Commentarii Mathematici Helvetici


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Volume 92, Issue 1, 2017, pp. 57–83
DOI: 10.4171/CMH/406

Published online: 2017-02-21

Fundamental domains and generators for lattice Veech groups

Ronen E. Mukamel[1]

(1) Rice University, Houston, USA

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.

Keywords: Riemann surfaces, Teichmüller theory, Veech groups

Mukamel Ronen: Fundamental domains and generators for lattice Veech groups. Comment. Math. Helv. 92 (2017), 57-83. doi: 10.4171/CMH/406