Algebraically Periodic Translation Surfaces

Translation surfaces arise in complex analysis, the classification of surface diffeomorphisms and the study of polygonal billiards. In recent years there have been some notable advances in understanding their properties. In particular Calta and McMullen have made significant advances in understanding translation structures in genus 2. This paper is motivated by our desire to reformulate and extend to higher genus some of the ideas that played a role in the work of Calta and McMullen. A fundamental principle in the analysis of translation surfaces is that properties of the geodesic flow on a given surface such as the growth of the number of closed geodesics or the distribution of long geodesic trajectories are closely linked to the behavior of the orbit of the surface under the natural SL(2,R) action on the moduli space of translation surfaces. The SL(2,R) orbit of a translation surface S is contained in a component of a stratum of moduli space. Let us call S an exceptional surface if the SL(2,R)-orbit of S is not dense in its stratum component. For a given surface S, the stabilizer in SL(2,R) of S is called the Veech group of S. If this stabilizer is a lattice in SL(2,R), then we say that S is a lattice surface. Lattice surfaces other than the torus provide examples of exceptional surfaces. The work of Calta and McMullen describes all exceptional surfaces of genus two. While many of the exceptional surfaces that arise in genus two are lattice surfaces there are classes of exceptional surfaces which are not lattice surfaces. The work in genus two leads to the following interesting conclusion. While the problem of determining the Veech group of a translation surface in genus two is quite delicate the problem of determining whether the Veech group is a lattice is considerably easier. Both Calta and McMullen introduce computable invariants of an algebraic nature which pick out the exceptional surfaces. McMullen shows that exceptional surfaces can be characterized in terms of the homological affine group which is an algebraic analog of the Veech group but is easier to determine. Calta shows that exceptional surfaces can be characterized in terms of algebraically periodic directions. A direction is algebraically periodic if the SAF invariant of the geodesic flow in that direction vanishes. We say a direction is completely periodic if all trajectories in that