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

Magnus pairs in, and free conjugacy separability of, limit groups Geom Dedicata (2018) 196:187–201 https://doi.org/10.1007/s10711-017-0314-1 ORIGINAL PAPER Magnus pairs in, and free conjugacy separability of, limit groups 1 1,2 Larsen Louder · Nicholas W. M. Touikan Received: 2 December 2016 / Accepted: 1 November 2017 / Published online: 6 January 2018 © The Author(s) 2018. This article is an open access publication Abstract There are limit groups having non-conjugate elements whose images are conjugate in every free quotient. Towers over free groups are freely conjugacy separable. Keywords Geometric group theory · Limit groups · Residual properties · Logic · Conjugacy Mathematics Subject Classification 20F65 · 20E05 · 57M07 1 Introduction This paper is concerned with the problem of finding free quotients of finitely generated groups in which non-conjugate elements have non-conjugate images. If a finitely generated group G is not residually free, then there will be non-trivial elements that will always be sent to (conjugates of) the identity. If G is residually free then it canonically embeds into a direct product of limit groups P = L × ··· × L and every homomorphism to a free group factors 1 n through one of the projections P  L . It is therefore natural to restrict our attention to the class of limit groups. A group is freely conjugacy separable if for any pair u,v ∈ G of non-conjugate elements there is a homomorphism G → F to a free group F such that the images of u and v in F are non- conjugate. Throughout this paper F will denote a non-abelian free group, F will denote a non-abelian free group of rank n,and F(X ) will denote the free group on the basis X. B Larsen Louder [email protected] Nicholas W. M. Touikan [email protected] Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK Stevens Institute of Technology, Hoboken, NJ, USA 123 188 Geom Dedicata (2018) 196:187–201 We give two different types of examples of limit groups which are not freely conjugacy separable for different reasons. In Sect. 2 we produce a limit group L with elements u,v such that the cyclic groups u, v are non-conjugate, but whose normal closures u  and v coincide. We call such a pair of elements a Magnus pair (see Definition 2.1.) Such elements must have conjugate images in any free quotient by a theorem of Magnus [17]. See [3,4]for earlier generalizations to closed surface groups. In Sect. 3 we construct a limit group which is a double of a free group over a cyclic group generated by a C-test word (see Definition 3.1). These limit groups, C-doubles,are lowrank and we are able to construct their Makanin-Razborov diagrams and observe the failure of free conjugacy separability directly. These groups were also found by Heil [8], who published a preprint while this paper was in preparation. Definition 1.1 A sequence of homomorphisms {φ : G → H } is discriminating if for every finite subset P ⊂ G \{1} there is some N such that for all j ≥ N , 1 ∈ / φ (P). Definition 1.2 A finitely generated group L is a limit group if there is a discriminating sequence of homomorphisms {φ : 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 [6]. Lioutikova [14], proved that iterated centralizer extensions (see Definition 4.3) of a free group F are freely conjugacy separable. It is a result of of Kharlampovich and Miasnikov [11]that all limit groups embed in to iterated centralizer extensions. Moreover by [7, Theorem 5.3], almost locally free groups [7, 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 results to the class of towers with the following strong free conjugacy separability result: Theorem B Let F be a non-abelian free group and let G be a tower over F (see Defini- tion 4.3). There is a discriminating sequence of retractions {φ : G  F}, such that for any finite subset S ⊂ G of pairwise non-conjugate elements, there is some N such that for all j ≥ N the elements of φ (S) are pairwise non-conjugate in F. Similarly for any indivisible γ ∈ L with cyclic centralizer there is some M such that for all k ≥ M, r (γ ) is indivisible. Theorem B also settles [7, Question 7.1], which asks if arbitrarily large collections of pair- wise nonconjugate elements can have pairwise nonconjugate images via a homomorphism to a free group. The proof of Theorem B is in Sect. 4 and follows from [12,19]. In Sect. 5, we analyze the failure in free conjugacy separability of our limit group with a Magnus pair and show that it is very different from the C-double constructed in Sect. 3.This motivates two natural questions about Magnus pairs in limit groups. Finally, we show that free conjugacy separability does not isolate towers within the class of limit groups. 2 A limit group with a Magnus pair Consider the fundamental group of the graph of spaces U given in Fig. 1. We pick elements u,v ∈ π (U) corresponding to the similarly labelled loops given in Fig. 1 and we also consider groups π ( ), π ( ) to be embedded into π (U). 1 u 1 v 1 123 Geom Dedicata (2018) 196:187–201 189 Fig. 1 The graph of spaces U. The attaching maps are of degree 1 and the black arrows show the orientations u v Definition 2.1 Let G be a group, and let ∼ be the equivalence relation g ∼ h if and only ± ± −1 if g is conjugate to h or h , and denote by [g] the ∼ equivalence class of g.A Magnus ± ± pair is a pair of ∼ classes [g] = [h] such that g  =  h . ± ± Note that if h ∼ g then g  =  h , but that the converse does not necessarily hold. The failure of the reverse implication is exactly witnessed by Magnus pairs. To save notation we will say that g and h are a Magnus pair if the classes [g] and [h] form a Magnus pair. Lemma 2.2 The elements u and v in π (U) are a Magnus pair. Proof The graph of spaces given in Fig. 1 gives rise to a cyclic graph of groups splitting D of π (U). The underlying graph X has 4 vertices and 8 edges where the vertex groups are u, v,π ( ),and π ( ). Now note that π ( ) can be given the presentation 1 u 1 v 1 u π ( ) =a, b, c, d | abcd = 1=a, b, c 1 u and that the incident edge groups have images a, b, c, abc=d. Without loss of ±1 ±1 generality v is conjugate to a,b,and c in π (U) and u is conjugate to d = abc in π (U) 1 1 which means that u ∈v  and, symmetrically considering  , v ∈u . On the other hand, the elements a, b, c, abc are pairwise non-conjugate in a, b, c. By inspecting the action on the Bass–Serre tree, u and v are clearly non-conjugate, and are therefre form a Magnus pair. 2.1 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: • Q = π () where  is a compact surface such that χ() ≤−1, with equality only if is orientable or ∂() =∅. • The images of the edge groups incident to Q correspond to the π -images of ∂() in π (). 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: 123 190 Geom Dedicata (2018) 196:187–201 • ρ is injective on the subgroup A 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, where R is constructed by first replacing each abelian vertex group A of D by A and then taking R 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 [5, Proposition 4.21] in our terminology. It is also given as Exercise 8 in [2,21]. Proposition 2.5 If L is a limit group, and G is a finitely generated group with a strict homomorphism ρ : G → L, then G is also limit group. 2.2 π (U) is a limit group but it is not freely conjugacy separable Consider the sequence of continuous maps given in Fig. 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 , H is homeomorphic to the space U. A continuous map from 1 2 U to the wedge of three circles is then constructed by filling in and collapsing the handles to arcs h , h , identifying the tori, and then mapping the resulting torus to a circle so that 1 2 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. u v Σ Σ v u u v u h Fig. 2 A continuous map from U to the wedge of three circles. The space on the top left is homeomorphic to U. This can be seen by cutting along the curves labelled u,v 123 Geom Dedicata (2018) 196:187–201 191 Lemma 2.6 The homomorphism π (U) → F given by the continuous map in Fig. 2 is onto, 1 3 the vertex groups π ( ), π ( ) have non-abelian image and the edge groups u, v are 1 v 1 u mapped injectively. Proof The surjectivity of the map π (U) → F as well as the injectivity of the restrictions 1 3 to u, v are obvious. Note moreover that the image of π ( ) contains (some conjugate 1 u −1 of) u, h uh  and is therefore non-abelian, the same is obviously true for the image of π ( ). 1 v The final ingredient is a classical result of Magnus. Theorem 2.7 [17] The free group F has no Magnus pairs. Proposition 2.8 π (U) is a limit group. For every homomorphism ρ : π (U) → F the images 1 1 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 (U). Proof Lemma 2.6 and Proposition 2.5 imply that π (U) is a Limit group. Lemma 2.2 and Theorem 2.7 imply that, for every homomorphism π (U) → F to a free group F,the image ±1 of u must be conjugate to the image of v even though u  v. 3 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. Definition 3.1 (C-test words [9]) A non-trivial word w(x ,..., x ) is a C-test word in n 1 n letters for F if for any two n-tuples (A ,..., A ), (B ,..., B ) of elements of F the m 1 n 1 n m equality w(A ,..., A ) = w(B ,..., B ) = 1 implies the existence of an element S ∈ F 1 n 1 n m −1 such that B = SA S for all i = 1, 2,..., n. i i Theorem 3.2 [9, Main Theorem] For arbitrary n ≥ 2 there exists a non-trivial indivisible word w (x ,..., x ) which is a C-test word in n letters for any free group F of rank m ≥ 2. n 1 n m Definition 3.3 (Doubles and retractions)Let F(x , y) denote the free group on two genera- ±1 tors, let w = w(x , y) denote some word in {x , y} . The amalgamated free product D(x , y; w) =F(x , y), F(r, s) | w(x , y) = w(r, s) is the double of F(x , y) along w. The homomorphism ρ : D(x , y; w)  F(x , y) given by r → x , s → y is the standard retraction. Definition 3.4 Let u ∈ F(x , y) ≤ D(x , y; w), but with u  w for any n,begiven 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. 123 192 Geom Dedicata (2018) 196:187–201 Proof w is not primitive in F(x , y) therefore by [20] L = D(x , y; w) is not free. Theorem 3.2 specifically states that w is not a proper power. It now follows from [15, Theorem 1.5] that D(w) cannot map onto F . The proof of the next theorem amounts to analyzing a Makanin-Razborov diagram. We refer the reader to [8] 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 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 , 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 ab ab π : D(x , y; w) → F(x , y) ∗ F(r, s) . ab 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. Indeed, since their image is nonabelian, their image is onto a non-abelian free group generated by two elements, therefore a free group of rank two; thus the restriction of the map is injective by the Hopf property. By Theorem 3.2,since w is a C-test word and φ(w(x , y)) = φ(w(r, s)), −1 −1 there is some S ∈ F such that Sφ(x )S = φ(r ) and Sφ(y)S = φ(s). Suppose now that w(x , y) mapped to a proper power, then by [1, 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 n n −n n −n S = φ(w) . Therefore φ(r ) = w φ(x )w and φ(s) = w φ(y)w ; so mirror pairs are mapped to conjugates and, in particular, mirror pairs in the commutator subgroup of F(x , y) and F(r, s) are mapped to conjugates of the same elements. 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 (F(x , y)/u(x , y)) ∗ F(r, s)/w n n where w =w∩u  and w is the image of w in w/w .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 n n F(r, s)/(u(r, s)) = F(r, s)/w , which implies by Theorem 2.7 that u(r, s) ∼ w , 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 Definition 4.1 Let G be a group. A regular quadratic extension of G is an extension G ≤ H such that 123 = Geom Dedicata (2018) 196:187–201 193 • H splits as a fundamental group of a graph of groups with two vertex groups: H = G and H = π () where H is a QH vertex group (See Definition 2.3.) v 1 v 2 2 • There is a retraction H  G such that the image of π () in G is non abelian. We say that  is the surface associated to the quadratic extension. And note that if ∂ =∅ then H = G ∗ π (). Definition 4.2 Let G be a group. An abelian extension by the free abelian group A is an extension G ≤ G ∗ (u⊕ A) = H where u ∈ G is such that either its centralizer Z (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 F = G ≤ G ≤ ··· ≤ G = G 0 1 n where G ≤ G is either a regular quadratic extension or an abelian extension. The G s i i +1 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. Definition 4.4 Let F = G ≤ ··· ≤ G = G be a tower decomposition of G.Wecall 0 n the graphs of groups decomposition of G with one vertex group G and the other vertex i i −1 group a surface group or a free abelian group as given in Definitions 4.1 and 4.2 the ith 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 [10]. 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 . G is also discriminated by retractions onto F. n−1 Following Definition 1.15 of [19]wehave: Definition 4.6 Let G be atower.A closure of G is another tower G with an embedding θ : G → G such that there is a commutative diagram ... G ≤ G ≤ ≤ G = 0 1 n ... ≤ G ≤ ≤ G = 0 G 1 n where the injections G → G are restrictions of θ and the horizontal lines are tower decompositions. Moreover the following must hold: 1. If G ≤ G is a regular quadratic extension with associated surface  such that ∂ is i i +1 “attached” to u ,..., u ≤ G then G ≤ G is a regular quadratic extension with 1 n i i i +1 associated surface  such that ∂ is “attached” to θ(u ),..., θ(u )≤ G ,insucha 1 n way that θ : G → G extends to a monomorphism θ : G → G which maps the i i +1 i i +1 vertex group π () surjectively onto the vertex group π () ≤ G . 1 1 i +1 123 194 Geom Dedicata (2018) 196:187–201 2. If G ≤ G is an abelian extension then G ≤ G is also an abelian extension. i i +1 i i +1 Specifically (allowing u = 1) if G = G ∗ (u ⊕ A ),then G = G ∗ i i +1 i u  i i θ(u ) i i +1 i i (θ(u )⊕ A ). Moreover we require the embedding θ : G → G to map u ⊕ A i i +1 i i i i +1 to a finite index subgroup of θ(u )⊕ A . We will now state one of the main results of [12,19] but first some explanations of termi- nology 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 [19]. We also note that our requirement on the Euler characteristic of the surface pieces given in Defini- tions 2.3 and 4.1 ensures that our towers are coordinate groups of normalized NTQ systems as described in the discussion preceding [12, Lemma 76] we also point out that a correcting embedding as described right before [12, Theorem 12] is in fact a closure in the terminology we are using. We now give an obvious corollary (in fact a weakening) of [19, Theorem 1.22], or [12, Theorem 12]; they are the same result. Let X, Y denote fixed tuples of variables. Lemma 4.7 [∀∃-lifting Lemma] Let F be a fixed non-abelian free group and let G =F, X | R(F, X ) be a standard finite presentation of a tower over F.Let W (X, Y, F) = 1 and V (X, Y, F) = 1 i i be (possibly empty) finite systems of equations and inequations (resp.) If the following holds: F | ∀X ∃Y R(F, X ) = 1 → W (X, Y, F) = 1 ∧ V (X, Y, F) = 1 i i i =1 then there is an embedding θ : G → G into some closure such that G | ∃Y W (θ (X ), Y, F) = 1 ∧ V (θ (X ), Y, F) = 1 i i i =1 where X and F are interpreted as the corresponding subsets of G =F, X | R(F, X ) In the terminology of [19]wehave G =F, X  and G =F, X, Z  for some collection of elements Z.Let Y = (y ,..., y ) be a tuple of elements in G that witness the existential 1 k sentence above. A collection of words y (F, X, Z ) = y is called a set of formal solution i G i in G .Accordingto[12, Definition 24] the tuple Y ⊂ G is an R-lift. Proposition 4.8 Let G be a tower over a non abelian free group Fand let S ⊂ Gbe a finite family of pairwise non-conjugate elements of G. There exists a discriminating family of retractions ψ : G  F such that for each ψ the elements of ψ (S) are pairwise non- i i i conjugate. 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 (F, X )} and {s (F, X )} (resp.) then we can express this as a sentence. Indeed, consider i j first the formula: ⎛ ⎡ ⎤ ⎡ ⎤ ⎞ −1 ⎝ ⎣ ⎦ ⎣ ⎦ ⎠ (F, X, t ) = p (F, X ) = 1 ∨ t s (F, X )t = s (F, X ) P,S i i j p ∈P (s ,s )∈(S) i i j 123 Geom Dedicata (2018) 196:187–201 195 where (S) ={(x , y) ∈ S × S | x = y)}. 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: F | ∀X (R(F, X )) = 1) →∃t  (F, X, t ) . (1) P,S It now follows by Lemma 4.7 that there is some closure θ : G → G such that G | ∃t  (F,θ(X ), t ). P,S Since 1 ∈ / P and θ is a monomorphisms none of the p (F, X ) are trivial so ⎡ ⎤ −1 ⎣ ⎦ G | ∃t t s (F, X )t = s (F, X ) . i j (s ,s )∈(S) i j 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 non-conjugate elements of G let G ≤ G be any −1 closure and suppose that there is some t ∈ G \ G such that tut = v. Let D be the nth 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 be the splitting induced by the action of G on T (G). By Definition 4.6 D is exactly the nth 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 nth level decomposition of G.If v is elliptic in the nth level decomposition of G then it is elliptic in the nth-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 nth level decom- −1 position of G.Let l , l denote the axes of u,v (resp.) in T (G) ⊂ T.Since tut = v,we u v must have t · l = l .Let e be some edge in l then by the previous paragraph t · e ⊂ l u v u 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 nth level decomposition of G to the edge groups of the nth 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 nth level decomposition of G. Suppose first that u,v were conjugate into G , then the result follows from the fact that there is a retraction n−1 G  G and by the induction hypothesis. Similarly by examining the induced splitting n−1 of G ≤ G , we see that u cannot be conjugate into G and v into the other vertex group n−1 of the nth-level decomposition. We finally distinguish two sub-cases. Case 2.1: G ≤ G is an abelian extension by the free abelian group A and u,v are n−1 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 ≤ G is a quadratic extension and u and v are conjugate in G into the vertex n−1 −1 group π (). Arguing as in Case 1 we find that if there is some t ∈ G such that tut = v 123 196 Geom Dedicata (2018) 196:187–201 then there is some g ∈ G such that gt fixes a vertex of T (G) ⊂ T whose stabilizer is conjugate to π (). 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 ⊂ S ⊂ S ⊂ ··· be an exhaustion of representatives of distinct 1 2 3 conjugacy classes of G by finite sets. For each S let {ψ } be the discriminating sequence given by Proposition 4.8.Wetake {φ } to be the diagonal sequence {ψ }. This sequence is necessarily discriminating and the result follows. It is worthwhile to point out that test sequences given in the proof of [19, Theorem 1.18], or the generic sequence givenin[12, Definition 44], because of their properties, must satisfy the conclusions of Theorem B. As an immediate consequence of the Sela’s completion construction ([19, Definition 1.12]) or canonical embeddings into NTQ groups ([13, Sect. 7]) Theorem B implies the following: Corollary 4.9 Let L be a limit group and suppose that for some finite set S ⊂ Lthere is a homomorphism f : L → F such that: • The elements of f (S) are pairwise non-conjugate. • There is a factorization f = f ◦ f ◦ ··· ◦ f m m−1 1 such that each f is a strict homomorphisms between limit groups (see Definition 2.4). Then there is a discriminating sequence ψ : L → F such that for all i the elements ψ (S) i i are pairwise non-conjugate. 5 Refinements 5.1 π (U) is almost freely conjugacy separable The limit group L constructed in Sect. 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 (U) are the only maximal cyclic subgroups of π (U) whose 1 1 conjugacy classes cannot be separated via a homomorphism to a free group π (U) → F. Proof We begin by embedding π (U) into a hyperbolic tower. Let ρ : π (U)  F be the 1 1 3 strict homomorphism given in Fig. 2. Consider the group −1 T =π (U), F , s | u = ρ(u), svs = ρ(v). 1 3 This presentation naturally gives a splitting D of T given in Fig. 3. We have a retraction ρ∗: T  F given by ⎨ g → ρ(g); g ∈ π (U) ρ∗: f → f ; f ∈ F s → 1 It therefore follows that T is a hyperbolic tower over F . 123 Geom Dedicata (2018) 196:187–201 197 Fig. 3 The splitting D of T u F v Claim: if α, β ∈ π (U) ≤ T are non-conjugate in π (U) and α, β are not both conjugate 1 1 to u or v in π (U) then they are not conjugate in T . If both α and β are elliptic, then this follows easily from the fact that the vertex groups are malnormal in T . 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 (U)) be the minimal π (U) invariant subtree. 1 1 −1 Suppose that there is some s ∈ T such that sαs = β, then as in the proof of Proposition 4.8 and Proposition we find that for some g ∈ π (U) either gs permutes two edges in T that are in distinct π (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 (U). Therefore we have a contradiction to the assumption that α, β are not conjugate in π (U). The claim is now proved. It therefore follows that if α, β ∈ π (U) ≤ T are as above, then by Theorem B there exists some retraction r : T  F such that r (α), r (β) are non-conjugate. This construction gives an alternative proof to the fact that π (U) is a limit group. The group T constructed is a triangular quasiquadratic group and the retraction ρ makes it non- degenerate, and therefore an NTQ group. T and therefore π (U) ≤ T are therefore limit groups by [10]. 5.2 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 Sect. 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 g h (u ,v ) 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, 123 198 Geom Dedicata (2018) 196:187–201 represented as an HNN extension L →F, t |tw(x , y) = w(x , y)t= F∗ x → x , y → y −1 −1 r → t xt, s → t yt 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 (x , y)a (r, s)... a (r, s) 1 2 l with a or a possibly trivial. The image of u in F∗ is 1 l −1 −1 u = a (x , y) t a (x , y)t ... t a (x , y)t . 1 2 l Consider the set of words of the form −1 −1 w (x , y) t w (x , y)t ...w (x , y) t w (x , y)t , 1 2 N −1 N with w or w possibly trivial. This set is clearly closed under multiplication, inverses and 1 N passing to F -normal form. It follows that we can identify the image of L with this set of −1 −1 words, which we call t ∗ t-syllabic words. Each factor w (x , t ) or t w (x , y)t is called i j −1 a t ∗ t-syllable. It is an easy consequence of Britton’s Lemma that if u is a hyperbolic, i.e. with cyclically −1 −1 −1 reduced syllable length more than 1, t ∗t-syllabic word and g ug is again t ∗t-syllabic t −1 for some g in F∗ then g must itself be t ∗t-syllabic. Indeed this can be seen by cyclically permuting the F∗ -syllables of a cyclically reduced word u. We refer the reader to [16, Sect. 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∗ .Now,since u  =v  , w L L in the bigger group F∗ we have: u  t =u   t = v   t =v  t F∗ L F∗ L F∗ F∗ w w w w By Theorem B or [14] centralizer extensions are freely conjugacy separable, therefore they cannot contain Magnus pairs. It follows that u,v must be conjugate in the bigger F∗ . −1 Let g ug = t v. Now both u and v must be hyperbolic so it follows that g must also F∗ −1 t be a t ∗ t-syllabic word; thus g is in the image of L of F∗ . Furthermore since the map L → F∗ is an embedding −1 −1 g ug = v ⇒ g ug = v, F∗ contradicting the fact that u,v are non conjugate in L. Perhaps the methods of the previous proof can be used and extended to address the following questions. Question 1 Does a limit group contain only finitely many Magnus pairs up to automorphism? In fact, an even simpler question: “Does a limit group only contain finitely many Magnus pairs?” is open. In particular the Magnus pair constructed in π (U) (see Sect. 2), viewed as an unordered pair (recall Definition 2.1), is Aut (π (U))-invariant. 123 Geom Dedicata (2018) 196:187–201 199 Question 2 Do Magnus pairs in limit groups arise from embedded groups of the form π (U), given in Sect. 2? More precisely, does every Magnus pair in a limit group always arise from an embedding of the fundamental group of a graph of orientable, genus-zero surfaces, amalgamated along their boundaries? 5.3 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. H = −1 −1 −2 −1 2 aba b , b a b a≤ F(a, b). And pick h ∈ H \[H, H ] such that H is rigid relative to h, i.e. H has no non-trivial cyclic or free splittings relative to h. Because h ∈[F, F] there is a quadratic extension F < F ∗ π () h 1 where  has one boundary component and has genus g = genus(h), in particular there is a retraction onto F. Consider now the subgroup L = H ∗ π (). h 1 Proposition 5.3 L as above is freely conjugacy separable. Proof Because H ≤ F was chosen to be malnormal, an easy Bass–Serre theory argument (e.g. apply [16, Theorem IV.2.8]) tells us that α, β ∈ L are conjugate if and only if they are conjugate in F ∗ π (). On the other hand by Theorem B, F ∗ π (), and hence L,are h 1 h 1 freely conjugacy separable. Definition 5.4 A splitting X is elliptic in a splitting Y if every edge group in X is conjugate into a vertex group of Y. Otherwise we say X is hyperbolic in Y. Theorem 5.5 [18, Theorem 7.1] Let G be an f.p. group with a single end. There exists a reduced, unfolded Z-splitting of G called a JSJ decomposition of G with the following properties: 1. Every canonical maximal QH (recall Definition 2.3) subgroup (CMQ) of G is conjugate to a vertex group in the JSJ decomposition. Every QH subgroup of G can be conjugated into one of the CMQ subgroups of G. Every non-CMQ vertex groups in the JSJ decomposition is elliptic in every Z-splitting of G. 2. An elementary Z-splitting G = A ∗ Bor G = A∗ which is hyperbolic in another C C elementary Z-splitting is obtained from the JSJ decomposition of G by cutting a 2-orbifold corresponding to a CMQ subgroup of G along a weakly essential simple closed curve (s.c.c.). 3. Let  be an elementary Z-splitting G = A ∗ Bor G = A∗ which is elliptic with C C respect to any other elementary Z splitting of G. There exists a G-equivariant simpli- cial map between a subdivision of T , the Bass–Serre tree corresponding to the JSJ JSJ decomposition, and T , the Bass–Serre tree corresponding to . 4. Let  be a general Z-splitting of G. There exists a Z-splitting  obtained from the JSJ decomposition by splitting the CMQ subgroups along weakly essential s.c.c. on their corresponding 2-orbifolds, so that there exists a G-equivariant simplicial map between a subdivision of the Bass–Serre tree T and T . 5. If JSJ is another JSJ decomposition of G, then there exists a G-equivariant simplicial map h from a subdivision of T to T , and a G-equivariant simplicial map h 1 JSJ JSJ 2 from a subdivision of T to T ,sothat h ◦ h and h ◦ h are G-homotopic to the JSJ JSJ 1 2 2 1 corresponding identity maps. 123 200 Geom Dedicata (2018) 196:187–201 We note that item 5. of the above theorem describes the canonicity of a JSJ decomposition, and that requiring the existence of an equivariant simplicial map from a subdivision of a tree S to a tree T is the same as requiring an equivariant continuous map from S to T that sends vertices to vertices. Lemma 5.6 The splitting L = H ∗ π () is a cyclic JSJ splitting. h 1 Proof This is an elementary Z splitting of L, let’s see how it can be obtained from the JSJ decomposition given in Theorem 5.5.Let T denote the Bass–Serre tree of the JSJ JSJ decomposition and let T denote the Bass–Serre tree of the splitting L = H ∗ π ().The h 1 factor π () is a QH subgroup, so by item 1. of Theorem 5.5, the JSJ decomposition must contain a CMQ vertex group π ( ) ≤ L where  is some surface with boundary. By 4. 1 M M of Theorem 5.5 π () ≤ L can be represented as a subsurface  ⊂  .Since L is not a 1 M closed surface group the JSJ decomposition has at least 2 vertex groups. By 4. of Theorem 5.5, we can cut the CMQ vertex group π ( ) ≤ L along simple closed 1 M curves to get a new splitting with Bass–Serre tree T such that T  T is obtained by JSJ 1 1 perhaps collapsing edges dual to the simple closed curves and there is an L-equivariant continuous (but perhaps not simplicial) map T  T . The subgroup π () is a vertex group of T , and in particular the element h ∈ π () acts elliptically on T . The subgroup 1 1 H must also act elliptically on T , for otherwise H has a cyclic or free splitting relative to h, contradicting rigidity. Since the vertex groups of T fix vertices of T ,there is also a continuous map T  T . It follows that T has at most 2 (conjugacy classes of) maximal 1 1 vertex groups so the map T  T is the identity (i.e. no CMQ subgroups cut along JSJ simple closed curves). It follows that we have L-equivariant maps T  T and T  T . JSJ JSJ Therefore L = H ∗ π () is a cyclic JSJ decomposition. h 1 Proposition 5.7 The limit group L = H ∗ π () does not admit a tower structure. h 1 Proof Suppose towards a contradiction that L was a tower, consider the last level: L < L = L . n−1 n Since L has no non-cyclic abelian subgroups L < L must be a hyperbolic extension. This n−1 means that L admits a cyclic splitting D with a vertex group L and a QH vertex group n−1 Q.Since L = H ∗ π () is a JSJ decomposition and π () is a CMQ vertex group. By h 1 1 1. and 4. of Theorem 5.5, the QH vertex group Q must be represented as π ( ),where 1 1 1 is a connected subsurface  ⊂ . It follows from 4. of Theorem 5.5 that the other vertex group must be L = H ∗ π ( ) where  =  \  . n−1 h 1 1 Since L < L is a quadratic extension there is a retraction L  L .Notehowever n−1 n−1 that because  has at least two boundary components H ∗ π ( ) = H ∗ F h 1 m where m =−χ( ). Now since we have a retraction L  L there is are x , y ∈ L n−1 i i n−1 such that h = [x , y ] i i i =1 But this would imply that h ∈[L , L ] which is clearly seen to be false by abelianizing n−1 n−1 H ∗ F and remembering that h ∈[ / H, H ]. 123 Geom Dedicata (2018) 196:187–201 201 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna- tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. References 1. Baumslag, G.: Residual nilpotence and relations in free groups. J. Algebra 2, 271–282 (1965) 2. Bestvina, M., Feighn, M.: Notes on Sela’s work: limit groups and Makanin-Razborov diagrams. 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Louder, L.: Scott complexity and adjoining roots to finitely generated groups. Groups Geom Dyn 7(2), 451–474 (2013) 16. Lyndon, R.C., Schupp, P.E.: Combinatorial group theory. In: Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1977 edition 17. Magnus, W.: Untersuchungen über einige unendliche diskontinuierliche Gruppen. Math. Ann. 105(1), 52–74 (1931) 18. Rips, E., Sela, Z.: Cyclic splittings of finitely presented groups and the canonical JSJ decomposition. Ann. Math. (2) 146(1), 53–109 (1997) 19. Sela, Z.: Diophantine geometry over groups. II. Completions, closures and formal solutions. Isr. J. Math. 134, 173–254 (2003) 20. Shenitzer, A.: Decomposition of a group with a single defining relation into a free product. Proc. Am. Math. Soc. 6(2), 273–279 (1955) 21. Wilton, H.: Solutions to Bestvina & Feighn’s exercises on limit groups. In: Geometric and Cohomological Methods in Group Theory, Volume 358 of London Mathematical Society, Lecture Note Series, pp. 30–62. Cambridge University Press, Cambridge (2009) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Geometriae Dedicata Springer Journals

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

Louder, Larsen; Touikan, Nicholas
Geometriae Dedicata , Volume 196 (1) – Jan 6, 2018
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Mathematics; Convex and Discrete Geometry; Differential Geometry; Algebraic Geometry; Hyperbolic Geometry; Projective Geometry; Topology
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10.1007/s10711-017-0314-1
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Geom Dedicata (2018) 196:187–201 https://doi.org/10.1007/s10711-017-0314-1 ORIGINAL PAPER Magnus pairs in, and free conjugacy separability of, limit groups 1 1,2 Larsen Louder · Nicholas W. M. Touikan Received: 2 December 2016 / Accepted: 1 November 2017 / Published online: 6 January 2018 © The Author(s) 2018. This article is an open access publication Abstract There are limit groups having non-conjugate elements whose images are conjugate in every free quotient. Towers over free groups are freely conjugacy separable. Keywords Geometric group theory · Limit groups · Residual properties · Logic · Conjugacy Mathematics Subject Classification 20F65 · 20E05 · 57M07 1 Introduction This paper is concerned with the problem of finding free quotients of finitely generated groups in which non-conjugate elements have non-conjugate images. If a finitely generated group G is not residually free, then there will be non-trivial elements that will always be sent to (conjugates of) the identity. If G is residually free then it canonically embeds into a direct product of limit groups P = L × ··· × L and every homomorphism to a free group factors 1 n through one of the projections P  L . It is therefore natural to restrict our attention to the class of limit groups. A group is freely conjugacy separable if for any pair u,v ∈ G of non-conjugate elements there is a homomorphism G → F to a free group F such that the images of u and v in F are non- conjugate. Throughout this paper F will denote a non-abelian free group, F will denote a non-abelian free group of rank n,and F(X ) will denote the free group on the basis X. B Larsen Louder [email protected] Nicholas W. M. Touikan [email protected] Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK Stevens Institute of Technology, Hoboken, NJ, USA 123 188 Geom Dedicata (2018) 196:187–201 We give two different types of examples of limit groups which are not freely conjugacy separable for different reasons. In Sect. 2 we produce a limit group L with elements u,v such that the cyclic groups u, v are non-conjugate, but whose normal closures u  and v coincide. We call such a pair of elements a Magnus pair (see Definition 2.1.) Such elements must have conjugate images in any free quotient by a theorem of Magnus [17]. See [3,4]for earlier generalizations to closed surface groups. In Sect. 3 we construct a limit group which is a double of a free group over a cyclic group generated by a C-test word (see Definition 3.1). These limit groups, C-doubles,are lowrank and we are able to construct their Makanin-Razborov diagrams and observe the failure of free conjugacy separability directly. These groups were also found by Heil [8], who published a preprint while this paper was in preparation. Definition 1.1 A sequence of homomorphisms {φ : G → H } is discriminating if for every finite subset P ⊂ G \{1} there is some N such that for all j ≥ N , 1 ∈ / φ (P). Definition 1.2 A finitely generated group L is a limit group if there is a discriminating sequence of homomorphisms {φ : 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 [6]. Lioutikova [14], proved that iterated centralizer extensions (see Definition 4.3) of a free group F are freely conjugacy separable. It is a result of of Kharlampovich and Miasnikov [11]that all limit groups embed in to iterated centralizer extensions. Moreover by [7, Theorem 5.3], almost locally free groups [7, 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 results to the class of towers with the following strong free conjugacy separability result: Theorem B Let F be a non-abelian free group and let G be a tower over F (see Defini- tion 4.3). There is a discriminating sequence of retractions {φ : G  F}, such that for any finite subset S ⊂ G of pairwise non-conjugate elements, there is some N such that for all j ≥ N the elements of φ (S) are pairwise non-conjugate in F. Similarly for any indivisible γ ∈ L with cyclic centralizer there is some M such that for all k ≥ M, r (γ ) is indivisible. Theorem B also settles [7, Question 7.1], which asks if arbitrarily large collections of pair- wise nonconjugate elements can have pairwise nonconjugate images via a homomorphism to a free group. The proof of Theorem B is in Sect. 4 and follows from [12,19]. In Sect. 5, we analyze the failure in free conjugacy separability of our limit group with a Magnus pair and show that it is very different from the C-double constructed in Sect. 3.This motivates two natural questions about Magnus pairs in limit groups. Finally, we show that free conjugacy separability does not isolate towers within the class of limit groups. 2 A limit group with a Magnus pair Consider the fundamental group of the graph of spaces U given in Fig. 1. We pick elements u,v ∈ π (U) corresponding to the similarly labelled loops given in Fig. 1 and we also consider groups π ( ), π ( ) to be embedded into π (U). 1 u 1 v 1 123 Geom Dedicata (2018) 196:187–201 189 Fig. 1 The graph of spaces U. The attaching maps are of degree 1 and the black arrows show the orientations u v Definition 2.1 Let G be a group, and let ∼ be the equivalence relation g ∼ h if and only ± ± −1 if g is conjugate to h or h , and denote by [g] the ∼ equivalence class of g.A Magnus ± ± pair is a pair of ∼ classes [g] = [h] such that g  =  h . ± ± Note that if h ∼ g then g  =  h , but that the converse does not necessarily hold. The failure of the reverse implication is exactly witnessed by Magnus pairs. To save notation we will say that g and h are a Magnus pair if the classes [g] and [h] form a Magnus pair. Lemma 2.2 The elements u and v in π (U) are a Magnus pair. Proof The graph of spaces given in Fig. 1 gives rise to a cyclic graph of groups splitting D of π (U). The underlying graph X has 4 vertices and 8 edges where the vertex groups are u, v,π ( ),and π ( ). Now note that π ( ) can be given the presentation 1 u 1 v 1 u π ( ) =a, b, c, d | abcd = 1=a, b, c 1 u and that the incident edge groups have images a, b, c, abc=d. Without loss of ±1 ±1 generality v is conjugate to a,b,and c in π (U) and u is conjugate to d = abc in π (U) 1 1 which means that u ∈v  and, symmetrically considering  , v ∈u . On the other hand, the elements a, b, c, abc are pairwise non-conjugate in a, b, c. By inspecting the action on the Bass–Serre tree, u and v are clearly non-conjugate, and are therefre form a Magnus pair. 2.1 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: • Q = π () where  is a compact surface such that χ() ≤−1, with equality only if is orientable or ∂() =∅. • The images of the edge groups incident to Q correspond to the π -images of ∂() in π (). 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: 123 190 Geom Dedicata (2018) 196:187–201 • ρ is injective on the subgroup A 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, where R is constructed by first replacing each abelian vertex group A of D by A and then taking R 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 [5, Proposition 4.21] in our terminology. It is also given as Exercise 8 in [2,21]. Proposition 2.5 If L is a limit group, and G is a finitely generated group with a strict homomorphism ρ : G → L, then G is also limit group. 2.2 π (U) is a limit group but it is not freely conjugacy separable Consider the sequence of continuous maps given in Fig. 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 , H is homeomorphic to the space U. A continuous map from 1 2 U to the wedge of three circles is then constructed by filling in and collapsing the handles to arcs h , h , identifying the tori, and then mapping the resulting torus to a circle so that 1 2 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. u v Σ Σ v u u v u h Fig. 2 A continuous map from U to the wedge of three circles. The space on the top left is homeomorphic to U. This can be seen by cutting along the curves labelled u,v 123 Geom Dedicata (2018) 196:187–201 191 Lemma 2.6 The homomorphism π (U) → F given by the continuous map in Fig. 2 is onto, 1 3 the vertex groups π ( ), π ( ) have non-abelian image and the edge groups u, v are 1 v 1 u mapped injectively. Proof The surjectivity of the map π (U) → F as well as the injectivity of the restrictions 1 3 to u, v are obvious. Note moreover that the image of π ( ) contains (some conjugate 1 u −1 of) u, h uh  and is therefore non-abelian, the same is obviously true for the image of π ( ). 1 v The final ingredient is a classical result of Magnus. Theorem 2.7 [17] The free group F has no Magnus pairs. Proposition 2.8 π (U) is a limit group. For every homomorphism ρ : π (U) → F the images 1 1 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 (U). Proof Lemma 2.6 and Proposition 2.5 imply that π (U) is a Limit group. Lemma 2.2 and Theorem 2.7 imply that, for every homomorphism π (U) → F to a free group F,the image ±1 of u must be conjugate to the image of v even though u  v. 3 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. Definition 3.1 (C-test words [9]) A non-trivial word w(x ,..., x ) is a C-test word in n 1 n letters for F if for any two n-tuples (A ,..., A ), (B ,..., B ) of elements of F the m 1 n 1 n m equality w(A ,..., A ) = w(B ,..., B ) = 1 implies the existence of an element S ∈ F 1 n 1 n m −1 such that B = SA S for all i = 1, 2,..., n. i i Theorem 3.2 [9, Main Theorem] For arbitrary n ≥ 2 there exists a non-trivial indivisible word w (x ,..., x ) which is a C-test word in n letters for any free group F of rank m ≥ 2. n 1 n m Definition 3.3 (Doubles and retractions)Let F(x , y) denote the free group on two genera- ±1 tors, let w = w(x , y) denote some word in {x , y} . The amalgamated free product D(x , y; w) =F(x , y), F(r, s) | w(x , y) = w(r, s) is the double of F(x , y) along w. The homomorphism ρ : D(x , y; w)  F(x , y) given by r → x , s → y is the standard retraction. Definition 3.4 Let u ∈ F(x , y) ≤ D(x , y; w), but with u  w for any n,begiven 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. 123 192 Geom Dedicata (2018) 196:187–201 Proof w is not primitive in F(x , y) therefore by [20] L = D(x , y; w) is not free. Theorem 3.2 specifically states that w is not a proper power. It now follows from [15, Theorem 1.5] that D(w) cannot map onto F . The proof of the next theorem amounts to analyzing a Makanin-Razborov diagram. We refer the reader to [8] 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 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 , 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 ab ab π : D(x , y; w) → F(x , y) ∗ F(r, s) . ab 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. Indeed, since their image is nonabelian, their image is onto a non-abelian free group generated by two elements, therefore a free group of rank two; thus the restriction of the map is injective by the Hopf property. By Theorem 3.2,since w is a C-test word and φ(w(x , y)) = φ(w(r, s)), −1 −1 there is some S ∈ F such that Sφ(x )S = φ(r ) and Sφ(y)S = φ(s). Suppose now that w(x , y) mapped to a proper power, then by [1, 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 n n −n n −n S = φ(w) . Therefore φ(r ) = w φ(x )w and φ(s) = w φ(y)w ; so mirror pairs are mapped to conjugates and, in particular, mirror pairs in the commutator subgroup of F(x , y) and F(r, s) are mapped to conjugates of the same elements. 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 (F(x , y)/u(x , y)) ∗ F(r, s)/w n n where w =w∩u  and w is the image of w in w/w .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 n n F(r, s)/(u(r, s)) = F(r, s)/w , which implies by Theorem 2.7 that u(r, s) ∼ w , 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 Definition 4.1 Let G be a group. A regular quadratic extension of G is an extension G ≤ H such that 123 = Geom Dedicata (2018) 196:187–201 193 • H splits as a fundamental group of a graph of groups with two vertex groups: H = G and H = π () where H is a QH vertex group (See Definition 2.3.) v 1 v 2 2 • There is a retraction H  G such that the image of π () in G is non abelian. We say that  is the surface associated to the quadratic extension. And note that if ∂ =∅ then H = G ∗ π (). Definition 4.2 Let G be a group. An abelian extension by the free abelian group A is an extension G ≤ G ∗ (u⊕ A) = H where u ∈ G is such that either its centralizer Z (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 F = G ≤ G ≤ ··· ≤ G = G 0 1 n where G ≤ G is either a regular quadratic extension or an abelian extension. The G s i i +1 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. Definition 4.4 Let F = G ≤ ··· ≤ G = G be a tower decomposition of G.Wecall 0 n the graphs of groups decomposition of G with one vertex group G and the other vertex i i −1 group a surface group or a free abelian group as given in Definitions 4.1 and 4.2 the ith 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 [10]. 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 . G is also discriminated by retractions onto F. n−1 Following Definition 1.15 of [19]wehave: Definition 4.6 Let G be atower.A closure of G is another tower G with an embedding θ : G → G such that there is a commutative diagram ... G ≤ G ≤ ≤ G = 0 1 n ... ≤ G ≤ ≤ G = 0 G 1 n where the injections G → G are restrictions of θ and the horizontal lines are tower decompositions. Moreover the following must hold: 1. If G ≤ G is a regular quadratic extension with associated surface  such that ∂ is i i +1 “attached” to u ,..., u ≤ G then G ≤ G is a regular quadratic extension with 1 n i i i +1 associated surface  such that ∂ is “attached” to θ(u ),..., θ(u )≤ G ,insucha 1 n way that θ : G → G extends to a monomorphism θ : G → G which maps the i i +1 i i +1 vertex group π () surjectively onto the vertex group π () ≤ G . 1 1 i +1 123 194 Geom Dedicata (2018) 196:187–201 2. If G ≤ G is an abelian extension then G ≤ G is also an abelian extension. i i +1 i i +1 Specifically (allowing u = 1) if G = G ∗ (u ⊕ A ),then G = G ∗ i i +1 i u  i i θ(u ) i i +1 i i (θ(u )⊕ A ). Moreover we require the embedding θ : G → G to map u ⊕ A i i +1 i i i i +1 to a finite index subgroup of θ(u )⊕ A . We will now state one of the main results of [12,19] but first some explanations of termi- nology 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 [19]. We also note that our requirement on the Euler characteristic of the surface pieces given in Defini- tions 2.3 and 4.1 ensures that our towers are coordinate groups of normalized NTQ systems as described in the discussion preceding [12, Lemma 76] we also point out that a correcting embedding as described right before [12, Theorem 12] is in fact a closure in the terminology we are using. We now give an obvious corollary (in fact a weakening) of [19, Theorem 1.22], or [12, Theorem 12]; they are the same result. Let X, Y denote fixed tuples of variables. Lemma 4.7 [∀∃-lifting Lemma] Let F be a fixed non-abelian free group and let G =F, X | R(F, X ) be a standard finite presentation of a tower over F.Let W (X, Y, F) = 1 and V (X, Y, F) = 1 i i be (possibly empty) finite systems of equations and inequations (resp.) If the following holds: F | ∀X ∃Y R(F, X ) = 1 → W (X, Y, F) = 1 ∧ V (X, Y, F) = 1 i i i =1 then there is an embedding θ : G → G into some closure such that G | ∃Y W (θ (X ), Y, F) = 1 ∧ V (θ (X ), Y, F) = 1 i i i =1 where X and F are interpreted as the corresponding subsets of G =F, X | R(F, X ) In the terminology of [19]wehave G =F, X  and G =F, X, Z  for some collection of elements Z.Let Y = (y ,..., y ) be a tuple of elements in G that witness the existential 1 k sentence above. A collection of words y (F, X, Z ) = y is called a set of formal solution i G i in G .Accordingto[12, Definition 24] the tuple Y ⊂ G is an R-lift. Proposition 4.8 Let G be a tower over a non abelian free group Fand let S ⊂ Gbe a finite family of pairwise non-conjugate elements of G. There exists a discriminating family of retractions ψ : G  F such that for each ψ the elements of ψ (S) are pairwise non- i i i conjugate. 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 (F, X )} and {s (F, X )} (resp.) then we can express this as a sentence. Indeed, consider i j first the formula: ⎛ ⎡ ⎤ ⎡ ⎤ ⎞ −1 ⎝ ⎣ ⎦ ⎣ ⎦ ⎠ (F, X, t ) = p (F, X ) = 1 ∨ t s (F, X )t = s (F, X ) P,S i i j p ∈P (s ,s )∈(S) i i j 123 Geom Dedicata (2018) 196:187–201 195 where (S) ={(x , y) ∈ S × S | x = y)}. 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: F | ∀X (R(F, X )) = 1) →∃t  (F, X, t ) . (1) P,S It now follows by Lemma 4.7 that there is some closure θ : G → G such that G | ∃t  (F,θ(X ), t ). P,S Since 1 ∈ / P and θ is a monomorphisms none of the p (F, X ) are trivial so ⎡ ⎤ −1 ⎣ ⎦ G | ∃t t s (F, X )t = s (F, X ) . i j (s ,s )∈(S) i j 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 non-conjugate elements of G let G ≤ G be any −1 closure and suppose that there is some t ∈ G \ G such that tut = v. Let D be the nth 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 be the splitting induced by the action of G on T (G). By Definition 4.6 D is exactly the nth 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 nth level decomposition of G.If v is elliptic in the nth level decomposition of G then it is elliptic in the nth-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 nth level decom- −1 position of G.Let l , l denote the axes of u,v (resp.) in T (G) ⊂ T.Since tut = v,we u v must have t · l = l .Let e be some edge in l then by the previous paragraph t · e ⊂ l u v u 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 nth level decomposition of G to the edge groups of the nth 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 nth level decomposition of G. Suppose first that u,v were conjugate into G , then the result follows from the fact that there is a retraction n−1 G  G and by the induction hypothesis. Similarly by examining the induced splitting n−1 of G ≤ G , we see that u cannot be conjugate into G and v into the other vertex group n−1 of the nth-level decomposition. We finally distinguish two sub-cases. Case 2.1: G ≤ G is an abelian extension by the free abelian group A and u,v are n−1 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 ≤ G is a quadratic extension and u and v are conjugate in G into the vertex n−1 −1 group π (). Arguing as in Case 1 we find that if there is some t ∈ G such that tut = v 123 196 Geom Dedicata (2018) 196:187–201 then there is some g ∈ G such that gt fixes a vertex of T (G) ⊂ T whose stabilizer is conjugate to π (). 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 ⊂ S ⊂ S ⊂ ··· be an exhaustion of representatives of distinct 1 2 3 conjugacy classes of G by finite sets. For each S let {ψ } be the discriminating sequence given by Proposition 4.8.Wetake {φ } to be the diagonal sequence {ψ }. This sequence is necessarily discriminating and the result follows. It is worthwhile to point out that test sequences given in the proof of [19, Theorem 1.18], or the generic sequence givenin[12, Definition 44], because of their properties, must satisfy the conclusions of Theorem B. As an immediate consequence of the Sela’s completion construction ([19, Definition 1.12]) or canonical embeddings into NTQ groups ([13, Sect. 7]) Theorem B implies the following: Corollary 4.9 Let L be a limit group and suppose that for some finite set S ⊂ Lthere is a homomorphism f : L → F such that: • The elements of f (S) are pairwise non-conjugate. • There is a factorization f = f ◦ f ◦ ··· ◦ f m m−1 1 such that each f is a strict homomorphisms between limit groups (see Definition 2.4). Then there is a discriminating sequence ψ : L → F such that for all i the elements ψ (S) i i are pairwise non-conjugate. 5 Refinements 5.1 π (U) is almost freely conjugacy separable The limit group L constructed in Sect. 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 (U) are the only maximal cyclic subgroups of π (U) whose 1 1 conjugacy classes cannot be separated via a homomorphism to a free group π (U) → F. Proof We begin by embedding π (U) into a hyperbolic tower. Let ρ : π (U)  F be the 1 1 3 strict homomorphism given in Fig. 2. Consider the group −1 T =π (U), F , s | u = ρ(u), svs = ρ(v). 1 3 This presentation naturally gives a splitting D of T given in Fig. 3. We have a retraction ρ∗: T  F given by ⎨ g → ρ(g); g ∈ π (U) ρ∗: f → f ; f ∈ F s → 1 It therefore follows that T is a hyperbolic tower over F . 123 Geom Dedicata (2018) 196:187–201 197 Fig. 3 The splitting D of T u F v Claim: if α, β ∈ π (U) ≤ T are non-conjugate in π (U) and α, β are not both conjugate 1 1 to u or v in π (U) then they are not conjugate in T . If both α and β are elliptic, then this follows easily from the fact that the vertex groups are malnormal in T . 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 (U)) be the minimal π (U) invariant subtree. 1 1 −1 Suppose that there is some s ∈ T such that sαs = β, then as in the proof of Proposition 4.8 and Proposition we find that for some g ∈ π (U) either gs permutes two edges in T that are in distinct π (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 (U). Therefore we have a contradiction to the assumption that α, β are not conjugate in π (U). The claim is now proved. It therefore follows that if α, β ∈ π (U) ≤ T are as above, then by Theorem B there exists some retraction r : T  F such that r (α), r (β) are non-conjugate. This construction gives an alternative proof to the fact that π (U) is a limit group. The group T constructed is a triangular quasiquadratic group and the retraction ρ makes it non- degenerate, and therefore an NTQ group. T and therefore π (U) ≤ T are therefore limit groups by [10]. 5.2 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 Sect. 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 g h (u ,v ) 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, 123 198 Geom Dedicata (2018) 196:187–201 represented as an HNN extension L →F, t |tw(x , y) = w(x , y)t= F∗ x → x , y → y −1 −1 r → t xt, s → t yt 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 (x , y)a (r, s)... a (r, s) 1 2 l with a or a possibly trivial. The image of u in F∗ is 1 l −1 −1 u = a (x , y) t a (x , y)t ... t a (x , y)t . 1 2 l Consider the set of words of the form −1 −1 w (x , y) t w (x , y)t ...w (x , y) t w (x , y)t , 1 2 N −1 N with w or w possibly trivial. This set is clearly closed under multiplication, inverses and 1 N passing to F -normal form. It follows that we can identify the image of L with this set of −1 −1 words, which we call t ∗ t-syllabic words. Each factor w (x , t ) or t w (x , y)t is called i j −1 a t ∗ t-syllable. It is an easy consequence of Britton’s Lemma that if u is a hyperbolic, i.e. with cyclically −1 −1 −1 reduced syllable length more than 1, t ∗t-syllabic word and g ug is again t ∗t-syllabic t −1 for some g in F∗ then g must itself be t ∗t-syllabic. Indeed this can be seen by cyclically permuting the F∗ -syllables of a cyclically reduced word u. We refer the reader to [16, Sect. 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∗ .Now,since u  =v  , w L L in the bigger group F∗ we have: u  t =u   t = v   t =v  t F∗ L F∗ L F∗ F∗ w w w w By Theorem B or [14] centralizer extensions are freely conjugacy separable, therefore they cannot contain Magnus pairs. It follows that u,v must be conjugate in the bigger F∗ . −1 Let g ug = t v. Now both u and v must be hyperbolic so it follows that g must also F∗ −1 t be a t ∗ t-syllabic word; thus g is in the image of L of F∗ . Furthermore since the map L → F∗ is an embedding −1 −1 g ug = v ⇒ g ug = v, F∗ contradicting the fact that u,v are non conjugate in L. Perhaps the methods of the previous proof can be used and extended to address the following questions. Question 1 Does a limit group contain only finitely many Magnus pairs up to automorphism? In fact, an even simpler question: “Does a limit group only contain finitely many Magnus pairs?” is open. In particular the Magnus pair constructed in π (U) (see Sect. 2), viewed as an unordered pair (recall Definition 2.1), is Aut (π (U))-invariant. 123 Geom Dedicata (2018) 196:187–201 199 Question 2 Do Magnus pairs in limit groups arise from embedded groups of the form π (U), given in Sect. 2? More precisely, does every Magnus pair in a limit group always arise from an embedding of the fundamental group of a graph of orientable, genus-zero surfaces, amalgamated along their boundaries? 5.3 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. H = −1 −1 −2 −1 2 aba b , b a b a≤ F(a, b). And pick h ∈ H \[H, H ] such that H is rigid relative to h, i.e. H has no non-trivial cyclic or free splittings relative to h. Because h ∈[F, F] there is a quadratic extension F < F ∗ π () h 1 where  has one boundary component and has genus g = genus(h), in particular there is a retraction onto F. Consider now the subgroup L = H ∗ π (). h 1 Proposition 5.3 L as above is freely conjugacy separable. Proof Because H ≤ F was chosen to be malnormal, an easy Bass–Serre theory argument (e.g. apply [16, Theorem IV.2.8]) tells us that α, β ∈ L are conjugate if and only if they are conjugate in F ∗ π (). On the other hand by Theorem B, F ∗ π (), and hence L,are h 1 h 1 freely conjugacy separable. Definition 5.4 A splitting X is elliptic in a splitting Y if every edge group in X is conjugate into a vertex group of Y. Otherwise we say X is hyperbolic in Y. Theorem 5.5 [18, Theorem 7.1] Let G be an f.p. group with a single end. There exists a reduced, unfolded Z-splitting of G called a JSJ decomposition of G with the following properties: 1. Every canonical maximal QH (recall Definition 2.3) subgroup (CMQ) of G is conjugate to a vertex group in the JSJ decomposition. Every QH subgroup of G can be conjugated into one of the CMQ subgroups of G. Every non-CMQ vertex groups in the JSJ decomposition is elliptic in every Z-splitting of G. 2. An elementary Z-splitting G = A ∗ Bor G = A∗ which is hyperbolic in another C C elementary Z-splitting is obtained from the JSJ decomposition of G by cutting a 2-orbifold corresponding to a CMQ subgroup of G along a weakly essential simple closed curve (s.c.c.). 3. Let  be an elementary Z-splitting G = A ∗ Bor G = A∗ which is elliptic with C C respect to any other elementary Z splitting of G. There exists a G-equivariant simpli- cial map between a subdivision of T , the Bass–Serre tree corresponding to the JSJ JSJ decomposition, and T , the Bass–Serre tree corresponding to . 4. Let  be a general Z-splitting of G. There exists a Z-splitting  obtained from the JSJ decomposition by splitting the CMQ subgroups along weakly essential s.c.c. on their corresponding 2-orbifolds, so that there exists a G-equivariant simplicial map between a subdivision of the Bass–Serre tree T and T . 5. If JSJ is another JSJ decomposition of G, then there exists a G-equivariant simplicial map h from a subdivision of T to T , and a G-equivariant simplicial map h 1 JSJ JSJ 2 from a subdivision of T to T ,sothat h ◦ h and h ◦ h are G-homotopic to the JSJ JSJ 1 2 2 1 corresponding identity maps. 123 200 Geom Dedicata (2018) 196:187–201 We note that item 5. of the above theorem describes the canonicity of a JSJ decomposition, and that requiring the existence of an equivariant simplicial map from a subdivision of a tree S to a tree T is the same as requiring an equivariant continuous map from S to T that sends vertices to vertices. Lemma 5.6 The splitting L = H ∗ π () is a cyclic JSJ splitting. h 1 Proof This is an elementary Z splitting of L, let’s see how it can be obtained from the JSJ decomposition given in Theorem 5.5.Let T denote the Bass–Serre tree of the JSJ JSJ decomposition and let T denote the Bass–Serre tree of the splitting L = H ∗ π ().The h 1 factor π () is a QH subgroup, so by item 1. of Theorem 5.5, the JSJ decomposition must contain a CMQ vertex group π ( ) ≤ L where  is some surface with boundary. By 4. 1 M M of Theorem 5.5 π () ≤ L can be represented as a subsurface  ⊂  .Since L is not a 1 M closed surface group the JSJ decomposition has at least 2 vertex groups. By 4. of Theorem 5.5, we can cut the CMQ vertex group π ( ) ≤ L along simple closed 1 M curves to get a new splitting with Bass–Serre tree T such that T  T is obtained by JSJ 1 1 perhaps collapsing edges dual to the simple closed curves and there is an L-equivariant continuous (but perhaps not simplicial) map T  T . The subgroup π () is a vertex group of T , and in particular the element h ∈ π () acts elliptically on T . The subgroup 1 1 H must also act elliptically on T , for otherwise H has a cyclic or free splitting relative to h, contradicting rigidity. Since the vertex groups of T fix vertices of T ,there is also a continuous map T  T . It follows that T has at most 2 (conjugacy classes of) maximal 1 1 vertex groups so the map T  T is the identity (i.e. no CMQ subgroups cut along JSJ simple closed curves). It follows that we have L-equivariant maps T  T and T  T . JSJ JSJ Therefore L = H ∗ π () is a cyclic JSJ decomposition. h 1 Proposition 5.7 The limit group L = H ∗ π () does not admit a tower structure. h 1 Proof Suppose towards a contradiction that L was a tower, consider the last level: L < L = L . n−1 n Since L has no non-cyclic abelian subgroups L < L must be a hyperbolic extension. This n−1 means that L admits a cyclic splitting D with a vertex group L and a QH vertex group n−1 Q.Since L = H ∗ π () is a JSJ decomposition and π () is a CMQ vertex group. By h 1 1 1. and 4. of Theorem 5.5, the QH vertex group Q must be represented as π ( ),where 1 1 1 is a connected subsurface  ⊂ . It follows from 4. of Theorem 5.5 that the other vertex group must be L = H ∗ π ( ) where  =  \  . n−1 h 1 1 Since L < L is a quadratic extension there is a retraction L  L .Notehowever n−1 n−1 that because  has at least two boundary components H ∗ π ( ) = H ∗ F h 1 m where m =−χ( ). 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