Canonical translation surfaces for computing Veech groups

TL;DR

This paper introduces a novel approach using an infinite translation surface to compute Veech groups of translation surfaces, providing explicit criteria and an algorithm for their determination.

Contribution

It presents a new method leveraging infinite surfaces and Voronoi staples to determine Veech groups, along with explicit hyperbolic bounds for Fuchsian groups.

Findings

Established a characterization of Veech groups via affine automorphisms of infinite surfaces.

Derived explicit hyperbolic bounds for Fuchsian groups stabilizing i.

Developed a new algorithm for computing Veech groups based on these results.

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 $(X, \omega)$ 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 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 $\mathrm{SL}(X,\omega)$ is a lattice we use this to give a condition guaranteeing that the full group $\mathrm{SL}(X,\omega)$ has been computed. Together, these results give rise to a new algorithm for computing Veech groups.