = {Deformation of tame admissible covers of curves},
      BOOKTITLE = {Aspects of {G}alois Theory},
      VENUE={{G}ainesville, {FL},
      1996},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      NUMBER = {256},
      PAGES = {239--282},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1999},
      MRCLASS = {14H30},
      MRNUMBER = {2001b:14048},
      ZBLNUMBER = {0995.14008},
      }
  • [Stefandiss] S. Wewers, "Construction of Hurwitz spaces," PhD Thesis , Essen, 1998.
    @phdthesis{Stefandiss,
      author={Wewers, Stefan},
      TITLE={Construction of Hurwitz spaces},
      TYPE={thesis},
      SCHOOL={Essen},
      YEAR={1998},
      NOTE={preprint no. 21 of the IEM},
      }
  • [Yo87] M. Yoshida, Fuchsian Differential Equations, Braunschweig: Friedr. Vieweg & Sohn, 1987, vol. 11.
    @book {Yo87, MRKEY = {986252},
      AUTHOR = {Yoshida, Masaaki},
      TITLE = {Fuchsian Differential Equations},
      SERIES = {Aspects of Mathematics},
      VOLUME={11},
      PUBLISHER = {Friedr. Vieweg \& Sohn},
      ADDRESS = {Braunschweig},
      YEAR = {1987},
      PAGES = {xiv+215},
      ISBN = {3-528-08971-7},
      MRCLASS = {32G13 (14D05 22E40 32C40 32G20 34A30 35A30)},
      MRNUMBER = {90f:32025},
      MRREVIEWER = {Takeshi Sasaki},
      ZBLNUMBER = {0618.35001},
      }
  • [Zo96] Go to document A. Zorich, "Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents," Ann. Inst. Fourier $($Grenoble$)$, vol. 46, iss. 2, pp. 325-370, 1996.
    @article {Zo96, MRKEY = {1393518},
      AUTHOR = {Zorich, Anton},
      TITLE = {Finite {G}auss measure on the space of interval exchange transformations. {L}yapunov exponents},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {46},
      YEAR = {1996},
      NUMBER = {2},
      PAGES = {325--370},
      ISSN = {0373-0956},
      CODEN = {AIFUA7},
      MRCLASS = {58F11 (28D05)},
      MRNUMBER = {97f:58081},
      MRREVIEWER = {Vadim A. Ka{\u\i}manovich},
      URL = {http://www.numdam.org/item?id=AIF_1996__46_2_325_0},
      ZBLNUMBER = {0853.28007},
      }

Authors

Irene I. Bouw

Institute of Pure Mathematics
Ulm University
D-89069 Ulm
Germany

Martin Möller

Institut für Mathematik
Goethe-Universität
Robert-Mayer-Str. 6-8
D-60054 Frankfurt am Main
Germany