Invariants of translation surfaces
Annales de l'Institut Fourier, Volume 51 (2001) no. 2, p. 461-495

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.

DOI : https://doi.org/10.5802/aif.1829
Classification:  30F60,  32G15
Keywords: flat surfaces, Teichmüller disks, billiards 
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {51},
     number = {2},
     year = {2001},
     pages = {461-495},
     doi = {10.5802/aif.1829},
     zbl = {0985.32008},
     mrnumber = {1824961},
     language = {en},
     url = {http://www.numdam.org/item/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. http://www.numdam.org/item/AIF_2001__51_2_461_0/

[A] P. Arnoux Ergodicité générique des billards polygonaux (d'après Kerckhoff, Masur, Smillie), Séminaire Bourbaki 1987/88 (Astérisque 161/162) Tome No 696-5 (1988), pp. 203-221 | Numdam | Zbl 0671.58023

[AF] P. Arnoux; A. Fathi Un exemple de difféomorphisme pseudo-Anosov, C. R. Acad. Sci. Paris, sér. I Math., Tome 312 (1991), pp. 241-244 | MR 1089706 | Zbl 0722.57015

[AH] P. Arnoux; P. Hubert Fractions continues sur les surfaces de Veech (To appear in J. Anal. Math.) | MR 1785277 | Zbl 1029.11035

[B] A. Beardon The geometry of discrete groups, Springer-Verlag, Berlin, Grad. Text Math., Tome 91 (1984) | MR 1393195 | Zbl 0528.30001

[BC] M. Boshernitzan; C. Carroll An extension of Lagrange's theorem to interval exchange tranformations over quadratic fields, J. Anal. Math., Tome 72 (1997), pp. 21-44 | Article | MR 1482988 | Zbl 0931.28013

[BL] C. Birkenhake; H. Lange Complex abelian varieties, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften, Tome vol. 302 (1992) | MR 1217487 | Zbl 0779.14012

[C] J.H. Conway Understanding groups like Γ 0 (N), Groups, difference sets, and the Monster (Columbus, OH, 1993), de Gruyter, Berlin (Ohio State Univ. Math. Res. Inst. Publ.) Tome 4 (1996), pp. 327-343 | Zbl 0860.11019

[EG] C.J. Earle; F.P. Gardiner; J.R. Quine And P. Sarnak, Eds. Teichmüller disks and Veech's ℱ-structures, Extremal Riemann surfaces, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 201 (1997), pp. 165-189 | Zbl 0868.32027

[EM] A. Eskin; H. Masur Pointwise asymptotic formulas on flat surfaces (1999) (Preprint)

[F] A. Fathi Some compact invariant sets for hyperbolic linear automorphisms of torii, Ergodic Theory Dynam. Systems, Tome 8 (1988), pp. 191-204 | MR 951268 | Zbl 0658.58028

[G] E. Gutkin; J.-M. Gambaudo, P. Hubert, P. Tisseur, S. Vaienti, Eds Branched coverings and closed geodesics in flat surfaces, with applications to billiards, Dynamical Systems from Crystal to Chaos, World Scientific, Singapore (2000), pp. 259-273

[GJ1] E. Gutkin; C. Judge The geometry and arithmetic of translation surfaces with applications to polygonal billiards, Math. Res. Lett., Tome 3 (1996), pp. 391-403 | MR 1397686 | Zbl 0865.30060

[GJ2] E. Gutkin; C. Judge Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J., Tome 103 (2000), pp. 191-213 | Article | MR 1760625 | Zbl 0965.30019

[Ha1] W. Harvey; J.R. Quine And P. Sarnak, Eds On certain families of compact Riemann surfaces, Mapping class groups and moduli spaces of Riemann surfaces, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 150 (1993), pp. 137-148 | Zbl 0793.32008

[Ha2] W. Harvey Drawings, triangle groups and algebraic curves (1997) (Preprint)

[He] H. Helling Bestimmung der Kommensurabilitätsklasse der Hilbertschen Modulgruppe, Math. Z., Tome 92 (1966), pp. 269-280 | Article | MR 228437 | Zbl 0143.30601

[HS] P. Hubert; T. Schmidt Veech groups and polygonal coverings, J. Physics and Geom., Tome 35 (2000), pp. 75-91 | Article | MR 1767943 | Zbl 0977.30027

[K] S. Katok Fuchsian groups, Univ. Chicago Press, Chicago, Chicago Lectures in Math. (1992) | MR 1177168 | Zbl 0753.30001

[KMS] S.P. Kerkhoff; H. Masur; J. Smillie Ergodicity of billiard flows and quadratic differentials, Ann. of Math., Tome 124 (1986), pp. 293-311 | Article | MR 855297 | Zbl 0637.58010

[Kr] I. Kra The Carathéodory metric on abelian Teichmüller disks, J. Anal. Math., Tome 40 (1981), pp. 129-143 | Article | MR 659787 | Zbl 0487.32017

[KS] R. Kenyon; J. Smillie Billiards in rational-angled triangles, Commentarii Math. Helvetici, Tome 75 (2000), pp. 65-108 | Article | MR 1760496 | Zbl 0967.37019

[KZ] A. Katok; A. Zemlyakov Topological transitivity of billiard flows in polygons, Math. Notes, Tome 18 (1975), pp. 760-764 | Article | Zbl 0323.58012

[L] A. Leutbecher Über die Heckeschen Gruppen G(λ), II, Math. Ann., Tome 211 (1974), pp. 63-86 | Article | MR 347736 | Zbl 0292.10020

[Mar] G.A. Margulis Discrete subgroups of semisimple Lie groups, Springer-Verlag, New York (1991) | MR 1090825 | Zbl 0732.22008

[Mas] H. Masur Closed geodesics for quadratic differentials with applications to billiards, Duke J. Math., Tome 53 (1986), pp. 307-314 | MR 850537 | Zbl 0616.30044

[MR] C. Maclachlan; G. Rosenberger Commensurability classes of two generator Fuchsian groups, Discrete groups and geometry, Cambridge University Press, Cambridge (London Math. Soc. Lecture Note Series) Tome 173 (1992), pp. 171-189 | Zbl 0849.30033

[MT] H. Masur; S. Tabachnikov Rational billiards and flat structures (Max-Planck-Institut für Mathematik, Bonn, preprint, 55) | MR 1928530 | Zbl 1057.37034

[S] B. Schindler Period matrices of hyperelliptic curves, Manuscripta Math., Tome 78 (1993), pp. 369-380 | Article | MR 1208647 | Zbl 0801.14008

[T] S. Tabachnikoff Billiards, Panoramas et Synthèses 1, Soc. Math. France, Paris (1995) | Zbl 0833.58001

[Tr] M. Troyanov Les surfaces euclidiennes à singularités coniques, Enseign. Math. (2), Tome 32 (1986), pp. 79-94 | MR 850552 | Zbl 0611.53035

[Ve1] W. Veech Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., Tome 97 (1989), pp. 553-583 | Article | MR 1005006 | Zbl 0676.32006

[Ve2] W. Veech The billiard in a regular polygon, Geom. Funct. Anal., Tome 2 (1992), pp. 341-379 | Article | MR 1177316 | Zbl 0760.58036

[Vo] Ya.B. Vorobets Plane structures and billiards in rational polyhedra: the Veech alternative (Russian), Uspekhi Mat. Nauk, Tome 51 (1996) | MR 1392678 | Zbl 0897.58029

[W] C. Ward Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergodic Theory Dynam. Systems, Tome 18 (1998), pp. 1019-1042 | Article | MR 1645350 | Zbl 0915.58059