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Billiards and Teichmüller curves on Hilbert modular surfaces

Author: Curtis T. McMullen
Journal: J. Amer. Math. Soc. 16 (2003), 857-885
MSC (2000): Primary 32G15; Secondary 37D50, 11F41, 14G35
Published electronically: June 19, 2003
MathSciNet review: 1992827
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Abstract: This paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two. These Teichmüller curves lie on Hilbert modular surfaces parameterizing Abelian varieties with real multiplication. Explicit examples, constructed from L-shaped polygons, give billiard tables with optimal dynamical properties.

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  • [Ah] L. Ahlfors.
    The complex analytic structure of the space of closed Riemann surfaces.
    In Analytic Functions, pages 45-66. Princeton Univ. Press, 1960. MR 23:A1798
  • [Ca] K. Calta.
    Veech surfaces and complete periodicity in genus 2.
    Preprint, 5/2002.
  • [CV] C. Ciliberto and G. van der Geer.
    Subvarieties of the moduli space of curves parametrizing Jacobians with nontrivial endomorphisms.
    Amer. J. Math. 114(1992), 551-570. MR 93d:14044
  • [CVT] C. Ciliberto, G. van der Geer, and M. Teixidor i Bigas.
    On the number of parameters of curves whose Jacobians possess nontrivial endomorphisms.
    J. Algebraic Geom. 1(1992), 215-229. MR 93c:14025
  • [CW] P. Cohen and J. Wolfart.
    Modular embeddings for some non-arithmetic Fuchsian groups.
    Acta Arith. 56(1990), 93-110. MR 92d:11039
  • [CFS] I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai.
    Ergodic Theory.
    Springer-Verlag, 1982. MR 87f:28019
  • [EO] A. Eskin and A. Okounkov.
    Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials.
    Invent. Math. 145(2001), 59-103. MR 2002g:32018
  • [FLP] A. Fathi, F. Laudenbach, and V. Poénaru.
    Travaux de Thurston sur les surfaces.
    Astérisque, volume 66-67, 1979.
  • [GR] B. H. Gross and D. E. Rohrlich.
    Some results on the Mordell-Weil group of the Jacobian of the Fermat curve.
    Invent. Math. 44(1978), 201-224. MR 58:10911
  • [GJ] E. Gutkin and C. Judge.
    Affine mappings of translation surfaces: geometry and arithmetic.
    Duke Math. J. 103(2000), 191-213. MR 2001h:37071
  • [HG] F. Hirzebruch and G. van der Geer.
    Lectures on Hilbert Modular Surfaces.
    Les Presses de l'Université de Montréal, 1981. MR 83i:10037
  • [KS] R. Kenyon and J. Smillie.
    Billiards on rational-angled triangles.
    Comment. Math. Helv. 75(2000), 65-108. MR 2001e:37046
  • [KMS] S. Kerckhoff, H. Masur, and J. Smillie.
    Ergodicity of billiard flows and quadratic differentials.
    Ann. of Math. 124(1986), 293-311. MR 88f:58122
  • [Ko] M. Kontsevich.
    Lyapunov exponents and Hodge theory.
    In The Mathematical Beauty of Physics (Saclay, 1996), pages 318-332. World Sci. Publishing, 1997. MR 99b:58147
  • [KZ] M. Kontsevich and A. Zorich.
    Connected components of the moduli spaces of Abelian differentials with prescribed singularities.
    Preprint, 2002.
  • [Kra] I. Kra.
    The Carathéodory metric on abelian Teichmüller disks.
    J. Analyse Math. 40(1981), 129-143.MR 83m:32027
  • [Li] D. Lind.
    The entropies of topological Markov shifts and a related class of algebraic integers.
    Ergod. Th. & Dynam. Sys. 4(1984), 283-300. MR 86c:58092
  • [Mas1] H. Masur.
    Transitivity properties of the horocyclic and geodesic flows on moduli space.
    J. Analyse Math. 39(1981), 1-10. MR 82k:30047
  • [Mas2] H. Masur.
    Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential.
    In Holomorphic Functions and Moduli I, pages 215-228. Springer-Verlag: MSRI publications volume 10, 1988. MR 90e:30046
  • [Mas3] H. Masur.
    Hausdorff dimension of the set of nonergodic foliations of a quadratic differential.
    Duke Math. J. 66(1992), 387-442. MR 93f:30045
  • [MT] H. Masur and S. Tabachnikov.
    Rational billiards and flat structures.
    In Handbook of Dynamical Systems, Vol. 1A, pages 1015-1089. North-Holland, 2002.
  • [Mc] C. McMullen.
    Teichmüller geodesics of infinite complexity.
    To appear, Acta Math.
  • [Pen] R. Penner.
    Bounds on least dilatations.
    Proc. Amer. Math. Soc. 113(1991), 443-450. MR 91m:57010
  • [Pu] J.-C. Puchta.
    On triangular billiards.
    Comment. Math. Helv. 76(2001), 501-505. MR 2002f:37060
  • [Rap] M. Rapoport.
    Compactifications de l'espace de modules de Hilbert-Blumenthal.
    Compositio Math. 36(1978), 255-335. MR 80j:14009
  • [Roy] H. L. Royden.
    Invariant metrics on Teichmüller space.
    In Contributions to Analysis, pages 393-399. Academic Press, 1974.MR 51:13290
  • [SW] P. S. Schaller and J. Wolfart.
    Semi-arithmetic Fuchsian groups and modular embeddings.
    J. London Math. Soc. 61(2000), 13-24. MR 2001a:11071
  • [Sch] T. A. Schmidt.
    Klein's cubic surface and a `non-arithmetic' curve.
    Math. Ann. 309(1997), 533-539. MR 98m:11036
  • [Tab] S. Tabachnikov.
    Société Mathématique de France, 1995.
  • [vG] G. van der Geer.
    Hilbert Modular Surfaces.
    Springer-Verlag, 1987. MR 89c:11073
  • [V1] W. Veech.
    Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards.
    Invent. Math. 97(1989), 553-583. MR 91h:58083a
  • [V2] W. Veech.
    Moduli spaces of quadratic differentials.
    J. Analyse Math. 55(1990), 117-171. MR 92e:32014
  • [V3] W. Veech.
    The billiard in a regular polygon.
    Geom. Funct. Anal. 2(1992), 341-379. MR 94a:11074
  • [V4] W. Veech.
    Geometric realizations of hyperelliptic curves.
    In Algorithms, Fractals and Dynamics (Okayama/Kyoto, 1992), pages 217-226. Plenum Publishing, 1995. MR 98f:14022
  • [Vo] Ya. B. Vorobets.
    Plane structures and billiards in rational polygons: the Veech alternative.
    Russian Math. Surveys 51(1996), 779-817. MR 97j:58092
  • [Wa] C. C. Ward.
    Calculation of Fuchsian groups associated to billiards in a rational triangle.
    Ergod. Th. & Dynam. Sys. 18(1998), 1019-1042. MR 2000b:30065

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Additional Information

Curtis T. McMullen
Affiliation: Mathematics Department, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138-2901

Received by editor(s): April 8, 2002
Published electronically: June 19, 2003
Additional Notes: Research supported in part by the NSF
Article copyright: © Copyright 2003 American Mathematical Society