On uniqueness of JSJ decompositions of finitely generated groups
Commentarii Mathematici Helvetici 78 (2003), 740-751

We give an example of two JSJ decompositions of a group that are not related by conjugation, conjugation of edge-inclusions, and slide moves. This answers the question of Rips and Sela stated in Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. 146 (1997), 53-109.

On the other hand we observe that any two JSJ decompositions of a group are related by an elementary deformation, and that strongly slide-free JSJ decompositions are genuinely unique. These results hold for the decompositions of Rips and Sela, Dunwoody and Sageev, and Fujiwara and Papasoglu, and also for accessible decompositions.