Canonical translation surfaces for computing Veech groups

Abstract

For each stratum of the space of translation surfaces, we show that there is an infinite area translation surface which contains in an appropriate manner a copy of every translation surface of the stratum. Given a translation surface $(X, \omega )$ in the stratum, a matrix is in its Veech group $\mathrm {SL}(X,\omega )$ if and only if an associated affine automorphism of the infinite surface sending each of a finite set, the “marked” Voronoi staples, arising from orientation-paired segments appropriately perpendicular to Voronoi 1-cells, to another pair of orientation-paired “marked” segments. We prove a result of independent interest. For each real $a\ge \sqrt{2}$ there is an explicit hyperbolic ball such that for any Fuchsian group trivially stabilizing i, the Dirichlet domain centered at i of the group already agrees within the ball with the intersection of the hyperbolic half-planes determined by the group elements whose Frobenius norm is at most a. Together, these results give rise to a new algorithm for computing Veech groups.