Cyclic splittings of finitely presented groups and the canonical JSJ deccomposition.

*(English)*Zbl 0910.57002The general program underlying the paper is to understand groups from the point of view of low dimensional topology. In a previous paper, the second author obtained a canonical splitting of Gromov hyperbolic groups which contains, in an appropriate sense, all cyclic splittings of the group, i.e. splittings of the group into free products with amalgamation or HNN-extensions over infinite cyclic subgroups. In the present paper, this JSJ-decomposition is extended to arbitrary (freely indecomposable) finitely presented groups \(G\); here the name comes from the Jaco-Shalen-Johannson decomposition of a Haken 3-manifold into a characteristic (or Seifert fibered) part which contains all essential annuli and tori, in particular contains all splittings of the fundamental group over cyclic and rank two abelian groups, and a remaining simple or hyperbolic part). The JSJ-splitting of a Gromov hyperbolic group is unique up to a sequence of elementary modifications (slidings, conjugation and conjugation of boundary monomorphisms); this remains open for arbitrary finitely presented groups.

A vocabulary is introduced which translates topological notions into algebraic ones. For example, for a separating (resp. nonseparating) simple closed curve the corresponding algebraic notion is a free product with amalgamation (resp. HNN-extension) over an infinite cyclic group. Then, like simple closed curves on a surface or a 2-orbifold, these algebraic s.c.c. may intersect or be disjoint. Given such an elementary \(\mathbb{Z}\)-splitting, an element is called elliptic if it is conjugate into a vertex group, otherwise hyperbolic. Giving two elementary \(\mathbb{Z}\)-splittings of the same group \(G\), the generator of the cyclic amalgam in one splitting may be elliptic or hyperbolic in the other splitting, so one has the three cases elliptic-elliptic, hyperbolic-hyperbolic (in which case the elementary splittings or algebraic s.c.c. are called intersecting) and elliptic-hyperbolic. One of the first results of the paper excludes the last case, and the main part of the paper is devoted to the hyperbolic-hyperbolic case of intersecting decompositions. In this last case, “maximal quadratically hanging” subgroups play an important role; these are planar subgroups \(Q\) of \(G\) (fundamental groups of 2-orbifolds with a finite number of punctures and cone points) such that \(G\) admits a \(\mathbb{Z}\)-splitting with \(Q\) as a vertex group and all edge groups (amalgams) involving the vertex \(Q\) are cyclic puncture subgroups of \(Q\). Note that each s.c.c. on the 2-orbifold defines an elementary \(\mathbb{Z}\)-splitting of \(Q\) and also of \(G\). The analysis of these subgroups leads to a “canonical quadratic decomposition” of a finitely generated group with a single end which is unique in the above sense and is the main step towards the JSJ-decomposition of a finitely presented group. In general, finitely generated groups do not admit a canonical JSJ-decomposition because Dunwoody has shown recently that the set of cyclic splittings of a finitely generated group may not be accessible; this implies that, in general, there is no finite splitting describing all cyclic splittings of a finitely generated group.

A vocabulary is introduced which translates topological notions into algebraic ones. For example, for a separating (resp. nonseparating) simple closed curve the corresponding algebraic notion is a free product with amalgamation (resp. HNN-extension) over an infinite cyclic group. Then, like simple closed curves on a surface or a 2-orbifold, these algebraic s.c.c. may intersect or be disjoint. Given such an elementary \(\mathbb{Z}\)-splitting, an element is called elliptic if it is conjugate into a vertex group, otherwise hyperbolic. Giving two elementary \(\mathbb{Z}\)-splittings of the same group \(G\), the generator of the cyclic amalgam in one splitting may be elliptic or hyperbolic in the other splitting, so one has the three cases elliptic-elliptic, hyperbolic-hyperbolic (in which case the elementary splittings or algebraic s.c.c. are called intersecting) and elliptic-hyperbolic. One of the first results of the paper excludes the last case, and the main part of the paper is devoted to the hyperbolic-hyperbolic case of intersecting decompositions. In this last case, “maximal quadratically hanging” subgroups play an important role; these are planar subgroups \(Q\) of \(G\) (fundamental groups of 2-orbifolds with a finite number of punctures and cone points) such that \(G\) admits a \(\mathbb{Z}\)-splitting with \(Q\) as a vertex group and all edge groups (amalgams) involving the vertex \(Q\) are cyclic puncture subgroups of \(Q\). Note that each s.c.c. on the 2-orbifold defines an elementary \(\mathbb{Z}\)-splitting of \(Q\) and also of \(G\). The analysis of these subgroups leads to a “canonical quadratic decomposition” of a finitely generated group with a single end which is unique in the above sense and is the main step towards the JSJ-decomposition of a finitely presented group. In general, finitely generated groups do not admit a canonical JSJ-decomposition because Dunwoody has shown recently that the set of cyclic splittings of a finitely generated group may not be accessible; this implies that, in general, there is no finite splitting describing all cyclic splittings of a finitely generated group.

Reviewer: B.Zimmermann (Trieste)