Recognition and constructive membership for discrete subgroups of $\mathrm{SL}_2(\mathbb{R})$

We provide algorithms to decide whether a finitely generated subgroup of $\mathrm{SL}_2(\mathbb{R})$ is discrete, solve the constructive membership problem for finitely generated discrete subgroups of $\mathrm{SL}_2(\mathbb{R})$, and compute a fundamental domain for a finitely generated Fuchsian group. These algorithms have been implemented in Magma for groups defined over real algebraic number fields.