Fundamental domains and generators for lattice Veech groups

Abstract

The moduli space $Q \mathcal M\_g$ of non-zero genus $g$ quadratic differentials has a natural action of $G=\mathrm {GL}\_2^+(\mathbb R)$ / $\langle ± \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$ $\rangle$. The Veech group PSL$(X,q)$ is the stabilizer of $(X,q) \in Q \mathcal M\_g$ in $G$. We describe a new algorithm for finding elements of PSL$(X,q)$ which, for lattice Veech groups, can be used to compute a fundamental domain and generators. Using our algorithm, we give the first explicit examples of generators and fundamental domains for non-arithmetic Veech groups where the genus of $\mathbb H$ / PSL$(X,q)$ is greater than zero.