Veech groups without parabolic elements

We prove that a ``bouillabaisse'' surface (translation surface which has two transverse parabolic elements) has totally real trace field. As a corollary, non trivial Veech groups which have no parabolic elements do exist. The proof follows Veech's viewpoint on Thurston's construction of pseudo-Anosov diffeomorphisms.

[1]  C. Ward.  Calculation of Fuchsian groups associated to billiards in a rational triangle , 1998, Ergodic Theory and Dynamical Systems.

[2]  W. Veech.  Gauss measures for transformations on the space of interval exchange maps , 1982 .

[3]  C. McMullen.  Teichmüller curves in genus two: Torsion divisors and ratios of sines , 2006 .

[4]  P. Cartier.  On random matrices, zeta functions, and dynamical systems , 2006 .

[5]  Richard Kenyon,et al.  Billiards on rational-angled triangles , 2000 .

[6]  C. McMullen.  Teichmüller curves in genus two: Discriminant and spin , 2005 .

[7]  Serge Tabachnikov,et al.  Rational billiards and flat structures , 2002 .

[8]  W. Veech.  The billiard in a regular polygon , 1992 .

[9]  Prime arithmetic Teichmuller discs in H(2) , 2004 .

[10]  W. Thurston.  On the geometry and dynamics of diffeomorphisms of surfaces , 1988 .

[11]  W. Veech.  Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards , 1989 .

[12]  J. M. Anderson,et al.  RIEMANN SURFACES (Graduate Texts in Mathematics, 71) , 1982 .

[13]  C. McMullen.  Billiards and Teichmüller curves on Hilbert modular surfaces , 2003 .

[14]  C. McMullen.  Teichmüller geodesics of infinite complexity , 2003 .

[15]  Séminaire Orsay,et al.  Travaux de Thurston sur les surfaces : Séminaire Orsay , 1979 .

[16]  Thomas A. Schmidt,et al.  INFINITELY GENERATED VEECH GROUPS , 2004 .

[17]  On groups generated by two positive multi-twists: Teichmuller curves and Lehmer's number , 2003, math/0304163.

[18]  C. McMullen.  Teichmüller curves in genus two: The decagon and beyond , 2005 .

[19]  N. Purzitsky.  A cutting and pasting of noncompact polygons with applications to fuchsian groups , 1979 .

[20]  Yaroslav Vorobets.  Planar structures and billiards in rational polygons: the Veech alternative , 1996 .

[21]  W. Veech.  Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards , 1989 .

[22]  E. Gutkin,et al.  Affine mappings of translation surfaces: geometry and arithmetic , 2000 .

[23]  Geometry of infinitely generated Veech groups , 2004, math/0410132.