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This paper and its companion arXiv:0911.3173 have been replaced byarXiv:1602.05139.  We define the compatibility JSJ tree of a group G over a class of subgroups.It exists whenever G is finitely presented and leads to a canonical tree (not adeformation space) which is invariant under automorphisms. Under acylindricityhypotheses, we prove that the (usual) JSJ deformation space and thecompatibility JSJ tree exist, and we describe their flexible subgroups. Weapply these results to finitely generated CSA groups, \Gamma-limit groups(allowing torsion), and relatively hyperbolic groups.

JSJ decompositions: definitions, existence, uniqueness. II. Compatibility and acylindricity