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

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 parameterize the collection of all discrete representations from an arbitrary but fixed finitely generated group G$G$ to PSL(2,R)${\rm PSL}(2, \mathbb {R})$. We define a new class of groups called PSL(2,R)${\rm PSL}(2, \mathbb {R})$-discrete limit groups and then use the factorization theorem to obtain useful information about this class of groups.