Universal JSJ-Decompositions for Finitely Presented Groups

Introduction A group G is said to split over a subgroup C if either G = A ∗C B, where A 6= C and B 6= C or G is an HNN-group G =< A ∗ C =< A, t|t−1at = θ(a) > where θ : C → A is an injective homomorphism. It is one of the basic results of Bass-Serre theory (see [DD] or [Se]), that a finitely generated group G splits over some subgroup C if and only if there is an action of G on a tree T , without inversions, such that for no vertex v ∈ V T is v fixed by all of G. Here the tree is a combinatorial tree, i.e. a connected graph with no cycles, and an action without inversions is one in which no element g ∈ G transposes the vertices of an edge. Tits [T] introduced the idea of an R-tree, which is a non-empty metric space in which any two points are joined by a unique arc, and in which every arc is isometric to a closed interval in the real line R. Alternatively an R-tree is a 0-hyperbolic space. A tree in the combinatorial sense can be regarded as a 1-dimensional simplicial complex. The polyhedron of this complex will be an R-tree called a simplicial R-tree. However not every R-tree is like this. A point p of an R-tree T is called regular if T − p has two components. An R-tree is simplicial if the points of T which are not regular form a discrete subspace of T . It is fairly easy to construct examples of R-trees where the set of non-regular points is not discrete. There are good introductory accounts of groups acting on R-trees in [Be] and [Sh]. We assume that all our actions are by isometries. It is a classical result that a group is free if and only if it has a free action on a simplicial tree. As the real line R is an R-tree and R acts on itself freely by translations, any free abelian group has a free action on an R-tree. Morgan and Shalen [MS] showed that the fundamental group of any compact surface other than the projective plane and the Klein bottle has a free action on an R-tree. Rips showed that the only finitely generated groups that act freely on an R-tree are free products of free abelian groups and surface groups. Rips never published his proof, but there are proofs of more general results by Bestvina -Feighn [BF] and by Gaboriau-Levitt-Paulin (see [P] or [C]). Shalen in [Sh, Question D] considers groups which act on an R-tree with small arc stabilizers. He asks if such a group must admit an action on a simplicial tree with small arc stabilizers. In [D] I proved that this is the case if G is finitely presented and the action on the R-tree has infinite cyclic arc stabilizers. In [D2] an example is given of a finitely generated group which acts on an R-tree with finite cyclic arc stabilizers but which does not split over a small subgroup. In his seminal work [St] Stallings showed that a finitely generated group with more than one end splits over a finite subgroup. In [D1] I showed that a finitely presented group is accessible. This means that a finitely presented group G has a decomposition as the fundamental group of a graph of groups in which vertex groups are one ended and edge groups are finite. This decomposition provides information about every action of G on a simplicial tree with finite edge groups. Thus, let S be the Bass-Serre G-tree associated with