On fundamental domains of arithmetic Fuchsian groups

Let $K$ be a totally real algebraic number field and ${\mathcal{O}}$ an order in a quaternion algebra ${\mathfrak{A}}$ over $K$. Assume that the group ${\mathcal{O}^{1}}$ of units in ${\mathcal{O}}$ with reduced norm equal to $1$ is embedded into $PSL_{2}({\mathbb{R})}$ as an arithmetic Fuchsian group. It is shown how Ford's algorithm can be effectively applied in order to determine a fundamental domain of ${\mathcal{O}^{1}}$ as well as a complete system of generators of ${\mathcal{O}^{1}}$.