Discrete representations of finitely generated groups into ${\rm PSL}(2,\mathbb{R})$

We prove a factorization theorem for Fuchsian groups similar to those proved by Agol and Liu for 3-manifold groups. As an application, we build Makanin-Razborov diagrams, which parametrize the collection of all discrete representations from an arbitrary but fixed finitely generated group $G$ to ${\rm PSL}(2, \mathbb{R})$. We define a new class of groups called ${\rm PSL}(2, \mathbb{R})$-discrete limit groups and then use the factorization theorem to obtain useful information about this class of groups.