and CHRIS JUDGE 1. Introduction. Translation surfaces naturally arise in the study of billiards in rational polygons (see [ZKa]). To any such polygon P , there corresponds a unique translation surface, S = S(P), such that the billiard flow in P is equivalent to the geodesic flow on S (see, e.g., [Gu2], [Gu3]). There is also a classical relation between translation surfaces and quadratic differentials on a Riemann surface S. Namely, each quadratic differential induces a translation structure on a finite puncturing of S or on a canonical double covering of S. Quadratic differentials have a natural interpretation as cotangent vectors to Teichmüller space, and this connection has proven useful in the study of billiards (see, e.g., [Ma2], [V1]). With a translation surface, S, one associates various algebraic and geometric objects: the induced affine structure of S and the group of affine diffeomorphisms, Aff(S); the holonomy homomorphism, hol : π 1 (S) → R 2 and the holonomy group Hol(S) = hol(π 1 (S)); the flat structure on S and the natural cell decompositions of its metric completion S. In the present paper, we study the relations between these objects, as well as relations among different translation surfaces. Our main focus is the group Aff(S) and the associated group of differentials, (S) ⊂ SL(2, R). The study of these groups began as part of W. Thurston's classification of surface diffeomorphisms in [Th2]. This study continued with the work of W. Veech in [V1] and [V2]. Veech produced explicit examples of translation surfaces S for which (S) is a nonarithmetic lattice. He showed that if (S) is a lattice, then the geodesic flow on S exhibits remarkable dynamical properties. For these reasons, we call (S) the Veech group of S, and if this group is a lattice, then we call S a Veech surface. We now describe the structure of the paper and our main results. In §2, we establish the setting. In particular, we recall the notion of a G-manifold and associated objects: the developing map, the holonomy homomorphism, and the holonomy group. We introduce the notion of the differential of a G-map with respect to a normal subgroup H ⊂ G. We also introduce the spinal triangulation, one of several cell decompositions canonically associated to a flat surface with cone points.
