A JSJ decomposition (or JSJ tree) of G over A is an A-tree T such that:

T is universally elliptic;

T dominates any other universally elliptic tree T’.

An A-tree is universally elliptic if its edge stabilizers are elliptic in every A-tree.

Recall that H is elliptic in T if it fixes a point in T (in terms of graphs of groups, H is contained in a conjugate of a vertex group).

Definition 2.1 (Ellipticity of trees). T1 is elliptic with respect to T2 if every edge stabilizer of T1 fixes a point in T2.

Note that T1 is elliptic with respect to T2 whenever there is a refinement \(\hat{T}1 \) of T1 that dominates T2. Edge stabilizers of T1 are elliptic in \(\hat{T}1 \), hence in T2. We show a converse statement.

Theorem: If G is finitely presented, then the JSJ deformation space D(JSJ) of G over A exists. It contains a tree whose edge and vertex stabilizers are finitely generated.