Martin Schmoll (Penn State)

On finite blocking properties for geodesics on translation surfaces

Abstract. One problem related to the asymptotic growth rate of various types of geodesic segments is to find $SL(2,\mathbb{R} )$ invariant subsets in moduli spaces of translation surfaces. We describe a way how to obtain $SL(2,\mathbb{R} )$invariant subsets for (branched) coverings of the flat torus by proving a finite blocking property, which roughly states that the every geodesic segment from the set of geodesic segments connecting two given points hits one of finitely many points (away from its endpoints). This simple fact for flat tori allows us to construct a lot of $SL(2,\mathbb{R} )$ invariant subsets for (branched) coverings of the flat torus. For torus coverings we give constructions, properties, some explanations and state questions related to the above ideas. As easy as results can be obtained for torus coverings as hard it seems to be, to make related statements for arbitrary translation surfaces.