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 $G={\mathrm{GL}}_2^{+}(\mathbb{R})$ / $⟨\pm \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ $⟩$. 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.