Abstract
It is well-known that Teichmüller discs that pass through “integer points” of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmüller discs is mostly unexplored: their number, genus, area, cusps, etc.
We prove that in genus two all translation surfaces in \(\mathcal{H}(2)\) tiled by a prime number n > 3 of squares fall into exactly two Teichmüller discs, only one of them with elliptic points, and that the genus of these discs has a cubic growth rate in n.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[Ca] K. Calta,Veech surfaces and complete periodicity in genus 2, Journal of the American Mathematical Society17 (2004), 871–908.
[CuPa] C. J. Cummins and S. Pauli,Congruence subgroups of PSL (2,Z) of genus less than or equal to 24, Experimental Mathematics12 (2003), 243–255.|http://www.math.tu-berlin.ed/pauli/congruence/|url
[EsMaSc] A. Eskin, H. Masur and M. Schmoll,Billiards in rectangles with barriers, Duke Mathematical Journal118 (2003), 427–463.
[EsOk] A. Eskin and A. Okounkov,Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Inventiones Mathematicae145 (2001), 59–103.
[Gu] E. Gutkin,Billiards on almost integrable polyhedral surfaces, Ergodic Theory and Dynamical Systems4 (1984), 569–584.
[GuHuSc] E. Gutkin, P. Hubert and T. Schmidt,Affine diffeomorphisms of translation surfaces: periodic points, Fuchsian groups, and arithmeticity, Annales Scientifiques de l'École Normale Supérieure (4)36 (2003), 847–866.
[GuJu1] E. Gutkin and C. Judge,The geometry and arithmetic of translation surfaces with applications to polygonal billiards, Mathematical Research Letters3 (1996), 391–403.
[GuJu2] E. Gutkin and C. Judge,Affine mappings of translation surfaces: geometry and arithmetic, Duke Mathematical Journal103 (2000), 191–213.
[Hu] P. Hubert,Étude géométrique et combinatoire de systèmes dynamiques d'entropie nulle, Habilitation à diriger des recherches, 2002.
[HL] P. Hubert and S. Lelièvre,Noncongruence subgroups in H(2), International Mathematics Research Notices2005, No. 1 (2005), 47–64.
[HuSc00] P. Hubert and T. Schmidt,Veech groups and polygonal coverings, Journal of Geometry and Physics35 (2000), 75–91.
[HuSc01] P. Hubert and T. Schmidt,Invariants of translation surfaces, Annales de l'Institut Fourier (Grenoble)51 (2001), 461–495.
[Ka] E. Kani,The number of genus 2 covers of an elliptic curve, Preprint (2003).
[KeMaSm] S. Kerckhoff, and H. Masur and J. Smillie,Ergodicity of billiard flows and quadratic differentials, Annals of Mathematics (2)124 (1986), 293–311.
[KeSm] R. Kenyon and J. Smillie,Billiards on rational-angled triangles, Commentarii Mathematici Helvetici75 (2000), 65–108.
[KoZo] M. Kontsevich and A. Zorich,Connected components of the moduli space of holomorphic differentials with prescribed singularities, Inventiones Mathematicae153 (2003), 631–678.
[Ma] H. Masur,Interval exchange transformations and measured foliations, Annals of Mathematics (2)115 (1982), 169–200.
[Mc] C. T. McMullen,Billiards and Teichmüller curves on Hilbert modular surfaces, Journal of the American Mathematical Society16 (2003), 857–885.
[Mc2] C. T. McMullen,Teichmüller curves in genus two: discriminant and spin, Mathematische Annalen, to appear.
[Mö] M. Möller,Teichmüller curves, Galois actions and ĜT-relations, Mathematische Nachrichten278 (2005), 1061–1077
[Ri] J. Rivat, Private communication.
[Schmi] G. Schmithüsen,An algorithm for finding the Veech group of an origami, Experimental Mathematics13 (2004), 459–472.
[Schmo] M. Schmoll,On the asymptotic quadratic growth rate of saddle connections and periodic orbits on marked flat tori, Geometric and Functional Analysis12 (2002), 622–649.
[Th] W. P. Thurston,On the geometry and dynamics of diffeomorphisms of surfaces, Bulletin of the American Mathematical Society (N.S.)19 (1988), 417–431.
[Ve82] W. A. Veech,Gauss measures for transformations on the space of interval exchange maps, Annals of Mathematics (2)115 (1982), 201–242.
[Ve87] W. A. Veech,Boshernitzan's criterion for unique ergodicity of an interval exchange transformation, Ergodic Theory and Dynamical Systems7 (1987), 149–153.
[Ve89] W. A. Veech,Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Inventiones Mathematicae97 (1989), 553–583.
[Ve92] W. A. Veech,The billiard in a regular polygon, Geometric and Functional Analysis2 (1992), 341–379.
[Ve95] W. A. Veech,Geometric realizations of hyperelliptic curves inAlgorithms, Fractals and Dynamics (Okayama/Kyoto, 1992) (Y. Takahashi, ed.), Plenum, New York, 1995, pp. 217–226.
[Vo] Ya. B. Vorobets,Planar structures and billiards in rational polygons: the Veech alternative, Russian Mathematical Surveys51 (1996), 779–817.
[Wa] C. C. Ward,Calculation of fuchsian groups associated to billiards in a rational triangle, Ergodic Theory and Dynamical Systems18 (1998), 1019–1042.
[Zo] A. Zorich,Square tiled surfaces and Teichmüller volumes of the moduli spaces of abelian differentials inRigidity in Dynamics and Geometry (Cambridge, 2000) (M. Burger and A. Iozzi, eds.), Springer, Berlin, 2002, pp. 459–471.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hubert, P., Lelièvre, S. Prime arithmetic Teichmüller discs in \(\mathcal{H}(2)\). Isr. J. Math. 151, 281–321 (2006). https://doi.org/10.1007/BF02777365
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02777365