# The fully residually $F$ quotients of $F *\langle x,y \rangle$

### Nicholas W. M. Touikan

Parc Scientifique et Technologique de Luminy, Marseille, France

## Abstract

We describe the fully residually $F$ groups, or limit groups relative to $F$, that are quotients of $F *\langle x,y \rangle$. 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.

## Cite this article

Nicholas W. M. Touikan, The fully residually $F$ quotients of $F *\langle x,y \rangle$. Groups Geom. Dyn. 6 (2012), no. 1, pp. 155–220

DOI 10.4171/GGD/154