Billiards, quadrilaterals and moduli spaces

Abstract

Totally geodesic subvarieties. Let Mg denote the moduli space of Riemann surfaces X of genus g. If we also record n unordered marked points on X, we obtain the moduli space Mg,n. A subvariety V of moduli space is totally geodesic if it contains every Teichmüller geodesic that is tangent to it. It is primitive if it does not arise from a lower– dimensional moduli space via a covering construction. The first family of primitive, totally geodesic varieties of dimension one in Mg was discovered by Veech in the 1980s [V2]. These rare and remarkable Teichmüller curves are related to Jacobians with real multiplication and polygonal billiard tables with optimal dynamical properties. A second family was discovered shortly thereafter [Wa]. To date only a handful of families of Teichmüller curves are known. The first known primitive, totally geodesic variety of dimension larger than one is the recently discovered flex surface F ⊂ M1,3 [MMW]. The surface F is closely related to a new type of SL2(R)–invariant subvariety ΩG in the moduli space of