On uniqueness of JSJ decompositions of finitely generated groups

Details On uniqueness of JSJ decompositions of finitely generated groups On uniqueness of JSJ decompositions of finitely generated groups

2001-10-17

arxiv.org/abs/math/0110176v2

Comment. Math. Helv. 78 (2003) 740-751

Mathematics

Group Theory

11 pages; shortened and reorganized; mathematical content unchanged

Scientific paper

10.1007/s00014-003-0780-y

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.

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