Constructing lattice surfaces with prescribed Veech groups: an algorithm

TL;DR

This paper introduces an algorithm to construct all translation surfaces with a specified lattice Veech group within any stratum, providing new insights into the finiteness and realization of such groups.

Contribution

It presents a novel algorithm for constructing translation surfaces with prescribed lattice Veech groups and offers a new proof of their finiteness in each stratum.

Findings

Finiteness of unit-area translation surfaces with the same lattice Veech group.

The square torus is the unique minimal stratum surface with Veech group SL(2,Z).

The algorithm can identify obstructions to certain lattices being Veech groups.

Abstract

The Veech group of a translation surface is the group of Jacobians of orientation-preserving affine automorphisms of the surface. We present an algorithm which constructs all translation surfaces with a given lattice Veech group in any given stratum. In developing this algorithm, we give a new proof of a finiteness result of Smillie and Weiss, namely that there are only finitely many unit-area translation surfaces in any stratum with the same lattice Veech group. Our methods can be applied to obtain obstructions of lattices being realized as Veech groups in certain strata; in particular, we show that the square torus is the only translation surface in any minimal stratum whose Veech group is all of $\mathrm{SL}_2\mathbb{Z}$.