An introduction to Veech groups of covering surfaces 

Gabriela Weitze-Schmithüsen
Department of Mathematik und Informatik
Universität des Saarlandes
Abstract:
Translation surfaces are Riemann surfaces with some extra structure. They can be defined in a fairly explicit way starting from polygons in the Euclidean plane which are glued along pairs of parallel edges of equal length via translations. Among other reasons, they have attracted a lot of attention within the last thirty years since under lucky circumstances they define complex algebraic curves in
\[M_g\]
, the moduli space of closed Riemann surfaces of genus
\[g\]
. These curves are called Teichmüller curves. They are (almost) determined by a discrete subgroup of 
\[SL(2,\mathbb{R})\]
called the Veech group. In this mini course we want to explain the relations between translation surfaces, Veech groups and Teichmüller curves and get to know these objects explicitly for special classes of translation surfaces.
References:
[1] Pascal Hubert and Thomas Schmidt, An introduction to Veech surfaces, Handbook of dynamical systems 1B (2006), 501-526.
[2] Howard Masur and Serge Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015–1089.
[3] Clifford J. Earle and Frederick P. Gardiner, Teichmüller disks and Veech’s F-structures, Extremal Riemann surfaces (San Francisco, CA, 1995), Contemp.Math., vol. 201, Amer. Math. Soc., Providence, RI, 1997, pp. 165–189.
[4] Ya. B. Vorobets, Billiards in rational polygons: periodic trajectories and d-stability, Mat. Zametki 62 (1997), no. 1, 66–75.