Canonical translation surfaces for computing Veech groups

Abstract

For each stratum of the space of translation surfaces, we introduce an infinite translation surface containing in an appropriate manner a copy of every translation surface of the stratum. Given a translation surface $(X,\ddot{I}‰)$ in the stratum, a matrix is in its Veech group $\mathrm{SL}(X,\ddot{I}‰)$ if and only if an associated affine automorphism of the infinite surface sends each of a finite set, the ``marked" {\em 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 $a$. %When $\mathrm{SL}(X,\ddot{I}‰)$ is a lattice we use this to give a condition guaranteeing that the full group $\mathrm{SL}(X,\ddot{I}‰)$ has been computed. Together, these results give rise to a new algorithm for computing Veech groups.

20 pages, 8 figures. New version has more appropriate title, and minor editing