JSJ-splittings for finitely presented groups over slender groups. (English) Zbl 0939.20047

One of the major themes in geometric group theory is to extend known results from compact 3-manifolds and their groups to finite 2-complexes and finitely presented groups. The paper at hand is a beautiful example of such a generalization. It describes JSJ-splittings for finitely presented groups over certain classes of small groups (such as virtually \(\mathbb{Z}\) or \(\mathbb{Z}\oplus\mathbb{Z}\) groups).
Broadly speaking, a JSJ-splitting for a group \(G\) over a class of small groups \(\mathcal C\) is a graph of groups decomposition of \(G\) whose vertex groups contain (up to conjugacy) all the \(\mathcal C\)-subgroups over which \(G\) splits as an amalgam. Furthermore, the vertex groups which contain such subgroups are of a special type (typically \(\mathcal C\)-by-2-orbifold groups).
The notion of a JSJ-splitting has its roots in the work of Jaco, Shalen and Johannson 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. One wishes to describe all embedded incompressible tori in \(M\). The Characteristic Submanifold Theorem of Jaco, Shalen and Johannson says that there exists a finite collection of embedded 2-sided incompressible tori, such that the pieces obtained by cutting \(M\) along these tori are either Seifert fibered spaces or simple manifolds (acylindrical and atoridal). This provides also a decomposition of the fundamental group \(\pi_1(M)\) as a graph of groups so that every free Abelian group of rank 2 over which \(\pi_1(M)\) splits as an amalgam is conjugate into one of the vertex groups, and those vertex groups that contain such splitting subgroups are fundamental groups of Seifert fibered spaces.
This paper presents a generalization of the work of Rips and Sela on splitting hyperbolic groups over infinite cyclic subgroups which relies on an intimate understanding of group actions on \(\mathbb{R}\)-trees. The approach presented here uses the theory of tracks, which are embedded 1-dimensional objects in a presentation 2-complex which capture geometrically the algebraic structure of a splitting of the ambient group as an amalgam (as a simple example one could think of a simple closed curve in a surface as a geometric realization of the decomposition of the surface group). Tracks have been used successfully on other problems in geometric group theory such as the accessibility of finitely presented groups and have also been used to give proofs of the Loop- and Sphere-Theorem in 3-manifold topology.
More recently, Fujiwara and Papasoglu have developed another approach, involving group actions on products of trees, which gives another proof of the main theorem of this article.


20F65 Geometric group theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E08 Groups acting on trees
57M05 Fundamental group, presentations, free differential calculus
57M07 Topological methods in group theory
57N10 Topology of general \(3\)-manifolds (MSC2010)
20F05 Generators, relations, and presentations of groups
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