Makanin–Razborov diagrams for hyperbolic groups

We give a detailed account of Zlil Sela’s construction of Makanin–Razborov diagrams describing Hom(G, Γ) whereG is a finitely generated group and Γ is a hyperbolic group. We also deal with the case where Γ has torsion. Les diagrammes Makanin–Razborov pour les groupes hyperboliques Résumé Nous proposons une présentation détaillée de la construction des diagrammes de Makanin–Razborov de Zlil Sela qui décrivent Hom(G, Γ) pour un groupe G de type fini et un groupe hyperbolique Γ. De plus, nous traitons le cas où Γ est un groupe ayant de la torsion.


Introduction
The question whether one can decide if a system of equations (with constants) in a free group has a solution was answered affirmatively by Makanin [31] who described an algorithm that produces such a solution if it exists and says No otherwise. In his groundbreaking work he introduced a rewriting process for systems of equations in the free semigroup. This process was later refined by Razborov to give a complete description of the set of solutions for a system of equations in a free group [32,33]. This description is now referred to as Makanin-Razborov diagrams. Rips recognized that the Makanin process can be adapted to study group actions on real trees which gave rise to what is now called the Rips machine, a structure theorem for finitely presented groups acting on real trees similar to Bass-Serre theory for groups acting on simplicial trees, see [5,17,21]. This has been generalized to finitely generated groups by Sela [38] and further refined by Guirardel [22]. Recently Dahmani and Guirardel [11] have in turn used the geometric ideas underlying the Rips theory to provide an alternative version of Makanin's algorithm.
Razborov's original description of the solution set has been refined independently by Kharlampovich and Myasnikov [26,27] and Sela [40]; this description has been an important tool in their solutions to the Tarski problems regarding the elementary theory of free groups. Kharlampovich

and Myasnikov modified Razborov's methods to obtain
Keywords Makanin -Razborov Diagrams. their own version of the rewriting process while Sela used the Rips machine extensively, bypassing much of the combinatorics of the process.
While it seems that the full potential of the ideas underlying the Makanin process has not yet been realized there are a number of generalizations of the above results. Sela [41] has shown the existence of Makanin-Razborov diagrams for torsion-free hyperbolic groups, which was then generalized to torsion-free relatively hyperbolic groups with finitely generated Abelian parabolic subgroups by Groves [19]. Makanin-Razborov diagrams for free products have been constructed independently by Jaligot and Sela following Sela's geometric approach and by Casals-Ruiz and Kazachkov [10] following the combinatorial appoach of Kharlampovich and Myasnikov. Casals-Ruiz and Kazachkov also constructed Makanin-Razborov diagrams for graph groups [10].
It is the purpose of this article to give a detailed account of Sela's construction of Makanin-Razborov diagrams for hyperbolic groups while also removing the torsionfreeness assumption. It should be noted that Dahmani and Guirardel's version of the Makanin algorithm also applies to hyperbolic groups with torsion.
We are only dealing with equations without constants, no new ideas are needed to deal with them. Thus we are interested in finding all tuples (x 1 , . . . , x n ) ∈ Γ n that satisfy equations w i (x 1 , . . . , x n ) = 1 for i ∈ I where w i is some word in the x ±1 j and Γ is a hyperbolic group. It is clear that these solutions are in 1-to-1 correspondence to homomorphisms from G = x 1 , . . . , x n | w i (x 1 , . . . , x n ), i ∈ I to Γ. Thus paramatrizing the set of solutions to the above system of equations is equivalent to parametrizing Hom(G, Γ). Theorem 1.1. Let Γ be a hyperbolic group and G be a finitely generated group. Then there exists a finite directed rooted tree T with root v 0 satisfying (1) The vertex v 0 is labeled by G.
(2) Any vertex v ∈ VT, v v 0 , is labeled by a group G v that is fully residually Γ.
(3) Any edge e ∈ ET is labeled by an epimorphism π e : G α(e) → G ω(e) such that for any homomorphism φ : G → Γ there exists a directed path e 1 , . . . , e k from v 0 to some vertex ω(e k ) such that where α i ∈ Aut G ω(e i ) for 1 ≤ i ≤ k and ψ : G ω(e k ) → Γ is locally injective.
Here a homomorphism is called locally injective if it is injective when restricted to 1-ended and finite subgroups, thus it is injective on the vertex groups of any Dunwoody/Linnell decompositon.
The proof broadly follows Sela's proof in the torsion-free case but is also partly inspired by Bestvina and Feighn's exposition of Sela's construction in the case of free groups [6] and by Groves's adaption of Sela's work to the relatively hyperbolic case. We further rely on Guirardel's version of the Rips machine.
When following Sela's strategy to prove Theorem 1.1 it is almost unavoidable to also prove the following results that are all proven in Section 7 and follow from the same stream of ideas. All results are due to Sela in the torsion-free case. Theorem 1 reconciles two related notions of Γ-limit groups in the case of hyperbolic Γ, see Section 2.1 for definitions.
Recall that a group G is called Hopfian if every epimorphism G → G is an isomorphism.

Corollary 7.9. Hyperbolic groups are Hopfian.
A group G is called equationally Noetherian if for every system of equations in G there exists a finite subsystem that has the same set of solutions. In Section 2 we introduce Γ-limit groups, these are the groups that occur as vertex groups in the Makanin-Razborov diagram. Moreover, we illustrate how Γ-limit groups admit actions on real trees if Γ is a hyperbolic group and study the stability properties of these actions. In Section 3 and Section 4 we discuss the Rips machine and the JSJdecomposition of Γ-limit groups before we give a detailed discussion of the shortening argument in Section 5. The shortening argument in particular implies that the Makanin-Razborov diagrams are locally finite. In Section 6 we then construct Makanin-Razborov diagrams for hyperbolic groups under the additional assumption that Γ is equationally Noetherian. In Section 7 we then discuss Sela's shortening quotients and prove that all hyperbolic groups are in fact equationally Noetherian, i.e. that the construction in Section 6 applies to all hyperbolic groups. The authors would like to thank Abderezak Ould Houcine for pointing out the content of Lemma 2.22 (3) and Lemma 6.1.
A first version of this paper circulated in 2010. We do not discuss the numerous more recent developments in the field.

Γ-limit groups and their actions on real trees
In this chapter we introduce the concept of a Γ-limit group. While we will later on solely be interested in the case where Γ is hyperbolic, the notion of a Γ-limit group can be formulated for any group Γ. In the hyperbolic case, a Γ-limit group naturally admits actions on metric R-trees, which arise as limits of group actions on the Cayley graph of Γ (w.r.t. a fixed finite generating set). These Γ-limit groups will occur in the Makanin-Razborov diagrams. In Section 2.2 we show in detail how these limit actions arise, and in Section 2.3 we prove important stability properties of these actions. We will then study the structure of virtually Abelian subgroups of Γ-limit groups. Most of the material is standard except that we have to deal with torsion.

Γ-limit groups
Throughout this section Γ is an arbitrary group. We will discuss sequences of homomorphisms from a given (f.g.) group G to Γ. We denote by Hom(G, Γ) the set of all such homomorphisms. Moreover, if (ϕ i ) i ∈N is a sequence of homomorphisms from G to Γ, we simply write (ϕ i ) ⊂ Hom(G, Γ).
With these notations we give a definition of a Γ-limit group which is analogous to the definition of Bestvina and Feighn [6] in the case where Γ is free. Definition 2.1. Let G be a group and (ϕ i ) ⊂ Hom(G, Γ). The sequence (ϕ i ) is stable if for any g ∈ G either ϕ i (g) = 1 for almost all i or ϕ i (g) 1 for almost all i. If (ϕ i ) is stable then the stable kernel of the sequence, denoted by ker − → (ϕ i ), is defined as ker − → (ϕ i ) := {g ∈ G | ϕ i (g) = 1 for almost all i}.
We then call the quotient G/ker − → (ϕ i ) the Γ-limit group associated to (ϕ i ) and the projection ϕ : G → G/ker − → (ϕ i ) the Γ-limit map associated to (ϕ i ).
We call a quotient map ϕ : G → G/N a Γ-limit map if it is the Γ-limit map associated to some stable sequence (ϕ i ) ⊂ Hom(G, Γ). We will denote the Γ-limit group G/N by L ϕ .

Remark 2.2.
Note that any subgroup H of a Γ-limit group L = L ϕ is a Γ-limit group. This can be seen by considering the stable sequence (ϕ i | ϕ −1 (H) ) ⊂ Hom(ϕ −1 (H), Γ) with associated Γ-limit group H. Cleary Γ and every subgroup of Γ are Γ-limit groups.
It turns out that Γ-limit groups are closely related to fully residually Γ groups: A group G is called residually Γ if for every non-trivial element g ∈ G there exists a homomorphism ϕ : G → Γ s.th. ϕ(g) 1. Further, G is called fully residually Γ if for any finite set S ⊂ G there exists a homomorphism ϕ : G → Γ such that ϕ| S is injective. Note that any subgroup of Γ is fully residually Γ. We can immediately verify that in the case of countable groups, every fully residually Γ group is a Γ-limit group: Lemma 2.3. If G is countable and fully residually Γ, then G is a Γ-limit group.
Proof. Choose a surjective map f : N → G. For each i ∈ N let M i := { f ( j) | j ≤ i} and choose ϕ i : G → Γ such that ϕ i | M i is injective. Clearly the sequence (ϕ i ) is stably injective. Thus G = G/ker − → (ϕ i ) is a Γ-limit group.
It turns out that in many situations, in particular in the case where Γ is hyperbolic, a partial converse is true as well, namely f.g. Γ-limit groups are fully residually Γ, see Corollary 7.6 below. This is more generally true whenever Γ is equationally Noetherian, which we will prove in Section 6.1.
From now on we will almost exclusively consider finitely generated groups.

Limit actions on real trees
Throughout this section G is a finitely generated group endowed with a finite generating set S G . Call a pseudo-metric d on G (left) G-invariant if d(g, h) = d(kg, k h) for all g, h, k ∈ G.
We consider A(G), the space of all G-invariant pseudo-metrics on G, with the compactopen topology. Thus a sequence (d i ) of G-invariant pseudo-metrics on G converges in A(G) iff the sequence (d i (1, g)) converges in R for all g ∈ G.
Lemma 2.4. Let (d i ) ⊂ A(G) be a sequence of G-invariant pseudo-metrics. If there exists λ ∈ R such that for each i ∈ N and s ∈ S G , d i (1, s) ≤ λ, then (d i ) has a subsequence which converges in A(G).
Proof. For k ∈ N, let B k := {g ∈ G | d S G (1, g) ≤ k} (where d S G denotes the word metric on G w.r.t. S G ). If λ is as above, it follows that for all k ∈ N and g ∈ B k , As B k is finite, the compactness of the cube [0, kλ] |B k | then implies that there is a subsequence (d i j, k ) j ∈N ⊂ (d i ) such that for all g ∈ B k , the sequence converges. Moreover, each sequence (d i j, k ) j ∈N may be chosen as a subsequence of (d i j, k−1 ) j ∈N . It is now obvious that the diagonal sequence d i k, k k ∈N converges in A(G).
Remark 2.5. Note that the condition that d i (1, s) ≤ λ for all s ∈ S and i ∈ N is essential, i.e. in general, sequences of pseudo-metrics do not have convergent subsequences. In the literature this is often bypassed by projectivizing the space A(G) and thereby compactifying it. We will not adopt this point of view.
In the following we study sequences of pseudo-metrics on a group G that are induced by actions on based hyperbolic G-spaces.
Throughout this article we use the following definition of δ-hyperbolicity: A pseudometric space (X, d) (resp. a pseudo-metric d) is called δ-hyperbolic for δ ≥ 0 if for any x, y, z, t ∈ X, where (x|y) t is the Gromov product of x and y with respect to t given by (x|y) t := 1 2 (d(x, t) + d(y, t) − d(x, y)) .

(2.2)
Note that the constant being δ 3 rather than δ is slightly non-standard. For a geodesic metric space this choice of constant implies the δ-thinness of geodesic triangles, i.e. that for any geodesic triangle [x, y] ∪ [y, z] ∪ [z, x] we have see [1]. The definition via the Gromov product has the advantage that it also applies to pseudo-metric spaces that are not geodesic.
In the following we study group actions. An action of a group G on a metric space X is a homomorphism ρ : G → Isom(X) from G to the isometry group of X. A (based) G-space is a tuple (X, x 0 , ρ) consisting of a metric space X, a base point x 0 ∈ X and an action ρ of G on X. If g ∈ G and x ∈ X, for convenience we will denote the element ρ(g)(x) ∈ X simply by gx if the action ρ is understood. Moreover, when we want to explicitly state the action, we use the notation ρ gx := ρ(g)(x) to improve readability. We will study pseudo-metrics that are induced by actions on metric spaces. Let X = (X, x, ρ) be a based G-space. Then the G-action ρ on X induces a pseudo-metric d x ρ : G × G → R ≥0 on G, given by d x ρ (g, h) = d X ( ρ gx, ρ hx). Note that this pseudo-metric is G-invariant, and that d x ρ is δ-hyperbolic if d X is δhyperbolic. We will denote d x ρ simply by d ρ if the basepoint is clear from the context. Lemma 2.6. For i ∈ N, let δ i ≥ 0 and X i = (X i , x i , ρ i ) a based δ i -hyperbolic G-space.
Assume that the sequence d x i ρ i converges in A(G) to a limit sequence d ∞ and Then there exists a based real G-tree (T, x, ρ) such that T is spanned by ρ Gx, and d x ρ = d ∞ .
2.1. A quadrilateral degenerating to a tree Proof. As lim i→∞ δ i = 0, it follows that d ∞ is 0-hyperbolic. We obtain a G-action on the 0-hyperbolic metric space ( G, d ∞ ) obtained from the pseudo-metric space (G, d ∞ ) by metric identification, i.e. by identifying points of distance 0. Then there exists a real G-tree T satisfying • T admints a G-equivariant isometric embedding η : G → T, • η( G) spans T, i.e. no proper subtree of T contains η( G).
We sketch the way T is constructed, for details we refer to Lemma 2.13 of [3]: Start with the metric space ( G, d ∞ ) and for any x, y ∈ G, add a segment of length d ∞ (x, y) between x and y. Finally, identify the initial segments of [x, y] and [x, z] of length (y|z) x for any x, y, z ∈ G. By construction, the induced pseudo-metric d x ρ of the G-action on T with basepoint x := η(1) is equal to d ∞ .
Later, we will need to scale the pseudo-metrics induced by sequences of actions of δ-hyperbolic G-spaces to ensure the existence of a converging subsequence. The scaling factor is the norm of the action as defined below. Note that the definition depends on the choice of the generating set S G . Definition 2.7. Let X = (X, x, ρ) a based G-space. The norm of the action ρ with respect to the base point x (and the fixed generating set S G ), denoted by | ρ| x , is defined as We may simply write | ρ| if the basepoint x is clear from the context. Using this definition, we obtain the following corollary as a consequence of Lemma 2.6.
Then there exists a based real G-tree (T, x, ρ) such that the induced sequence 1 |ρ i | d ρ i of scaled pseudo-metrics on G has a subsequence converging in A(G) to d ρ , and T is spanned by ρ Gx.
Proof. For each s ∈ S G and i ∈ N, we have 1 So the claim follows from Lemma 2.6.
Note that the action of G on the limit tree T may not be minimal. However, Theorem 2.11 below shows that the minimality of the action can be guaranteed if the basepoints are chosen appropriately. Before we show this, we introduce the useful concept of approximating sequences. Definition 2.9. Let (X i ) = ((X i , x i , ρ i )) be a sequence of metric G-spaces. Assume that the sequence (d ρ i ) converges to a pseudo-metric d ρ induced by the based G-space It is easy to see that every point ρ gx in the orbit of the basepoint x is approximated by the sequence ( ρ i gx i ). In particular, the sequence (x i ) of basepoints approximates x. However, in general the limit space may contain points which do not admit an approximating sequence. But the following lemma implies that this cannot occur under the hypothesis of Lemma 2.6.
Then the following hold.
(1) Every t ∈ T has an approximating sequence.
(2) If (t i ) and (t i ) are approximating sequences for some t ∈ T, then is an approximating sequence for t then ( ρ i gt i ) is an approximating sequence for ρ gt.
(4) If (t i ) and (y i ) are approximating sequences for t and y then Proof. Let t ∈ T. As T is spanned by ρ Gx there exist g 1 , This choice clearly implies that (2.3) holds for g ∈ {g 1 , g 2 }. Now consider an arbitrary g ∈ G. To prove (1) we need to show that 2)). Since T is 0-hyperbolic, this implies w.l.o.g. (possibly after exchanging g 1 and g 2 ) that ( ρ g 2 x| ρ gx) t = 0 as in Figure 2 2. An approximating sequence (t i ) of t.
Thus (t i ) is an approximating sequence of t and (1) is established.
To prove (2) note first that it suffices to deal with the case where (t i ) is constructed as in the proof of (1), in particular t i ∈ [ ρ i g 1 x i , ρ i g 2 x i ] for all i and some fixed g 1 , g 2 ∈ G. As (t i ) is an approximating sequence for t it follows that (3) is trivial and (4) follows from (2) and the fact that we can construct approximating sequences for t and y as in the proof of (1) by choosing g 1 , g 2 such that both t and y lie on [ ρ g 1 x, ρ g 2 x].
We conclude the section with the following theorem, it guarantees that a limit action is minimal provided that base points are centrally located, meaning that the norm is smallest for the chosen base points. Theorem 2.11. Let (X i , x i , ρ i ) for i ∈ N and (T, x, ρ) be as in Lemma 2.6. If for any i and any y ∈ X i ,

4)
then the limit action of G on T is minimal.
Proof. The proof is by contradiction. Assume that T ⊂ T is a proper G-invariant subtree. Recall that T is spanned by the orbit of the base point x. This implies that x T as otherwise the orbit ρ Gx and therefore T would be contained in T . Let p x be the nearest point projection of x to T . It follows that for any g ∈ G we have ρ gp for all g ∈ G. Moreover for some s ∈ S G we have ρ sx x as otherwise ρ Gx = x and therefore T = {x} as T is spanned by ρ Gx. This implies in particular that for some contradicting the assumption on the x i . We conclude this section by explaining how the previous statements imply that most Γ-limit groups act on real trees if Γ is a hyperbolic group.
Let Γ be a hyperbolic group with finite generating set S Γ and X := Cay(Γ, S Γ ). As before G is a group with finite generating set S G . Let δ be the hyperbolicity constant of X.
Recall that Γ is a subgroup of Isom(X) if we consider the natural left action of Γ on X. Thus any ϕ ∈ Hom(G, Γ) defines an action of G on X, namely It follows that (X, 1, ϕ) is a based G-space for any ϕ ∈ Hom(G, Γ). This action induces the δ-hyperbolic pseudo-metric d ϕ := d 1 ϕ on G given by We can now define the norm of φ, cf. Definition 2.7: Definition 2.12. We call |ϕ| := |ϕ| 1 = s ∈S G d X (1, ϕ(s)) the norm of ϕ.
For any g ∈ Γ let c g : Γ → Γ be the inner automorphism given by c g (h) for all h, k ∈ G. It follows in particular that |c g • ϕ| = |ϕ| g .
(2) (ϕ i ) does not contain a subsequence (ϕ j i ) such that ker ϕ j i = ker ϕ j i for all i, i ∈ N.
The second condition implies that we may assume that the ϕ i are pairwise distinct after passing to a subsequence. It follows in particular that lim i→∞ |ϕ i | = ∞ as for any K there are only finitely many homomorphisms of norm at most K.
Let now X i = (X i , d X i ) be the metric space obtained from X by scaling with the factor 1 |ϕ i | , thus X i = X (the underlying sets) and Clearly G also acts on X i by isometries where the action on the underlying sets X = X i coincide, hence we obtain a based G-space The following theorem is a consequence of Theorem 2.6 and the above discussion. Here we call a G-space non-trivial if it does not have a global fixed point. Theorem 2.13. Let (ϕ i ) ⊂ Hom(G, Γ) be a sequence of conjugacy-short homomorphisms. Then one of the following holds.

Proof. It follows from Lemma 2.4 that
for all s ∈ S G . Assume that (ϕ i ) does not contain a constant subsequence. Then d ρ i is δ i -hyperbolic with lim δ i = 0, and it follows from Lemma 2.6 that the limit metric concides with d ρ for some real based G-tree (T, x, ρ). As |ϕ i | ≤ |ϕ i | g for all i ∈ N and g ∈ Γ it further follows from Theorem 2.11 that this action is minimal. Thus we need to argue that T has more than one point. This however follows from the fact that Remark 2.14. Let (ϕ i ) and (T, x, ρ) be as in the conclusion of Theorem 2.13 (2). By Lemma 2.10 any t ∈ T admits an approximating sequence, i.e. a sequence (t i ) with t i ∈ X i = X such that for all g ∈ G we have As lim i→∞ |ϕ i | = ∞ this clearly implies that for each i ∈ N, t i may be chosen to be a vertex in X.
We conclude this section with an important example that shows how distinct sequences of homomorphisms in Hom(Z 2 , Z) give rise to distinct actions on real trees. We consider G = Z 2 with the generating set S G = {(1, 0), (0, 1)} and Γ = Z with the generating set S Γ = {1}.

Stability of limit actions of limit groups
Throughout this section let G be a f.g. group and Γ be an infinite hyperbolic group, both equipped with word metrics relative to fixed finite generating sets S G and S Γ respectively.
Call a sequence (ϕ i ) ⊂ Hom(G, Γ) convergent if the sequence d ϕ i |ϕ i | converges in A(G). Now let (ϕ i ) be a convergent sequence and assume that (ϕ i ) does not contain a constant subsequence. Then we obtain a G-tree (T, x, ρ) as in Corollary 2.8. We will simply say that the sequence (ϕ i ) converges to the limit G-tree T.
Moreover, if (ϕ i ) is stable with Γ-limit map ϕ, then the action ρ factors through ϕ, i.e. we obtain an action ρ ∞ : L ϕ → Isom(T) s.th. ρ = ρ ∞ • ϕ. We call the space (T, x, ρ ∞ ) the limit L ϕ -tree of the sequence (ϕ i ), resp. (T, x, ρ) the limit G-tree. (We may simply speak of the limit tree, or limit action, of (ϕ i ) if it is clear from the context whether we refer to the G-action or the L ϕ -action.) Now consider an arbitrary stable sequence (ϕ i ) ⊂ Hom(G, Γ) with associated Γ-limit map ϕ. We may assume that ϕ i is conjugacy-short for all i ∈ N as replacing ϕ i with c g • ϕ i for some g ∈ Γ does not change the kernel and therefore preserves the stability of (ϕ i ).
If (ϕ i ) contains a constant subsequence, i.e. contains a homomorphism ϕ infinitely many times, then the stability of (ϕ i ) implies that ker ϕ = ker − → (ϕ i ). It then follows that If such a sequence does not exist, then Theorem 2.13 implies that (ϕ i ) has a subsequence (ϕ j i ) such that the sequence of pseudo-metrics ( 1 |ϕ j i | d ϕ j i ) converges to a pseudo-metric d x ρ for some minimal non-trivial G-tree (T, x, ρ). Thus, (ϕ j i ) is a converging subsequence. We will call a sequence with the above properties strict: (1) ϕ i is conjugacy-short for all i ∈ N.
Thus the above discussion implies that any strict sequence has a subsequence that converges to a limit G-tree T.
Remark. Note that the limit action does depend on the choice of the subsequence. Indeed, consider the sequences (ϕ i ) ⊂ Hom(Z 2 , Z) and (ϕ i ) ⊂ Hom(Z 2 , Z) discussed at the end of the previous section. Clearly both sequences are stable and is not convergent the two subsequences (ϕ i ) and (ϕ i ) are. However they converge to distinct actions of Z 2 on R.
Definition 2.16 ([5]). Let T be a G-tree. A non-degenerate subtree S ⊂ T is called stable if for every non-degenerate subtree S ⊂ S, stab G (S ) = stab G (S). Otherwise S is called unstable. The tree T is stable if every non-degenerate subtree of T contains a stable subtree.
The following theorem is the main result of this section. Recall that a tripod is a tree spanned by three points and a tripod is called non-degenerate if it is not spanned by two points.
If P is a class of groups then we say that a group G is finite-by-P if G contains a finite normal subgroup N such that G/N is in P.
Theorem 2.17. Let (ϕ i ) ⊂ Hom(G, Γ) be a convergent strict sequence with induced Γ-limit map ϕ, and L = L ϕ . Let T be the limit L-tree. Then the following hold for the action of L on T.
(1) The stabilizer of any non-degenerate tripod is finite.
(2) The stabilizer of any non-degenerate arc is finite-by-Abelian.
(3) Every subgroup of L which leaves a line in T invariant and fixes its ends is finite-by-Abelian.
(4) The stabilizer of any unstable arc is finite.
Before we proceed with the proof of Theorem 2.17 we recall some useful facts about torsion subgroups of hyperbolic groups. Proposition 2.18. Let Γ be a hyperbolic group. Then the following hold.
(1) There exists a constant N = N(Γ) such that every torsion subgroup of Γ has at most N elements.
(2) There exists a constant L = L(Γ) such that for every subgroup H ≤ Γ, one of the following holds.
(a) H is a finite group (of order at most N(Γ)).
(b) For any generating set S of H there exists a hyperbolic element γ ∈ H such that |γ| S ≤ L, where | · | S denotes the word length on H relative to S.
Proof. Note first that torsion subgroups of hyperbolic groups are finite, see e.g. [18,Chapitre 8,Corollaire 36]. Thus (1) follows from the fact that for any hyperbolic group Γ there exists N(Γ) such that any finite group is of order at most N(Γ), see [7,8].
(2) is essentially due to M. Koubi [28]. Proposition 3.2 of [28] implies that there exists a finite setS ⊂ Γ such that any set S ⊂ Γ is either conjugate to a subset ofS or that there exists a word w in S ∪ S −1 of length at most 2 such that w represents a hyperbolic element of Γ. Now for each subset S ofS let L(S) = 0 if S is finite and let L(S) be the length of a shortest word in S ∪ S −1 that represents a hyperbolic element otherwise.
It turns out that the existence of a uniform bound on the order of finite subgroups of a hyperbolic group Γ ensures the same for Γ-limit groups. Lemma 2.19. Let Γ be hyperbolic and L be a Γ-limit group. Then the following hold.
(1) Every torsion subgroup of L has at most N(Γ) elements.
Proof. Choose a stable sequence (ϕ i ) ⊂ Hom(G, Γ) with induced Γ-limit map ϕ : G → L = L ϕ . We prove (1) by contradiction. Thus assume there exists a torsion subgroup E ≤ L that contains N(Γ) + 1 pairwise distinct elements g 0 , . . . , g N (Γ) . For each k = 0, . . . , N(Γ), The stability of (ϕ i ) implies that ϕ i ( g m ) ϕ i ( g n ) for large i and 0 ≤ n m ≤ N(Γ). Thus Proposition 2.18 (1) implies that ϕ i ( g 0 ), . . . , ϕ i ( g N (Γ) ) is infinite for large i. Proposition 2.18 (2) then implies that for large i there exists a word w i in g 0 , . . . , g N (Γ) of length at most L(Γ) such that ϕ i (w i ) is of infinite order. Now there are only finitely many such words. Thus there exists a word w such that w = w i for infinitely many i. As E is assumed to be a torsion group it follows that ϕ(w) k = 1 for some k, i.e. that w k ∈ ker − → (ϕ i ) and therefore w k ∈ ker ϕ i for almost all i, a contradiction. Thus (1) is proven.
We now show (2). Clearly, if A is finite-by-Abelian, so are all f.g. subgroups. Thus we need to show that if the commutator subgroup of A is infinite, i.e. contains N(Γ) + 1 distinct elements g 0 , . . . , g N (Γ) , then the same is true for some f.g. subgroup of A. This however is obvious as any element of the commutator subgroup of some group is the product of finitely many commutators and therefore lies in the commutator subgroup of a finitely generated subgroup.
Let now Γ be a hyperbolic group and X := Cay(Γ, S Γ ) its Caleygraph. For any hyperbolic element γ ∈ Γ we denote its fixed points in ∂ X by p γ + and p γ − and we denote by A γ the union of all geodesics connecting p γ + and p γ − . We call A γ the axis of γ. A γ is a γ-invariant subset of X that is contained in the 2δ-neighborhood of any geodesic connecting p γ + and p γ − .
Lemma 2.20. Let Γ be a hyperbolic group and X := Cay(Γ, S Γ ). Then there exists a constant K such that for any hyperbolic element γ ∈ Γ and x ∈ X we have Proof. If the translation length of γ is greater than 10δ then the definition of hyperbolicity easily implies that this inequality holds for K = 0 as the path [x, is a (1, 4δ)-quasigeodesic joining x and γx. For any fixed hyperbolic element γ the existence of such a constant K is also easily established. As there are only finitely many conjugacy classes of elements with translation length at most 10δ this guarantees the existence of a uniform K as above.
The following proposition is the main step in the proof of Theorem 2.17 (2) and (3).
Then there exist constants C, ε ∈ R >0 such that for any set Y ⊂ Γ the following holds.
then Y is either finite or finite-by-Z.
Proof. Let K be the constant from Lemma 2.20. We may assume that K ≥ δ where δ is the hyperbolicity constant of X. Let M be the number of elements of Γ of word length at most 20δ. We will show that the claim of the proposition holds for C := 1000 · K and ε := 1 1000(M ·L(Γ)+1) . Let Y ⊂ Γ and assume that x 1 , x 2 ∈ X satisfy the conditions of the proposition with C, ε as above. If Y is finite there is nothing to show. Thus we may assume that Y is infinite. By Proposition 2.18 (2) there exists a word w 0 of length at most L(Γ) in Y ∪ Y −1 that represents a hyperbolic element of Γ. As before we denote by p w 0 + and p w 0 − the fixed points of w 0 in ∂ X. We will show that any y ∈ Y also fixes p w 0 + and p w 0 − . The triangle inequality and the assumption of the proposition imply that for any w ∈ Y we have thus x i is contained in the d(x 1 , x 2 ) 500 -neighborhoods of both A w 0 and w j 0 yA w 0 = A (w j 0 y)w 0 (w j 0 y) −1 . Let now z ∈ V X = Γ be the midpoint of a geodesic segment [x 1 , x 2 ] between x 1 and x 2 . The 2δ-thinness of geodesic triangles imply that the 2δ-neighborhoods of both A w 0 and w j 0 y A w 0 contain the geodesic segment [y 1 , y 2 ] ⊂ [x 1 , x 2 ] that is centered at z and of length 99 100 d(x 1 , x 2 ). It now follows easily that As there are M points in the 20δ-neighborhood of z it follows that there are j 1 < j 2 ∈ {0, . . . , M } such that After conjugation with w As [w 0 , y] commutes with w j 0 0 it follows that [w 0 , y] preserves the fixed points of w j 0 0 , i.e. preserves p w 0 + and p w 0 − . As [w 0 , y] = w 0 · yw 0 y −1 and as w 0 preserves p w 0 + and p w 0 − it follows that yw 0 y −1 also preserves p w 0 + and p w 0 − , thus y preserves {p w 0 + , p w 0 − } as a set. It now easily follows from assumption (2) that y in fact preserves p w 0 + and p w 0 − pointwise. This implies that Y preserves p w 0 + and p w 0 − . Thus Y is a 2-ended group that acts end-preservingly on itself, thus Y is finite-by-Z by Lemma 4.1 of [43].
We can now prove Theorem 2.17.
Proof of Theorem 2.17. We first prove (1). Let D be a non-degenerate tripod spanned by x, y, z ∈ T. By Lemma 2.19 it suffices to show that H := stab L (D) is a torsion group. Let h ∈ H and pick h ∈ G such that ϕ( h) = h. We need to show that ϕ i ( h) is of finite order for large i as this implies that h N (Γ)! ∈ ker − → (ϕ i ) which implies that h is a torsion element. We follow the argument from the proof of Lemma 4.1 in [35]. We argue by contradiction, thus after passing to a subsequence we may assume that ϕ i ( h) is hyperbolic for all i ∈ N.
Let (x i ), (y i ) and (z i ) be approximating sequences of x, y and z. By Remark 2.14 we may assume that the x i , y i and z i are vertices of X, i.e. elements of Γ. By Lemma 2.20 there exists a constant K such that for all i ∈ N and therefore x lies on A h . The same argument shows that y and z lie on A h contradicting the assumption that x, y and z span a non-degenerate tripod as A h is a line.
To prove (2), assume that H ≤ G stabilizes a non-degenerate arc [x, y] ⊂ T. Let (x i ) and (y i ) be approximating sequences of x, respectively y. Clearly lim i→∞ d X (x i , y i ) = ∞ and  (2).
The proof of (3) is similar to that of (2). Assume that H ≤ G acts orientationpreservingly on a line Y ⊂ T with ends x and y. Choose sequences (x k ) k ∈N and (y k ) k ∈N of points on Y that converge to x, respectively y.
Clearly lim k→∞ d T (x k , y k ) = ∞ and therefore is just the translation length of ϕ(h) and therefore independent of k.
For each k choose approximating sequences (x k i ) i ∈N of x k and (y k i ) i ∈N of y k . Now fix h ∈ H and k ∈ N. It follows from the definition of approximating sequences that As the right hand sides of these equations tend to 0 as k tends to ∞ if follows that for some subsequence (ϕ m i ) of (ϕ i ) we get .
As H is countable a diagonal argument shows that we can assume that this holds for all h ∈ H after passing to a subsequence. Thus we can argue as in the proof of (2). To prove (4), let [y 1 , y 2 ] [y 3 , y 4 ] and As γ does not fix both y 3 and y 4 , we may assume γ(y 3 ) y 3 .

Virtually Abelian subgroups of Γ-limit groups
Call a group almost Abelian if it contains a finite-by-Abelian subgroup of finite index. It is trivial that any virtually Abelian group is almost Abelian and the converse holds for finitely generated groups, see Theorem 8.40 of [9].
While we will later see that almost Abelian subgroups of finitely generated Γ-limit groups (with Γ hyperbolic) are finitely generated and therefore virtually Abelian, this is can also be verified directly exploiting the ideas from the previous section.
Note that subgroups of almost Abelian groups are almost Abelian and that almost Abelian subgroups of hyperbolic groups are 2-ended. Throughout this section Γ is a hyperbolic group. We study not necessarily finitely generated almost Abelian Γ-limit groups.

Lemma 2.22. Let
A be an infinite Γ-limit group. Then the following hold.
(1) If A is almost Abelian, it is either finite-by-Abelian or contains a unique finite-by-Abelian subgroup A + of index 2.
(3) A is virtually Abelian iff A is almost Abelian.
(4) A is virtually Abelian iff all f.g. subgroups of A are virtually Abelian.
Proof. Assume that A = a 0 , a 1 , . . . . As A is infinite it contains an element of infinite order by Proposition 2.18, thus we may assume that a 0 is of infinite order. Choose a stable sequence (ϕ i ) ⊂ Hom(G, Γ) with induced Γ-limit map ϕ : G → A = L ϕ . For each i pick a liftã i ∈ G such that ϕ(ã i ) = a i . For k ∈ N put A k = a 0 , . . . , a k and A k := ã 0 , . . . ,ã k , clearly A = k ∈N A k . Note that all A k and A k are infinite as a 0 and therefore alsoã 0 is of infinite order.
(1). Let A be almost Abelian. Note first that all A k are finitely generated and almost Abelian and therefore finitely presented. Choose relators r 1 , . . . , r m k ∈ F(a 0 , . . . , a k ) such that A k = a 0 , . . . , a k | r 1 , . . . , r m k .
Let r l be the word obtained from r l by replacing occurences of a ±1 j byã ±1 j . As ϕ( r l ) = 1 for all l it follows that r l ∈ ker − → (ϕ i ) and therefore ϕ i ( r l ) = 1 for all l and large i. This implies that there exists m k such that A k,i := ϕ i ( A k ) is a quotient of A k for i ≥ m k and therefore almost Abelian. We may further assume that for i ≥ m k ϕ i (ã 0 ) is of infinite order. For i ≥ m k the group A k,i is therefore an infinite almost Abelian subgroup of some hyperbolic group and hence 2-ended. For i ≥ m k let A + k,i be the subgroup of A k,i consisting of all elements that preserve the ends of A k,i . Clearly, | A k,i : As A k is finitely generated it contains only finitely many subgroups of index 2. A diagonal argument therefore shows that after passing to a subsequence we can assume that for each k there exists a subgroup A + k ≤ A k and n k ≥ m k ∈ N such that A + k = A + k,i for all i ≥ n k . As the images ϕ i ( A + k ) act orientation preservingly on an axis of Γ for i ≥ n k it follows from Proposition 2.21 that A + k,i : . It follows as in the proof of Theorem 2.17 (2) that [A + k , A + k ] is finite, i.e. that also A + k is finite-by-Abelian.
Clearly, A + k is of index at most 2 in A k . It is further easily verified that A + k ≤ A + k+1 as for contains an infinite dihedral group and cannot be finite-by-Abelian. As this holds for all (large enough) i, it follows easily that U is not finite-by-Abelian, which is a contradiction. This proves (1).
(2). If A is finite-by-Abelian and all choices are as in the proof of (1), then A k,i = A + k,i is finite-by-Z for each k and i ≥ n k . As the order of finite subgroups is bounded by N(Γ) there are only finitely many isomorphism classes of subgroups that are met by the A k,i . In particular there exists a constant M such that for all k and i the center as there are only finitely many subgroups of given (finite) index. Thus we can assume, after passing to a subsequence, that . After passing to a subsequence we can assume that | A k : (3). If A is almost Abelian then A is virtually Abelian by (1) and (2). Moreover if A is virtually Abelian then the proof of (1) goes through unchanged as images of virtually Abelian groups are virtually Abelian, it follows that A is almost Abelian.
(4). Clearly, if A is virtually Abelian, so is every f.g. subgroup. Conversely, assume that all finitely generated subgroups of A are virtually Abelian. This implies in particular that A k is virtually Abelian for all k. If infinitely many A k are finite-by-Abelian, then each f.g. subgroup is finite-by-Abelian as a subgroup of some A k , and the claim follows from Lemma 2.19. So assume that (for large enough k) A k is not finite-by-Abelian. By (1) A k contains a unique finite-by-Abelian subgroup U k of index 2. The uniqueness of U k implies that U k ≤ U k+1 , as U k+1 ∩ A k is a finite-by-Abelian subgroup of A k of index 2 and therefore equal to U k . It follows that U = k ∈ NU k is of index 2 in A, and finite-by-Abelian by Lemma 2.19. The assertion follows.
We are now able to establish the following properties of virtually Abelian subgroups which will be important later on. Lemma 2.23. Let L be a Γ-limit group. Then the following hold.
(1) If L is finite-by-Abelian then the subgroup E ≤ L that is generated by the torsion elements of L is finite and therefore of order at most N(Γ).
is finite or finite-by-Z for sufficiently large i. Assume that ϕ i ( E k ) is finite-by-Z for large i. Hence it acts invariantly on an axis in Γ and this action is orientation-preserving. If the image is infinite, it is therefore isomorphic to an HNN-extension (with surjective boundary monomorphisms), i.e.
But this HNN-extension is not generated by torsion elements, which is a contradiction. Thus ϕ i ( E k ) is finite for large i, hence of order at most N(Γ). This implies that ker ϕ i ∩ E k is of index at most N(Γ) in E k . As E k is f.g., there are only finitely many such kernels. The stability of (ϕ i ) then implies that ker ϕ i ∩ E k eventually stabilizes. It follows that Define A k := ã 0 , . . . ,ã k and B k := b 0 , . . . ,b k . The same argument as in the proof of Lemma 2.22 shows for each k and sufficiently large i both ϕ i ( A k ) and ϕ i ( B k ) are virtually Abelian, hence 2-ended. Now for large i the element We get the following immediate consequences.

Corollary 2.24.
Let L be a Γ-limit group and a ∈ L be an element of infinite order. Then

a is virtually Abelian}
is the unique maximal virtually Abelian subgroup of L containing a.
Proof. Let {a 0 , a 1 , . . .} be the set of those elements that satisfy that a, a i is virtually Abelian. Applying Lemma 2.23 (2) repeatedly implies that for each k, A k := a, a 0 , . . . , a k is virtually Abelian. Thus A is virtually Abelian by Lemma 2.22 (4). The uniqueness and maximality of A are trivial.

Corollary 2.25. Let A be a maximal virtually Abelian subgroup of a Γ-limit group L and
Proof. Suppose that g Ag −1 ∩ A is infinite. It follows from Lemma 2.23 that A, g Ag −1 is virtually Abelian and therefore equal to A as A is maximal. Choose an element a ∈ A of infinite order. Then a, gag −1 ≤ A is virtually Abelian. Pick lifts g,ã of g, a in L. Then ϕ i ( ã, gã g −1 ) is virtually Abelian and therefore 2-ended for large i. Thus ϕ i ( g) preserves or exchanges the ends of ϕ i (ã) . It follows that ϕ i ( g), ϕ i (ã) is 2-ended for large i, thus a, g is virtually Abelian. The statement follows now from Corollary 2.24.

The structure of groups acting on real trees
Bass-Serre theory clarifies the algebraic structure of groups acting on simplicial trees. The structure of groups acting on real trees is more complicated but still fairly well understood provided that the action satisfies certain properties. This theory is mainly based on ideas of Rips who in turn applied ideas from the Makanin-Razborov rewriting process. Rips (unpublished) described the structure of finitely presented groups acting freely on real trees, see [17] for an account of his ideas. This was then generalized to stable actions by Bestvina and Feighn [5]. Sela [38] then proved a version for finitely generated groups under stronger stability assumptions; the version we present is a generalization of Sela's result due to Guirardel [22]. We first fix notations for graphs of groups and recall the notion of a graph of actions. We then briefly study 2-orbifolds and describe certain actions of 2-orbifold groups on real trees. We then formulate the structure theorem of [22] in those terms.

Graphs of groups and the Bass-Serre tree
In this section we fix the notations for basic Bass-Serre theory as we will need precise language later on. For details see Serre's book [42] or [24] for more similar notation.
A graph A is understood to consist of a vertex set V A, a set of oriented edges E A, a fixed point free involution −1 : E A → E A and a map α : E A → V A which assigns to each edge e its initial vertex α(e). Moreover, we will denote α(e −1 ) alternatively by ω(e) and call ω(e) the terminal vertex of e.
A graph of groups A then consists of an underlying graph A and the following data.
(2) For each e ∈ E A, an edge group A e = A e −1 .
(3) For each e ∈ E A, an embedding α e : A e → A α(e) .
Again, the embedding α e −1 will alternatively be denoted by ω e . The maps α e and ω e are called the boundary monomorphisms of the edge e.
where e 1 , . . . , e k is an edge path in A from v to w, a 0 ∈ A v and a i ∈ A ω(e i ) for i = 1, . . . , k. For two A-paths p = a 0 , e 1 , . . . , e k , a k and q = b 0 , f 1 , . . . , f l , b l satisfying that ω(e k ) = α( f 1 ), we define a product pq by An equivalence relation ∼ on the set of A-paths is defined as the relation generated by the elementary equivalences a, e, b ∼ aα e (c), e, ω e (c −1 )b and a, e, ω e (c), e −1 , b ∼ aα e (c)b. We denote the ∼-equivalence class of a an A-path p by [p]. We call an A-path reduced if it cannot be shortened by an elementary equivalence.
Given a base vertex v 0 ∈ V A, the fundamental group of A with respect to v 0 , π 1 (A, v 0 ), is the set of equivalence classes of A-paths from v 0 to v 0 , with the multiplication given by Recall that the edge and vertex groups of A embed into π 1 (A). These embeddings are unique up to conjugacy.
If p = a 0 , e 1 , . . . , e k , a k then we say that p ≈ q if q ∼ a 0 , e 1 , . . . , e k , a k a for some a ∈ A v , this defines an equivalence relation on the set of A-path starting at v 0 . We denote the ≈-equivalence class of p by p . The ≈-equivalence classes are precisely the vertices of the Bass-Serre tree (A, v 0 ). We will usually simply write A rather than (A, v 0 ). For any vertexṽ of A, we will denote the projection ofṽ to V A by ↓ṽ.
If we choose for each e ∈ E A a set C e of left coset representatives of α e (A e ) in A α(e) , then each A-path q is equivalent to a unique reduced A-path q = a 0 , e 1 , . . . , e k , a k such that a i−1 ∈ C e i for 1 ≤ i ≤ k. We say that q is in normal form (relative to the set {C e | e ∈ E A}, which we usually don't mention explicitly).
Any vertexṽ ∈ V A is represented by a unique reduced A-path pṽ = a 0 , e 1 , a 1 , . . . , a k−1 , e k , 1 which is in normal form. We call pṽ the representing path ofṽ. Note that any normal form A-path p representingṽ is of the form p = pṽa for a unique a ∈ A ↓ṽ .
With the above notations, we obtain a natural action of π 1 (A, v 0 ) on A in the following way. For [q] ∈ π 1 (A, v 0 ) put With this G-action on A, for everyṽ ∈ V A the map is an isomorphism between the vertex group A ↓ṽ and the stabilizer of the vertexṽ in A.
Likewise, for an edge e = (ṽ 1 ,ṽ 2 ) ∈ E A, the map where a 1 is as in (3.1) and c a 1 denotes conjugation by a 1 , is an isomorphism between the edge group A ↓ e and the stabilizer of the edge e. It follows easily from (3.1) that θ e = θ e −1 .
Sometimes we will need to refine a given splitting of a group, i.e. increase the complexity of a graph of groups decomposition by splitting some vertex group in a way that is compatible with the existing splitting. The following is obvious.
Moreover for each edge e ∈ E A the attaching map α and boundary monomorphism α e are as follows.
(3) If e ∈ E A and α(e) = v then α (e) = w e and α e : If A is a refinement of A then π 1 (A ) π 1 (A). The operation inverse to a refinement is called a collapse.

Graphs of actions
In this section we recall the notion of a graph of actions, see [29]. This is a way of decomposing an action of a group on a real tree into pieces. In the structure theorem these pieces will be of very simple types.

Definition 3.2. A graph of actions is a tuple
• for each e ∈ E A, p α e ∈ T α(e) is a point fixed by α e (A e ), • l : E A → R ≥0 is a function satisfying l(e) = l(e −1 ) for all e ∈ E A.
We call l(e) the length of e. If l = 0 then we omit l, i.e. we write The points p α e are called attaching points. In the following we will denote p α e −1 alternatively by p ω e . Note that p ω e ∈ T ω(e) and that p ω e is fixed by ω e (A e ). Associated to any graph of actions G is a real tree T G obtained by replacing the vertices of the Bass-Serre tree (A, v 0 ) by copies of the trees T v and any lift e ∈ E A of e ∈ E A by a segment of length l(e). T G comes with a natural π 1 (A, v 0 )-action. In the remainder of this section we will give a detailed description of the construction of T G and its natural metric.
Let G be a graph of actions as above. Choose a base vertex v 0 ∈ V A and sets of left coset representatives C e of α e (A e ) in A α(e) for all e ∈ E A.
For any e ∈ EA define T e := [0, l(↓ e)] × { e} to be a copy of the real interval [0, l(↓e)]. Let d e be the standard metric on T e , i.e. d e ((x, e), (y, e)) = |x − y| for all (x, e), (y, e) ∈ T e . We define Note that T e consists of a single point if l(↓ e) = 0. Now for any (x, e) ∈ T E G and g ∈ π 1 (A, v 0 ), put g(x, e) := (x, g e).
This clearly defines an action of π 1 (A, v 0 ) on T E G , which satisfies d e (x, y) = d g e (gx, gy) for all x, y ∈ T e and g ∈ π 1 (A, v 0 ).
We can now define the tree T G . We put where ∼ is the equivalence relation generated by The definition of T G ensure that the images of T α( e) and T ω( e) in T G are joint by a segment of length l(↓ e) for any e ∈ E A, see  It is clear from the above observations that this equivalence relation is preserved by If y 1 , y 2 ∈ T G do not lie in the same edge or vertex space then d G (y 1 , y 2 ) is computed as follows: G and e 1 , . . . , e k is a reduced path in A fromṽ 1 toṽ 2 then Recall that the restriction of d G to any vertex tree Tṽ, resp. edge segment T e , equals dṽ, resp. d e . It therefore follows from (3.3) that the distance of two points in T G can be computed entirely in terms of the metrics d v of the vertex trees of G and its length function l. The case we are mostly interested in is case 1 above, i.e. the case where y 1 and y 2 are contained in vertex trees Tṽ 1 and Tṽ 2 .
We say that a G-tree T splits as a graph of actions G(A) if G π 1 (A) and there is a G-equivariant isometry from T to T G . Remark 3.3. Let G be a graph of actions, e ∈ E A and g ∈ A α(e) . Assume that G is the graph of actions obtained from G by replacing the attaching point p α e ∈ T α(e) by gp α e and the embedding α e : A e → A α(e) by i g • α e . Then T G also splits as the graph of actions G .
It follows from the remark that in a graph of actions splitting of a tree T we are free to alter the attaching points within their orbits of the vertex actions. In particular, if a vertex group A v acts with dense orbits on T v , the attaching points in A v can be chosen to be arbitrarily close to each other. This will turn out useful in Section 5.

2-orbifolds
We will only be interested in 2-orbifolds that are non-spherical. Thus for us a 2-orbifold O is a quotient of R 2 by a group Γ of homeomorphisms that acts properly discontinuously. In particular all surfaces different from the 2-sphere and the projective plane occur this way. Our discussion of 2-orbifolds is very brief and informal; for a more detailed discussion the reader is referred to the article of P. Scott [37] or the book of Zieschang, Vogt and Coldewey [46]. Note that we can assume that R 2 is either endowed with the standard Euclidean metric or with a metric of constant curvature -1 making it isometric to the hyperbolic plane, and that Γ acts by isometries.
A 2-orbifold O = R 2 /Γ is topologically a surface but points whose lifts in R 2 have non-trivial stabilizers are endowed with a marking that reflects the structure of the stabilizer. The stabilizers are in Homeo(R 2 ) conjugate to a finite group of Euclidean isometries and are therefore of one of the following types: (1) Z p and generated by a rotation of order p.
(2) A dihedral group D 2p and generated by two reflections whose product is a rotation of order p with p ∈ N.
(3) A group of order two being generated by a reflection.
In the quotient, points of the first type are depicted as points labeled by p and are referred to as cone points, points of the last type are depicted as points of fat lines and are referred to as points of the reflection boundary and points lifting to points with dihedral stabilisers occur as points incident to two reflection lines and are labeled by p if the dihedral group was of type D 2p . Thus 2-orbifolds can locally be depicted as in Figure 3.2. We further say that a 2-orbifold is of cone type if it has no reflection boundary.
There is a well-known definition for the fundamental group of a 2-orbifold which generalizes the case of a surface, see [37]. In the case of an orbifold O of cone type with cone points of orders p 1 , . . . , p k this is easily done as follows. First let S O be the surface (with boundary) obtained by replacing each cone point with a boundary component. Then define π 1 (O) to be the quotient of π 1 (S O ) obtained by adding the relations γ p i i where γ i is the homotopy class represented by the boundary component corresponding to the ith cone point. This fundamental group recovers the group Γ just as the fundamental group of a manifold recovers the group of Decktransformations on the universal covering, in fact a completely analogous theory of coverings can be developed for 2-orbifolds, see [37].
We do not consider points on the reflection boundary to lie in the boundary of a 2-orbifold. Thus 2-orbifolds as we defined them are open and have no boundary. If however Γ is finitely generated then we can, as in the surface case, find a compact suborbifold of O = R 2 /Γ with boundary whose complement consist of ends of type (0, 1) × S 1 or belong to the reflection boundary. This compact suborbifold is a natural compactification of the original orbifold and we will ignore the difference between the orbifold and this compact subspace. Thus orbifolds have boundary components which are either loops or segments joining points on the reflection boundary.  Given a simple closed curve on a 2-orbifold that does not bound a disk with at most one cone point, cutting the orbifold along this line corresponds to decomposing the orbifold group into an HNN-extension or amalgamated product over the infinite cyclic group. We call a simple closed curve of this type essential. Similarly cutting along a segment joining two points on the reflection boundary that does not bound a disk without cone points in the interior and at most one exceptional point on the reflection boundary corresponds to a splitting along an infinite dihedral group. Again we call such segments essential.  Figure 3.5 are orbifolds without boundary, however a similar statement holds for orbifolds with boundary: Given an essential simple closed curve or an essential segment joining two points on the reflection boundary that does not meet the boundary of the orbifold, this curve induces a splitting of the orbifold group along an infinite cyclic or infinite dihedral group such that all peripheral subgroups are elliptic in this splitting.
In the remainder of this section we will describe an important class of actions of fundamental groups of cone type orbifold groups on real trees. We start with actions of fundamental groups of surfaces with boundary. While these groups are algebraically just free groups the action is constructed using the surface geometry. The orbifold group actions are then constructed as quotients of those actions.
Surfaces with boundary can be described by unions of bands that have a foliation with a transverse measure. We briefly describe their structure, see [5] for a detailed discussion. Suppose that the following data is given.
are isometric embeddings such that the following hold: (a) All but finitely many points of Γ lie in the image of precisely two of the From this data we construct a surface with boundary by gluing bands of width r i to Γ using the maps f + i and f − i as follows. For each i define the band B i as The associated band complex is then defined as Clearly Σ is a surface with boundary, see Figure 3.6 for a once-puncured torus. F 3.6. A union of (two) bands homeomorphic to a puctured torus Now each band B i has a natural foliation where the leaves are the sets {t} × [0, 1] with t ∈ J i . These foliations of the B i extend to a foliation of Σ where leaves of the B i are joint along common endpoints in Γ to leaves of Σ. Thus the leaves of the foliation of Σ are graphs with vertices of valency at most 3. Moreover there are at most 2m vertices of valency 3 and each boundary component is a subset of some leaf. In particular all but finitely many leaves are homeomorphic to R or S 1 . We will be mostly interested in cases where the foliation of Σ has the property that every leaf is dense and either homeomorphic to R or consists of a single boundary component of Σ with infinite rays attached to them. Such foliations are called arational. If we assume that the foliation depicted in Figure 3.6 is arational then there is precisely one leaf that is not homeomorphic to R, it consists of the single boundary component with two rays attached to it. Now the foliation of Σ induces a foliation of the universal covering Σ and the action of π 1 (Σ) on Σ by deck transformations respects this foliation, i.e. gives an action on the leaf space T. It is clear that any element corresponding to a boundary component of Σ acts with a fixed point on T as it preserves some lift of the leaf containing the corresponding boundary component. T comes with a pseudo-metric where the distance between two leaves x and y is the length of the shortest path starting in x and ending in y. The length of the curve is here measured with respect to the natural transverse measure of the foliation, in particular for any curve contained in the lift of some band the transverse measure is simply the length of the projection to the base; see [5] for details. If the foliation is arational then the pseudo-metric is a metric turning T into a real tree; for the remainder of this section we assume this to be the case.
Let γ 1 , . . . , γ k be homotopy classes representing the boundary components. By construction the γ i act with a fixed point on T. Recall that we obtain a 2-orbifold group H = π 1 (O) from π 1 (Σ) by factoring out relations of the type γ r i i for 1 ≤ i ≤ k with r i ∈ N 0 . Denote the kernel of the projection π 1 (Σ) → H by N. Now H clearly acts on T := T/N and T is again a real tree as N is generated by elliptic elements. We say that this action of H is dual to an arational foliation on the cone type orbifold O.

The structure theorem
In this section we state the structure theorem for finitely generated groups acting on R-trees as it appears in [22]. This theorem (and its relatives) are usually simply referred to as the Rips machine.
We recall from [22] that a G-tree T satisfies the ascending chain condition if for any sequence of arcs I 1 ⊃ I 2 ⊃ . . . in T whose lengths converge to 0, the sequence of the stabilizers of the segments is eventually constant.
The following theorem is a generalization of Sela's version of the Rips machine for finitely generated groups [38]. Other than Sela's original version it allows us to deal with torsion.
Theorem 3.4 (Main Theorem of [22]). Consider a non-trivial action of a f.g. group G on an R-tree T by isometries. Assume that • T satisfies the ascending chain condition, • for any unstable arc J ⊂ T, Then either G splits over the stabilizer of an unstable arc or over the stabilizer of an infinite tripod, or T splits as a graph of actions where each vertex action of A v on the vertex tree T v is either • simplicial: a simplicial action on a simplicial tree, • of orbifold (or Seifert) type: the action of A v has kernel N v and the faithful action of A v /N v is dual to an arational measured foliation on a closed 2-orbifold with boundary, or is a finitely generated group acting with dense orbits on T v .
Note that if the G-tree T admits a splitting as a graph of actions G as in Theorem 3.4 and if A contains a nondegenerate simplicial vertex tree, we get a refined splitting of T as a graph of actions , l) such that any vertex tree is either of axial type, or of orbifold type or is degenerate, i.e. consists of a single point. This is easily achieved by decomposing each simplicial vertex tree using Bass-Serre theory, possibly after subdiving some edges to ensure that the original attaching points are vertices. Note that if an edge e of the refined graph of actions has non-zero length then both T α(e) and T ω(e) are degenerate.

The virtually Abelian JSJ-decomposition of Γ-limit groups
Throughout this chapter Γ is a hyperbolic group. We first study basic properties of virtually Abelian splittings of Γ-limit groups and then discuss virtually Abelian JSJ-decompositions of finitely generated Γ-limit groups, splittings that reveal all virtually Abelian splittings simultaneously. We closely follow Sela's construction of the JSJ-decomposition [38] but similar to the discussion of Bestvina and Feighn [6] do not require the JSJ to be unfolded. The fact that the JSJ can be chosen to be unfolded can be established using the acylindricity of the splittings. This has been observed by Rips and Sela [36] in the case of splittings over cyclic groups. That this can also be done in the current setting has been established independently by Guiradel and Levitt [23] and the first author in his PhD thesis [34].

Modifying splittings
Recall that a graph of groups is minimal if the corresponding Bass-Serre tree contains no proper invariant subtree. A finite graph of groups is minimal iff there is no surjective boundary monomorphism into a vertex group of a valence 1 vertex. A graph of groups is called reduced if no boundary monomorphism is surjective. Thus any reduced finite graph of groups is minimal.
The JSJ-decompositions we construct reveal virtually Abelian splittings only up to certain modifications. We start by introducing these modifications, they all leave the fundamental group unchanged. (1) Boundary slide: Let e ∈ E A. A boundary slide (of the boundary monomorphism α e ) is the replacement of α e by c g • α e for an element g ∈ A α(e) .
Then we first perform a boundary slide such that ω e 1 (A e 1 ) ≤ α e 2 (A e 2 ) and then replace e 1 with an edge e 1 with edge group A e 1 = A e 1 such that (a) α(e 1 ) = α(e 1 ) and α e 1 = α e 1 .
• ω e 1 . The combination of the initial boundary slide and the subsequent modification is called an edge slide of e 1 over e 2 .
(3) Folding/Unfolding: Let e ∈ E A and α e (A e ) < C < A α(e) . A folding along e is the replacement of A ω(e) by C * A e A ω(e) , A e by C and the corresponding replacement of the boundary monomorphisms α e and ω e . The inverse of a folding along e is an unfolding along e.
Note that we usually only consider graphs of groups up to boundary slides. An edge slide can be alternatively defined on the Bass-Serre tree: Given two inequivalent edges f 1 and f 2 such that ω( f 1 ) = α( f 2 ) and stab f 1 ≤ stab f 2 we slide f 1 over f 2 in an equivariant way. One can also think of it as first (equivariantly) subdividing f 1 into f 1 and f 1 and then (equivariantly) identifying f −1 1 with f 2 . Note that the action on the vertex set of the tree is unchanged under this operation.
1. An edge slide as seen in the tree The following lemma is trivial as the existence of a surjective boundary monomorphism is preserved by boundary slides and edge slides.

Lemma 4.2. Let
A be a graph of groups. Assume that A is obtained from A by boundary slides and edge slides. Then A is reduced and minimal iff A is reduced and minimal.

Virtually Abelian splittings of Γ-limit groups
In this section we study splittings of Γ-limit groups as fundamental groups of graphs of groups with virtually Abelian edge groups. We call such splittings virtually Abelian splittings.
In the following we call a virtually Abelian group large if it contains a one-ended subgroup. Note that for finitely generated virtually Abelian groups, being large is equivalent to being one-ended. A crucial observation in this section will be that any finite virtually Abelian splitting of a Γ-limit group can be modified by boundary slides and some further simple modifications such that all large virtually Abelian subgroups are elliptic.    The conclusion now follows from the observation that an edge is slid over another edge only if their edge groups are contained in the same maximal virtually Abelian subgroup. Thus the above process for one maximal virtually Abelian subgroup does not affect the validity of the conclusion of the proposition for the other. Thus we can use finitely many edge slides to obtain the desired conclusion for all maximal large virtually Abelian subgroups simultaneously. This process is finite as there are only finitely many edge groups and therefore only finitely many conjugacy classes of large maximal virtually Abelian subgroups that act non-elliptically.
We now explain how splittings that satisfy the conclusion of Proposition 4.4 can be modified such that afterwards all large virtually Abelian subgroups are elliptic. We call a virtually Abelian splitting with this property compatible.
Let A be a finite virtually Abelian splitting of L satisfying the conclusion of Proposition 4.4. Let M be a large maximal virtually Abelian subgroup of L that is not elliptic.
Assume first that M satisfies (2) of Proposition 4.4. Then we subdivide the edge e into edge e 1 and e 2 such that α(e 1 ) = α(e), ω(e 2 ) = ω(e) and ω(e 1 ) = α(e 2 ) = v is a new vertex. Moreover A e 1 = A 1 , A e 2 = A 2 and A v = A 1 * A e A 2 and the boundary monomorphisms are the natural ones, see  We now argue that for a finitely generated Γ-limit grou L there exists an upper bound for the complexity of a minimal (and reduced) virtually Abelian splitting of L, which only depends on the rank of L and N(Γ). For a given graph of groups A, we define its   The following theorem from [45] provides a bound for the complexity of (k, C)acylindrical splittings. It is a generalization of Sela's acylindrical accessibility theorem [38] which deals with the case C = 1, see also [44].  Proof. Note first that for any e ∈ E A the representatives of MA(A e ) are elliptic. If the representatives are large this holds as A is assumed to be compatible; otherwise they contain an elliptic subgroup of finite index (a conjugate of A e ) which also implies that they are elliptic.
For any conjugacy class of maximal virtually Abelian elliptic subgroups . By a finite sequence of edge slides any edge e can be slid such that it is incident to v MA(A e ) . The (2, N(Γ))-acylindricity of the obtained splitting is easily verified: Otherwise there exists a segment Y of length 3 in the Bass-Serre tree fixed by some infinite virtually Abelian group H. The construction implies that any edge of Y must be incident to a vertex that is a lift of v [M] . As Y is of length 3 there must be two distinct such vertices, say v 1 and v 2 . Choose g ∈ L such that gv 1 = v 2 . It follows that H ⊂ stab(v 1 ) ∩ stab(v 2 ) = stab(v 1 ) ∩ g stab(v 1 )g −1 . As H is infinite and stab(v 1 ) is maximal virtually Abelian it follows from Corollary 2.25 that g ∈ stab(v 1 ), a contradiction. Thus the obtained graph of groups is (2, N(Γ))-acylindrical.
The construction of the (2, N(Γ))-acylindrical graph of groups in the proof of Lemma 4.6 depends on the choice of the vertices v [M] and the output is therefore not canonical. We will obtain a canonical splitting (up to boundary slides) by performing the following normalization process for a given compatible virtually Abelian splitting A of L. As before we assume that all edge groups are infinite.
This process is only a slight modification of the proof of Lemma 4.6. Choose Finally, we minimize the obtained graph of groups by removing unnecessary valence 1 vertices and corresponding edges.
It is clear that, up to boundary slides, the obtained graph of groups does not depend on the choice of the v i . We say that A is in normal form if it is the output of the normalization process for some graph of groups A (or equivalently, if the normalization process of A reproduces A). We further call the vertices v 1 , . . . , v k in the above construction the characteristic vertices of the normal form. By construction any edge of a graph of groups in normal form is incident to exactly one characteristic vertex, thus the underlying graph is bipartite. A graph of groups in normal form is (2, N(Γ))-acylindrical by the same argument as in the proof of Lemma 4.6 and minimal by construction. Note that a graph of groups in normal form may be non-reduced as some edge groups may be maximal virtually abelian subgroups of L. It is however easily verified that any graph of groups in normal form can be obtained by applying the normalization process to some reduced, minimal and (2, N(Γ))-acylindrical graph of groups A . As by construction C(A) ≤ 2C(A ) it follows, using Theorem 4.5, that there is a global upper bound for the complexity of all normal form splittings of a given finitely generated one-ended Γ-limit group L.

Morphisms of graphs of groups
For a based simplicial G-tree T = (T,ṽ 0 ) and a based simplicial H-tree Y = (Y, u 0 ) a morphism from T to Y is a pair (ϕ, f ) where ϕ : G → H is a homomorphism and f : T → Y is a simplicial map such that f (ṽ 0 ) = u 0 and that for all x ∈ T and g ∈ G. Any such morphism can be encoded on the level of the associated graphs of groups. We will discuss such morphisms for graphs of groups and make some basic observations.
A morphism from a graph of groups A to a graph of groups B is a tuple (3) ψ e is a homomorphism from A e to B f (e) for all e ∈ E A.
• ψ e for all e ∈ E A.
We will denote o −1 e −1 alternatively by t e . A morphism from A to B induces a homomorphism f * : given by We will write ψ f v or ψ f e instead of ψ v or ψ e if we want to make explicit that the maps come from the morphism f. We will further say that a morphism f is surjective if f * is surjective.
The morphism f determines a morphism f : (A, v 0 ) → (B, u 0 ) (that maps the base pointṽ 0 to u 0 ). The pair (f * , f ) is a morphism from the π 1 (A, v 0 )-tree (A, v 0 ) to the π 1 (B, u 0 )-tree (B, u 0 ), see [24] for details. Moreover any morphism from a G-tree T to an H-tree Y occurs this way.
The following simple proposition will be usefull in subsequent sections. In the statement of the proposition we identify the fundamental group of A and the fundamental group of

Then there exists a graph of groupsĀ obtained from A by collapses of edges followed by subdivisions of edges and a morphism
Proof. Let T A = (A, v 0 ) and T B = (B, u 0 ) be the Bass-Serre trees with base pointsṽ 0 and u 0 . It suffices to show that, after (equivariant) collapses and subdivisions of edges of T A , there exists a morphism (η, f ) from the π 1 (A, v 0 )-tree T A to the π 1 (B, u 0 )-tree T B such that f (ṽ 0 ) = u 0 and that p B ( f (v)) = h(p A (v)) for all v ∈ VT A where p A : T A → A and p B : T B → B are the canonical quotient maps.
Pick a maximal tree Y A in A and a lift Y A to T A such that the lift of v 0 isṽ 0 . By assumption we can choose for each vertex v ∈ Y A a vertex w v ∈ T B such that η(stab v) ≤ stab w v and ; we may further assume that wṽ 0 = u 0 . We now define a map f : VT A → VT B by gv → η(g)w v for all v ∈ Y A and g ∈ π 1 (A, v 0 ). The map is easily extended to the edges of T A by mapping an edge e = (v 1 , v 2 ) to the reduced edge path p from f (v 1 ) to f (v 2 ). Here we apply k − 1 subdivisions to the edge e if the length k of p is greater than 1, or collapse e in case k = 0.
To make the map simplicial we need to collapse

The virtually Abelian JSJ-decomposition of a Γ-limit group
In this section we establish the existence of virtually Abelian JSJ-decompositions of finitely generated Γ-limit groups. A virtually Abelian JSJ-decomposition of a group G is a splitting of G in which all compatible virtually Abelian splittings of G are apparent.
In the following we say that a vertex group A v of a graph of groups A is a QH-vertex group (quadratically hanging vertex groups) if the following hold.
(1) A v is finite-by-orbifold, i.e. there exists an orbifold O with fundamental group O = π 1 (O), some finite group E and a short exact sequence (2) For any edge e ∈ E A s.t. α(e) = v there exists a peripheral subgroup O e of O such that α e (A e ) is in A v conjugate to a finite index subgroup of π −1 (O e ).
We will also say that a subgroup of G is a QH-subgroup if it is conjugate to a QH-vertex group of some splitting A of G.
It is a trivial but important observation that any essential simple closed curve or essential segment joining two points on the reflection boundary of O induces a splitting of G = π 1 (A) over a 2-ended group. We call such a splitting geometric with respect to the QH-subgroup A v .
We can depict the splitting A by drawing the orbifold O for the vertex group A v and depicting all non-QH vertex groups as balls joint by edges. Recall that a 1-edge splitting A 1 , i.e. a splitting as an amalgamated product or an HNN-extension, of a group G is called elliptic with respect to another splitting A 2 if the edge group of A 1 is conjugate to a subgroup of a vertex group of A 2 . Otherwise A 1 is called hyperbolic with respect to A 2 . It is an important observation [36] that if G is one-ended and A 1 and A 2 are 1-edge splittings of G over 2-ended groups then the two splittings are either both elliptic or both hyperbolic with respect to each other. In the first case we say that A 1 and A 2 are elliptic-elliptic and in the latter that they are hyperbolic-hyperbolic.
The following theorem is the key step in the proof of the JSJ-decomposition, it implies in particular the existence of a splitting of a Γ-limit group that encodes all hyperbolic-hyperbolic splittings over 2-ended subgroups.

Theorem 4.9.
Let G be a f.g. one-ended group. Assume that there exists N such that any finite subgroup of G has order at most N.
Then there exists a reduced, minimal (2, N)-acylindrical graph of groups decomposition A of G such that the following hold: (1) Any 1-edge splitting of G over a 2-ended group that is hyperbolic-hyperbolic with respect to another 1-edge splitting over a 2-ended group is geometric with respect to some QH-vertex group of A.
(2) If B is a virtually Abelian splitting of G such that no edge group of B is 2-ended and hyperbolic-hyperbolic with respect to another splitting over a 2-ended group then the QH-vertex groups of A are elliptic with respect to B.
(3) Any QH-subgroup of G is conjugate to a subgroup of a QH-vertex group A v , moreover this subgroup corresponds to a suborbifold of the orbifold corresponding to v. In particular every maximal QH-subgroup is conjugate to a vertex group of A.
Proof. For the proof we can essentially refer to the construction of the JSJ-decomposition of Dunwoody and Sageev [15] and Fujiwara and Papasoglu [16], but we need a few adaptions. First, note that any given splitting can be modified to be (2, N)-acylindrical while preserving the conjugacy classes of the QH-vertex groups. Indeed, we first refine the splitting by replacing each QH-vertex group A v by a tree of groups consisting of a vertex x v with vertex group A v and for each peripheral subgroup P an edge e P with edge group P and a vertex x P with vertex group P such that α(e P ) = x v and ω(e P ) = x P with the boundary monomorphisms being the obvious ones. We may refine in such a way that all previous edges are incident to one of the vertices of type x P . We then collapse all edges not incident to a vertex of type x v . The (2, N)-acylindricity of the obtained splitting follows from the fact that for any QH-vertex group A v and peripheral subgroups P 1 and P 2 corresponding to distinct boundary components the following hold.
(2) gP 1 g −1 ∩ P 2 is finite for all g ∈ A v . This observation implies in particular that the number of QH-vertex groups of any splitting of G is bounded in terms of N and the rank of G because of Theorem 4.5. This also bounds the complexity (genus and number of exceptional points) of the orbifolds corresponding to the QH-vertex groups as orbifolds of large complexity can be cut along essential simple closed curves or segments to produce a splitting with a higher number of QH-vertex groups that is again (2, N)-acylindrical. Now, Dunwoody and Sageev [15] and Fujiwara and Papasoglu [16] construct splittings such that arbitrary finite collections of hyperbolic-hyperbolic splittings over 2-ended groups can be seen as geometric splittings in QH-vertex groups. In their proofs they then assume that G is finitely presented so they can apply Bestvina-Feighn accessibility [4] to guarantee termination of the construction. In our case we can exploit the fact that the obtained splittings are (2, N)-acylindrical after the modifications discussed above. The same argument was applied in the construction of the quadratic decomposition in [36].
To see (2) suppose that a QH-vertex group A v of A is non-elliptic in B. Then there exists a geometric 1-edge splitting D corresponding to some essential simple closed curve or essential segment on the orbifold such that the single edge group D e of D is non-elliptic in B, see Corollary 4.12 of [16]. It follows that D is hyperbolic with respect to the 1-edge splitting corresponding to some edge f ∈ E B, i.e. the splitting obtained by collapsing all edges of B but f . Note that B f is elliptic in D: If B f is 2-ended this holds by assumption on B and if B f is large then B f must act on D with infinite pointwise stabilizer contradicting the (2, N)-acylindricity of D. It follows, see Remark 2.3 of [16] that G splits over a subgroup of D e of infinite index, i.e. that G splits over a finite group. This contradicts the one-endedness of G.
Before we proceed with the virtually Abelian JSJ-decomposition, we introduce some notation concerning virtually Abelian vertex groups of splittings. Recall that if M is an infinite virtually Abelian subgroup of a Γ-limit group then M + denotes the unique maximal finite-by-Abelian subgroup of M which is of index at most 2.

Definition 4.10. Let
A be a graph of groups decomposition of a finitely generated 1-ended Γ-limit group L. Let v ∈ V A such that A v is virtually Abelian.
Let ∆ be the set of homomorphisms η : A + v → Z such that η(α e (A + e )) = 0 for all e ∈ E A with α(e) = v. We then define A simple homology argument shows that A + v /P + v is a finitely generated free Abelian group whose rank is bounded from above in terms of rank L and the valence of v in A. Together with Theorem 4.9 this implies in particular that JSJ-decompositions in the sense of the following definition exist. Definition 4.11. Let L be a finitely generated one-ended Γ-limit group and A be a virtually Abelian compatible splitting of L. Then A is called a virtually Abelian JSJ-decomposition of L if the following hold.
(1) Every splitting over a 2-ended group that is hyperbolic-hyperbolic with respect to another splitting over a 2-ended group is geometric with respect to a QH-subgroup of A.
(2) Any edge group of A that can be unfolded to be finite-by-Abelian is finite-by-Abelian.
(3) For any virtually Abelian vertex group A v , the rank of A + v /P + v cannot be increased by unfoldings.
(4) A is in normal form and of maximal complexity among all virtually Abelian compatible splittings of L that satisfy (1)-(3).
In the following we refer to vertex groups of a virtually Abelian JSJ-decomposition which are neither QH nor virtually Abelian as rigid. Note that any virtually Abelian JSJ-decomposition can be obtained from a splitting as in Theorem 4.9 by refinements of non-QH-subgroups, unfoldings and the normalization process. This is true as the maximal QH-subgroups must be elliptic by Definition 4.11 (2).
The following theorem describes the basic properties of virtually Abelian JSJdecompositions of Γ-limit groups. In the following we say that a graph of groups B is visible in A if B is obtained from A by a sequence of collapses. Theorem 4.12. Let L be a finitely generated one-ended Γ-limit group and let A be a virtually Abelian JSJ-decomposition of L. Then the following hold.

(1) Let B be a virtually Abelian compatible splitting of L such that all maximal QH-subgroups are elliptic. Assume further that B is either in normal form or a 1-edge splitting. Then B is visible in a graph of groups obtained from A by unfoldings followed by foldings and edge slides.
(2) Any other JSJ-decomposition B of L can be obtained from A by a sequence of unfoldings and foldings.
Remark. While Theorem 4.12 (2) implies that any finite collection of JSJ-decompositions has a common unfolding we do not claim that the JSJ can be chosen to be unfolded, i.e. such that no further unfolding is possible.
Proof. We first prove (1). Let T A and T B be the respective Bass-Serre-trees of A and B. For each v ∈ V A, by restricting the G-action on T B to A v , we obtain a (possibly trivial) splitting A v of A v corresponding to a minimal A v -invariant subtree. Note that by assumption these splittings are trivial for QH-vertex groups. Denote by A the graph of groups obtained by refining A in each vertex v by A v , and normalizing this refined graph of groups. By construction, neither the Betti number nor the number of edges of A decrease by the refinement. As the complexity cannot increase by the maximality assumption, it follows that both the Betti number and the number of edges remain unchanged, in particular C(A ) = C(A). We show that A can be obtained from A by unfoldings and that B is visible in A after foldings and edge slides.
By construction, all vertex groups of A are elliptic in A. Therefore, by Lemma 4.7, there is a graph of groupsĀ obtained from A by collapses and subdivisions and a morphism f :Ā → A with f * = id G , which maps the characteristic vertices of A to the characteristic vertices of A. The (2, N(Γ))-acylindricity of A implies that the stabilizer of a segment of length 2 in the Bass-Serre tree A can only have infinite stabilizer if its midpoint corresponds to a characteristic vertex. Thus the proof of Proposition 4.7 implies that no subdivision is necessary as one endpoint of each edge path that occurs as an image of an edge must correspond to a characteristic vertex and as characteristic and non-characteristic vertices alternate.
Moreover, as A is minimal, T A does not contain a proper G-invariant subtree, so f is surjective. This implies that no edges are collapsed as #E A = #E A . It follows that A = A , and we obtain a morphism, again called f, from A to A whose underlying graph morphism f is a graph isomorphism.
For each e ∈ E A , we have A e ≤ A f (e) (we identify A with f(A ), in particular we write A e instead of ψ e (A e )). Assume that for some e, A e is a proper subgroup of A f (e) , and assume w.l.o.g. that α(e) is the characteristic vertex of MA(A e ). Then we can perform a fold along e, replacing A e by A f (e) and A ω(e) by A f (e) * A e A ω(e) . After applying finitely many such folds we get that A e A f (e) for all e ∈ E A . As f * : π 1 (A ) → π 1 (A) is an isomorphism, it follows that A , after the foldings, is isomorphic to A. Conversely, A can be obtained from A by unfoldings.
We now show that B is visible in a splitting obtained from A by foldings and edge slides. By construction all vertex groups of A are elliptic in B. Again there is a graph of groupsĀ, which is obtained from A by collapses of edges, and a morphism f :Ā → B. The argument that no subdivision is necessary is the same as above.
After foldings that replace the edge groupsĀ e with B f (e) we can assume that f is bijective on edge groups. Assume now that f (e 1 ) = f (e 2 ) for some e 1 , e 2 ∈ EĀ, in particularĀ e 1 =Ā e 2 = B f (e 1 ) . Possibly after changing the orientation of e 1 and e 2 we can assume that α(e 1 ) = α(e 2 ) = v MA(A e 1 ) .
We can now alterĀ by identifying e 1 and e 2 by a Stallings fold of type IA or IIIA, see [4], clearly f factors through this fold. Note that this fold can also be thought of as first sliding e 1 over e 2 and then collapsing e 1 . After finitely many such operations we obtain a graph of groups A and a morphism f : A → B such that f is a graph isomorphism, that is bijective on edge groups and induces an isomorphism on the level of the fundamental group. Thus A is isomorphic to B. As A has been obtained from A by edge collapses, foldings and edge slides it follows that B is visible in a splitting obtained from A by foldings and edge slides as the collapses can be performed last. This concludes the proof of (1).
If B is itself a JSJ-decomposition of L then the same argument that showed that A can be obtained from A by foldings shows that B can be obtained from A by foldings. This proves (2).
We conclude this section by defining the modular group of a one-ended Γ-limit group with respect to a given virtually Abelian splitting, using the following definition of natural extensions of vertex automorphisms and the subsequent definition of Dehn twists. Definition 4.13. Let G be a group with a splitting G = π 1 (A, v 0 ), and v ∈ V A. Assume that σ v ∈ Aut(A v ) is an automorphism such that for each e ∈ E A with α(e) = v there exists an element γ e ∈ A v such that σ v (a) = γ e aγ −1 e for all a ∈ α e (A e ). Then the map φ : G → G, [a 0 , e 1 , a 1 , . . . , e n , a n ] → [ā 0 , e 1 ,ā 1 , . . . , e n ,ā n ] = γ e n+1 = 1) is a well-defined automorphism of G. We call it a natural extension of σ v (with respect to the base vertex v 0 ), and say that σ v is naturally extendable.
Note that a natural extension of a vertex automorphism σ v is not unique as the γ e are not uniquely determined by σ v . Definition 4.14. Let A be a graph of groups, e ∈ E A and g ∈ A α(e) such that gag −1 = a for all a ∈ ω e (A e ). Then we call the automorphism f * of π 1 (A) induced by the morphism with o −1 e −1 = t e = g and o f = 1 for f e −1 the Dehn twist along e by g. Note that in the case of 1-edge splittings this just recovers the usual Definition of a Dehn twist. In the case of an amalgamated product, i.e. a splitting with a single non-loop edge e, the automorphism f * : π 1 (A, α(e)) → π 1 (A, α(e)) is given by f * ([a 0 , e, a 1 , e −1 , a 2 , . . . , a 2k−2 , e, a 2k−1 , e −1 , a 2k ]) = [a 0 , e, ga 1 g −1 , e −1 , a 2 , . . . , a 2k−2 , e, ga 2k−1 g −1 , e −1 , a 2k ]. (1) Inner automorphisms of L.
(2) If e ∈ E A with A e finite-by-Abelian and A e is the 1-edge splitting obtained from A by collapsing all edges but e: A Dehn twist along e by an elliptic (with respect to A) element g ∈ Z(M + ) where M is the maximal virtually Abelian subgroup containing A e .

(4) Natural extensions of automorphisms of maximal virtually Abelian vertex groups
A v which restrict to the identity on P + v and to conjugation on each virtually Remark. Note that the extendable automorphisms of the QH-subgroups all correspond to automorphisms of the corresponding orbifold. We also note that the above Dehn twists along e by g can be realized in the following way: If g is contained in A v , we may slide the edge e (by a finite sequence of edge slides) to obtain a graph of groups A where v = α (e). This induces an isomorphism ϕ from π 1 (A) to π 1 (A ) for the obtained graph of groups A . Now, consider f as in Definition 4.14, inducing the Dehn twist along e by g in A . Then ϕ −1 • f * • ϕ is the desired map.
It turns out that if A is a virtually Abelian JSJ-decomposition of L then Mod A (L) contains all other modular groups.

Lemma 4.16. Let A be a virtually Abelian splitting of a one-ended Γ-limit group L and assume that A is obtained from A by edge slides and boundary slides. Then
Proof. It is obvious that boundary slides preserve the modular group. So assume that A is obtained from A by an edge slide of e 1 over e 2 , i.e. e 1 is replaced by e 1 , using the notations of Definition 4.1. It is easy to see that all natural extensions of vertex automorphisms, as well as any Dehn twist along an edge distinct from e 2 are unaffected by the edge slide. Now assume that α is a Dehn twist along e 2 by g ∈ L. Then, in A , α appears as the product of the Dehn twists by g along e 2 and by ω −1 e 1 • ω e 2 (g) along e 1 . First, assume that φ is a Dehn twist along an edge e ∈ E B. By Theorem 4.12, the induced 1-edge splitting of L with edge e and edge group B e is visible in A after unfoldings, foldings and edge slides. As unfolding and folding the edge group does not alter the maximal virtually Abelian subgroup M that contains A e , M + is also unchanged. Thus the presence of the Dehn twist is preserved by foldings and unfoldings unless an edge is folded such that the corresponding edge group ceases to be finite-by-Abelian. But since the edge groups of the JSJ-decomposition are finite-by-Abelian if possible they do exist in the JSJ if they exist in any splitting. Moreover, edge slides preserve the Dehn twist by Lemma 4.16. Now assume that φ is a natural extension of an automorphism σ ∈ Aut(B v ) for some v ∈ V B. If v is a QH-subgroup, there is nothing to show as these automorphisms lift to automorphisms of the QH-vertex group of A containing the QH-vertex group of B. Thus we can assume that B v is virtually Abelian. We may assume that B v is maximal virtually Abelian, as otherwise the maximal virtually Abelian subgroup B v would be conjugate into another vertex group B v as B is assumed to be compatible and we may assume that φ is a natural extension of an automorphism of B v . It follows that there is a vertex u ∈ V A such that A u = B v . We show that φ arises as a natural extension of σ in A. It clearly suffices to show that P + u ⊂ P + v . It follows from Theorem 4.12 that after unfolding A we get a graph of groups A such that there exists a morphism from A to B. Denote the image of the vertex u in A by u . The existence of this morphism implies that P + u ⊂ P + v , thus it suffices to show that P + u = P + u . Now both A + u /P + u and A + u /P + u are finitely generated free Abelian groups and as A is a JSJ-decomposition it follows that rank A + u /P + u ≥ rank A + u /P + u . As the quotient map θ : A + u → A + u /P + u factors through A + u /P + u this implies that θ is an isomorphism as f.g. free Abelian groups are hopfian. Thus P + u ⊂ P + v . If B is not compatible, we can use edge slides to assure that B satisfies the conclusion of  where A is an arbitrary virtually Abelian JSJ-decomposition of L.

Γ-factor sets of one-ended groups
The term of factor sets was coined in [6] in the context of free groups. In this chapter we define Γ-factor sets for an arbitrary group Γ, which are a natural generalization of factor sets. The goal of this chapter is the proof of Theorem 5.2, which states that if Γ is hyperbolic, any finitely generated one-ended group G admits a Γ-factor set. This is the main step in the construction of Makanin-Razborov diagrams.

Γ-Factor sets
Definition 5.1. Let G and Γ be groups and H ≤ Aut(G). A Γ-factor set of G relative to H is a finite set of proper quotient maps {q i : G → Γ i } such that for each non-injective homomorphism q : G → Γ, there exists some α ∈ H such that q • α factors through some q i .
The following is the main theorem of Section 5.

Theorem 5.2.
Let G be a f.g. one-ended group and Γ be a hyperbolic group. Then the following hold.
(1) If G is not fully residually Γ then there exists a Γ-factor set of G relative to {id}.
(2) If G is fully residually Γ then there exists a Γ-factor set of G relative to Mod(G).
The proof of the first part of Theorem 5.2 is trivial. Indeed, if G is not fully residually Γ, then there is a finite set S ⊂ G \ {1} such that for any ϕ ∈ Hom(G, Γ), S ∩ ker ϕ ∅. Thus the set of quotient maps is a Γ-factor set relative to {id}.
If G is fully residually Γ the argument is significantly more involved and it turns out to be crucial to allow for precomposition with modular automorphisms as it allows us to only consider short homomorphisms in the sense of Definition 5.3 below. Recall that by Lemma 2.3 G is a Γ-limit group, thus Mod(G) is defined (cf. Corollary 4.18). For the remainder of Section 5, we fix a f.g. one-ended group G and a hyperbolic group Γ with fixed finite generating sets S G and S Γ respectively. Definition 5.3. A homomorphism ϕ : G → Γ is called short relative to H ≤ Aut(G), if for every α ∈ H and g ∈ Γ, |ϕ| ≤ |c g • ϕ • α| (where c g denotes conjugation by g and | · | is as in Definition 2.12).
Remark 5.4. It follows that a homomorphism that is short relative to any subgroup of Aut(G) is conjugacy short and therefore satisfies (2.4) of Theorem 2.11. Thus every convergent sequence of pairwise distinct short homomorphisms from G to Γ yields a non-trivial limit action of G on a real tree.
For the remainder of this chapter we will not always explicitly mention Mod(G), i.e. short will always mean short relative to Mod(G) and Γ-factor set will mean Γ-factor set relative to Mod(G). It will always be obvious that the constructed automorphisms are indeed modular automorphisms.
The proof of the second claim of Theorem 5.2 is by contradiction, i.e. we assume that G does not admit a Γ-factor set. This assumption implies the following. Proof. For i ∈ N, let B i ⊂ G be the ball of radius i with center 1 in G (with respect to the word metric). For each i, there is a non-injective ϕ i ∈ Hom(G, Γ) such that B i ∩ ker ϕ i = {1}, as otherwise the set of quotient maps would be a Γ-factor set of G. Moreover, since the definition of the factor set allows precomposition by a modular automorphism of G, each ϕ i can be chosen to be short.
Clearly, the obtained sequence (ϕ i ) of short homomorphisms is stably injective. Since each ϕ i is non-injective, it occurs only finitely many times in the sequence. Thus (ϕ i ) has a convergent subsequence of pairwise distinct non-injective homomorphisms, see Lemma 2.6.
In the remainder of this chapter we prove the following proposition, which yields a contradiction to the conclusion of Lemma 5.5 and therefore implies Theorem 5.2. Proposition 5.6. Let (ϕ i ) be a stably injective convergent sequence of pairwise distinct homomorphisms from G to Γ. Then the ϕ i are eventually not short. Remark 5.7. If (ϕ i ) ⊂ Hom(G, Γ) is a stably injective sequence, we can associate to (ϕ i ) a sequence ( ϕ i ) where ϕ i = c g i • ϕ i • α i , for some α i ∈ Mod(G) and g i ∈ Γ, and ϕ i is short for each i ∈ N. After passing to a subsequence we can again assume that ( ϕ i ) is stable. Proposition 5.6 implies that Q := G/ker − → ( ϕ i ) is a proper quotient of G. This is an instance of a shortening quotient discussed in Section 7.1.
Section 5.2 is entirely devoted to the proof of Proposition 5.6.

The shortening argument
Let (ϕ i ) be as in Proposition 5.6. By Theorem 2.6, (ϕ i ) converges to a non-trivial action of G on an R-tree T = (T, x 0 ).
Since ker − → (ϕ i ) = 1 it follows that the action of G = G/ker − → (ϕ i ) on T satisfies the stability assertions of Theorem 2.17. This implies that T satisfies the assumptions of Theorem 3.4. As G is assumed to be one-ended and the stabilizers of unstable arcs are finite, G does not split over the stabilizer of an unstable arc. It follows that T splits as a graph of actions. We will use this decomposition of the action of G on T to show that for large i, the homomorphisms ϕ i are not short, more precisely we will construct modular automorphisms α i ∈ Mod(G) such that |ϕ i • α i | < |ϕ i | for large i.
Let G = G(A) be the graph of actions decomposition of T given by Theorem 3.4. We identify G with π 1 (A, v 0 ) and assume that the basepoint x 0 is given by x 0 = (x 0 ,ṽ 0 ), whereṽ 0 = 1 is the base vertex of A = (A, v 0 ) andx 0 ∈ T v 0 . Moreover, we denote the metric d G of G simply by d. By Remark 3.3 we can assume that the following hold for any e, f ∈ E A.
(2) If α(e) = α( f ) and p α e is A α(e) -equivalent to p α f then p α e = p α f . In the construction of the shortening automorphisms we need to deal with each of the different types of vertex trees of G, we will do this in the following three sections before plugging it all together to conclude. The shortening argument first appeared in [35] but the underlying ideas are also implicit in the work of Razborov. In order to deal with torsion a number of additional issues need to be addressed.

Axial components
The purpose of this section is to prove the following.
Theorem 5.8. Let v 1 ∈ V A be an axial vertex. For any finite subset S ⊂ G there exists some φ ∈ Mod(G) such that for any g ∈ S the following hold.
• If [x 0 , gx 0 ] has a nondegenerate intersection with a vertex space Tṽ and ↓ṽ = v 1 , The main step is to construct an automorphism of the axial vertex group that shortens the action on its vertex tree and that can be extended to an automorphism of G. We start by studying the algebraic structure of axial groups.
Let G A = A v 1 be a vertex group with an axial action on a tree T A = T v 1 . We assume that the group G A does not preserve the ends of T A , i.e. that G A contains elements that act by reflections. We denote the index 2 subgroup of G A which preserves the ends by G + A . By Theorem 2.17, G + A is finite-by-Abelian. The case where G A preserves the ends follows by considering only G + A . Let E := {g ∈ G + A | |g| < ∞} . E is normal in G A of order at most N(Γ) by Lemma 2.23. Put H := G A /E and denote by π the corresponding quotient map As E is contained in the kernel of the action of G A on T A , the action factors through π and induces an action of H on T A . Let H + = π(G + A ). As G + A is finite-by-Abelian and any finite subgroup of G + A lies in E, H + is Abelian. Moreover, H + is torsion-free as for any g ∈ G + A s.t. g k ∈ E for some k ≥ 1 we have g k · |E | = 1, and so g ∈ E by construction. Since the image of G + A (and therefore also the image of H) in Isom(T A ) is f.g. by Theorem 3.4, there is a decomposition where A is f.g. free Abelian and B is the torsion-free Abelian kernel of the action of H + on T A . Put A = π −1 (A) and B = π −1 (B). Proof. Let s be an arbitrary element of H \ H + and lets be a lift of s to G A , clearlỹ s ∈ G A \ G + A . It follows as in the proof of Lemma 2.23 that for large i the element ϕ i (s) is of finite order as it either lies in a finite group or in a 2-ended group exchanging the ends. Thuss is of finite order, i.e.s 2 ∈ E. It follows that s 2 = π(s 2 ) = 1. This proves that We show that the action is as desired. Let h ∈ H + . Choose h such that π( h) = h. For large i the group ϕ i ( h,s ) is 2-ended with ϕ i (s) exchanging ends and h preserving ends.
Here it is easily verified that ϕ i ( hs hs −1 ) is of finite order for large i; thus hs hs −1 ∈ E, i.e. hshs −1 = 1. It follows that shs −1 = h −1 as desired.
As B is the kernel of the action of G A on T A it follows that A normalises B, i.e. A acts on B by conjugation. We can immediately see that the kernel of this action is of finite index in A: Let nows be a lift of s to G A and Auts(G A ) ≤ Aut(G A ) be the subgroup consisting of those automorphisms that restrict to the identity on B,s and preserve A. Let Aut * s (G A ) ≤ Auts(G A ) be the subgroup consisting of those automorphisms α that conjugate all point stabilizers pointwise. Thus for any α ∈ Aut * s (G A ) and x ∈ T A there exists some g x such that α(g) = g x gg −1 x for all g ∈ stab x.
Proof. Note first that for any x ∈ T A either stab x = B or stab x = B, ws for some w ∈ A. As all elements of Auts(G A ) act trivially on B by definition we can ignore the first case. Note further that there are only finitely many conjugacy classes of stabilizers of type B, ws ; this follows from the fact that if A is free Abelian of rank n then A, s has precisely 2 n conjugacy classes of reflections. As Auts(G A ) acts on conjugacy classes of stabilizers this implies that there exists a finite index subgroup Aut s (G A ) of Auts(G A ) that preserves conjugacy classes of stabilizers.
For any point stabilizer C = B, ws let Aut C s (G A ) ≤ Aut s (G A ) be the subgroup consisting of those automorphisms α that conjugate C, i.e. for which α(c) = gcg −1 for all c ∈ C and some fixed g.
To prove the claim of the lemma it suffices to show that for any such C the group Aut C s (G A ) is of finite index in Aut s (G A ). Indeed as there are only finitely many conjugacy classes and the intersection of finitely many subgroups of finite index is of finite index, this proves the claim.
Let now C = B, ws . Suppose that there exists a sequence ( Such elements f i and e i exist as by assumption α i (w) ∈ A and C is conjugate to α i (C). After passing to a subsequence we can assume that e i = e for all i ∈ N and some fixed e ∈ E. Moreover, we may assume that , this follows as the kernel of the action of A on B by conjugation is of finite index in A.
It follows that for all i, j we have Any α ∈ Auts(G A ) restricts to an automorphism of A and therefore induces an automorphism of A = A/E. Denote the subgroup of Aut(A) induced in this fashion by Ks. Moreover let K * s be the subgroup of Aut(A) induced by Aut * s (G A ) Lemma 5.12. Lets be as above. Then Ks is of finite index in Aut(A).
Proof. Suppose that A is free Abelian of rank n and let a 1 , . . . , a n be a basis of A. The proof is by contradiction thus we assume that Ks is of infinite index in Aut(A). Choose a sequence (α i ) of elements of Aut(A) that represent pairwise distinct cosets of Ks, i.e.
After passing to a subsequence we can assume that for all i, j ∈ N and 1 ≤ k, l ≤ n the following hold: (2) The actions of x i k and x j k on B coincide.
This however implies that for i, j the map x j k → x i k for 1 ≤ k ≤ n extends to an automorphism α ∈ Aut(G A , B,s ). Now this automorphism induces α i • α −1 j on A contradicting our assumption that α i • α −1 j

Ks.
As an immediate consequence of Lemma 5.11 and Lemma 5.12 we get the following. Corollary 5.13. Lets be as above. Then K * s is of finite index in Aut(A). The following proposition is the main technical result of this section. Proposition 5.14. Let G A be as above and x, x 1 , . . . , x k ∈ T A . For each finite S ⊂ G A and > 0, there exist elements γ 1 , . . . , γ k ∈ G A and an automorphism σ of G A such that the following hold.
We can moreover assume that Proof. Possibly after choosing a different reflection s we can assume that a lifts of s fixes a point ps such that d(x, ps) ≤ /4. Let a 1 , . . . , a n be a basis of the free Abelian group A; recall that A acts on T A by translations with dense orbits. The Euclidean algorithm guarantees the existence of a sequence (α i ) ⊂ Aut(A) such that the translation length of α i (a k ) (1 ≤ k ≤ n) converges to 0 for i → ∞. This implies that the translation lengths of α i (a) converge to 0 for any a ∈ A. As |Aut(A) : K * s | < ∞ we can choose (α i ) ⊂ K * s . For any i ∈ N letα i be a lift of α i to Aut * s (G A ). Now any element g ∈ S can be written asã gbgs η g whereã g ∈ A,b g ∈ B and η g ∈ {0, 1}. Asα i ∈ Auts(G A ) it follows thatα i (g) =α i (ã g )b gs η g for all i ∈ N and g ∈ S. Moreover asα i (ã g ) is a lift of α i (π(ã g )) it follows that the translation length of α i (ã g ) converges to 0. Thus we get for all g ∈ S. This implies that for sufficiently large i assertions (1) and (2) are satisfied for σ =α i .
If stab x i = B then γ i can be replaced by γ i h with h ∈ K B A while preserving (2). As

K B
A acts on T A with dense orbits this ensures the existence of some γ i such that both (2) and (3) are satisfied.
If stab x i is of type B,ãs for someã ∈ A then the fixed point ofα i (stab x i ) = B,α i (ã)s converges to ps as the translation length ofα i (ã) converges to 0. As this fixed point equals γ i x i and as d(x, ps) ≤ /4 assertion (3) follows for large i. Now assume that x = x i for some i. Then either stab x = B or we can chooses such that stab x = B,s . As in both cases theα i restrict to the identity on stab x = stab x i we can choose γ i = 1. Moreover, it is trivial that we can choose γ i = γ j whenever x i = x j . The claim follows.
Proof of Theorem 5.8. For any g ∈ S andṽ with ↓ṽ = v 1 the intersection Tṽ ∩ [x 0 , gx 0 ] is either empty or a (possibly degenerate) segment. If all such intersections are degenerate for all g ∈ S, there is nothing to show as the theorem holds for φ = id G . Thus we can assume that at least one such intersection is non-degenerate. Let r > 0 be the length of the shortest non-degenerate segment that occurs this way.
Recall from (3.4) that if q is a normal form A-path q = a 0 , e 1 , a 1 , e 2 , . . . , e k , a k (5.2) and g = [q] then Choose a point p ∈ T v 1 , if v 0 = v 1 then we choose p :=x 0 . Let S v 1 be the set of elements of A v 1 that occur in the normal forms of the elements in S. By Proposition 5.14, there is an automorphism σ of A v 1 and for each e ∈ E A with α(e) = v 1 an element γ e ∈ A v 1 such that the following hold.
• For e ∈ E A with α(e) = v 1 , the restriction of σ to α e (A e ) ≤ stab p α e is conjugation by γ e , and Fix such an automorphism σ and let φ ∈ Aut(G) be a natural extension of σ (cf. Definition 4.13). Thus if g = [a 0 , e 1 , . . . , e k , a k ] as before we get φ(g) = [ā 0 , e 1 ,ā 1 , e 2 , . . . , e k ,ā k ] In the following we compare the summands occuring in (5.3) to those occuring in (5.5). If a i =ā i the corresponding summands clearly coincide. Thus we can assume that a i ā i ∈ A v 1 . We distinguish two cases.
If moreover d ω(e i ) (p ω e i , a i p α e i+1 ) = 0 then p ω e i and p α e i+1 are A v 1 -equivalent and therefore p ω e i = p α e i+1 by assumption, thus a i ∈ stab p ω e i and γ e −1 i = γ e i+1 , by assumption this implies that σ(a i ) = γ e i+1 a i γ −1 e i+1 . The above computation therefore implies that Case 2. If i = 0 (the case i = k is analogous) then we get If moreover d v 1 (p, a 0 p α e 1 ) = 0 then p and p α e 1 are A v 1 equivalent and therefore p = p α e 1 by assumption. Thus a 0 ∈ stab p and therefore γ e 1 ∈ stab p. Thus Comparing the summands in (5.3) to those occuring in (5.5) shows that each summand is preserved unless it corresponds to a non-degenerate intersection with some vertex tree Tṽ with ↓ṽ = v 1 (of length at least r) in which case it replaced by at most r 2 . This proves the claim.

Orbifold components
Analogous to Theorem 5.8, we prove the existence of a shortening automorphism provided there exists a vertex of orbifold type.

Theorem 5.15.
Let v 1 ∈ V A be an orbifold type vertex. For any finite subset S ⊂ G there exists some φ ∈ Mod(G) such that the following hold for any g ∈ S.
• If [x 0 , gx 0 ] has a nondegenerate intersection with a vertex space Tṽ and ↓ṽ = v 1 , The proof of Theorem 5.15 follows from the following proposition in exactly the same way as Theorem 5.8 follows from Proposition 5.14.

Proposition 5.16.
Let v ∈ V A be an orbifold type vertex and x, x 1 , . . . , x k ∈ T v . For each finite S ⊂ A v and > 0, there exist elements γ 1 , . . . , γ k ∈ A v and an automorphism σ ∈ Aut(A v ) such that the following hold.
We can moreover assume that The remainder of this section is decicated to the proof of Proposition 5.16. The argument in this case is essentially due to Rips and Sela [35] who give a proof of Proposition 5.16 in the case where the action of A v on T v has trivial kernel. Thus we only need to address the case where this kernel is non-trivial.
If H is a family of subgroups of G then we will denote by Aut H (G) the subgroup of Aut(G) consisting of those automorphisms that act on each H ∈ H by conjugation with an element of G.
Then S has finite index in Aut H (G).
Proof. The proof is in two steps. We first prove that the subgroup S 1 of Aut H (G) consisting of automorphisms that lift to automorphisms of G is of finite index in Aut H (G). We then show that S is a finite index subgroup of S 1 .
Assume that |Aut H (G) : S 1 | = ∞. Then there exists a sequence (α i ) ⊂ Aut(G) of automorphisms such that α i S 1 α j S 1 for i j. Let S E | R E be a presentation of E and s 1 , . . . , s m r 1 , . . . , r g be a finite presentation of G. Every automorphism α i of G gives rise to a (not unique) presentation of G as i α j ∈ S 1 and therefore α i S 1 = α j S 1 , contradicting the above assumption. Thus |Aut H (G) : S 1 | < ∞.
Let now S 2 be the subgroup of Aut( G) consisting of all lifts of automorphisms of S 1 . It clearly suffices to show that S 2 ∩ Aut H ( G) is of finite index in S 2 .
For anyα ∈ S 2 and i = 1, . . . , k we have α( H i ) = c i H i c −1 i for some c i ∈ G (but the conjugation is not pointwise in general). Since H i is malnormal, c i is unique up to elements of H i . It follows that α induces a well-defined outer automorphism σ i (α) of H i represented by the automorphism Out( H i ), α → (σ 1 (α), . . . , σ k (α)).
This clearly proves the assertion as Out(H i ) is finite for all i.
We can now proceed with the proof of Proposition 5.16.
Proof of Proposition 5.16. Denote by E the kernel of the action of A v on T v . Since T v is not a line, E stabilizes a tripod, and therefore is finite by Theorem 2.17. Denote by π : A v → P := A v /E the quotient map. Then P acts faithfully on T v , and by Theorem 3.4, P is the fundamental group of a compact 2-orbifold Σ with boundary. In particular, P is finitely presented. Now let x be an arbitrary point of T v . Applying Proposition 5.2 of [35], we get an infinite sequence (α i ) of automorphisms of P satisfying the following.
• For each (infinite cyclic) peripheral subgroup Z of P the restriction α i | Z is conjugation with an element c (e) i ∈ P.
• For any peripheral subgroup Z of P the distance between x and the (unique) fixed point of α i (Z) tends to 0.
Let F ⊂ E A be the set of edges whose initial vertex is v. For each e ∈ F, the image Z e of α e (A e ) under π in P = π(A v ) = π 1 (Σ) correspond to a loop in Σ that is homotopic to a boundary component. This implies that π(α e (A e )) is malnormal in P.
Define S ≤ Aut H (P) as in Lemma 5.17 with respect to the collection of subgroups H := {Z e | e ∈ F} of P and H := { Z e := π −1 (Z e ) | e ∈ F}.
Then by Lemma 5.17, |Aut H (P) : S| < ∞. It follows that there is a subsequence (α i j ) ⊂ (α i ) such that all α i j are in the same left coset C of S. Fix a representative γ ∈ C. Then the sequence (α i ) given by α j := γ −1 α i j is in S and lim i→∞ |α i | = 0. Choosing i large enough and extending α i to A v , gives the desired automorphism.

Simplicial components
Assume now that G has a nondegenerate simplicial vertex tree. Thus G can be refined in a simplicial type vertex yielding a (refined) graph of actions with non-zero length function l such that all vertices that are adjacent to edges of non-zero length have degenerate vertex trees. We denote this graph of actions again by G. We can still assume that the base point x 0 is contained in a vertex treeṽ 0 = 1 , i.e. that x 0 = [x 0 ,ṽ 0 ]. Indeed, if x 0 is contained in the interior of an edge segment T e , we can split the corresponding edge ↓ e ∈ E A by introducing a valence 2 vertex with vertex group A ↓ e and degenerate vertex tree such that x 0 is precisely a lift of this vertex tree.
We will construct a Dehn twist automorphism on an edge with non-zero length such that powers of this Dehn twist shorten the action of G on X induced by ϕ i (for large enough i). Other than in the axial and orbifold cases these automorphisms do not shorten the action on the limit tree.
The following proposition is the key observation needed for the construction of the shortening automorphisms. In the following d i := d X |ϕ i | is the scaled metric on the Cayley graph X. Thus for all g, h ∈ G. Recall that (X, d i ) is δ i -hyperbolic with lim δ i = 0. Also recall from (3.2) that for any c ∈ A e and lift e of e the element θ e (c) is a natural lift of c to the stabilizer of e. If (y i ), (z i ) ⊂ X are approximating sequences of T α( e) and T ω( e) (which are single points) respectively, then Proof. We assume that ε = 1, the case where ε = −1 is an immediate consequence due to the equivariance of the action. Fix some lift e of e and put c e := θ e (c). For large i the element ϕ i (c e ) is hyperbolic and we define A i to be the axis of ϕ i (c e ) in X, i.e. the union of all geodesics joining the ends fixed by ϕ i (c e ). A i is easily seen to be in the 4δ i -neighbourhood of any of these geodesics with respect to the metric d i . For each i let y i and z i be points on A i closest to y i and z i respectively. It is clear that lim i→∞ d i (y i , y i ) = 0 and lim i→∞ d i (z i , z i ) = 0 as c e fixes Tṽ 1 and Tṽ 2 in the limit action.
Moreover, there are integers m i such that where l(ϕ i (c e )) denotes the translation length of ϕ i (c e ) (note that we used the non-scaled metric d here).
As d(y i , ϕ i (c e )) m i z i ) is globally bounded from above, it follows that To conclude, it suffices to show that the choice of the m i does not depend on the choice of the lift of e. Indeed this follows immediately from the fact that if e = h e is another for sufficiently large i. Otherwise, d i (1, ϕ i • σ m i (g)) = d i (1, ϕ i (g)).
As the tree T G is minimal it follows that for any generating set S of G the normal form of at least one element of S contains an edge e ±1 . Thus we obtain the following immediate corollary.
Proof of Proposition 5.19. Let q = a 0 , e 1 , a 1 , e 2 , . . . , a n be a normal form A-path s.th. g = [q], and e 1 , . . . , e n be the reduced edge path in A fromṽ 0 to gṽ 0 . Then + d gṽ 0 (p ω e n , gx 0 ).  (1) and (ϕ i (g)) are approximating sequences of x 0 and gx 0 respectively. Thus by Lemma 2.10, for any > 0 and large enough i, we have For each i ∈ N we putā i 0 := a 0 and for k ∈ {1, . . . , n} put Moreover for 0 ≤ k ≤ n and i ∈ N we define q k := a 0 , e 1 , a 1 , . . . , a k , and it is easy to verify thatq i k q −1 k ∼q i k−1 q −1 k−1 θ e k (c) ε ·m i . Therefore by Proposition 5.18 lim Comparing this to (5.9) we obtain

The shortening automorphism
In view of the previous sections, we are now able to conclude the proof of Proposition 5.6. Let (ϕ i ) ⊂ Hom(G, Γ) a converging stable sequence of pairwise distinct homomorphisms with associated Γ-limit map ϕ. Assume that all ϕ i are short (with respect to fixed finite generating sets of G and Γ). Then, by Theorems 2.6 and 2.11 we obtain a non-trivial limit G-tree T, which splits as a graph of actions G by Theorem 3.4.
If G contains an axial vertex space or an orbifold type vertex space it follows from Theorem 5.8, respectively Theorem 5.15, that we can shorten the action on T by precomposing with an automorphism α ∈ Aut(G). As this action is being approximated by the action on X via the ϕ i it follows that for large i these actions can be shortened likewise by precomposing with α. This proves that for large i the ϕ i are not short, which is the claim of the theorem.
In the remaining case G contains a simplicial vertex space and the Proposition follows immediately from Corollary 5.20.

Makanin-Razborov diagrams
In this chapter we use the existence of Γ-factor sets proven in the previous chapter to give a complete description of the set of all homomorphisms from a finitely generated group G to an equationally Noetherian hyperbolic group Γ. In Section 7.2 we will see that this assumption was vacuous as all hyperbolic groups are equationally Noetherian.
Γ is called equationally Noetherian if for every n ∈ N and any subset S ⊂ Γ[x 1 , . . . , x n ] there exists a finite subset S 0 ⊂ S such that rad(S) = rad(S 0 ).
It was shown by Guba [20] that f.g. free groups are equationally Noetherian. His proof exploited the fact that free groups are linear which allows him to appeal to some classical algebraic geometry. This fact was used in [2] to show that large classes of linear groups are equationally Noetherian. Note however that hyperbolic groups are not necessarily linear [25]. Thus we can in general not appeal to linearity to establish that hyperbolic groups are equationally Noetherian.
The following simple fact was pointed out to the authors by Abderezak Ould Houcine, it says that to check whether a finitely generated group Γ is quationally Noetherian it suffices to check systems of equations without constants: given by β(y i ) = g i for 1 ≤ i ≤ k and β(x i ) = x i for 1 ≤ i ≤ n. Put S = β −1 (S). By hypothesis there exists a finite set S 0 ⊂ S such that rad( S) = rad( S 0 ). Clearly S 0 := β( S 0 ) is a finite subset of S, as further rad(S) = rad(S 0 ) this implies the claim.
Choose presentations x 1 , . . . , and G ∞ := x 1 , . . . , x n | R ∞ , i.e. G ∞ is the direct limit of the G i . As Γ is equationally Noetherian it follows that rad(R ∞ ) = rad(R i ) for some i. The claim follows.

Corollary 6.3. If Γ is equationally Noetherian then any sequence
of epimorphisms of finitely generated groups that are residually Γ eventually stabilizes.
Proof. Because of Lemma 6.2 it suffices to show that if two groups G and G are residually Γ and π : G → G is a non-injective epimorphism then π * : Hom(G , Γ) → Hom(G, Γ) is not surjective. This is obvious as for k ∈ ker π \ {1} there is a homomorphism φ : G → Γ such that φ(k) 1 as G is residually Γ. Clearly φ does not lie in the image of π * . Corollary 6.4. Suppose that Γ is equationally Noetherian. Then any Γ-limit group is fully residually Γ.
Proof. Let L = F k /ker − → (ϕ i ) be a Γ-limit group. Choose a sequence of finitely presented groups such that L is their direct limit. By Corollary 6.3 there exists some G i 0 such that any homomorphism ϕ : G i 0 → Γ factors through L. After passing to a subsequence we can further assume that any ϕ i factors through G i 0 as G i 0 is finitely presented. Thus the sequence factors in fact through L, i.e. there exists a stable sequence (η i ) ⊂ Hom(L, Γ) such that ker − → (η i ) = 1, i.e. that L = L/ker − → (η i ). This clearly implies that for any finite set M ⊂ L, η i | M is injective for sufficiently large i.
We will need the following simple lemma, its proof is identical to that in the case of a free group, see [6]. Lemma 6.5. Let Γ be an equationally Noetherian group and G be a finitely generated group. Then there exist finitely many groups L 1 , . . . , L k and epimorphisms q i : G → L i such that the following hold.
Proof. Let G be the universal residually Γ quotient of G, i.e. G = G/N where N is the intersection of all kernels of homomorphisms from G to Γ. Clearly any homomorphism φ : G → Γ factors through the canoncial projection π : G → G = G/N. It therefore suffices to show that there exists a finite collection of groups L 1 , . . . , L k that are fully residually Γ and epimorphisms q i : G → L i for 1 ≤ i ≤ k such that any homomorphism φ : G → Γ factors through some q i . If G is fully residually Γ then the claim is trivial. Thus we can assume that G is not fully residually Γ. It follows that there exists a finite set M = {g 1 , . . . , g k } ⊂ G \ {1} such that ker φ ∩ M ∅ for any homomorphism φ : G → Γ. It follows that any φ : G → Γ factors through one of the epimorphisms q i : G → L i where L i is the universal residually Γ quotient of G/ g i for 1 ≤ i ≤ k and q i is the canonical quotient map.
If L i is not fully residually Γ we repeat this construction for L i , after finitely many iterations this must teminate by Lemma 6.2. Thus we get a finite directed tree of epimorphisms such that any homomorphism factors through one branch, the assertion follows by choosing as epimorphisms the compositions of epimorphisms along maximal (directed) branches of this tree.

Dunwoody decompositions
Recall that a group G is called accessible if there exists a reduced graph of groups A such that the following hold.
(2) Any edge group of A is finite.
(3) Any vertex group of A is one-ended or finite.
We call any such A a Dunwoody decompositon of G. Note that the graph of groups A is far from being unique for a given accessible group G. However the maximal vertex groups are unique up to conjugacy; indeed they are precisely the maximal one-ended subgroups of G.
It is the Dunwoody accessibility theorem [12] that states that all finitely presented groups are accessible. It turns out that f.g. groups are in general not accessible but the particular case that we will need is covered by the following theorem of P. Linnell [30] of which Theorem 4.5 is a generalization. Theorem 6.6. Let G be a f.g. group. Suppose that there exists some constant C such that any finite subgroup H of G if of order at most C. Then G is accessible.
Let now Γ be a hyperbolic group and L a Γ-limit group. As the order of finite subgroups of L is bounded by N(Γ) it follows that Theorem 6.6 applies to L, i.e. L admits a Dunwoody decomposition D. As modular automorphisms of the vertex groups of D restrict to the identiy on all finite subgroups it follows that they extend to automorphisms of L.
In the following we call the subgroup of Aut L consisting of those automorphisms that restrict (up to conjugation) to modular automorphism of the vertex groups of D the modular group of L and denote it by Mod(L). In the case of a one-ended group this recovers our original definition.
Let now G be an accessible group and Γ be a group. Then we call a homomorphism ψ : G → Γ locally injective if ψ is injective when restriced to the vertex groups of some (and therefore all) Dunwoody decomposition of G. Note that this is equivalent to saying that ψ is injective when restricted to 1-ended and finite subgroups of G.

MR-diagrams for equationally Noetherian hyperbolic groups
In this section we give a proof of the main theorem of this article, i.e. the description of Hom(G, Γ) for some finitely generated group G and some hyperbolic group Γ under the additional assumption that Γ is equationally Noetherian. It will then be the purpose of Section 7 to establish that all hyperbolic groups have this property. Theorem 6.7. Let Γ be an equationally Noetherian hyperbolic group and G be a finitely generated group. Then there exists a finite directed rooted tree T with root v 0 satisfying (1) The vertex v 0 is labeled by G.
(2) Any vertex v ∈ VT, v v 0 , is labeled by a Γ-limit group G v .
(3) Any edge e ∈ ET is labeled by an epimorphism π e : G α(e) → G ω(e) such that for any homomorphism φ : G → Γ there exists a directed path e 1 , . . . , e k from v 0 to some vertex ω(e k ) such that where α i ∈ Mod G ω(e i ) for 1 ≤ i ≤ k and ψ is locally injective. Remark 6.8. In case G is fully residually Γ, the factorization of homomorphisms from G to Γ as in Theorem 6.7 requires modular automorphisms of G before the first proper quotient map. Thus in this case the diagram has precisely one edge e satisfying α(e) = v 0 , and π e : G = G α(e) → G ω(e) is an isomorphism.
Proof of Theorem 6.7. In view of Lemma 6.5 and Corollary 6.3 it clearly suffices to show that any group G that is fully residually Γ admits a finite set {q i : G → Γ i } of proper quotient maps such that any homomorphism ϕ : G → Γ which is not locally injective factors through some q i after precomposition with an element of Mod G.
Choose a Dunwoody decomposition D of G. For each vertex group D v there is a factor set We now define the factor set for G to be To see that this is a factor set let ϕ : G → Γ be a non-locally injective homomorphism. Choose As α extends to an automorphism α ∈ Mod(G) it follows that ϕ • α factors through Q v i .

Shortening quotients and applications
In the previous section we have constructed Makanin-Razborov diagrams for equationally Noetherian hyperbolic groups. It is the purpose of this last chapter to establish that all hyperbolic groups are equationally Noetherian, i.e. that the construction of the Makanin-Razborov diagrams applies to all hyperbolic groups.

Shortening quotients
In Section 5, see Remark 5.7, we have seen that if (ϕ i ) ⊂ Hom(G, Γ) is a stable sequence such that ker − → (ϕ i ) = 1, i.e. that L = G/ker − → (ϕ i ) = G, then we can construct a proper quotient G/ker − → ( ϕ i ) of G = L where the ϕ i are the shortened ϕ i . This quotient is clearly again a Γ-limit group and is called a shortening quotient.
This construction only works if G is fully residually Γ. It is the main purpose of this section to construct shortening quotients for arbitrary Γ-limit groups. In the end, see Corollary 7.6, it will turn out that all Γ-limit groups are fully residually Γ. We will first treat one-ended Γ-limit groups and then deal with the general case.
Let L = G/ker − → (ϕ i ) be a one-ended Γ-limit group and A be an almost Abelian JSJdecomposition of L. Lemma 7.1 below guarantees that we can approximate L by a sequence of groups (W i ) that are endowed with splittings that appoximate A. Lemma 7.1. Let G be a finitely presented group and L = G/ker − → (ϕ i ) be a one-ended Γ-limit group with associated Γ-limit map ϕ : G → L. Let A be an almost Abelian JSJ-decomposition of L, in particular L = π 1 (A, v 0 ).
Then there exists a sequence of graphs of groups (A i ) with underlying graph A and finitely presented fundamental groups W i = π 1 (A i , v 0 ), surjective morphisms f i : A i → A i+1 and h i : A i → A and an epimorphism γ : G → W 0 such that the following hold.
(3) L is the direct limit of the sequence W i , i.e.
Proof. This is a simple application of foldings as discussed in [4] and Dunwoody's vertex morphisms [13]. Let T = A be the Bass-Serre tree corresponding to A, thus T is an L-tree. The Dunwoody Resolution Lemma guarantees that there is a G-tree Y with finitely generated edge and vertex stabilizers and a surjective morphism (id, p) from Y to the L-tree T, see [12,14] or [4]. After applying finitely many folds to the G-tree Y we obtain a G-tree Y such that the induced map on the graphs of groups Y /G → T/L = A is bijective on the level of graphs and surjective for the edge and vertex groups of A that are finitely generated, see [4].
We now apply vertex morphisms to quotient out the kernels of the homomorphisms of edge groups and of the vertex groups whose targets are QH-subgroups or almost Abelian groups. This clearly adds only finitely many relations. Denote the resulting graph of groups by A 0 , the morphism from A 0 to A clearly satisfies (4)- (6).
We can now continue to apply folds of type IIA and IIB, see [4], and get a sequence of graphs of groups satisfying (7) and (8). At each step we further add all relators to edge groups and vertex group mapped to almost Abelian vertex groups which makes this sequence preserve properties (4)-(6). Finally we add at each step the shortest relator to the vertex groups that are mapped to rigid vertex groups that does not yet hold. This then implies that all relations of L will hold eventually, i.e. that (3) holds. Parts (1) and (2) hold by construction.
Let now G be finitely presented with fixed finite generating set S, (ϕ i ) ⊂ Hom(G, Γ) a stable sequence such that L = G/ker − → (ϕ i ) is a one-ended Γ-limit group. Let A be an almost Abelian JSJ-decomposition of L. Choose sequences (A i ), (h i ) and ( f i ), (W i ) and γ as in Lemma 7.1.
G → W k be the epimorphism induced by the f i and γ. Then S i := ξ i (S) is a generating set of W i . After replacing (ϕ i ) by a subsequence we can assume that ϕ i factors through ξ i for all i, i.e. that ϕ i = λ i • ξ i for some λ i : W i → Γ. This is clearly possible as the finitely many defining relations of W i lie in ker − → (ϕ i ) and therefore in the kernel of ϕ j for sufficiently large j.
Let now λ i : W i → Γ be the homomorphism obtained from shortening λ i by precomposition with elements of Mod A i (W i ) and postcomposition with an inner automorphism. Here shortness is measured with respect to the generating set S i . We then put η i := λ i • ξ i . After passing to a subsequence we can assume that (η i ) is stable. We then put and call Q a shortening quotient of L. It is clear from the construction that Q is a quotient of L. Indeed if g ∈ ker − → (ϕ i ) then g ∈ ker ξ i for large i as we assume that (3) of Lemma 7.1 is satisfied. Thus g ∈ ker η i = ker λ i • ξ i for large i and therefore g ∈ ker − → (η i ). We denote the projection from L to Q by π, thus we have η = π • ϕ if η and ϕ are the Γ-limit maps associated to the sequences (η i ) and (ϕ i ). (1) The epimorphism π : L → Q is injective on rigid vertex groups of A.
(2) If (η i ) is not contained in finitely many conjugacy classes then Q is a proper quotient of L.
(3) If all almost Abelian subgroups of Q are finitely generated then the following hold: (a) If a subsequence of (η i ) factors through η : G → Q then a subsequence of (ϕ i ) factors through ϕ : G → L. (b) All almost Abelian subgroups of L are finitely generated.
Proof. (1). Let g ∈ L be an element that is conjugate to an element h of a rigid vertex group A v of A such that π(g) = 1. We need to show that g = 1.
As we assume that (7) of Lemma 7.1 is satisfied it follows that for some i 0 there exists Choose g ∈ G such that ξ i 0 ( g) = g i 0 and put g i = ξ i ( g) for all i. Note that for i ≥ i 0 the element g i is conjugate to some element k i ∈ A i v . Now π(g) = 1 means that η i ( g) = λ i • ξ i ( g) = λ i (g i ) = 1 for large i, in particular any element conjugate to g i lies in the kernel of λ i . Recall that λ i = λ i • α i for some modular automorphism α i of W i . As g i is conjugate to the element k i ∈ A i v and as modular automorphisms act on rigid groups by conjugation it follows that α i (g i ) is also conjugate to k i and therefore conjugate to g i . Thus (2). Assume to the contrary that (η i ) contains infinitely many conjugacy classes and that L = Q, i.e. that L = Q = G/ker − → (η i ). After passing to a subsequence we can assume that (η i ) converges to an action (T, x, ρ) of L on an R-tree T satisfying the assumptions of Theorem 3.4. Let now G be the graph of actions decomposition corresponding to this action. As in Section 5 we distinguish 3 different cases.
If the graph of actions has an orbifold type vertex then there is an automorphism of its vertex group that extends to a modular automorphism α of L which shortens the action of L on T, i.e. for which | ρ • α| x ≤ | ρ| x . Recall that η = h k * • ξ k for all k. Now this orbifold type vertex group corresponds to a suborbifold of one of the orbifold type vertices of the JSJ-decomposition of L. Thus α can be lifted to any W i as the morphisms h i are isomorphisms when restricted to QH-subgroups. Thus there exists contradicting the shortness of the λ i . If the action has an axial type vertex then we can choose α as in the case of an orbifold type vertex but the lifting is slightly more subtle. Note first that the vertex group corresponding to this axial vertex space is also a vertex group of the JSJ-decomposition of L. This is true as the group must be elliptic in the JSJ as it is an almost Abelian subgroup that is not 2-ended. We can assume that it is a vertex group as we could otherwise refine the JSJ contradicting its maximality. Now the morphisms ψ h i v : A i v → A v are not necessarily surjective on almost Abelian vertex groups. However for large i the group ψ h i v (A i v ) ≤ A v contains all generators, and therefore all elements, of A v that act non-trivially on the axial tree. This means we can define an automorphism of A i v that extends to an automorphism α i of W i such that λ i • α i is shorter than λ i by analyzing the action of A i v on the axial tree via ψ h i v . Note that this is easier than in Section 5 as the group A i v is finitely generated. In the simplicial case the edge group along which the Dehn twist is performed either corresponds to an edge group of the JSJ or to a simple closed curve in a QH-subgroup of the JSJ. In both cases we can simply lift the Dehn twists to W i and thereby shorten the homomorphism η i . In the case where the edge group corresponds to a simple closed curve of a QH-subgroup this is obvious, in the other case it follows as an element that is central in the edge group of the graph of actions is also central in the corresponding edge group of A i .
(3). Note first that any edge group of A, the JSJ-decomposition of L, is contained in either a rigid or an orbifold type vertex group. It follows that all edge groups of A are finitely generated. Indeed if the edge group is contained in a rigid vertex group then it follows from (1) that the edge group embeds into Q and is therefore finitely generated. Otherwise the edge group is virtually cyclic and the modular automorphisms act on the group by conjugation, it therefore follows as in the proof of (1) that it is embedded into Q and is therefore finitely generated.
As L = π 1 (A) is finitely generated and all edge groups of A are finitely generated it follows that also all vertex groups of A are finitely generated. Thus there exist i 0 such that for i ≥ i 0 the morphism h i is bijective on all edge groups and non-rigid vertex groups. On the rigid vertex groups h i is surjective, i.e. the morphism consists just of vertex morphisms in the sense of Dunwoody [13]. As almost all ξ i factor through W i 0 we can pass to a subsequence and assume that i 0 = 0.
Suppose now that η i factors through Q. For any v ∈ V A denote the kernel of the map As η i factors through Q and therefore through L it follows that K v ⊂ ker η i for any vertex group A v . As the η i and the ϕ i only differ by precomposition with an automorphism that acts by conjugation on rigid vertex groups this implies that K v ⊂ ker ϕ i for all rigid vertex groups A v . Thus ϕ i factors through L as all other relations of L already hold in W 0 . The second assertion follows immediately from the proof.
The above construction only works for one-ended Γ-limit groups as we need the existence of an almost Abelian JSJ-decomposition. In the remainder of this section we will show that the concept of a shortening quotient generalizes naturally to all Γ-limit groups.
Let now G be a finitely presented group and (ϕ i ) ⊂ Hom(G, Γ) be a stable sequence. Put L := G/ker − → (ϕ i ) and denote the associated Γ-limit map by ϕ. Let D be a Dunwoody decomposition of L, i.e. L = π 1 (D, v 0 ), all edge groups of D are finite and no vertex group splits over finite groups. Thus every vertex group is either finite or one-ended.
As in the proof of Lemma 7.1 we see that there is a graph of groups D whose underlying graph D is the same graph that is underlying D, a morphism f : D → D and an epimorphism γ : G → π 1 (D , v 0 ) such that the following hold.
(3) The morphism f is bijective on edge groups, thus f only consists of a collection of vertex morphisms on some vertices. Now as π 1 (D , v 0 ) is finitely presented almost all ϕ i factor through γ. Thus after omitting finitely many elements from (ϕ i ) we can assume that for all i there exists ϕ i : π 1 (D , v 0 ) → L such that ϕ i =φ i • γ.
For each vertex v ∈ V D we get a stable sequence (φ v i ) whereφ v i : D v → Γ is the restriction ofφ i to D v . Note that this restriction is only unique up to inner automorphisms of Γ unless we choose a preferred conjugate of D v in π 1 (D , v 0 ). Independently of these conjugacy factors the obtained sequence is stable for all v ∈ V D and we have . Now for every one-ended D v we can apply the construction of the shortening quotient to the sequence (φ v i ) and obtain (after passing to a subsequence) a new stable sequence (η v i ) ⊂ Hom(D v , Γ) such that ker − → (φ v i ) ≤ ker − → (η v i ) and that all conclusions of Proposition 7.2 hold for the quotient map for all i. Now as the shortening automorphisms act on finite subgroups by conjugation it follows that for each i there exists a (not unique) homomorphismη i : π 1 (D , v 0 ) → Γ such that the restriction ofη i to D v is conjugate toη v i for all v ∈ V D. We put η i =η i • γ : G → Γ. After passing to a subsequence we can assume that (η i ) is stable and we put Q := G/ker − → (η i ). As ker − → (ϕ i ) ≤ ker − → (η i ) by construction we have a natural epimorphism π : L → Q. As in the one-ended case it follows that η = π • ϕ if η and ϕ are the Γ-limit maps associated to the sequences (η i ) and (ϕ i ), respectively.
It is clear that the epimorphism π maps the vertex groups D v of D to subgroups of Q that are isomorphic to their shortening quotients Q v , but we do not claim that the Q v are vertex groups of the Dunwoody decomposition of Q. Theorem 7.3. Let G be a finitely presented group and L = G/ker − → (ϕ i ) a Γ-limit group. Let (η i ) be as above and Q = G/ker − → (η i ). Let π : L → Q be the natural quotient map. Then one of the following holds.
(2) A subsequence of (η i ) factors through η and all almost Abelian subgroups of Q are finitely generated.
If moreover all almost Abelian subgroups of Q are finitely generated then the following hold.
(a) If a subsequence of (η i ) factors through η : G → Q then a subsequence of (ϕ i ) factors through ϕ : G → L.
(b) Almost Abelian subgroups of L are finitely generated.
Proof. We first prove (1). Assume that ker π = 1, i.e. that L = G/ker − → (ϕ i ) = G/ker − → (η i ) = Q. Thus for each v ∈ V D, the epimorphism π v : D v → Q v is an isomorphism, hence by Proposition 7.2, (η v i ) contains only finitely many conjugacy classes. After passing to a subsequence we can assume that for each v, allη v i are conjugate, i.e. that ker − → (η v i ) = kerη v i for all i and all v ∈ V D, in particular D v Q v = G/ker − → (η v i ) η v i (G) ≤ Γ. As almost Abelian subgroups of hyperbolic groups are 2-ended it follows that all almost Abelian subgroups of vertex groups of D and therefore of π 1 (D, v 0 ) = L = Q are finitely generated.
We will now show that a subsequence of (η i ) factors through η = π • ϕ where η is the limit map associated to (η i ) = (η i • γ). As all η i fator through γ if follows that the associated Γ-limit map η also factors through γ. Chooseη such that η =η • γ. As ker π = 1 it follows that ker η = ker π • ϕ = ker ϕ. Thus the kernel ofη is normally generated by the stable kernels ker − → (η v i ). By the above remark there is a subsequence of (η i ) for which ker − → (η v i ) = kerη v i for all v ∈ V D and i, it follows that this subsequence factors through η. Now suppose that all almost Abelian subgroups of Q are finitely generated. The shortening quotient Q v of D v embeds into Q for all v, thus all almost Abelian subgroups of Q v are finitely generated. It thus follows from Proposition 7.2 that all almost Abelian subgroups of D v and therefore also L = π 1 (D v ) are finitely generated, this proves (b). The proof of (a) is similar to the proof of the first part; it suffices to show for a subseqence of (ϕ i ) we have ker − → (φ v i ) ⊂ kerφ v i . By assumption a subsequence of (η j i ) of (η i ) factors through η which implies that the sequences (η v j i ) factor through D v → Q v = D v /ker − → (η v i ) for all v ∈ DV. As almost Abelian subgroups of Q v are finitely generated it follows from Proposition 7.2 that a subsequence of (φ v i ) factors through D v → D v = D v /ker − → (φ v i ) for all v ∈ V D, thus a subsequence of (φ i ) factors π 1 (D , v 0 ) → L = π 1 (D, v 0 ), i.e. a subsequence of (ϕ i ) = (φ i • γ) factors through ϕ : G → L.
If L = G/ker − → (ϕ i ) and Q = G/ker − → (η i ) are as in Theorem 7.3 then we call Q a shortening quotient of L and we say that (η i ) is obtained from (ϕ i ) by shortening or by the shortening procedure. It will be important in the next section that η factors through ϕ if η and ϕ are the Γ-limit maps corresponding to (η i ) and (ϕ i ).

Hyperbolic groups are equationally Noetherian
In this chapter we show that hyperbolic groups are equationally Noetherian. We fix a hyperbolic group Γ. Crucial to the argument is a partial order on Γ-limit maps defined as follows.
Fix a f.g. group G. By definition, every homomorphism η : G → Γ is a Γ-limit map, arising from the constant sequence (η). Note further that the relation ≤ on the set of all Γ-limit maps from G is transitive. We will show that there are only finitely many maximal Γ-limit maps with respect to ≤. The main technical step is the proof of the following theorem.
As an immediate consequence of Theorem 7.5 we get the following. Corollary 7.6. Finitely generated Γ-limit groups are fully residually Γ.
Choose E = { g 1 , . . . , g k } ⊂ G such that η( g j ) = g j for j = 1, . . . , k. As η| E is injective, there exists an i 0 ∈ N such that for i ≥ i 0 , η i | E is injective. Moreover, by Theorem 7.5, there is an i ≥ i 0 such that η i =η i • η for someη i ∈ Hom(L, Γ). Clearly, η i | E is injective. Theorem 7.5 is an immediate consequence of Lemma 7.7 and Lemma 7.8. Lemma 7.7. Let (ϕ i ) ⊂ Hom(F k , Γ) be a stable sequence and ϕ its associated Γ-limit map. Then one of the following holds.
Proof. Assume that (1) does not hold. Let (η 1 i ) be a stable sequence obtained from (ϕ i ) by shortening (and passing to a subsequence) and let η 1 be the corresponding Γ-limit map. If η 1 < ϕ then we choose a sequence (η 2 i ) with Γ-limit map η 2 by shortening (η 1 i ) and so on. By assumption this process terminates, i.e. for some s we have ker η s = ker η s+1 . By Theorem 7.3 (2) a subsequence of (η s+1 i ) factors through η s+1 and all almost Abelian subgroups of L η s+1 are finitely generated. Applying Theorem 7.3 (a) and (b) s + 1 times implies that a subsequence of (ϕ i ) factors through ϕ, i.e. that (2) occurs. Lemma 7.8. There exists no infinite descending sequence of Γ-limit maps.
Proof. Assume that an infinite descending sequence of Γ-limit maps exists. For each k ∈ N, choose a stable sequence (η k i ) ⊂ Hom(G, Γ) with associated Γ-limit map η k : G → G/ker − → (η k i ) such that (1) η 1 > η 2 > . . . is an infinite descending sequence of Γ-limit maps, (2) for each n > 1, ifη n is a Γ-limit map such thatη n < η n−1 and there is an infinite descending sequence η n−1 >η n > . . . of Γ-limit maps, then |kerη n ∩ B n | ≤ |ker η n ∩ B n |, where B n is the Ball of radius n in G around the identity with respect to some fixed finite generating set.
It is clear that such a sequence exists, as the η i can be chosen inductively to satisfy property 2. For each n choose an index i n such that (3) ker η n i n ∩ B n = ker η n ∩ B n , (4) ker η n+1 ker η n i n .
As η n+1 < η n , it is clear that these conditions are satisfied if i n is chosen sufficiently large. By construction, the diagonal sequence (η n i n ) n∈N ⊂ Hom(G, Γ) is stable. Denote its Γ-limit map by η ∞ , clearly η ∞ < η n for all n.
In [39] proved that torsion-free hyperbolic groups are Hopfian, essentially the same argument works in our situtation; see also [19] for the case of torsion-free toral relatively hyperbolic groups. Proof. Let Γ be a hyperbolic group. We need to show that any epimorphism η : Γ → Γ is an isomorphism, i.e. has trivial kernel.
Note that η n : Γ → Γ (the nth power of η) is also an epimorphism. If η has non-trivial kernel then ker η n+1 < ker η n for all n. Thus we have an infinite sequence id > η > η 2 > η 3 > . . . of Γ limit maps (recall that all homomorphisms to Γ are Γ-limit maps coming from the constant sequence), a contradiction to Lemma 7.8.
be an infinite ascending sequence of Γ-limit maps. There exists a Γ-limit map η such that for every n ∈ N, η n < η.
as any homomorphism from F n to Γ that factors through ϕ must factor through some maximal Γ-limit map and therefore through some η i . Assume that for each i there is a finite set S i 0 ⊂ S such that rad(ker η i ∪ S i 0 ) = rad(ker η i ∪ S). Putting S 0 := S i 0 , we get rad(ker ϕ ∪ S 0 ) = rad(ker η i ∪ S 0 ) = rad(ker η i ∪ S) = rad(S) ∩ rad(ker η i ) = rad(S) ∩ rad(ker ϕ) = rad(S ∪ ker ϕ), which is a contradiction. Thus for some i 0 such a set S i 0 0 does not exist and the conclusion follows by putting η = η i 0 . Proof. Let Γ be a hyperbolic group. Because of Lemma 6.1 it suffices to check that for any set S ⊂ F n = F(x 1 , . . . , x n ) ⊂ Γ[x 1 , . . . , x n ] there exists a finite subset S 0 ⊂ S such that rad(S) = rad(S 0 ). Assume that n ∈ N and S = {w 1 , w 2 , . . .} ⊂ F n such that rad(S) rad(S 0 ) for every finite S 0 ⊂ S. We show that this implies the existence of an infinite descending sequence of Γ-limit maps, contradicting Lemma 7.8. Let ϕ 1 : F n → F n / w 1 and η 1 a Γ-limit map with ker ϕ 1 ≤ ker η 1 as in Lemma 7.12. Then inductively for each i, pick w j i ∈ S \ ker η i−1 and put ϕ i : F n / ker η i ∪ w j i and apply Lemma 7.12 to obtain a Γ-limit map η i with ker ϕ i ≤ ker η i . Then all η i are Γ-limit maps and η 1 > η 2 > η 3 > . . . .