Jsj-splittings for Finitely Presented Groups over Slender Groups

We generalize the JSJ-splitting of Rips and Sela to give decompositions of nitely presented groups which capture splittings over certain classes of small subgroups. Such classes include the class of all 2-ended groups and the class of all virtually ZZ groups. The approach, called \track zipping", is relatively elementary, and diiers from the Rips-Sela approach in that it does not rely on the theory of R-trees but rather on an understanding of certain embedded 1-complexes (called patterns) in a presentation 2-complex for the ambient group. The purpose of this paper is to describe JSJ-splittings for nitely presented groups over certain classes of small groups. Broadly speaking, a JSJ-splitting for a group G over a class of groups C is a graph of groups decomposition of G whose vertex groups contain (up to conjugacy) all the C-subgroups over which G splits as an amalgam. Furthermore, the vertex groups which contain such subgroups are of a special type. As the name suggests , the notion of a JSJ-splitting has its roots in the work of Jaco and Shalen 10] and Johannson 11] on characteristic submanifolds of 3-manifolds. Their work plays a key role in Thurston's geometrization program for 3-manifolds. Here is a rough description of the JSJ-splitting in the 3-manifold setting. Suppose M is a closed, irreducible 3-manifold. We wish to describe all embedded incompressible tori in M.