Classification of Fuchsian groups with torsion

In their recent paper, Bergfalk and Smythe prove that the isometry equivalence relation on hyperbolic surfaces with finitely-generated fundamental group is concretely classifiable, and ask whether the same result holds true for 2-dimensional hyperbolic orbifolds, or equivalently, whether the action of $\PSL_2(\mathbb{R})$ on its space of finitely-generated discrete subgroups is concretely classifiable.

In this note we answer this question in the affirmative.

We then use the result to prove that a nonsingular ergodic $\PSL_2(\mathbb{R})$-space with nonelementary finitely-generated stabilizers is homogeneous, in similarity with a result of Stuck-Zimmer for lattices in semisimple lie groups.

The main ingredients of our proof are Selberg's lemma and a result of Greenberg on commensurators.