  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.
