Magnus pairs in, and free conjugacy separability of, limit groups

There are limit groups having non-conjugate elements whose images are conjugate in every free quotient. Towers over free groups are freely conjugacy separable.

Definition 1.1. A sequence of homomorphisms {φ i : G → H} is discriminating if for every finite subset P ⊂ G\ {1} there is some N such that for all j ≥ N, 1 ∈ φ j (P ).
Definition 1.2. A finitely generated group L is a limit group if there is a discriminating sequence of homomorphisms {φ i : L → F}, where F is a free group.
Theorem A. The class of limit groups is not freely conjugacy separable.
This should be seen in contrast to the fact that limit groups are conjugacy separable [CZ07]. Furthermore Lioutikova in [Lio03] proves that iterated centralizer extensions (see Definition 4.3) of a free group F are F-conjugacy separable. It is a result of of Kharlampovich and Miasnikov [KM98b] that all limit groups embed in to iterated centralizer extensions. Moreover by [GLS09, Theorem 5.3] almost locally free groups [GLS09,Definition 4.2] cannot have Magus pairs. This class includes the class of limit groups which are ∀∃-equivalent to free groups. The class of iterated centralizer extensions and the class of limit groups ∀∃-equivalent to free groups are contained in the class of towers, also known as NTQ groups. We generalize these previous results to the class of towers with the following strong F-conjugacy separability result: Theorem B. Let F be a non-abelian free group and let G be a tower over F (see Definition 4.3). There is a discriminating sequence of retractions {φ i : G ։ F}, such that for any finite subset S ⊂ G of pairwise non-conjugate elements, there is some positive integer N such that for all j ≥ N the elements of φ j (S) are pairwise non-conjugate in F. Similarly for any indivisible γ ∈ L with cyclic centralizer there is some positive integer M such that for all k ≥ M , r k (γ) is indivisible.
This theorem also settles [GLS09, Question 7.1], which asks if arbitrarily large collections of pairwise nonconjugate elements can have pairwise nonconjugate images via a homomorphism to a free group. The proof of Theorem B is in Section 4 and follows from results of Sela [Sel03] and Kharlampovich and Myasnikov [KM05], which form the first step in their (respective) systematic studies of the ∀∃-theory of free groups.
Finally, in Section 5 we analyze the failure of free conjugacy separability of our limit group with a Magnus pair and show that this is very different from C-double constructed in Section 3. We then show that the free conjugacy separability does not isolate the class of towers within the class of limit groups.
Throughout this paper, unless mentioned otherwise, F will denote a nonabelian free group, F n will denote a non-abelian free group of rank n, and F(X) will denote the free group on the basis X. given in Figure 1 and we also consider groups π 1 (Σ u ), π 1 (Σ v ) to be embedded into π 1 (U).
Definition 2.1. Let G be a group, and let ∼ ± be the equivalence relation g ∼ ± h if and only if g is conjugate to h or h −1 , and denote by [g] ± the ∼ ± equivalence class of g. A Magnus pair is a pair of Note that if h ∈ [g] ± then g = h , and that the relation "have the same normal closure" is coarser than ∼ ± , and if a group has a Magnus pair then it is strictly coarser than ∼ ± . To save notation we will say that g and h are a Magnus pair if their corresponding equivalence classes are.
Lemma 2.2. The elements u and v in π 1 (U) are a Magnus pair.
Proof. The graph of spaces given in Figure 1 gives rise to a cyclic graph of groups splitting D of π 1 (U). The underlying graph X has 4 vertices and 8 edges where the vertex groups are u , v , π 1 (Σ u ), and π 1 (Σ v ). Now note that π 1 (Σ u ) can be given the presentation and that the incident edge groups have images a , b , c , abc = d . Without loss of generality v ±1 is conjugate to a,b, and c in π 1 (U) and u ±1 is conjugate to d = abc in π 1 (U) which means that u ∈ v and, symmetrically considering On the other hand, the elements a , b , c , abc are pairwise non-conjugate in a, b, c and we now easily see that u and v are non-conjugate by considering the action on the Bass-Serre tree. u and v therefore form a Magnus pair.

Strict homomorphisms to limit groups
Definition 2.3. Let G be a finitely generated group and let D be a 2-acylindrical cyclic splitting of G. We say that a vertex group Q of D is quadratically hanging (QH) if it satisfies the following: • The images of the edge groups incident to Q correspond to the π 1 -images of ∂(Σ) in π 1 (Σ).
Definition 2.4. Let G be torsion-free group. A homomorphism ρ : G → H is strict if there some 2-acylindrical abelian splitting D of G such that the following hold: • ρ is injective on the subgroup A D generated by the incident edge groups of each each abelian vertex group A of D.
• ρ is injective on each edge group of D.
• ρ is injective on the "envelope"R of each non-QH, non-abelian vertex group R of D, whereR is constructed by first replacing each abelian vertex group A of D by A D and then takingR to be the subgroup generated by R and the centralizers of the edge groups incident to R.
• the ρ-images of QH subgroups are non-abelian.
This next Proposition is a restatement of Proposition 4.21 of [CG05] in our terminology. It is also given as Exercise 8 in [BF09,Wil09].
Proposition 2.5. If L is a limit group, G is some finitely generated group such that there is a strict homomorphism ρ : G → L, then G is also limit group.
2.2 π 1 (U) is a limit group but it is not freely conjugacy separable.
Consider the sequence of continuous maps given in Figure 2. The space on the top left obtained by taking three disjoint tori, identifying them along the longitudinal curves as shown, and then surgering on handles H 1 , H 2 is homeomorphic to the space U. A continuous map from U to the wedge of three circles is then constructed by filling in and collapsing the handles to arcs h 1 , h 2 , identifying the tori, and then mapping the resulting torus to a circle so that the image of the longitudinal curve u (or v, as they are now freely homotopic inside a torus) maps with degree 1 onto a circle in the wedge of three circles.
Lemma 2.6. The homomorphism π 1 (U) → F 3 given by the continuous map in Figure 2 is onto, the vertex groups π 1 (Σ v ), π 1 (Σ u ) have non-abelian image and the edge groups u , v are mapped injectively. Proof. The surjectivity of the map π 1 (U) → F 3 as well as the injectivity of the restrictions to u , v are obvious. Note moreover that the image of π 1 (Σ u ) contains (some conjugate of) u, h 1 uh −1 1 and is therefore non-abelian, the same is obviously true for the image of π 1 (Σ v ).
The final ingredient is a classical result of Magnus.
Theorem 2.7 ( [Mag31]). The free group F has no Magnus pairs. Proposition 2.8. π 1 (U) is a limit group. For every homomorphism ρ : π 1 (U) → F the images ρ(u), ρ(v) of the elements u, v given in Lemma 2.2 are conjugate in F even though the pair u, v are not conjugate in π 1 (U).
Proof. Lemma 2.6 and Proposition 2.5 imply that π 1 (U) is a Limit group. Lemma 2.2 and Theorem 2.7 imply that, for every homomorphism π 1 (U) → F to a free group F, the image of u must be conjugate to the image of v ±1 even though u ∼ ± v.

A different failure of free conjugacy separability
We now construct another limit group L that is not freely conjugacy separable, but for a completely different reason.
Theorem 3.2 ([Iva98, Main Theorem]). For arbitrary n ≥ 2 there exists a non-trivial indivisible word w n (x 1 , . . . , x n ) which is a C-test word in n letters for any free group F m of rank m ≥ 2.
Definition 3.3 (Doubles and retractions). Let F(x, y) denote the free group on two generators, let w = w(x, y) denote some word in {x, y} ±1 . The amalgamated free product Definition 3.4. Let u ∈ F(x, y) ≤ D(x, y; w), but with u ∼ ± w n for any n, be given by a specific word u(x, y). Its mirror image is the distinct element u(r, s) ∈ F(r, s) ≤ D(x, y; w). u(x, y) and u(r, s) form a mirror pair.
It is obvious that mirror pairs are not ∼ ± -equivalent. Let w be a C-test word and let L = D(x, y; w). It is well known that any such double is a limit group. We will call L a C-double.
Lemma 3.5. The C-double L cannot map onto a free group of rank more than 2.
Proof. w is not primitive in F(x, y) therefore by [She55] L = D(x, y; w) is not free. Theorem 3.2 specifically states that w is not a proper power. It now follows from [Lou13, Theorem 1.5] that D(w) cannot map onto F 3 .
The proof of the next theorem amounts to analyzing a Makanin-Razborov diagram. We refer the reader to [Hei16] for an explicit description of this diagram.
Theorem 3.6. For any map φ : L → F from a C-double to some free group, if u(x, y) ∈ F(x, y) lies in the commutator subgroup [F(x, y), F(x, y)], but is not conjugate to w n for any n, then the images φ (u(x, y)) and φ (u(r, s)) of mirror pairs are conjugate. In particular the limit group L is not freely conjugacy separable. Furthermore mirror pairs u(x, y), u(r, s) do not form Magnus pairs.
Proof. To answer this question we must analyze all maps for L to a free group. By Lemma 3.5, any such map factors through a surjection onto F 2 , or factors through Z.
Case 1: φ(w) = 1. In this case the factor F(x, y) does not map injectively, it follows that its image is abelian. It follows that φ factors through the free product In this case all elements of the commutator subgroups of F(x, y) and F(r, s) are mapped to the identity and therefore have conjugate images.
Case 2: φ(w) = 1. In this case the factors F(x, y), F(r, s) ≤ D(x, y; w) map injectively. By Theorem 3.2, since w is a C-test word and φ(w(x, y)) = φ(w(r, s), there is some S ∈ F 2 such that Sφ(x)S −1 = φ(r) and Sφ(y)S −1 = φ(s). Suppose now that w(x, y) mapped to a proper power, then by [Bau65,Main Theorem] w(x, y) ∈ F(x, y) is part of a basis, which is impossible. It follows that the centralizer of φ (w) is φ(w) so that S = φ(w) n . Therefore φ(r) = w n φ(x)w −n and φ(s) = w n φ(y)w −n and the result follows in this case as well.
We now show that a mirror pair u(x, y) and u(r, s) is not a Magnus pair. Consider the quotient D(x, y; w)/ u(x, y) . By using a presentation with generators and relations, the group canonically splits as the amalgamated free product where w n = w ∩ u and w is the image of w in w / w n . Now if u(x, y) = u(r, s) then we must have D(x, y; w)/ u(r, s) = D(x, y; w)/ u(x, y) . This implies F(r, s)/ (u(r, s)) = F(r, s)/ w n , which implies by Theorem 2.7 that u(r, s) ∼ ± w n , which is a contradiction.
It seems likely that failure of free conjugacy separability should typically follow from C-test word like behaviour, rather than from existence of Magnus pairs. 4 Towers are freely conjugacy separable.
• H splits as a fundamental group of a graph of groups with two vertex groups: • There is a retraction H ։ G such that the image of π 1 (Σ) in G is non abelian.
We say that Σ is the surface associated to the quadratic extension. And note that if ∂Σ = ∅ then H = G * π 1 (Σ).
Definition 4.2. Let G be a group. An abelian extension by the free abelian group A is an extension G ≤ G * u ( u ⊕A) = H where u ∈ G is such that either its centralizer Z G (u) = u , or u = 1. In the case where u = 1 the extension is G ≤ G * A and it is called a singular abelian extension.
Definition 4.3. Let F be a (possibly trivial) free group. A tower of height n over F is a group G obtained from a sequence of extensions is either a regular quadratic extension or an abelian extension. The G ′ i s are the levels of the tower G and the sequence of levels is a tower decomposition. A tower consisting entirely of abelian extensions is an iterated centralizer extension.
We call the graphs of groups decomposition of G i with one vertex group G i−1 and the other vertex group a surface group or a free abelian group as given in Definitions 4.1 and 4.2 the i th level decomposition.
Towers appear as NTQ groups in the work of Kharlampovich and Miasnikov, and as ω-residually free towers, as well as completions of strict resolutions in the the work of Sela. It is a well known fact that towers are limit groups [KM98a]. This also follows easily from Proposition 2.5 and the definitions.
Proposition 4.5. Let G be a tower of height n over F. Then G is discriminated by retractions G → G n−1 . G is also discriminated by retractions onto F.
Following Definition 1.15 of [Sel03] we have: Definition 4.6. Let G be a tower. A closure of G is another tower G ⋆ with an embedding θ : G ֒→ G ⋆ such that there is a commutative diagram i are restrictions of θ and the horizontal lines are tower decompositions. Moreover the following must hold: 1. If G i ≤ G i+1 is a regular quadratic extension with associated surface Σ such that ∂Σ is "attached" to u 1 , . . . , u n ≤ G i then G ⋆ i ≤ G ⋆ i+1 is a regular quadratic extension with associated surface Σ such that ∂Σ is "attached" to θ(u 1 ) , . . . , θ(u n ) ≤ G ⋆ i , in such a way that θ : G i ֒→ G ⋆ i extends to a monomorphism θ : G i+1 ֒→ G ⋆ i+1 which maps the vertex group π 1 (Σ) surjectively onto the vertex group π 1 (Σ) ≤ G ⋆ i+1 .
2. If G i ≤ G i+1 is an abelian extension then G ⋆ i ≤ G ⋆ i+1 is also an abelian extension. Specifically (allowing u i = 1) We will now state one of the main results of [KM05] and [Sel03] but first some explanations of terminology are in order. Towers are groups that arise as completed limit groups corresponding to a strict resolution and the definition of closure corresponds to the one given in [Sel03]. We also note that our requirement on the Euler characteristic of the surface pieces given in Definitions 2.3 and 4.1 ensures that our towers are coordinate groups of normalized NTQ systems as described in the discussion preceding [KM05, Lemma 76], we also point out that a correcting embedding as described right before [KM05, Theorem 12] is in fact a closure in the terminology we are using. We then there is an embedding θ : G ֒→ G ⋆ into some closure such that where X and F are interpreted as the corresponding subsets of G = F, X | R(F, X) In the terminology of [Sel03] we have G = F, X and G ⋆ = F, X, Z for some collection of elements Z. Let Y = (y 1 , . . . , y k ) be a tuple of elements in G ⋆ that witness the existential sentence above. A collection of words y i (F, X, Z) = G * y i is called a set of formal solution in G ⋆ . According to [KM05, Definition 24] the tuple Y ⊂ G ⋆ is an R-lift. Proof. Suppose towards a contradiction that this was not the case. Then either there exists a finite subset P ⊂ G \ {1} such that for every retraction r : G ։ F, 1 ∈ r(P ) or the elements of r(S) are not pairwise non-conjugate. If we write elements of P and S as fixed words {p i (F, X)} and {s j (F, X)} (resp.) then we can express this as a sentence. Indeed, consider first the formula: In English this says that either some element of P vanishes or two distinct elements of S are conjugated by some element t. We therefore have: It now follows by Lemma 4.7 that there is some closure θ : G ֒→ G ⋆ such that G ⋆ |= ∃tΦ P,S (F, θ(X), t).
Since 1 ∈ P and θ is a monomorphisms none of the p i (F, X) are trivial so In particular there are elements u, v ∈ G which are not conjugate in G but are conjugate in G ⋆ . We will derive a contradiction by showing that this is impossible. We proceed by induction on the height of the tower. If the tower has height 0 then G = F and the result obviously holds. Suppose now that the claim held for all towers of height m ≤ n. Let G have height n and let u, v be nonconjugate elements of G let G ≤ G ⋆ be any closure and suppose that there is some t ∈ G ⋆ \ G such that tut −1 = v.
Let D be the n th level decomposition of G ⋆ and let T be the corresponding Bass-Serre tree. Let T (G) be the minimal G-invariant subtree and let D G be the splitting induced by the action of G on T (G). By Definition 4.6 D G is exactly the n th level decomposition of G and two edges of T (G) are in the same G-orbit if and only if they are in the same G ⋆ -orbit. We now consider separate cases: Case 1: Without loss of generality u is hyperbolic in the n th level decomposition of G. If v is elliptic in the n th level decomposition of G then it is elliptic in the n th -level decomposition of G ⋆ and therefore cannot be conjugate to u which acts hyperbolically on T .
It follows that both u, v must be hyperbolic elements with respect to the n th level decomposition of G. Let l u , l v denote the axes of u, v (resp.) in T (G) ⊂ T . Since tut −1 = v, we must have t · l u = l v . Let e be some edge in l u then by the previous paragraph t·e ⊂ l v must be in the same G-orbit as e, which means that there is some g ∈ G such that gt·e = e, but again by Definition 4.6 the inclusion G ≤ G ⋆ induces a surjection of the edge groups of the n th level decomposition of G to the edge groups of the n th level decomposition of G ⋆ , it follows that gt ∈ G which implies that t ∈ G contradicting the fact that u, v were not conjugate in G.
Case 2: The elements u, v are elliptic in the n th level decomposition of G. Suppose first that u, v were conjugate into G n−1 , then the result follows from the fact that there is a retraction G ։ G n−1 and by the induction hypothesis. Similarly by examining the induced splitting of G ≤ G ⋆ , we see that u cannot be conjugate into G n−1 and v into the other vertex group of the n th -level decomposition. We finally distinguish two sub-cases.
Case 2.1: G n−1 ≤ G is an abelian extension by the free abelian group A and u, v are conjugate in G into some free abelian group w ⊕ A. Any homomorphic image of w ⊕ A in F must lie in a cyclic group, since u = v in G ⋆ and G ⋆ is discriminated by retractions onto F, there must be some retraction r : G ⋆ → F such that r(u) = r(v) which means that u, v are sent to distinct powers of a generator of the cyclic subgroup r( w ⊕ A). It follows that their images are not conjugate in F so u, v cannot be conjugate in G ⋆ .
Case 2.2: G n−1 ≤ G is a quadratic extension and u and v are conjugate in G into the vertex group π 1 (Σ). Arguing as in Case 1 we find that if there is some t ∈ G ⋆ such that tut −1 = v then there is some g ∈ G such that gt fixes a vertex of T (G) ⊂ T whose stabilizer is conjugate to π 1 (Σ). Again by the surjectively criterion in item 1. of Definition 4.6, gt ∈ G contradicting the fact that u, v were not conjugate in G. All the possibilities have been exhausted so the result follows.
proof of Theorem B. Let S 1 ⊂ S 2 ⊂ S 3 ⊂ . . . be an exhaustion of representatives of distinct conjugacy classes of G by finite sets. For each S j let {ψ j i } be the discriminating sequence given by Proposition 4.8. We take {φ i } to be the diagonal sequence {ψ i i }. This sequence is necessarily discriminating and the result follows. Corollary 4.9. Let L be a limit group and suppose that for some finite set S ⊂ L there is a homomorphism f : L → F such that: • The elements of f (S) are pairwise non-conjugate.
• There is a factorization such that each f i is a strict homomorphisms between limit groups (see Definition 2.4).
Then there is a discriminating sequence ψ i : L → F such that for all i the elements ψ i (S) are pairwise non-conjugate. 5 Refinements 5.1 π 1 (U) is almost freely conjugacy separable.
The limit group L constructed in Section 3 had an abundance of pairs of nonconjugate elements whose images had to have conjugate images in every free quotient. The situation is completely different for our Magnus pair group.
Proposition 5.1. u , v ≤ π 1 (U) are the only maximal cyclic subgroups of π 1 (U) whose conjugacy classes cannot be separated via a homomorphism to a free group π 1 (U) → F. Proof. We begin by embedding π 1 (U) into a hyperbolic tower. Let ρ : π 1 (U) ։ F 3 be the strict homomorphism given in Figure 2. Consider the group This presentation naturally gives a splitting D of L given in Figure 3. We have a retraction ρ * : L ։ F 3 given by It therefore follows that L is a hyperbolic tower over F 3 .
Claim: if α, β ∈ π 1 (U) ≤ L are non-conjugate in π 1 (U) and α, β are not both conjugate to u or v in π 1 (U) then they are not conjugate in L. If both α and β are elliptic, then this follows easily from the fact that the vertex groups are malnormal in L. Also α cannot be elliptic while β is hyperbolic. Suppose now that α, β are hyperbolic. Let T be the Bass-Serre tree corresponding to D and let T ′ = T (π 1 (U)) be the minimal π 1 (U) invariant subtree. Suppose that there is some s ∈ L such that sαs −1 = β, then as in the proof of Proposition 4.8 and Proposition we find that for some g ∈ π 1 (U) either gs permutes two edges in T ′ that are in distinct π 1 (U)-orbits or it fixes some edge in T ′ . The former case is impossible and it is easy to see that the latter case implies that gs ∈ π 1 (U). Therefore we have a contradiction to the assumption that α, β are not conjugate in π 1 (U). The claim is now proved.
It therefore follows that if α, β ∈ π 1 (U) ≤ L are as above, then by Theorem B there exists some retraction r : L ։ F 3 such that r(α), r(β) are non-conjugate.
This construction gives an alternative proof to the fact that π 1 (U) is a limit group. The group L constructed is a triangular quasiquadratic group and the retraction ρ * makes it non-degenerate, and therefore an NTQ group. L and therefore π 1 (U) ≤ L are therefore limit groups by [KM98a].

C-doubles do not contain Magnus pairs.
Theorem B enables us to examine a C-double L more closely.
Proposition 5.2. The C-double L constructed in Section 3 does not contain a Magnus pair.
Proof. We need to show that if two elements u, v of L have the same normal closure in L then they must be conjugate. Suppose that u, v are both elliptic with respect to the splitting (as a double) of L but not conjugate. By Theorem 3.2 if they are conjugate to a mirror pair (u g , v h ) for some g, h ∈ L then they do not form a Magnus pair, i.e. they have separate normal closures. Otherwise there are homomorphisms L → F in which u, v have non-conjugate images, therefore by Theorem 2.7 the normal closures of their images are distinct; so u = v as well.
Suppose now that u or v is hyperbolic in L. Recall the generating set x, y, r, s for L given in Definition 3.3. Let F = F(x, y) and consider the embedding into a centralizer extension, represented as an HNN extension The stable letter t makes mirror pairs conjugate in this bigger group. A hyperbolic element of L can be written as a product of syllables u = a 1 (x, y)a 2 (r, s) · · · a l (r, s) with a 1 or a l possibly trivial. The image of u in F * t w is u = a 1 (x, y) t −1 a 2 (x, y)t · · · t −1 a l (x, y)t .
Consider the set of words of the form with w 1 or w N possibly trivial. This set is clearly closed under multiplication, inverses and passing to F t w -normal form. It follows that we can identify the image of L with this set of words, which we call t −1 * t-syllabic words. Each factor w i (x, t) or t −1 w j (x, y)t is called a t −1 * t-syllable.
It is an easy consequence of Britton's Lemma that if u is a hyperbolic, i.e. with cyclically reduced syllable length more than 1, t −1 * t-syllabic word and g −1 ug is again t −1 * t-syllabic for some g in F * t w then g must itself be t −1 * tsyllabic. Indeed this can be seen by cyclically permuting the F * t w -syllables of a cyclically reduced word u. We refer the reader to [LS01, §IV.2] for further details about normal forms and conjugation in HNN extensions.
Suppose now that u, v are non conjugate in L, but have the same normal closure in L. Since at least one of them is hyperbolic in L, it is clear from the embedding that its image must also be hyperbolic with respect to the HNN splitting F * t w . Now, since u L = v L , in the bigger group F * t w we have: By Theorem B or [Lio03] centralizer extensions are freely conjugacy separable, therefore they cannot contain Magnus pairs. It follows that u, v must be conjugate in the bigger F * t w . Let g −1 ug = F * t w v. Now both u and v must be hyperbolic so it follows that g must also be a t −1 * t-syllabic word; thus g is in the image of L of F * t w . Furthermore since the map L ֒→ F * t w is an embedding g −1 ug = F * t w v ⇒ g −1 ug = L v, contradicting the fact that u, v are non conjugate in L.

A non-tower limit group that is freely conjugacy separable
In this section we construct a limit group that is freely conjugacy separable but which does not admit a tower structure. Let H ≤ [F, F] be some f.g. malnormal subgroup of F, e.g. where Σ has one boundary component and has genus g = genus(h), in particular there is a retraction onto F. Consider now the subgroup L = H * h π 1 (Σ).
map ρ from T JSJ to the Bass-Serre tree T corresponding to H * h π 1 (Σ) in which H stabilizes a vertex v. It follows that H acts on φ −1 ({v}) = T H ⊂ T JSJ . Since H is rigid relative to h and h acts elliptically on T JSJ , T H cannot be infinite, since that would imply that H admits an essential cyclic splitting relative to h. T H must in fact be a point. Otherwise T H is a finite tree tree and there must be a "boundary" vertex u ∈ T H such that H ≥ L u . Since φ(u) = v, L-equivariance implies that L u fixes v so that L u ≤ H, which is a contradiction. It follows that in this case H is actually a vertex group of the JSJ decomposition and π 1 (Σ) must be a CMQ vertex group. The second case is that h is hyperbolic in some other Z-splitting D of L. Since H is rigid relative to h, H must be hyperbolic with respect to D. Now by 2. of Theorem 5.5 the splitting L = H * h π 1 (Σ) can be obtained from the JSJ splitting of L by cutting along a simple closed curve on some CMQ vertex group, and this curve is conjugate to h. But this means that H admits a cyclic splitting as a graph of groups with a QH vertex group π 1 (Σ ′ ) such that the π 1 -image of some connected component of ∂Σ ′ is conjugate to h , in particular H must have a cyclic splitting relative to h, which contradicts the fact that H is rigid relative to h.
Proposition 5.7. The limit group L = H * h π 1 (Σ) does not admit a tower structure.
Proof. Suppose towards a contradiction that L was a tower, consider the last level: L n−1 < L n = L.
Since L has no non-cyclic abelian subgroups L n−1 < L must be a hyperbolic extension. This means that L admits a cyclic splitting D with a vertex group L n−1 and a QH vertex group Q. Since L = H * h π 1 (Σ) is a JSJ decomposition and π 1 (Σ) is a CMQ vertex group. By 1. and 4. of Theorem 5.5, the QH vertex group Q must be represented as π 1 (Σ 1 ), where Σ 1 is a connected subsurface Σ 1 ⊂ Σ. It follows from 4. of Theorem 5.5 that the other vertex group must be L n−1 = H * h π 1 (Σ ′ ) where Σ ′ = Σ \ Σ 1 . Since L n−1 < L is a quadratic extension there is a retraction L ։ L n−1 . Note however that because Σ ′ has at least two boundary components H * h π 1 (Σ ′ ) = H * F m where m = −χ(Σ ′ ). Now since we have a retraction L ։ L n−1 there is are x i , y i ∈ L n−1 such that But this would imply that h ∈ [L n−1 , L n−1 ] which is clearly seen to be false by abelianizing H * F m and remembering that h ∈ [H, H].