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

Abstract

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 to . We define a new class of groups called ‐discrete limit groups and then use the factorization theorem to obtain useful information about this class of groups.