O ct 2 00 7 Algebraically Periodic Translation Surfaces

A translation surface is a compact surface with a geometric structure. Translation surfaces arise in complex analysis, the classification of surface diffeomorphisms and the study of polygonal billiards. This structure can be defined via complex analytic methods (where it is called an abelian differential). It can also be described as a pair of measured foliations but for our purposes the following description is more convenient. A translation surface can be obtained by gluing a finite number of polygons P1, . . . , Pn ⊂ R 2 along parallel edges in such a way that the result is a compact surface. In this paper we synthesize and further develop an algebraic theory of translation surfaces that builds upon the work of many authors. ( To name a very few: [GJ1], [C], [KS], [A1], [A2], [M2]). Many of the results in this paper are of a purely algebraic nature but they are all motivated by a desire to understand the geometry and dynamics of translation surfaces. Given a translation surface S and an α ∈ SL(2,R) we obtain a new translation surface αS by letting α act on the polygons P1, . . . , Pn that comprise S. The Veech group of S consists of those α ∈ SL(2,R) for which αS is equivalent to S as a translation surface. If the Veech group of S is a lattice in SL(2,R), then we say that S is a lattice surface. Algebraic tools play a role in several recent advances in the study of translation surfaces. Kenyon and Smillie [KS] introduced a translation surface invariant, the J-invariant, which takes values in R∧QR . They used the J-invariant as a tool in the classification ofl lattice surfaces that arise from acute, rational triangular billiard tables. ([KS],[P]) Calta ([C]) and McMullen ([M2]) both use algebraic methods in their independent work on genus two translation structures. In [C], the Jinvariant and projections of the J-invariant are used to develop algebraic criteria for the flow on a surface in a given direction to be periodic. A projection of the J-invariant in a direction v is in fact the SAF invariant ([A1]) associated to an interval exchange transformation induced by the flow in the direction v. These ideas played an important role in her discovery of infinitely many lattice surfaces in genus two.