The fully residually \(F\) quotients of \(F*\langle x,y\rangle\).
The author describes the fully residually \(F\) groups, or limit groups relative to a free group \(F\), that are quotients of \(F*\langle x,y\rangle\). He shows the following: Theorem 1.1. Let \(F_{R(S)}\), a fully residually \(F\) quotient of \(F*\langle x,y\rangle\), be freely indecomposable modulo \(F\). The underlying graph \(X\) of its cyclic JSJ decomposition modulo \(F\) has at most 3 vertices, and \(X\) has at most two cycles. All vertex groups except maybe the vertex group \(F\leq\widetilde F\leq F_{R(S)}\) are either free of rank 2 or free Abelian. In all cases the vertex group \(\widetilde F\) can be generated by \(F\) and two other elements. Finally \(F_{R(S)}\) has uniform hierarchical depth relative to \(F\) at most 4.
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