Subgroups of PSL(2,R): criteria for discreteness

Question

Let $G$ be a subgroup of $SL_2(\mathbb R)$ generated by a set of matrices $\mathcal M=(M_i)_{i\in I}$.

Is there an effective criterion on $\mathcal M$ ensuring that $G$ is discrete?

Answer 1

The question can be reduced to the case that $G$ is $2$-generated. In this case there is an effective algorithm to decide discreteness by Jane Gilman, The non-Euclidean Euclidean algorithm. In particular see Theorem $3.1$ and the following.