We definite invariants of translation surfaces which refine Veech groups. These aid in exact determination of Veech groups. We give examples where two surfaces of isomorphic Veech group cannot even share a common tree of balanced affine coverings. We also show that there exist translation surfaces of isomorphic Veech groups which cannot affinely cover any common surface. We also extend a result of Gutkin and Judge and thereby give the first examples of noncompact Fuchsian groups which cannot appear as Veech groups. We give an infinite family of these.
Nous définissons, pour une surface de translation, un invariant de revêtement affine. Cet invariant est un raffinement du groupe de Veech. Il nous permet de construire un exemple de deux surfaces de translation qui ont le même groupe de Veech et qui ne sont pas dans le même arbre de revêtements affines.
Keywords: flat surfaces, Teichmüller disks, billiards
Mot clés : surfaces plates, disques de Teichmüller, billards
Hubert, Pascal 1; Schmidt, Thomas A. 2
@article{AIF_2001__51_2_461_0, author = {Hubert, Pascal and Schmidt, Thomas A.}, title = {Invariants of translation surfaces}, journal = {Annales de l'Institut Fourier}, pages = {461--495}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {51}, number = {2}, year = {2001}, doi = {10.5802/aif.1829}, zbl = {0985.32008}, mrnumber = {1824961}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1829/} }
TY - JOUR AU - Hubert, Pascal AU - Schmidt, Thomas A. TI - Invariants of translation surfaces JO - Annales de l'Institut Fourier PY - 2001 SP - 461 EP - 495 VL - 51 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1829/ DO - 10.5802/aif.1829 LA - en ID - AIF_2001__51_2_461_0 ER -
%0 Journal Article %A Hubert, Pascal %A Schmidt, Thomas A. %T Invariants of translation surfaces %J Annales de l'Institut Fourier %D 2001 %P 461-495 %V 51 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1829/ %R 10.5802/aif.1829 %G en %F AIF_2001__51_2_461_0
Hubert, Pascal; Schmidt, Thomas A. Invariants of translation surfaces. Annales de l'Institut Fourier, Volume 51 (2001) no. 2, pp. 461-495. doi : 10.5802/aif.1829. https://aif.centre-mersenne.org/articles/10.5802/aif.1829/
[A] Ergodicité générique des billards polygonaux (d'après Kerckhoff, Masur, Smillie), Séminaire Bourbaki 1987/88 (Astérisque 161/162), Volume No 696-5 (1988), pp. 203-221 | Numdam | Zbl
[AF] Un exemple de difféomorphisme pseudo-Anosov, C. R. Acad. Sci. Paris, sér. I Math., Volume 312 (1991), pp. 241-244 | MR | Zbl
[AH] Fractions continues sur les surfaces de Veech (To appear in J. Anal. Math.) | MR | Zbl
[B] The geometry of discrete groups, Grad. Text Math., 91, Springer-Verlag, Berlin, 1984 | MR | Zbl
[BC] An extension of Lagrange's theorem to interval exchange tranformations over quadratic fields, J. Anal. Math., Volume 72 (1997), pp. 21-44 | DOI | MR | Zbl
[BL] Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften, vol. 302, Springer-Verlag, Berlin, 1992 | MR | Zbl
[C] Understanding groups like , Groups, difference sets, and the Monster (Columbus, OH, 1993) (Ohio State Univ. Math. Res. Inst. Publ.), Volume 4 (1996), pp. 327-343 | Zbl
[EG] Teichmüller disks and Veech's -structures, Extremal Riemann surfaces (Contemp. Math.), Volume 201 (1997), pp. 165-189 | Zbl
[EM] Pointwise asymptotic formulas on flat surfaces (1999) (Preprint)
[F] Some compact invariant sets for hyperbolic linear automorphisms of torii, Ergodic Theory Dynam. Systems, Volume 8 (1988), pp. 191-204 | MR | Zbl
[G] Branched coverings and closed geodesics in flat surfaces, with applications to billiards, Dynamical Systems from Crystal to Chaos (2000), pp. 259-273
[GJ1] The geometry and arithmetic of translation surfaces with applications to polygonal billiards, Math. Res. Lett., Volume 3 (1996), pp. 391-403 | MR | Zbl
[GJ2] Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J., Volume 103 (2000), pp. 191-213 | DOI | MR | Zbl
[Ha1] On certain families of compact Riemann surfaces, Mapping class groups and moduli spaces of Riemann surfaces (Contemp. Math.), Volume 150 (1993), pp. 137-148 | Zbl
[Ha2] Drawings, triangle groups and algebraic curves (1997) (Preprint)
[He] Bestimmung der Kommensurabilitätsklasse der Hilbertschen Modulgruppe, Math. Z., Volume 92 (1966), pp. 269-280 | DOI | MR | Zbl
[HS] Veech groups and polygonal coverings, J. Physics and Geom., Volume 35 (2000), pp. 75-91 | DOI | MR | Zbl
[K] Fuchsian groups, Chicago Lectures in Math., Univ. Chicago Press, Chicago, 1992 | MR | Zbl
[KMS] Ergodicity of billiard flows and quadratic differentials, Ann. of Math., Volume 124 (1986), pp. 293-311 | DOI | MR | Zbl
[Kr] The Carathéodory metric on abelian Teichmüller disks, J. Anal. Math., Volume 40 (1981), pp. 129-143 | DOI | MR | Zbl
[KS] Billiards in rational-angled triangles, Commentarii Math. Helvetici, Volume 75 (2000), pp. 65-108 | DOI | MR | Zbl
[KZ] Topological transitivity of billiard flows in polygons, Math. Notes, Volume 18 (1975), pp. 760-764 | DOI | Zbl
[L] Über die Heckeschen Gruppen , II, Math. Ann., Volume 211 (1974), pp. 63-86 | DOI | MR | Zbl
[Mar] Discrete subgroups of semisimple Lie groups, Springer-Verlag, New York, 1991 | MR | Zbl
[Mas] Closed geodesics for quadratic differentials with applications to billiards, Duke J. Math., Volume 53 (1986), pp. 307-314 | MR | Zbl
[MR] Commensurability classes of two generator Fuchsian groups, Discrete groups and geometry (London Math. Soc. Lecture Note Series), Volume 173 (1992), pp. 171-189 | Zbl
[MT] Rational billiards and flat structures (Max-Planck-Institut für Mathematik, Bonn, preprint, 55) | MR | Zbl
[S] Period matrices of hyperelliptic curves, Manuscripta Math., Volume 78 (1993), pp. 369-380 | DOI | MR | Zbl
[T] Billiards, Panoramas et Synthèses 1, Soc. Math. France, Paris, 1995 | Zbl
[Tr] Les surfaces euclidiennes à singularités coniques, Enseign. Math. (2), Volume 32 (1986), pp. 79-94 | MR | Zbl
[Ve1] Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., Volume 97 (1989), pp. 553-583 | DOI | MR | Zbl
[Ve2] The billiard in a regular polygon, Geom. Funct. Anal., Volume 2 (1992), pp. 341-379 | DOI | MR | Zbl
[Vo] Plane structures and billiards in rational polyhedra: the Veech alternative (Russian), Uspekhi Mat. Nauk, Volume 51 (1996) | MR | Zbl
[W] Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergodic Theory Dynam. Systems, Volume 18 (1998), pp. 1019-1042 | DOI | MR | Zbl
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