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Veech groups without parabolic elements. (English) Zbl 1101.30044
The authors prove that any translation surface with the Veech group containing two transverse parabolic elements has a totally real trace field. It follows from the above result, that there are nontrivial Veech groups that has no parabolic elements.

MSC:
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
30F50 Klein surfaces
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
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References:
[1] P. Arnoux and J.-C. Yoccoz, Construction de difféomorphismes pseudo-Anosov , C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 75–78. · Zbl 0478.58023
[2] K. Calta, Veech surfaces and complete periodicity in genus two , J. Amer. Math. Soc. 17 (2004), 871–908. · Zbl 1073.37032 · doi:10.1090/S0894-0347-04-00461-8
[3] A. Douady, A. Fathi, D. Fried, F. Laudenbach, V. PoéNaru, and M. Shub, Travaux de Thurston sur les surfaces , Astérisque 66 – 67 , Soc. Math. France, Montrouge, 1979.
[4] H. M. Farkas and I. Kra, Riemann Surfaces , 2nd ed., Grad. Texts in Math. 71 , Springer, New York, 1992. · Zbl 0764.30001
[5] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic , Duke Math. J. 103 (2000) 191–213. · Zbl 0965.30019 · doi:10.1215/S0012-7094-00-10321-3
[6] J. Hubbard, “Homeomorphisms of surface” in Dynamique dans l’espace de Teichmüller et applications aux billards rationnel (Marseille, 2003), lecture notes.”
[7] P. Hubert and S. LelièVre, Prime arithmetic Teichmüller discs in \(\mathcal H(2)\) , Israel J. Math. 151 (2006), 281–321. · Zbl 1138.37016 · doi:10.1007/BF02777365
[8] P. Hubert and T. A. Schmidt, Infinitely generated Veech groups , Duke Math. J. 123 (2004), 49–69. · Zbl 1056.30044 · doi:10.1215/S0012-7094-04-12312-8
[9] -, Geometry of infinitely generated Veech groups , Conform. Geom. Dyn. 10 (2006), 1–20. · Zbl 1095.30033 · doi:10.1090/S1088-4173-06-00120-2
[10] S. Katok, Fuchsian Groups , Chicago Lectures in Math., Univ. of Chicago Press, Chicago, 1992. · Zbl 0753.30001
[11] R. Kenyon and J. Smillie, Billiards on rational-angled triangles , Comment. Math. Helv. 75 (2000), 65–108. · Zbl 0967.37019 · doi:10.1007/s000140050113
[12] C. J. Leininger, On groups generated by two positive multi-twists: Teichmüller curves and Lehmer’s number , Geom. Topol. 8 (2004), 1301–1359. · Zbl 1088.57002 · doi:10.2140/gt.2004.8.1301 · emis:journals/UW/gt/GTVol8/paper36.abs.html · eudml:126060
[13] H. Masur and S. Tabachnikov, “Rational billiards and flat structures” in Handbook of Dynamical Systems, Vol. 1A , North-Holland, Amsterdam, 2002, 1015–1089. · Zbl 1057.37034
[14] C. T. Mcmullen, Billiards and Teichmüller curves on Hilbert modular surfaces , J. Amer. Math. Soc. 16 (2003), 857–885. · Zbl 1030.32012 · doi:10.1090/S0894-0347-03-00432-6
[15] -, Teichmüller geodesics of infinite complexity , Acta Math. 191 (2003), 191–223. · Zbl 1131.37052 · doi:10.1007/BF02392964
[16] -, Teichmüller curves in genus two: Discriminant and spin , Math. Ann. 333 (2005), 87–130. · Zbl 1086.14024 · doi:10.1007/s00208-005-0666-y
[17] -, Teichmüller curves in genus two: The decagon and beyond , J. Reine Angew. Math. 582 (2005), 173–199. · Zbl 1073.32004 · doi:10.1515/crll.2005.2005.582.173
[18] -, Teichmüller curves in genus two: Torsion divisors and ratios of sines , to appear in Invent. Math. · Zbl 1103.14014 · doi:10.1007/s00222-006-0511-2
[19] M. MöLler, Variations of Hodge structures of a Teichmüller curve , J. Amer. Math. Soc. 19 (2006), 327, –344. · Zbl 1090.32004 · doi:10.1090/S0894-0347-05-00512-6
[20] N. Purzitsky, A cutting and pasting of noncompact polygons with applications to Fuchsian groups , Acta Math. 143 (1979), 233–250. · Zbl 0427.30039 · doi:10.1007/BF02392095
[21] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces , Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417–431. · Zbl 0674.57008 · doi:10.1090/S0273-0979-1988-15685-6
[22] W. A. Veech, Gauss measures for transformations on the space of interval exchange maps , Ann. of Math. (2) 115 (1982), 201–242. JSTOR: · Zbl 0486.28014 · doi:10.2307/1971391 · links.jstor.org
[23] -, Teichmüller curves in modular space, Eisenstein series, and an application to triangular billiards , Invent. Math. 97 (1989), 553–583. · Zbl 0676.32006 · doi:10.1007/BF01388890 · eudml:143714
[24] -, The billiard in a regular polygon , Geom. Funct. Anal. 2 (1992), 341–379. · Zbl 0760.58036 · doi:10.1007/BF01896876 · eudml:58107
[25] Y. B. Vorobets, Planar structures and billiards in rational polygons: The Veech alternative , Russian Math. Surveys 51 (1996), 779–817. · Zbl 0897.58029 · doi:10.1070/RM1996v051n05ABEH002993
[26] C. C. Ward, Calculation of Fuchsian groups associated to billiards in a rational triangle , Ergodic Theory Dynam. Systems 18 (1998), 1019–1042. · Zbl 0915.58059 · doi:10.1017/S0143385798117479
[27] A. Zorich, “Flat surfaces” in Frontiers in Number Theory, Physics and Geometry, Vol. 1: On Random Matrices, Zeta Functions and Dynamical Systems (Les Houches, France, 2003) , Springer, Berlin, 2006, 439–586. · Zbl 1129.32012 · doi:10.1007/978-3-540-31347-2_13
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