Invariants of translation surfaces
[Invariants des surfaces de translation]
Annales de l'Institut Fourier, Tome 51 (2001) no. 2, pp. 461-495.

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.

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.

DOI : 10.5802/aif.1829
Classification : 30F60, 32G15
Keywords: flat surfaces, Teichmüller disks, billiards
Mot clés : surfaces plates, disques de Teichmüller, billards
Hubert, Pascal 1 ; Schmidt, Thomas A. 2

1 Institut de Mathématiques de Luminy, Case 907, 163 avenue de Luminy, 13288 Marseille Cedex 09 (France)
2 Oregon State University, Department of Mathematics, Kidder Hall 368, Corvallis OR 97331-4605 (USA)
@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, Tome 51 (2001) no. 2, pp. 461-495. doi : 10.5802/aif.1829. https://aif.centre-mersenne.org/articles/10.5802/aif.1829/

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

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

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

[BC] M. Boshernitzan; C. Carroll 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] C. Birkenhake; H. Lange Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften, vol. 302, Springer-Verlag, Berlin, 1992 | MR | Zbl

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

[EG] C.J. Earle; F.P. Gardiner; J.R. Quine and P. Sarnak, eds. Teichmüller disks and Veech's -structures, Extremal Riemann surfaces (Contemp. Math.), Volume 201 (1997), pp. 165-189 | Zbl

[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, Volume 8 (1988), pp. 191-204 | MR | Zbl

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

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

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

[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 (Contemp. Math.), Volume 150 (1993), pp. 137-148 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[Ve1] W. Veech 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] W. Veech The billiard in a regular polygon, Geom. Funct. Anal., Volume 2 (1992), pp. 341-379 | DOI | MR | Zbl

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

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

Cité par Sources :