Veech surface

In mathematics, a Veech surface is a translation surface (X,ω) (a Riemann surface X with a holomorphic 1-form ω) whose group SL(X,ω) of affine diffeomorphisms is a lattice in SL₂(R) (a discrete subgroup of cofinite volume). The image of SL(X,ω) in PSL₂(R) is called the Veech group.

The Veech dichotomy says that geodesic flow on a Veech surface is either periodic or ergodic. Veech surfaces, the Veech group, and the Veech dichotomy are named after William A. Veech.

The group SL₂(R) acts on the set of translation surfaces as follows. A translation surface has coordinate maps to the complex numbers, which can be identified with R² and is therefore acted on by SL₂(R). Composing the coordinate charts of a surface with an action of this group gives a new set of coordinate charts for a new translation surface. The group SL(X,ω) is the subgroup of SL₂(R) fixing the translation surface (X,ω).

References

• Hubert, Pascal; Schmidt, Thomas A. (2006), "An introduction to Veech surfaces", Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, pp. 501–526, doi:10.1016/S1874-575X(06)80031-7, MR 2186246 
• Masur, Howard (2006), "Ergodic theory of translation surfaces", Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, pp. 527–547, doi:10.1016/S1874-575X(06)80032-9, MR 2186247 
• Veech, W. A. (1989), "Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards", Inventiones Mathematicae 97 (3): 553–583, doi:10.1007/BF01388890, ISSN 0020-9910, MR 1005006