Limit groups over coherent right-angled Artin groups are cyclic subgroup separable
aa r X i v : . [ m a t h . G R ] J a n Limit groups over coherent right-angled Artingroups are cyclic subgroup separable
Jonathan Fruchter
Abstract
We prove that cyclic subgroup separability is preserved under exponential completion forgroups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolicgroups; we do so by exploiting the structure of these completions as iterated free products withcommuting subgroups. From this we deduce that the cyclic subgroups of limit groups overcoherent RAAGs are separable, answering a question of Casals-Ruiz, Duncan and Kazachov.We also discuss relations between free products with commuting subgroups and the wordproblem, and recover the fact that limit groups over coherent RAAGs and toral relativelyhyperbolic groups have a solvable word problem.
Contents C . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Exponential groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Limit groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 A -completions and cyclic subgroup separability 64 Free products with commuting subgroups and the word problem 11 In a recent paper, [CDK20], Casals-Ruiz, Duncan and Kazachov defined a new class of groups C , which was carefully designed to serve as a general framework for studying limit groups over(coherent) right-angled Artin groups (RAAGs). They succeed in showing that the limit groupsover any coherent right-angled Artin group G embed in the Z [ t ] -completion of G , and that thiscompletion can be built by repeatedly extending centralisers. Casals-Ruiz, Duncan and Kazachov[CDK20] then pose the question: are cyclic subgroups of limit groups over coherent RAAGs closedin the profinite topology? Our main purpose here is to give a positive answer to this question.Constructing completions of groups using extensions of centralisers is not a new idea; in the1960s, Baumslag gave a construction of the Q -completion of a group with unique roots, whichuses extensions of centralisers. Many would argue that the solution to Tarski’s question about thefirst order theory of non-abelian free groups epitomized the use of extensions of centralisers. As a1rst step towards showing that all non-abelian free groups share the same first order theory, Selainvestigated the structure of limit groups (over free groups) in [Sel01] and showed that they admita hierarchical structure. This structure implies that limit groups embed in the Z [ t ] -completionof a free group, or in other words F Z [ t ] serves as a universe for limit groups over the free group F . Similar results were obtained by Kharlampovich and Miasnikov, and in [Kha12] they extendedtheir argument to prove that the limit groups over any toral relatively hyperbolic group G embedin G Z [ t ] (the definition of a limit group over any group G is recalled in Subsection 2.5 below).The latter work was carried out in the context of groups in the class CSA: groups whosemaximal abelian subgroups are malnormal. The authors showed that extending a centraliser of aCSA group yields a CSA group, which allowed them to use induction in the process of repeatedlyextending centralisers. RAAGs, even coherent ones, do not necessarily lie in the class CSA. Thisdeficit motivated Casals-Ruiz, Duncan and Kazachov to seek a broader setting in which similarlystructured proofs would work, leading them to the class C . They show in [CDK20] that this class C contains all coherent RAAGs, as well as all toral relatively hyperbolic groups. In addition, theyprove that if G is in the class C (and satisfies the technical condition R of Definition 3.5) then G Z [ t ] can be built as an iterated centraliser extension over G and is fully residually G . They also showthat if G is a coherent RAAG, then every limit group over G embeds in G Z [ t ] . They suggested thatthis fact, combined with the relatively simple structure of G Z [ t ] , ought to provide fertile ground foraddressing algorithmic problems and establishing residual properties of limit groups over RAAGs,and highlighted some specific challenges of this type, which we address in this paper.As well as building on [CDK20], the results that we shall present here rely crucially on thesimple observation that direct extensions of centralisers, which are the basis for the constructionof Z [ t ] -completions, have a much tamer structure than more general amalgamated products. Bydefinition, if C G ( u ) denotes the centraliser of u in G , then the direct extension of C G ( u ) by C G ( u ) × B is the quotient of the free product G ∗ B by relations that force the subgroups C G ( u ) and B to commute. This is an example of a free product with commuting subgroups : such aproduct is obtained from pairs of groups L ≤ G and M ≤ H by forming the amalgamated freeproduct G ∗ L ( L × M ) ∗ M H , which is abbreviated to ⟨ G, H ∣ [
L, M ] = ⟩ . We shall exploit the wayin which G Z [ t ] is built from extensions of centralizers to prove the following theorems, repeatedlyemploying a criterion developed by Loginova [Log00] for the cyclic subgroup separability of certainfree products with commuting subgroups. When G is a coherent RAAG, the special combinatorialstructure of its defining graph (see Subsection 2.3) also plays an important part in our proof. Theorem 1.
Let G be a group in the class C which satisfies condition R , and let A be a ring. If G is cyclic subgroup separable, then the A -completion of G , G A , is cyclic subgroup separable. Theorem 2.
Limit groups over coherent RAAGs are cyclic subgroup separable.
In the classic (and more restrictive) setting of limit groups over free groups, Wilton showedthat all finitely generated subgroups are separable [Wil08]. However, limit groups over coherentRAAGs are not necessarily subgroup separable: the coherent RAAG given by the presentation L = ⟨ x, y, z, w ∣ [ x, y ] = [ y, z ] = [ z, w ] = ⟩ , and whose defining graph isis not subgroup separable [NW00, Theorem 1.2].In the last section of the paper we shall discuss the word problem for free products withcommuting subgroups. Given L ≤ G and M ≤ H , if the word problem is solvable both in G and in H , and the membership problem is solvable for L in G and for M in H , then there is a solutionto the word problem in ⟨ G, H ∣ [
L, M ] = ⟩ . We shall use once again the way in which exponentialcompletions of groups from the class C can be built by iterating centraliser extensions to prove thefollowing: 2 roposition 3. Let G be a group in the class C . If G satisfies condition R and has a solvableword problem, then every finitely generated subgroup H of G A has a solvable word problem. In particular, we recover the fact that limit groups over coherent RAAGs and toral relativelyhyperbolic groups have a solvable word problem.
Acknowledgements
I would like to thank Martin Bridson for his warm encouragement andgenerous feedback.
Given a group G , we denote by Z ( G ) its center, and by C G ( g ) the centraliser of g ∈ G . Throughoutthis paper, we assume that all rings are associative, have a free abelian additive group and amultiplicative identity 1. For such a ring A , the additive group generated by 1 is denoted char ( A ) ≅ Z and whenever we refer to Z ⊂ A we in fact mean char ( A ) . For a graph Γ, we denote by VΓ theset of vertices of Γ and by EΓ ⊂ VΓ × VΓ the set of edges of Γ.We recall the notion of a free product with commuting subgroups : let G and H be groups, andsuppose that L and M are subgroups of G and H respectively. The free product of G and H withcommuting subgroups L and M is the quotient of the free product G ∗ H by the normal closureof the set of relations {[ ℓ, m ]∣ ℓ ∈ L and m ∈ M } . We often abbreviate and refer to this group as ⟨ G, H ∣ [
L, M ] = ⟩ . A subgroup H of a group G is called separable if it is an intersection of finite index subgroups of G . If H is separable in G , then every g ∉ H can be separated from H in a finite quotient of G ,that is, there is a homomorphism f ∶ G → Q where Q is finite and f ( g ) ∉ f ( H ) . Another elegantdescription of a separable subgroup is in terms of the profinite topology on G . Recall that in thistopology, a local base of the identity in G is the set of finite index normal subgroups of G . A localbase at any g ∈ G is then obtained by taking g -cosets of the finite index normal subgroups of G .Furthermore, every finite index normal subgroup of G is also closed in this topology. Therefore, asubgroup H ≤ G is separable if and only if it is closed in the profinite topology on G . Definition 2.1.
A group G is called cyclic subgroup separable if each of its cyclic subgroups isseparable. C We remind the reader of the definition of a right-angled Artin group:
Definition 2.2.
Let Γ be a simple graph; the right-angled Artin group (or in short, RAAG) G ( Γ ) is the group with presentation ⟨ VΓ ∣ [ v, u ] , ( v, u ) ∈ EΓ ⟩ . We refer to the graph Γ as the defining graph of G ( Γ ) .Note that in the definition above we do not restrict ourselves to finite graphs. Recall that agroup is called coherent if all of its finitely generated subgroups are finitely presented. In [Dro87,Theorem 1], Droms shows that a finitely generated RAAG G ( Γ ) is coherent if and only if its3efining graph Γ is chordal : every subgraph of Γ that is a cycle of more than 3 vertices admits achord, i.e., an edge that connects two vertices of the cycle. Note that if Γ is an infinite chordalgraph, then every finitely generated subgroup H of G ( Γ ) is a subgroup of a RAAG G ( Γ ′ ) , whereΓ ′ is a finite and full subgraph of Γ; Γ ′ is chordal, which implies that H is finitely presented andtherefore G ( Γ ) is coherent. The class of coherent RAAGs includes free groups, free abelian groupsand RAAGs which are fundamental groups of 3-manifolds (see [Dro87, Theorem 2]).In [CDK20], Casals-Ruiz, Duncan and Kazachov define a new class of groups C which wascrafted specifically to satisfy the following property: the Z [ t ] -completion (see Subsection 2.4) of agroup G in the class C can be built by iterating extensions of centralisers (see Section 3), and isfully residually G . The definition of the class C is long and technical, and is beyond the scope of thispaper; in this paper we do not use the definition of the class C directly, but rather use propertiesof groups in the class C proven in [CDK20]. We briefly mention that a group G in the class C istorsion-free, has unique roots and satisfies the Big Powers (BP) property: for every g , . . . , g k ∈ G such that [ g i , g i + ] ≠
1, there exists a positive integer N such that for every n , . . . , n k > N , g n ⋯ g n k k ≠ . The BP property is used to show that extending centralisers of G yields a group which is fullyresidually G . For further details, we refer the reader to [CDK20, Section 3]. Free groups, freeabelian groups and more generally coherent RAAGs all lie in the class C ; in addition, toral relativelyhyperbolic groups (torsion-free groups which are hyperbolic relative to a set of free abelian groups)are also in C . Recall that throughout this paper we assume that rings are associative, have a free abelian ad-ditive group and a multiplicative identity 1; as a consequence, characteristic subrings are alwaysisomorphic to Z . The definitions appearing below are simplified versions of the originals: for everydefinition which involves a ring A and a subring A , we assume that A = char ( A ) ≅ Z . Furtherdetail can be found in [MR94] and [MR95]. Definition 2.3.
Let A be a ring. A group G is called an A -group if there is a map G × A → G which satisfies the following (below, g a denotes the image of ( g, a ) under the map G × A → G ):1. g = g , g = a = g ∈ G and a ∈ A ,2. g a + b = g a g b and ( g a ) b = g ab for every g ∈ G and a, b ∈ A ,3. ( hgh − ) a = hg a h − for every g, h ∈ G and a ∈ A , and4. for every g, h ∈ G and a ∈ A , if [ g, h ] = ( gh ) a = g a h a .We call G a partial A -group if there exists P ⊂ G × A such that g a is defined whenever ( g, a ) ∈ P ,and all the properties above hold whenever the arguments belong to P .A homomorphism f ∶ G → H where G and H are A -groups is called an A -homomorphism if f ( g a ) = ( f ( g )) a for every g ∈ G and a ∈ A . If H ≤ G and G is a partial A -group we say that H is a full A -subgroup of G if h a is defined, and lies in H , for every h ∈ H and a ∈ A . Within our limitedsettings, an A -completion of a group G is defined as follows: Definition 2.4.
Let G be a group. An A -completion of G is an A -group G A which satisfies thefollowing: 4. there is a homomorphism τ ∶ G → G A such that no proper full A -subgroup of G A contains τ ( G ) , and2. if f ∶ G → H is a homomorphism and H is an A -group, then f factors via G A ; in other words,there exists a unique A -homomorphism f ∶ G A → H such that f = f ○ τ .By [MR94, Theorems 1 and 2], every group admits an A -completion and this completion isunique (up to A -isomorphism). We also remark that if G is abelian, then G A is abelian andcoincides with G ⊗ Z A as the two groups satisfy the same universal property. Limit groups (over free groups) were first defined by Sela in [Sel01]; this class of groups coincideswith the class of finitely generated fully residually free groups, which has been extensively studiedsince the 1960s. In this paper we deal with limit groups over groups in the class C , and forexpository purposes, throughout this subsection, adopt the approach of Champetier and Guirardel(see [CG05]).Given a group K , a limit group over K is, simply put, a limit of finitely generated markedsubgroups of K in the space of marked groups. A marked group is a pair ( G, S ) such that G isa group and S is a finite generating set of G . We say that two marked groups ( G, { s , . . . , s n }) and ( G ′ , { s ′ , . . . , s ′ n }) are isomorphic (as marked groups) if the map which sends each s i ∈ G to s ′ i ∈ G ′ extends to an isomorphism G → G ′ . Fixing a positive integer n , we define G n to be the setof marked groups ( G, S ) such that ∣ S ∣ = n .The set G n can be viewed as a topological space, where the topology is induced by the followingpseudometric: given ( G, S ) , ( G ′ , S ′ ) ∈ G n , set v (( G, S ) , ( G ′ , S ′ )) to be the maximal integer N suchthat w ( S ) = G if and only if w ( S ′ ) = G ′ for every word w of length at most N . If ( G, S ) and ( G ′ , S ′ ) are isomorphic as marked groups, set v (( G, S ) , ( G ′ , S ′ )) = ∞ . The distance between ( G, S ) and ( G ′ , S ′ ) in G n is d n (( G, S ) , ( G ′ , S ′ )) = e − v (( G,S ) , ( G ′ ,S ′ )) . Definition 2.5.
Let G be a group. A group H is called a limit group over G if there is an integer n such that H is the limit of a sequence of marked groups ( H i , S i ) ∈ G n (with respect to the topologydefined above), and such that H i ≤ G for every i ∈ N .In many cases, and in particular the cases that interest us (namely coherent RAAGs andtoral relatively hyperbolic groups), limit groups admit a simpler description. A group G is called equationally Noetherian if the following holds: given a tuple x = ( x , . . . , x n ) of variables, and aset Σ ⊂ G ∗ F ( x ) of equations over G , there is a finite subset Σ of Σ such that { g = ( g , . . . , g n ) ∈ G n ∣ σ ( g ) = ∀ σ ∈ Σ } = { g = ( g , . . . , g n ) ∈ G n ∣ σ ( g ) = ∀ σ ∈ Σ } where σ ( g ) is the element of G obtained by replacing each x i with g i . In other words, every systemof equations over G is equivalent to a finite subsystem. If G is equationally Noetherian, then H is a limit group over G if and only if H is finitely generated and fully residually G . The fact thatevery finitely generated fully residually G group is a limit group is easy to see, and is true for anygroup. We sketch a proof for the converse below:Suppose that ( H, S ) is the limit of a sequence ( H i , S i ) in G n and that H i ≤ G for every i ∈ N .In the free group F ( S ) , let Σ ⊂ F ( S ) be the kernel of the homomorphism F ( S ) → H induced bythe inclusion of S . Since G is equationally Noetherian, the system of equations Σ is equivalent to5 finite subsystem Σ ⊂ Σ in G . As σ ( S ) = H for every σ ∈ Σ , σ ( S i ) = H i for sufficientlylarge i . This implies that the map which sends S to S i extends to a homomorphism f i ∶ H → H i for sufficiently large i . Finally, for any finite subset E of H , ( H i , S i ) is sufficiently close to ( H, S ) in G n (as long as i is large enough) to ensure that f i ∶ H → H i is injective on E .Groves showed in [Gro05, Theorem 5.16] that toral relatively hyperbolic groups are equationallyNoetherian. The fact that coherent RAAGs are equationally Noetherian follows from their linearityand is mentioned in [CRK15].If G is a coherent RAAG or a toral relatively hyperbolic group, then limit groups over G admityet another description: they are the finitely generated subgroups of the Z [ t ] -completion of G (see[CDK20, Corollary 6.12 and Theorem 8.1] and [Kha12, Theorems D. and E.]). We further explorethis characterisation of limit groups in the following section. A -completions and cyclic subgroup separability Exponential completions of (certain) groups in the class C exhibit a fairly friendly structure: theycan be built, ”from the group G up”, by iterating extensions of centralisers . Definition 3.1.
Let G be a group and let u ∈ G . Let H be another group, and let ϕ ∶ C G ( u ) → H be an injective homomorphism such that ϕ ( u ) ∈ Z ( H ) . The extension of the centraliser C G ( u ) by H is the group G ( u, H ) = G ∗ C G ( u ) = ϕ ( C G ( u )) H. If ϕ ( C G ( u )) is a direct factor of H , then the extension is said to be direct . If, furthermore, H = ϕ ( C G ( u )) × Z , the extension is said to be free .Direct extensions of centralisers have a particularly nice structure. If G ( u, C G ( u ) × B ) is adirect extension of the centraliser C G ( u ) by C = C G ( u ) × B , then G ( u, C G ( u ) × B ) = G ∗ C G ( u ) ( C G ( u ) × B ) = G ∗ C G ( u ) ⟨ C G ( u ) , B ∣ [ C G ( u ) , B ] = ⟩ = ⟨ G, B ∣[ C G ( u ) , B ] = ⟩ , or in other words G ( u, C G ( u ) × B ) is the free product of G and B with commuting subgroups C G ( u ) and B .In [Log00], the author gives a criterion under which free products with commuting subgroupsare cyclic subgroup separable. This was later generalized in [Sok14]. Theorem 3.2 ([Log00, Main Theorem], [Sok14, Theorem 2.1]) . Let G and H be cyclic subgroupseparable groups and let L ≤ G and M ≤ H . If the group with presentation ⟨ G, H ∣ [
L, M ] = ⟩ isresidually finite, then it is cyclic subgroup separable. This criterion will be used throughout this section; as a warm-up we prove the following lemmawhich easily follows from Theorem 3.2:
Lemma 3.3.
Let G be a group in the class C and let u ∈ G be such that C G ( u ) is abelian. Let B be a free abelian group and write C = C G ( u ) × B . If G is cyclic subgroup separable, then so is thedirect centraliser extension G ( u, C ) . roof. From the discussion preceding this lemma, G ( u, C ) is the free product of G and B withcommuting subgroups C G ( u ) and B . Since B is free abelian, its cyclic subgroups are separable;by our assumption, G is also cyclic subgroup separable. Therefore, by Theorem 3.2, it is enoughto show that G ( u, C ) is residually finite.By [CDK20, Theorem 4.2], G ( u, C ) is fully residually G . Since G is cyclic subgroup separableit is residually finite, and hence G ( u, C ) is residually finite. Remark . It is worth mentioning that in an earlier paper, Loginova shows that a free productwith commuting subgroups ⟨ G, H ∣ [
L, M ] = ⟩ is residually finite if and only if G and H areresidually finite, L is separable in G and M is separable in H (see [Log99, Theorem 1]). It followsthat if G is a residually finite group in the class C , then abelian centralisers in G are separable.In particular, abelian centralisers in graph towers over coherent RAAGs (see [CDK20, Section 7])are separable.The remainder of this section is devoted to proving Theorems 1 and 2. We do so by analysingthe construction of the A -completion of a group G from the class C in steps, following [CDK20], andproving that each step yields a cyclic subgroup separable group. Note that by [CDK20, Proposition6.1], if G is abelian then G A is abelian, and therefore cyclic subgroup separable. We thereforerestrict our attention to non-abelian groups in the class C . We also remark that in [CDK20], theauthors assume that a non-abelian group G ∈ C satisfies an additional condition, named condition R, in order to show that G A enjoys the structure of an iterated centraliser extension. We makethis assumption too. Definition 3.5.
A group G ∈ C is said to satisfy condition R if it is a partial A -group, and forevery u ∈ G , if C G ( u ) is non-abelian, then the centre Z ( C G ( u )) of C G ( u ) is a full A -subgroup.We remind the reader that throughout this paper we assume that all rings are associative, havea free abelian additive subgroup and a multiplicative identity 1. We also recall the statement ofTheorem 1 for the convenience of the reader: Theorem 1.
Let G be a group in the class C which satisfies condition R , and let A be a ring. If G is cyclic subgroup separable, then the A -completion of G , G A , is cyclic subgroup separable. The strategy behind the construction of the A -completion of G is rather straightforward: werepeatedly extend centralisers in G to obtain a group G ∗ such that G is a full A -subgroup of G ∗ ; inother words, for every g ∈ G , the action of A on g within G ∗ is defined. Iterating this construction,we eventually obtain the A -completion of G . We recall the following construction from [MR95]which allows us to extend multiple centralisers at once: Definition 3.6 ([MR95, Definition 8], [CDK20, Definition 4.7]) . Let C = { C G ( u i )} i ∈ I be a set ofcentralisers in a group G and let { ϕ i ∶ C G ( u i ) → H i } i ∈ I be injective homomorphisms such that ϕ i ( u i ) ∈ Z ( H i ) for every i ∈ I . Let T be the tree whose vertex set is { v } ∪ { v i } i ∈ I and whose edgeset is { e i = ( v, v i )} i ∈ I . Let T G be the graph of groups whose underlying graph is T , and whosevertex groups, edge groups and edge maps are as follows:1. G v = G ,2. G v i = H i ,3. G e i = C G ( u i ) ,4. the map which maps G e i into G v is the inclusion, and5. the map which maps G e i into G v i is ϕ i . 7he fundamental group of this graph of groups is called a tree extension of centralisers . It isdenoted by G ( C , H , Φ ) where H = { H i } i ∈ I and Φ = { ϕ i } i ∈ I .We have already seen (Lemma 3.3) that direct extensions of abelian centralisers of cyclic sub-group separable groups in C are cyclic subgroup separable. The following lemma, which relies on[CDK20, Proposition 4.8], shows that the same holds for tree extensions of centralisers: Lemma 3.7.
Let G be a group in the class C and let C = { C G ( u i )} i ∈ I be a set of abelian centralisersin G such that no two of them are conjugate. For each i ∈ I , let H i be a free abelian group and let ϕ i ∶ H i → C G ( u i ) be an injective homomorphism such that H i = ϕ i ( C i ) × K i for some K i ≤ H i .Keeping the notation of Definition 3.6, if G is cyclic subgroup separable, then the tree extension ofcentralisers G ( C , H , Φ ) is cyclic subgroup separable.Proof. Let < be a well-ordering of the set I ∪ { } (assuming 0 ∉ I and 0 < i for every i ∈ I ). Weconstruct, by recursion, a direct system of groups { G i } i ∈ I ∪{ } over I , along with inclusion maps f i,j ∶ G i → G j (for i < j ) and retractions r j,i ∶ G j → G i (for i < j ). For readability, we refer to themaps r j,i as retractions, but formally we mean that r j,i ( f i,j ( g )) = g for every g ∈ G i . In addition,1. G = G ,2. for every j < i ∈ I , the centraliser of u i in G j coincides with the centraliser of u i in G , thatis C G j ( f ,j ( u i )) = f ,j ( C G ( u i )) ,3. every G i is cyclic subgroup separable,4. ⋃ i ∈ I G i = G ( C , H , Φ ) .Suppose first that i ∈ I is a successor ordinal, that is i = j +
1; suppose in addition that for every k ≤ j the groups G k have been defined, along with the suitable inclusion maps and retractions.Set G i to be the direct extension of the centraliser C G j ( f ,j ( u i )) by H i . Note that this is well-defined, since we assume that C G j ( f ,j ( u i )) = f ,j ( C G ( u i )) so C G j ( f ,j ( u i )) is a direct factorof H i . By Lemma 3.3, G i is cyclic subgroup separable. Let f j,i be the obvious inclusion map G j → G i , and for every k < j set f k,i = f j,i ○ f k,j . Define the retraction r i , j ∶ G i → G j by mappingevery element of K i to f ,j ( u i ) . Similarly, for every k ≤ j set r i,k = r j,k ○ r i,j . In addition, forevery k > i , the fact that f ,i ( u k ) is not a conjugate of any element in C G j ( f ,j ( u i )) implies that C G i ( f ,i ( u k )) = f ,i ( C G ( u k )) .Suppose now that i is a limit ordinal, and that for every k < i the groups G k , along with thesuitable inclusion maps and retractions, have been defined. Consider the directed system of groups { G j } j < i and let G i be its direct limit. Denote by f j,i ∶ G j → G i the canonical embedding of G j in G i for j < i . To define retractions r i,j ∶ G i → G j , consider the cofinal system { G k } j < k < i ; its directlimit is G i . For every j < k ≤ ℓ < i we have the following commuting diagram: G k r k,j ❇❇❇❇❇❇❇❇ f k,ℓ (cid:15) (cid:15) G ℓ r ℓ,j / / G j and by the universal property of direct limits we obtain a map r i,j ∶ G i → G j along with thefollowing commuting diagrams (for j < k < i ): G k f k,i / / r k,j ❆❆❆❆❆❆❆❆ G ir i,j (cid:15) (cid:15) G j g ∈ G j , r i,j ( f j,i ( g )) = r i,j ○ f k,i ( f j,k ( g )) = r k,j ( f j,k ( g )) = g so r i,j is a retraction. In addition, the fact that C G j ( f ,j ( u k )) = f ,j ( C G ( u k )) for every j < i and k ≥ i implies that C G i ( f ,i ( u k )) = f ,i ( C G ( u k )) for every k ≥ i . The existenceof these retractions also implies that G i is cyclic subgroup separable. Let g, h ∈ G i be such that h ∉ ⟨ g ⟩ ; there is j < i and g ′ , h ′ ∈ G j such that f j,i ( g ′ ) = g , f j,i ( h ′ ) = h and h ′ ∉ ⟨ g ′ ⟩ . Since G j iscyclic subgroup separable, there is a map q ∶ G j → Q such that q ( h ′ ) ∉ ⟨ q ( g ′ )⟩ and Q is finite. Themap q ○ r i,j ∶ G i → Q separates h from ⟨ g ⟩ .We now define G i to be the direct extension of the centraliser C G i ( f ,i ( u i )) by H i . By Lemma3.3, G i is cyclic subgroup separable. We also set f i ∶ G i → G i to be the inclusion map, and r i ∶ G i → G i to be the retraction which maps K i to f ,i ( u i ) . Finally, define f j,i = f i ○ f j,i and r i,j = r i,j ○ r i . The desired properties of G i , the maps f j,i and the retractions r i,j can be verifiedas in the successor stage.The cyclic subgroup separability of ⋃ i ∈ I G i = G ( C , H , Φ ) also follows, as in either the successoror the limit stage, depending on the order type of { } ∪ I .With Lemma 3.7 in our arsenal, we are ready to describe the construction of the A -completion ofa group G in C and prove Theorem 1. Assume in addition that G satisfies condition R; recall that,as in the paragraph preceding Theorem 1, we first construct a group G ∗ which contains G , and suchthat G is a full A -subgroup of G ∗ . We begin by choosing a set of centralisers C ( G ) = { C G ( u i )} i ∈ I in G which satisfies the following:1. every centraliser in C is abelian, and not a full A -subgroup of G ,2. no two centralisers in C ( G ) are conjugate, and3. any abelian centraliser in G which is not a full A -subgroup is conjugate to a centraliser in C ( G ) .Note that the existence of a set C which satisfies the conditions above is guaranteed by Zorn’sLemma. Recall that as in Subsection 2.4, for every C G ( u i ) ∈ C ( G ) we have that C G ( u i ) A = C G ( u i ) ⊗ Z A ; in addition, C G ( u i ) is a direct summand of C G ( u i ) A . Setting H( G ) = { C G ( u i ) A } i ∈ I and Φ ( G ) = { ϕ i ∶ C G ( u i ) → C G ( u i ) A } i ∈ I where each ϕ i is the canonical embedding, we define G ∗ = G ( C ( G ) , H( G ) , Φ ( G )) . By [CDK20, Lemma 6.5] G is a full A -subgroup of G ∗ . In addition, by [CDK20, Lemma 6.6], G ∗ satisfies condition R and we can iterate this construction. As in [CDK20, Subsection 6.2], wedefine a directed system of groups G = G ( ) < G ( ) < ⋯ < G ( n ) < ⋯ G ( n + ) = ( G ( n ) ) ∗ = G ( n ) ( C ( G ( n ) ) , H( G ( n ) ) , Φ ( G ( n ) )) and the maps f i,j ∶ G ( i ) → G ( j ) are the inclusion maps. The direct limit of this system ⋃ n ∈ N G ( n ) is called an iterated centraliser extension of G by A , or in short an ICE of G by A . Note that ⋃ n ∈ N G ( n ) is an A -group, since every g ∈ G lies in G ( n ) for some n , and therefore the action of A on g is already defined in G ( n + ) . As a matter of fact, ⋃ n ∈ N G ( n ) is the A -completion of G as evidentin [CDK20, Theorem 6.3]. Theorem 1 now follows: Proof of Theorem 1.
By Lemma 3.7, G ( n + ) retracts onto G ( n ) for every n ∈ N ; composing theseretractions we obtain retractions r n,m ∶ G ( n ) → G ( m ) for every m < n . As in the proof of Lemma3.7, these retractions imply the existence of retractions from the direct limit ⋃ n ∈ N G ( n ) onto each G ( n ) . In addition, each G ( n ) is cyclic subgroup separable.Let g, h ∈ ⋃ n ∈ N G ( n ) be such that h ∉ ⟨ g ⟩ ; g and h lie in some G ( n ) . Since G ( n ) is cyclic subgroupseparable, there is a homomorphism q ∶ G ( n ) → Q such that q ( h ) ∉ ⟨ q ( g )⟩ and Q is finite. Thecomposition q ○ r n ∶ ⋃ n ∈ N G ( n ) → Q separates h from ⟨ g ⟩ . Corollary 3.8.
Limit groups over cyclic subgroup separable toral relatively hyperbolic groups arecyclic subgroup separable.Proof.
Let G be a toral relatively hyperbolic group; in particular G lies in C and satisfies conditionR. By Theorem 1, the Z [ t ] -completion of G is cyclic subgroup separable, and by [Kha12, TheoremsD. and E.] limit groups over G are exactly the finitely generated subgroups of G Z [ t ] .With a bit more work, we can also deduce the following: Theorem 2.
Limit groups over coherent RAAGs are cyclic subgroup separable.Proof.
Let G ( Γ ) be a coherent RAAG. By [CDK20, Corollary 6.12 and Theorem 8.1], limit groupsover G ( Γ ) are exactly the finitely generated subgroups of G ( Γ , Z [ t ]) Z [ t ] (where G ( Γ , Z [ t ]) is thegraph product whose underlying graph is Γ, and whose vertex groups are all Z [ t ] ). In light ofTheorem 1, since G ( Γ , Z [ t ]) lies in C and satisfies condition R, it is enough to show that G ( Γ , Z [ t ]) is cyclic subgroup separable. Let g, h ∈ G ( Γ , Z [ t ]) be such that h ∉ ⟨ g ⟩ ; there is a finite full subgraph∆ of Γ such that g, h ∈ G ( ∆ , Z [ t ]) . Note that G ( Γ , Z [ t ]) retracts onto G ( ∆ , Z [ t ]) by killing eachvertex group G v for v ∉ V∆. Hence it is sufficient to show that G ( ∆ , Z [ t ]) is cyclic subgroupseparable for every finite full subgraph ∆ of Γ.Since Γ is chordal, so is ∆. It is a famous result that every finite chordal graph, and in particular∆, admits a perfect elimination ordering (see [RTL76]), that is an ordering ( v , v , . . . , v n ) of V∆such that the following holds: for every i , the neighbours of v i amongst v , v , . . . , v i − form aclique. We show by induction on n that G ( ∆ , Z [ t ]) is cyclic subgroup separable. For every i ≤ n denote by ∆ i the full subgraph of ∆ whose vertices are v , v , . . . , v i ; denote by G i the copy of Z [ t ] which corresponds to the vertex v i of ∆. For n = G ( ∆ , Z [ t ]) is free abelian and thereforecyclic subgroup separable. Suppose now that G ( ∆ i , Z [ t ]) is cyclic subgroup separable and thatthe neighbours of v i + in ∆ are v i , . . . , v i k . We have that G ( ∆ i + , Z [ t ]) = ⟨ G, G i + ∣ [ G i + , G i j ] , j = , . . . , k ⟩ = ⟨ G, G i + ∣ [ G i + , ⟨ G i , . . . , G i k ⟩]⟩ where the last equality follows from the fact that v i , . . . , v i k form a clique in ∆ i and therefore ⟨ G i , . . . , G i k ⟩ = G i × ⋯ × G i k . In other words, G ( ∆ i + , Z [ t ]) is the free product of G ( ∆ i , Z [ t ]) and G i + with commuting subgroups ⟨ G i , . . . , G i k ⟩ and G i + . Since G ( ∆ i , Z [ t ]) and G i + are cyclicsubgroup separable and since G ( ∆ i + , Z [ t ]) is residually finite as the graph product of residuallyfinite groups (see, for example, [Gre90]), it follows that G ( ∆ i , Z [ t ]) is cyclic subgroup separableby Theorem 3.2. 10 Free products with commuting subgroups and the wordproblem
It is well-known that an amalgamated product G ∗ K H admits a solution to the word problem if theword problem is solvable in G and in H and there is a solution to the membership problem for K in both G and H . A similar statement can be made for free products with commuting subgroups: Lemma 4.1.
Let G and H be groups with a solvable word problem and let L ≤ G and M ≤ H .Suppose that the membership problem is solvable for L in G and for M in H . Then there is asolution to the word problem in ⟨ G, H ∣ [
L, M ] = ⟩ . Recall that a free centraliser extension is a centraliser extension of the form G ( u, C G ( u ) × Z ) = ⟨ G, t ∣ [ C G ( u ) , t ] = ⟩ , where u ∈ G . Using Lemma 4.1 above, we obtain: Proposition 3.
Let G be a group in the class C . If G satisfies condition R and has a solvableword problem, then every finitely generated subgroup H of G A has a solvable word problem.Proof. The fact that H is finitely generated and embeds in G A implies that H embeds in a groupobtained from G by taking finitely many free extensions of centralisers; that is, there are groups G , G , . . . , G n such that G = G , G i + = ⟨ G i , t i ∣ [ C G i ( u i ) , t i ] = ⟩ for some u i ∈ G i and H ≤ G n .We prove that G n , and hence H , has a decidable word problem by induction on n . Supposethat G i has a solvable word problem. By Lemma 4.1, a solution to the membership problem for C G i ( u i ) in G i would imply that G i + has a solvable word problem. But checking whether g ∈ G i lies in C G i ( u i ) is equivalent to asking whether [ g, u i ] = G i , which is solvable by the inductionhypothesis. Hence the word problem in G i + is solvable, which completes the proof. Corollary 4.2.
Limit groups over coherent RAAGs and toral relatively hyperbolic groups have asolvable word problem.Proof. If G ( Γ ) is a coherent RAAG and H is a limit group over G ( Γ ) , then by [CDK20, Theorem8.1] H is a finitely generated subgroup of G ( Γ , Z [ t ]) Z [ t ] . The group G ( Γ , Z [ t ]) lies in C , satisfiescondition R and admits an algorithm which checks whether a given word in the canonical generatorsis trivial or not. Similarly, if G is toral relatively hyperbolic and H is a limit group over G , thenby [Kha12, Theorems D. and E.] H is a finitely generated subgroup of G Z [ t ] . G satisfies conditionR and by [GW18, Subsection 2.7] has a solvable word problem.The fact that limit groups over coherent RAAGs have a decidable word problem was alreadymentioned in [CDK20], and follows from these groups being finitely presented and residually finite.The following proposition, which we record here for the sake of completeness, gives a solution to theword problem for limit groups over toral relatively hyperbolic groups (toral relatively hyperbolicgroups are equationally Noetherian [Gro05, Theorem 5.16] and limit groups over toral relativelyhyperbolic groups are recursively presented since they embed in finitely presented groups): Proposition 4.3.
Let G be an equationally Noetherian group. If G has a solvable word problem,then so does every finitely generated, recursively presented, residually G group.Proof. Let H be a finitely generated, recursively presented, residually G group. We execute thefollowing two algorithms in parallel: first, since H is recursively presented there is an algorithmwhich takes a word g ∈ H as its input, and returns ’yes’ if g = S be a finite generating set of H and let g be a word in the alphabet S ∪ S − . Theequational Noetherianity of G implies that there is an algorithm which checks, within finite time,whether a map S → G extends to a homomorphism f ∶ H → G . Given such a homomorphism f ,using a solution to the word problem in G the algorithm can further verify whether or not f ( g ) ≠ H is residually G , the algorithm described will return ’no’ whenever g ≠ H .11 eferences [CDK20] Montserrat Casals-Ruiz, Andrew Duncan, and Ilya Kazachkov, Limit groups over coher-ent right-angled Artin groups , arXiv e-prints (2020), arXiv:2009.01899.[CG05] Christophe Champetier and Vincent Guirardel,
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