group is the Fuchsian group which is the image in of the subgroup of transformations such that is isomorphic (as a translation surface) to . Equivalently, is the group of derivatives of affine diffeomorphisms (where affine is defined locally outside the singularities, with respect to the affine structure induced by the translation structure). Veech groups have the following properties: They are discrete subgroups in ; They are never cocompact. Veech groups can be either finitely generated or not. Veech surfaces A Veech surface is by definition a translation surface whose Veech group is a lattice in , equivalently its action on the hyperbolic plane admits a fundamental domain of finite volume. Since it is not cocompact it must then contain parabolic elements. Examples of Veech surfaces are the square-tiled surfaces, whose Veech groups are commensurable to the modular group . The square can be replaced by any parallelogram (the translation surfaces obtained are exactly those obtained as ramified covers of a flat torus). In fact the Veech group is arithmetic (which amounts to it being commensurable to the modular group) if and only if the surface is tiled by parallelograms. There exists Veech surfaces whose Veech group is not arithmetic, for example the surface obtained from two regular pentagons glued along an edge: in this case the Veech group is a non-arithmetic Hecke triangle group. On the other hand, there are still some arithmetic constraints on the Veech group of a Veech surface: for example its trace field is a number field that is totally real. Geodesic flow on translation surfaces Geodesics A geodesic in a translation surface (or a half-translation surface) is a parametrised curve which is, outside of the singular points, locally the image of a straight line in Euclidean space parametrised by arclength. If a geodesic arrives at a singularity it is required to stop there. Thus a maximal geodesic is a curve defined on a closed interval, which is the whole real line if it does not meet any singular point. A geodesic is closed or periodic if its image is compact, in which case it is either a circle if it does not meet any singularity, or an arc between two (possibly equal) singularities. In the latter case the geodesic is called a saddle connection. If (or in the case of a half-translation surface) then the geodesics with direction theta are well-defined on : they