A New Algorithm for Computing the Veech Group of a Translation Surface

Graduation date: 2017We give a new characterization of elements in the Veech group of a translation surface. This provides a computational test for Veech group membership. We use this computational test in an algorithm that detects when the Veech group is a lattice (has co-finite area), and in this case computes a fundamental polygon for the action of the Veech group on the hyperbolic plane. A standard result, essentially due to Poincaré, provides that a complete set of generators for the Veech group can then be obtained from the side pairings associated to this fundamental polygon.\ud Our approach introduces a new computational framework used to formulate a membership criterion for the Veech group of a compact translation surface (X,ω). We represent (X,ω) on a certain non-compact translation surface O that can be used to represent any translation surface within the SL(2,ℝ) orbit of the translation equivalence class of (X,ω). The surface O has an easily computed SL(2,ℝ)-action. When this action is restricted to the translation surface representations mentioned above, it corresponds to the usual SL(2,ℝ)-action on the set of equivalence classes of translation surfaces. The Veech group of a compact translation surface is therefore the stabilizer of its representation on O