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 [RS]. 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.
Cite this article
Max Forester, On uniqueness of JSJ decompositions of finitely generated groups. Comment. Math. Helv. 78 (2003), no. 4, pp. 740–751DOI 10.1007/S00014-003-0780-Y