JSJ decompositions of groups Vincent Guirardel, Gilbert Levitt Abstract This is an account of the theory of JSJ decompositions of finitely generated groups, as developed in the last twenty years or so. We give a simple general definition of JSJ decompositions (or rather of their Bass- Serre trees), as maximal universally elliptic trees. In general, there is no preferred JSJ decomposition, and the right object to consider is the whole set of JSJ decompositions, which forms a contractible space: the JSJ deformation space (analogous to Outer Space). We prove that JSJ decompositions exist for any finitely presented group, without any assumption on edge groups. When edge groups are slender, we describe flexi- ble vertices of JSJ decompositions as quadratically hanging extensions of 2-orbifold groups. Similar results hold in the presence of acylindricity, in particular for splittings of torsion-free CSA groups over abelian groups, and splittings of relatively hyperbolic groups over virtually cyclic or parabolic subgroups. Using trees of cylinders, we obtain canonical JSJ trees (which are invariant under automorphisms). We introduce a variant in which the property of being universally elliptic is replaced by the more restrictive and rigid property of being universally compatible. This yields a canonical compatibility JSJ tree, not just a deformation space. We show that it exists for any finitely presented group. We give many examples, and we work throughout with relative decompositions (restricting to trees where certain subgroups are elliptic). Introduction JSJ decompositions first appeared in 3-dimensional topology with the theory of the char- acteristic submanifold by Jaco-Shalen and Johannson [JS79, Joh79] (the terminology JSJ was popularized by Sela). We start with a quick review (restricting to manifolds without boundary). From 3-manifolds to groups Let M be a closed orientable 3-manifold. Given a finite collection of disjoint embedded 2-spheres, one may cut M open along them, and glue balls to the boundary of the pieces to make them boundaryless. This expresses M as a connected sum of closed manifolds Mi. arXiv:1602.05139v2 [math.GR] 28 Jul 2017 The prime decomposition theorem (Kneser-Milnor) asserts that one may choose the spheres 3 so that each Mi is either irreducible (Mi = S , and every embedded 2-sphere bounds a 2 1 6 ball) or homeomorphic to S S ; moreover, up to a permutation, the summands Mi are uniquely determined up to homeomorphism.× On the group level, one obtains a decomposition of G = π1(M) as a free product G = G1 Gp Fq, with Gi the fundamental group of an irreducible Mi and Fq a free ∗ · · · ∗ ∗ group of rank q coming from the S2 S1 summands. This decomposition is a Grushko × decomposition of G, in the following sense: each Gi is freely indecomposable (it cannot be written as a non-trivial free product), non-trivial, and not isomorphic to Z (non-triviality of Gi is guaranteed by the Poincaré conjecture, proved by Perelman). Any finitely generated 1 group has a Grushko decomposition, with q well-defined and the Gi’s well-defined up to conjugacy (and a permutation). The prime decomposition implies that one should focus on irreducible manifolds. Since all spheres bound balls, one now considers embedded tori. In order to avoid trivialities (such as a torus bounding a tubular neighborhood of a curve), tori should be incompressible: the embedding T 2 M induces an injection on fundamental groups. ! The theory of the characteristic submanifold now says that, given an irreducible M, there exists a finite family of disjoint non-parallel incompressible tori such that each T component Nj of the manifold (with boundary) obtained by cutting M along is either atoroidal (every incompressible torus is boundary parallel) or a Seifert fiberedT space, i.e. a 3-manifold having a singular fibration by circles over a 2-dimensional surface Σ (better viewed as a 2-dimensional orbifold).1 Moreover, any incompressible torus may be isotoped to be disjoint from . With groups inT mind, let us point out an important feature of this decomposition. If two incompressible tori cannot be made disjoint by an isotopy, they may be isotoped to be contained in a Seifert piece. Conversely, in a Seifert fibered space, preimages of intersecting simple curves on Σ are intersecting tori. We thus see that the presence of intersecting tori in M forces some surface Σ to appear. One of the remarkable facts about JSJ theory for groups is that a similar phenomenon occurs. For instance, if a finitely generated one-ended group admits two splittings over 2 Z that “intersect” each other in an essential way, then it must contain the fundamental group of a compact surface, attached to the rest of the group along the boundary (see Theorem 4 below and Section 6, in particular Proposition 6.25). We also point out that the family mentioned above is unique up to isotopy. On the other hand, a family of spheres definingT the prime decomposition is usually not unique. Similarly, the Grushko decompositions of a group usually form a large outer space [CV86, GL07b], whereas one may often construct canonical splittings of one-ended groups, which are in particular invariant under automorphisms (see Theorems 6 and 8 below). These topological ideas were carried over to group theory by Kropholler [Kro90] for some Poincaré duality groups of dimension at least 3, and by Sela for torsion-free hy- perbolic groups [Sel97b]. Constructions of JSJ decompositions were given in more general settings by many authors (Rips-Sela [RS97], Bowditch [Bow98], Dunwoody-Sageev [DS99], Fujiwara-Papasoglu [FP06], Dunwoody-Swenson [DS00], Scott-Swarup [SS03], Papasoglu- Swenson [PS09]. ). This has had a vast influence and range of applications, from the iso- morphism problem and the structure of the group of automorphisms of hyperbolic groups, to diophantine geometry over groups, and many others. In this group-theoretical context, one has a finitely generated group G and a class of subgroups (such as cyclic groups, abelian groups, ...), and one tries to understand A splittings (i.e. graph of groups decompositions) of G over groups in . The family of tori A of the 3-manifold is replaced by a splitting of G over groups in . The authors construct aT splitting enjoying a long list of properties, rather specific to eachA case. Our first goal is to give a simple general definition of JSJ decompositions stated by means of a universal maximality property, together with general existence and unique- ness statements in terms of deformation spaces (see below). The JSJ decompositions constructed in [RS97, Bow98, DS99, FP06] are JSJ decompositions in our sense (see Sub- section 2.6). The regular neighbourhood of [SS03] is of a different nature. In [DS00, SS03], one looks 1 Thurston’s geometrization conjecture, whose proof was completed by Perelman, asserts that each Nj has a geometric structure. In particular, every atoroidal Nj with infinite fundamental group is hyperbolic (its interior admits a complete metric with finite volume and constant curvature −1). 2 More precisely: the edge groups of each splitting should be hyperbolic in the Bass-Serre tree of the other splitting. 2 at almost invariant sets rather than splittings, in closer analogy to a 3-manifold situation where one wants to understand all immersed tori, not just the embedded ones. One obtains a canonical splitting of G rather than just a canonical deformation space. See Parts IV and V for canonical splittings, and [GL10] for the relation between [SS03] and usual JSJ decompositions. Definition of JSJ decompositions To motivate the definition, let us first consider free decompositions of a group G, i.e. decompositions of G as the fundamental group of a graph of groups with trivial edge groups, or equivalently actions of G on a simplicial tree T with trivial edge stabilizers. Let G = G1 Gp Fq be a Grushko decomposition, as defined above (Gi is non- ∗ · · · ∗ ∗ trivial, not Z, freely indecomposable). One may view G as the fundamental group of one of the graphs of groups pictured on Figure 1. The corresponding Bass-Serre trees have trivial edge stabilizers, and the vertex stabilizers are precisely the conjugates of the Gi’s; we call a tree with these properties a Grushko tree (if G is freely indecomposable, Grushko trees are points). G2 t1 b G1 G1 b t2 . G G G4 . 2 b 3 b . b tq p = 5, q = 6 Gp G5 Figure 1: Graph of groups decompositions corresponding to two Grushko trees. Since the Gi’s are freely indecomposable, Grushko trees T0 have the following maxi- mality property: if T is any tree on which G acts with trivial edge stabilizers, Gi fixes a point in T , and therefore T0 dominates T in the sense that there is a G-equivariant map T0 T . In other words, among free decompositions of G, a Grushko tree T0 is as far as possible! from the trivial tree (a point): its vertex stabilizers are as small as possible (they are conjugates of the Gi’s). This maximality property does not determine T0 uniquely, as it is shared by all Grushko trees; we will come back to this key fact later, when discussing uniqueness. When more general decompositions are allowed, for instance when one considers split- tings over Z, there may not exist a tree with the same maximality property. The funda- mental example is the following. Consider an orientable closed surface Σ, and two simple closed curves c1; c2 in Σ with non-zero intersection number. Let Ti be the Bass-Serre tree of the associated splitting of G = π1(Σ) over Z π1(ci). Since c1 and c2 have positive ' intersection number, π1(c1) is hyperbolic in T2 (it does not fix a point) and vice-versa.