The fully residually $F$ quotients of $F\ast ⟨x,y⟩$

Abstract

We describe the fully residually $F$ groups, or limit groups relative to $F$, that are quotients of $F\ast ⟨x,y⟩$. We use the structure theory of finitely generated fully residually free groups to produce a finite list of possible types of cyclic JSJ decompositions modulo $F$ that can arise. We also give bounds on uniform hierarchical depth.