Residually hyperbolic groups ^{∗}

G.Baumslag, A. Myasnikov, V. Remeslennikov

1 Introduction

The method of approximations was used a lot in group theory, for example, to solve the classical algorithmical problems. Thus, if a finitely presented group G is residually finite, then it has solvable word problem. More general, one can consider a system of arbitrary equations w(x) = 1 over G, in this event, solutions of it can be interpreted as homomorphisms from the quotient group G(w) = G ∗ F (x)/ < w > to G which are identical on G (here < w > is the normal closure of w in the free product of G and the free group F (x)). It turned out to be very important to describe those groups G(w) which are fully residually G (in other terminology, ω-residually G). In general, this question looks completely untractable, but under some natural restrictions on G, such as hyperbolicity or admitting free action on a Λ-tree, the problem seems not that hopeless.

In this paper we describe different types of residually G groups for an arbitrary given torsion-free hyperbolic group G. The main technical theorem states that under some natural conditions an extension of a centralizer of G is fully residually G. It generalizes a theorem of G.Baumslag [GB4] for a free group G. He proved the ω-residual freeness of surface groups (with few exeptions) by embedding them in an extension of a centralizer of a free group. The idea of constructing this type of examples of ω-residually G groups has proved to be very fruitfull. Moreover, it turned out that all known finitely generated ω-residually G groups are subgroups of groups which can be obtained from G by finitely many extensions of centralizers. The conjecture is that there are no others! (see for details conjecture 1) The ω-residuality is much stronger that the ordinary residuality, it allows one to approximate any finite number of elements at once, by one and the same homomorphism.. Quite surprisingly, we found out that for many groups H both types of approximations coinside. For example, if H is a torsion-free hyperbolic group or a group acting freely on some Λ-tree then for an arbitrary group G if H is residually G then H is ω-residually G. Similar results for ω-residually free groups were obtained by B.Baumslag [BB], in several places we followed his ideas.

∗Preprint N 24 of IITAM SB RAS. - Omsk: OmSU, 1995. - 35 p.

R.Lyndon [L] introduced and investigated the tensor Z[x]-completi-on F^{Z[x]} of a free group F and applied it to describe sets of solutions of equations of one variable over free groups. G.Baumslag [GB1] invented a technique to deal with tensor Q-completion F^{Q} of the group F. In [MR2] A.Myasnikov and V.Remeslennikov using extensions of centralizers constructed tensor A-completions G^{A} of groups G from a very wide class and arbitrary abelian group A. It turned out that Baumslag’s and Lyndon’s completions are the basic examples of the most principal types of completions which we call correspondingly Baumslag’s type and Lyndon’s type. As a corollary of our results on extensions of centralizers we prove that for torsion-free hyperbolic group G and torsion-free abelian group A the completion G^{A} is ω-residually G^{Q}, moreover, in this event every finitely generated subgroup of G^{A} is ω-residually hyperbolic, this approximation is effective and one can precisely describe the approximating hyperbolic groups. In addition to the statement above, if G^{A} is happen to be of Lyndon’s type then any finitely generated subgroup of G^{A} is ω-residually G.

Intuitevely, it was vaguely clear, that if H is ω-residually G, then H reflects a lot of properties of G. Fortunately, it turned out one can describe explicitely what properties H actually inherites from G. V.Remeslennikov proved [R1] that a finitely generated group G is ωresidually free if and only if it satisfies exactly the same universal sentences as a nonabelian free group F (i.e. G is universally equivalent to F). We generalize this result to arbitrary Equationally Noetherian groups. Really, let G and H be finitely generated groups and at least one of them is equationally Noetherian. Then G and H are universally equivalent if and only if they are mutually ω-residually in each other. A minor modification gives a criterion for universal equivalence in the infinitely generated case. As a corollary of G.Makanin result for free groups we prove that the universal theory of the completion F^{A} of Lyndon’s type is decidable. Equationally Noetherian groups are interesting objects in themselves. They are analogs of Noetherian rings and can be described as those groups where any (infinite) system of equations is equivalent to some finite part of it. For a while, all known examples of finitely generated equationally Noetherian groups came naturally from matrix groups over Noetherain rings. At the end of the paper we investigate some basic properties of equationally Noetherian groups and formulate several open problems.

2 Preliminaries and ω-residuallity of abelian groups

In this section we will prove that if A is a torsion-free abelian group then any residually A group is an ω-residually A group. It is not the case for arbitrary abelian groups and we show some typical counterexamples.

We start with definitions and some simple general facts.

Definition 1 Let H be a group and G be a family of groups.

1) A homomorphism of groups ψ : H ⟶ G separates a nontrivial element h ∈ H if ψ(h) ≠ 1;

2) A family of homomorphisms Ψ = {ψ : H ⟶ G | G ∈ G} is called a separating (ω-separating ) family of homomorphisms if any nontrivial h ∈ H (any finite number of nontrivial elements h_{1}, …, h_{n} ∈ H) can be separated by some ψ ∈ Ψ. In this case H is called a residually G group (ω-residually G group).

If the family G consists only of one group G we will say that H is residually (ω-residually) G group.

Note, that we do not assume separating homomorphisms to be onto. It makes no difference when the family G is closed under taking subgroups, but in general, this notions are different, for instance, for residually simple groups.

One can define a notion of n-residually G which means that one can separate in G any finite set of n elements. It turned out, that for a very wide class of groups, for example, for CSA-groups (see the next section) 2-residually G allready implies ω-residually G.

Residually G groups inherit some properties of groups from G, such as to be torsion-free , to be abelian or, more general, to satisfy a system of identities. That follows from the remark below.

Remark 1 A group H is a residually G group if and only if H is isomorphic to a subgroup of a cartesian product \overline{∏}G_{i}, where G_{i} ∈ G (diagonal embedding).

In the rest of the section we will discuss when the property “to be residually G” implies the property “to be ω-residually G”.

Propostion 1 Let G be a group. Then the following conditions are equivalent to one another:

1)every residually G group is an ω-residually G group;

2) G × G is an ω-residually G group.

Proof. Let us suppose the group G × G is an ω-residually G group. Let H be an arbitrary residually G group. By the previous remark we can assume that H ≤ \overline{∏}G. Let h_{1}, …, h_{n} be nontrivial elements in H. There exists a projection π from the cartesian product \overline{∏}G onto a finite product G × … × G such that all h_{i}‘s have nontrivial images. By induction on the number of the factors in this finite product one can reduce the problem to the case G × G. In opposite direction the statement is obvious. □

There are a lot of abelian groups A such that A × A is, of course, residually A but is not an ω-residually A group. For example, this is true for any finite group. Simply because A × A has more elements then A (note, if B is ω-residually A then for any finite subset X < B there exists a homomorphism B ⟶ A which is injective on X). The next proposition tells one how to generalize this idea and to construct more groups of this type.

Propostion 2 Let A be an abelian group which has a finite nontrivial p-component for some prime p. Then A × A is not ω-residually A.

Proof. Suppose, in contrary, that A × A is ω-residually A. At first, let us note that if A is an abelian group and B is ω-residually A group, then B is an abelian group and the torsion subgroup T (B) of B is ω-residually T (A) group. In our case, the p-component T_{p}(A × A) of A × A is ω-residually T_{p}(A). But T_{p}(A × A) = T_{p}(A) × T_{p}(A) is finite and contains more elements then T_{p}(A), so it can not be embedded into T_{p}(A) - contradiction to the ω-residuallity of T_{p}(A × A) by T_{p}(A). □

Nevertheless, for torsion-free abelian groups the both types of residuality coinside.

Theorem 1 Let A be an arbitrary torsion-free abelian group. Then any residually A group is an ω-residually A group.

Proof. By the previuos proposition one have to prove only that the group A × A is an ω-residually A group. Apparently, the map ψ_{m} : (b, c) ⟶ b − mc from A × A to A is a homomorphism of abelian groups for any integer m. Suppose a finite set of nontrivial elements in A × A is chosen:

X = {(b_{1}, c_{1}), …, (b_{n}, c_{n})}

.

In a torsion-free group if c_{i} ≠ 1 an equation b_{i} − xc_{i} = 0 has at most one solution in integers , let’s denote it by m_{i} . Therefore, if m ≠ m_{1}, ..., m_{n} then ψ_{m} separates X in A. □

Of course, the proof above does not work in nonabelian case because the map a ⟶ a^{m}, a ∈ A, is not an endomorphism of a nonabelian group A.

Corollary 1 A free abelian group B is an ω-residually G group for an arbitrary torsion-free group G.

3 Residuality of CSA-groups

In [MR2] a notion of a CSA-group was introduced. It turned out that CSA-groups constitute a very large class of groups. Indeed, all abelian groups, all torsion-free hyperbolic groups [MR2], and all groups acting freely on Λ-trees [Bas] are CSA-groups . In [GKM] all one relator CSA-groups (as well as commutative transitive one relator groups) were described.

In this section we will show that for a nonabelian CSA-group H and arbitrary group G if H is residually G then H is an ω-residually G group. Furthermore, if G is a CSA-group then ”2-residually G” allways implies ”ω-residually G”. From other hand, if G is nonabelian CSA-group then G × G is, of course, residually G group, but not ω-residually G group.

Some of the results here are a straightforward generalization of the B. Baumslag results [BB] for residually free groups.

Definition 2 A subgroup H of a group G is called malnormal (or conjugately separated ) if H ∩ H^{x} = 1 for any x ∈ G − H.

Definition 3 A group G is called a CSA-group if all its maximal abelian subgroups are conjugately separated.

Definition 4 A group G is called commutative transitive if the commutation is a transitive relation on all nontrivial elements of G.

Remark 2 [MR2] CSA-groups are commutative transitive.

Propostion 3 Let H be a group and G be a family of CSA-groups. Then:

1. H is a 2-residually G group ⟹ H is a CSA-group;

2. H is commutative transitive and residually G ⟹ H is CSAgroup.

Proof. By the remark above CSA-groups are commutative transitive. To write down the definition of commutative transitivity one needs only two inequalities, so every 2-residually CSA-group is commutative transitive. Let A be an arbitrary maximum abelian subgroup in H. Due to commutative transitivity A is a centralizer of some nontrivial element, say u. Suppose A is not a malnormal subgroup in H. Then there exist nontrivial elements a, b ∈ A and an element x ∈ H such that a^{x} = b and x does not belong to A. One can express it as follows: [u, a] = 1, [u, b] = 1, [u, x] ≠ 1, a^{x} = b, a ≠ 1. If G ∈ G and ϕ : H ⟶ G is a homomorphism separating a and [u, x] from 1, then images of elements a, b, x, u satisfy exactly the conditions above, but in G - contradiction to the CSA-property of G.

For the second statement of the proposition one needs only to prove that H is 2-residually CSA-group. If H is an abelian group, then it is CSA-group by definition. Suppose H is nonabelian group. From the transitivity of commutation the center of H is trivial. Moreover, H has no abelian normal subgroups. Because, if A is a such subgroup, then [x, a] ≠ 1 for some a ∈ A, x ∈ H and one can separate the element [x, a] in some G ∈ G, but this implies that the image of H in G contains a nontrivial abelian normal subgroup - contradiction to the CSA-property of G. Now, let us consider two arbitrary nontrivial elements g, h ∈ H. If for some x ∈ H we have [g, h^{x}] ≠ 1 then any homomorphism which separates [g, h^{x}] also separates both elements g, h. Otherwise, [g, h^{x}] = 1 for any x ∈ H, but it means (due to commutative transitivity) that the normal closure of h is abelian - contradiction to the absence of normal abelian sub! groups in H. It proves that H is 2-residually CSA and hence CSA according to the statement 1. □

Corollary 2 Let G be a nonabelian CSA-group. Then G × G is a residually G but not an ω-residually G group.

Proof. If G is a nonabelian group then G × G is not a commutation transitive and therefore it is not a CSA-group. Hence, it can not be ω-approximated by CSA-groups.□

Theorem 2 Let H be a non-abelian CSA-group, G be a family of groups. Then if H is a residually G group, then H is an ω-residually G group.

Proof. Let h_{1}, …, h_{n} be arbitrary nontrivial elements in H. We have to prove that all of them can be separated from identity by some homomorphism from H to some G ∈ G.

We can consider H as a subgroup of the cartesian product \overline{G} = \overline{∏}G_{i}, where G_{i} ∈ G, i ∈ I. There could be two cases for the elements h_{i}-s.

Case 1) There exists an index i ∈ I such that all i-components of all elements h_{1}, …, h_{n} are nontrivial. In that case, the i-projection π_{i} : \overline{∏}G_{i} ⟶ G_{i} is a homomorphism separating all h_{i}-s from identity.

Case 2). There is no such index i as in the first case. It means that for any index i ∈ I there is h_{k} with a trivial i-component. Let us introduce some notations. For element g ∈ \overline{G} by I(g) we denote the set of all indexes where components of g are trivial, and for a subset X ⊂ \overline{G}, I(X) = ⋂ I(g), where g runs over X. It is clear that our assumption is equivalent to I(h_{1}) ∪ … ∪ I(h_{k}) = I. It is easy to see that in the cartesian product \overline{G} one has the following inclusions:

a) I(g) ∩ I(f) ⊂ I(gf);

b) I(g) ⊂ I(x^{−1}gx) ;

c) I(g) ∪ I(f) ⊂ I([g, f]);

d) I(h) ⊂ I(< h >\overline{G}) for any h ∈ \overline{G} (here < h >\overline{G} is the normal closure of h in \overline{G}).

Let N_{i} =< h_{i} >^{H} be a normal closure of an element h_{i} in H. We claim that I(h_{1}) ∪ … ∪ I(h_{n}) ⊂ I([N_{1}, …, N_{n}]). Indeed, from c) by induction we have

I(g_{1}) ∪ … ∪ I(g_{n}) ⊂ I([g_{1}, …, g_{n}]),

¿From d) I(h_{i}) ⊂ I(x_{i}) for any x_{i} ∈< h_{i} >\overline{G}. Therefore,

I(h_{1}) ∪ … ∪ I(h_{n}) ⊂ I(x_{1}) ∪ … ∪ I(x_{n}) ⊂ I([x_{1}, …, x_{n}]),

for any x_{i} ∈< h_{i} >\overline{G}. By a)

I(h_{1}) ∪ … ∪ I(h_{n}) ⊂ I([x_{1}, …, x_{n}][y_{1}, …, y_{n}]),

for any x_{i}, y_{i} ∈< h_{i} >\overline{G}. It follows

I(h_{1}) ∪ … ∪ I(h_{n}) ⊂ I([N_{1}, …, N_{n}]).

But according to our assumptions I(h_{1})∪…∪I(h_{n}) = I, therefore, I([N_{1}, …, N_{n}]) = I, or in other words, [N_{1}, …, N_{n}] = 1. Let k be the minimal positive integer with this property. Let us define S = N_{1} if k = 2; and S = [N_{1}, …, N_{k−1}] if k > 2. In both cases S ≠ 1, but [S, N_{k}] = 1. The subgroup N_{k} is a normal subgroup in a CSA-group H, hence it is non-abelian subgroup in H (otherwise H is an abelian group - in contrary to the conditions of the theorem). Thus, we have two commuting nontrivial subgroups S and N_{k} in H and at least one of them is not abelian. It follows that commutation in H is not transitive -contradiction to the properties of CSA-group. Therefore, the case 2 is impossible. The theorem has been proved. □

Corollary 3 Let G be a family of CSA-groups. Then

1 Any 2-residually G group is ω-residually G;

2 Any commutative transitive residually G group is ω-residually G.

Proof. Let a group H satisfies the conditions 1 or 2 of the corollary. By proposition 3 the group H is a CSA-group and one can apply theorem 2. □

Corollary 4 Let H_{1}, H_{2} be groups and G be a family of CSA-groups. The following conditions are equivalent to one another:

1) H_{1} ∗ H_{2} is residually G;

2) H_{1} ∗ H_{2} is ω-residually G.

Proof. 2) evidently implies 1). For reverse, it is enough (according to the corollary above) to prove that H = H_{1} ∗ H_{2} is 2-residually G. Again, one can consider two arbitrary elements g, h ∈ H and commutators [g, h^{x}]. At least one of those commutators are nontrivial, because the centralizer of the normal closure of any nontrivial element in the free product H is trivial. Therefore, to separate the pair g, h it is sufficient to separate a nontrivial commutator [g, h^{x}]. □

In the next section we will prove that under some natural restrictions the conditions 1) and 2) in the proposition are both equivalent to the condition that the groups H_{1} and H_{2} are both ω-residually G.

4 Approximation retracts for extension of centralizers of hyperbolic groups

In this section we construct concrete families of separating retracts for some special kind of HNN-extensions - so called extensions of centralizers. It turned out (see [Bas], [GB1], [GB4,][GKM], and [MR2]) that extensions of centralizers play an important role in the theory of groups acting on trees, exponential groups and one relator groups. From another hand, extensions of centralizers of a group G (and their subgroups) are a good and up to now the unique source of ω-residually G groups.

Let G be a group. The centralizer of an element u ∈ G is denoted by C_{G}(u) or C(u) or C if the group G and the element u can be determined from the context.

Definition 5 Let G be a group, C(u) the centralizer of an element u ∈ G. Suppose C(u) is abelian and ϕ : C(u) ⟶ A is a monomorphism of C(u) into abelian group A. Then the group

G(u, A) = ⟨G ∗ A | C(u) = C(u)^{ϕ}⟩

is called an extension of the centralizer C(u) by the group A with respect to ϕ.

Some particular cases of this construction are of special interest. We will classify them with respect to the type of the embedding ϕ : C(u) → A. Thus, the extension is:

1) direct if A = C(u)^{ϕ} × B;

2) free (of rank n) if A = C(u)^{ϕ} × B and B is a free abelian group (of rank n).

Note: from now on we will identify the centralizer C with it’s image C^{ϕ} in A, so C ≤ A and extension of the centralizer is made with respect to identical embedding of C in A.

Free extensions of centralizers of the rank 1 are the corner stone of the whole business of centralizers extensions, so let us discuss it in more details.

Let G = ⟨X | R⟩ is a presentation of G. It is easy to see that a free rank 1 extension of centralizer C(u) of a group G has the following presentation ⟨X, t | R, [y, t]⟩_{y∈C(u)} (here t is a generator of a cyclic direct complement of C(u) in A). We will write this presentation as

G(u, t) = ⟨G, t | [C(u), t] = 1⟩.

Note, that G(u, t) can be obtained from G by an HNN-extension with respect to the identity isomorphism C(u) → C(u) :

G(u, t) = ⟨G, t | t^{−1}at = a, a ∈ C(u)⟩.

Let C ≤ A be abelian groups and G(u, A) an extension of a centralizer C(u) = C of a group G by the group A. By definition, an element g ∈ G(u, A) is in a reduced form

g = g_{1}a_{1} … g_{n}a_{n}g_{n+1}  (1)

if a_{i} ∈ A − C, i = 1, …, n, g_{j} ∈ G − C, j = 2, …, n.

For a direct extension of a centralizer C by a group B , i.e. when A = C × B, it is more convenient to use semicanonical forms of elements.

A reduced form

g = g_{1}b_{1} ⋯ b_{n}g_{n+1}

is semicanonical, iff 1 ≠ b_{i} ∈ B for all i.

It is not difficult to see that any two semicanonical forms of g can be transformed from one to another by a finite sequence of commutations of the form bc = cb, where c ∈ C, b ∈ B.

In our treatment with extension of centralizers we will use very special types of separating homomorphisms.

By definition if G ≤ H then any homomorphism H ⟶ G is called a retract on G if it is identical on G.

Definition 6 Let H be a group and G be a family of subgroups of H. The group H is called strongly residually (strongly ω-residually) G if there exists a separating (ω-separating) family of homomorphisms {ψ : H ⟶ G | G ∈ G} such that every ψ is a retract on G.

Retracts appear very naturally in extensions of centralizers, it goes directly from the universal properties of free products with amalgamation:

Lemma 1 Let G(u, A) be an extension of a centralizer C = C(u) by an abelian group A. If ψ : A ⟶ C is a retract on C, then the map λ_{ψ} : G(u, A) ⟶ G such that λ_{ψ} = id on G and λ_{ψ} = ψ on A can be extended up to a homomorphism G(u, A) ⟶ G (we will denote it by the same λ_{ψ}). Moreover, λ_{ψ} is a retract on G.

The following lemma shows the importance of direct extensions of centralizers.

Lemma 2 Let C ≤ A be torsion-free abelian groups. Then the following statements are equivalent to one another:

1) A is strongly ω-residually C;

2) A = C × B and B is ω-residually C.

Proof. Suppose we have 2). Let Ψ be a separating family of homomorphisms from B to C, id_{C} the identical map on C. Then {id_{C} + mψ | m ∈ Z, ψ ∈ Ψ} is a separating family of retracts from C × B to C (see theorem 1). by the universal property of direct products. Let us prove the implication 1) ⟹ 2). Let Ψ be a separating family of retracts from A to C and ψ_{0} ∈ Ψ, then A = C × B, where B = kerψ_{0}. Moreover, all the restrictions of homomorphisms from Ψ on B give one a separating family for B in C.□

Thus, we are interesting now in direct extensions of centralizers. Suppose that A = C × B and B is strongly ω-residually C with Ψ as an ω-separating family of retracts. Our goal is to formulate necessary conditions on a group G(u, A) to be strongly ω-residually G with the separating family Λ_{Ψ} = {λ_{ψ} | ψ ∈ Ψ} defined above.

Let u_{1}, …, u_{n} be elements of infinite order in a group G, and g_{1}, …, g_{n+1} be arbitrary elements in G satisfying the following conjugate separation (CS) condition:

[u_{i}^{g_{i+1}}, u_{i+1}] ≠ 1, i = 1, …, n − 1.

For such u_{i}’s and g_{i}’s one can consider the set of elements

S(u, g, m) = {g_{1}u_{1}^{m_{1}}g_{2}u_{2}^{m_{2}} … u_{n}^{m_{n}}g_{n+1} | |m_{i}| ≥ m}

Now we are able to formulate the main condition of separation.

Definition 7 A group G satisfies the condition of separation (S) if for any elements u_{1}, …, u_{n} and g_{1}, …, g_{n+1} in G satisfying the condition (CS) there exists a constant m such that the set S(u, g, m) does not contain the identity element.

G.Baumslag was the first who considered similar type of separation conditions for a free group F. He proved [GB4] that if u_{1} = u_{2} = … = u_{n} = u are equal and nontrivial elements in F, then the set S(u, g, m) does not contain identity for a some number m.

Remark 3 1) in any CSA-group in the case u_{1} = … = u_{n} the (CS) condition is equivalent to [u, g_{i}] ≠ 1 for i = 2, … n;

2) any abelian group satisfies the separation condition (S);

3) any torsion-free hyperbolic group satisfies the separation condition (S) (Ol’shanskii [Ol]).

In the proof of the main theorem below we will need the separation condition not for arbitrary tuples (u_{1}, …, u_{n}), (g_{1}, …, g_{n+1}) satisfying the condition (CS), but only for the tuples where u_{1} = … = u_{n}.).

Let us return to the group G(u, A), where A = C×B. Willing to use the separation condition (S) in G we need to be able to approximate elements from B by some “large” powers of elements from C. To this end we introduce the following

Definition 8 Let B and C are abelian groups. If Ψ is an ω-separating family of homomorphisms from B to C, then adc(Ψ) is the subgroup in Hom(B, C) generated by Ψ.

The family adc(Ψ) also ω-separates B in C but in the case when C is torsion-free it satisfies an extra pleasant property: if ψ ∈ Ψ separates an element b ∈ B then homomorphism mψ ∈ adc(Ψ) separates b in C by at least m-s power.

Let us formulate the main technical result of this section.

Theorem 3 Let G be a CSA-group satisfying the separation condition (S), G(u, A) a direct extension of a centralizer C(u) = C of an element u ∈ G of infinite order by an abelian group A = C × B. Then if B is ω-residually C with separating family Ψ then G(u, A) also is a CSAgroup satisfying the separation condition (S) and the group G(u, A) is strongly ω-residually G. Moreover, the family

Λ_{adc(Ψ)} = {λ_{ψ} : G(u, A) ⟶ G | ψ ∈ adc(Ψ)}

is an ω-separating family of retracts.

A proof of the theorem is founded on the following two lemmas.

Lemma 3 Let G be a group satisfying the separation condition (S). Then any group H which is ω-separated by G also satisfies the separation condition (S).

Proof. Let tuples W = (w_{1}, …, w_{n}) and X = (x_{1}, …, x_{n+1}) of elements in H satisfy the condition (CS). We have to prove that there exists a number m such that all elements in the set S(W, X, m) are nontrivial. The (CS) condition means that x_{i+1}^{−1}w_{i}x_{i+1} does not belong to the centralizer C(w_{i+1}) or, in other words, the commutator z_{i} = [x_{i+1}^{−1}w_{i}x_{i+1}, w_{i+1}] is nontrivial for any i = 1, …, n − 1. By our condition H is ω-separated by G. Hence there exists a homomorphism ψ : H ⟶ G which separates all commutators z_{i}‘s from zero in G. Therefore, the emages W^{ψ}, X^{ψ} of W and X, correspondingly, also satisfy the separation condition (CS) in the group G. It is clear that the set S(W^{ψ}, X^{ψ}, m) is the homomorphic image of the set S(W, X, m) in H. By the conditions of the lemma, G satisfies the separation condition (S), it implies that the set S(W^{ψ}, X^{ψ}, m) does not contaian the identity in G. Consequently, it‘s preimage S(W, X, m) does not has the identity of H. Thus, the separation condition (S) holds in the group H.□

Lemma 4 Let G be a group satisfying the conditions of the theorem. Then the extension of centralizer G(u, A) is strongly ω-residually G. Moreover, the family of homomorphisms Λ_{adc(Ψ)}, defined above, is an ω-separating family of retracts.

Proof. Let g be an element in G(u, A). One can write it in the semicanonical form

g = x_{1}b_{1}x_{2} ⋯ b_{n}x_{n+1}

where 1 ≠ b_{i} ∈ B, x_{i} ∈ G for all i and x_{i} ∉ C_{G}(u) for i = 2, …, n.

We claim, the tuples X = (x_{1}, …, x_{n+1}) and U = (u, …, u) satisfy the separation condition (CS) in G. Indeed, the group G is a CSAgroup, therefore by remark 3 the condition (CS) is equivalent to [x_{i}, u] ≠ 1 that is exactly x_{i} ∉ C_{G}(u).

Let ψ ∈ Ψ separates in C all elements b_{1}, …, b_{n}. By (S)-condition the set S(X, U^{ψ}, m) has no trivial elements in G for some large enough m. Therefore, if we will take mψ and the corresponding homomorphism λ_{mψ} the image of g under homomorphism λ_{mψ} is nontrivial in G. Thus, one can separate any single element g by any λ_{mψ} where m is large enough, say, m ≥ m(g). Therefore, if we have a finite number of elements g_{1}, …, g_{k} ∈ G(u, A) then λ_{mψ} will separate all of them provided m ≥ max{m(g_{1}), …, m(g_{k})}. It proves that G(u, A) is strongly ω-residually G and Λ_{adc(Φ)} is an ω-separating family of retracts.□

The proof of the theorem. According to the previous lemmas one needs only to prove that the group G(u, A) is a CSA-group. But it is proved in [MR2].□

Corollary 5 Let G be a non-abelian CSA-group satisfying the separation condition (S), A be a torsion-free Abelian group and G(u, A) an extension of centralizer of Lyndon’s type. The following is true:

1. (Global approximations) If A is free abelian, then G(u, A) is strongly ω-residually G;

2. (Local approximations) Any finitely generated subgroup of G^{A} is strongly ω-residually G.

Proof. Any extension of a centralizer by a free abelian group is a free extension (see lemma below). It proves 1 (due to the theorem 3). Let’s prove 2.

Any finitely generated subgroup H of G(u, A) is contained in a subgroup G(u, B) for a suitable finitely generated subgroup B ∈ A. In this case B is a free Abelian group, hence by the theorem above G(u, B) is strongly ω-residually G, so is H.□

As a particular case of corollary 5 we have

Corollary 6 Let G be a torsion-free hyperbolic group and A be a torsion-free Abelian group and G(u, A) an extension of centralizer of Lyndon’s type. The following is true:

1. If A is free abelian, then G(u, A) is strongly ω-residually G;

2. Any finitely generated subgroup of G^{A} is strongly ω-residually G.

The following result can be obtained simply iterating the corollary above.

Corollary 7 Let G be a torsion-free hyperbolic group and a group H is constructed from G by a sequence of free extensions of centralizers. Then H is a strongly ω-residually G group.

Apparently, not all extensions of centralizers are direct. We need some terminology to deal with arbitrary extensions.

Let us recall that a subgroup T is an isolated subgroup in a group H if for any h ∈ H one has h^{n} ∈ T for some nonzero ineteger n then h ∈ T. The minimal isolated subgroup in H containing T is called the isolator of T in H and is denoted by \sqrt{T}. The isolator \sqrt{T} always exists because an intersection of isolated subgroups is again an isolated subgroup. If C ≤ A are abelian groups then \sqrt{C} is the preimage in A of the torsion subgroup of the quotient A/C under the canonical epimorphism from A to A/C. For a torsion-free abelian group A a subgroup C is isolated iff C is a pure subgroup of A.

Definition 9 Let G(u, A) be an A-extension of a centralizer C(u). Then:

G(u, A) is termed to be of Lyndon’s type if C(u) is an isolated subgroup of A;

G(u, A) is termed to be of Baumslag’s type if √\overline{C(u)} = A.

The A-extension G(u, A) of a centralizer C(u) = C can be obtained in two steps: firstly, G(u, \sqrt{C}) is an \sqrt{C}-extension of centralizer C(u) and it is of Baumslag’s type; secondly, G(u, A) is an A-extension of the centralizer C_{G(u,\sqrt{C})}(u) = \sqrt{C} in G(u, \sqrt{C}) and it is of Lyndon’s type.

Definition 10 The sequence of extensions

G ≤ G(u, \sqrt{C}) ≤ G(u, A)

is called the canonical Baumslag-Lyndon decomposition (shortly, BLdecomposition).

The usefullness of the notion of isolator in our context is clear from the following lemma, which is a simple exercise from the theory of abelian groups.

Lemma 5 Let C ≤ A be torsion-free abelian groups. Then: 1) If A/C is a free abelian group then A = C × B and C = \sqrt{C};

2) If A is a finitely generated then A = \sqrt{C} × B and B is free of finite rank;

3) If C is a divisible subgroup in A then A = C × B and C = \sqrt{C}.

Thus, starting from a group G with the separation condition (S) in the BL-decomposition G ≤ G(u, \sqrt{C}) ≤ G(u, A) one can hope to get a direct A-extension of a centralizer G(u, A) of the group G(u, \sqrt{C}), but to apply theorem 3 we need to have the group G(u, \sqrt{C}) to satisfy the condition (S)! Fortunately, extensions of centralizers of Baumslag’s type do not spoil the condition (S), at least for torsion-free hyperbolic groups G. Moreover, using BL-decompositions and the lemma above one can prove the following general result.

Theorem 4 Let G be a torsion-free hyperbolic group and G(u, A) be an arbitrary A-extension of a centralizer by a torsion-free abelian group A. Then G(u, A) satisfies the separation condition (S).

Proof. Let G(u, A) be an A-extension of a centralizer C(u) = C by a torsion-free abelian group A.

In the proof we will use the fact that the condition (S) is a local property, it means that a group satisfies (S) if and only if any finitely generated subgroup of it satisfies (S). Every finitely generated subgroup of G(u, A) is contained in a subgroup of the type G(u, A_{0}), where C ≤ A_{0} and A_{0} is a finitely generated subgroup of A.

Let G ≤ G(u, \sqrt{C}) ≤ G(u, A_{0}) be a canonical BL-decomposition of the extension G(u, A). Note that C is an infinite cyclic group [Gr], therefore, \sqrt{C} is a group of rank 1, i.e. \sqrt{C} is embeddable in the group Q of rational numbers.

Case 1). A = \sqrt{C}, i.e. we have a completion of the Baumslag’s type. Then \sqrt{C} is embeddable in Q and therefore, G(u, \sqrt{C}) is embeddable in G(u, Q). But if G is a torsion-free hyperbolic group then G(u, Q) is the union of a chain of it’s hyperbolic subgroups (see [KM]). It implies that any finitely generated subgroup of G(u, \sqrt{C}) is contained in some hyperbolic group, hence it satisfies the separation condition (S) (by the local nature of this condition it is inherited by subgroups). This proves the statement in the case 1).

Case 2). A > \sqrt{C}. By lemma 5 we have A = \sqrt{C} × B for some free abelian group B. Therefore, G(u, A_{0}) is a free extension of a centralizer of the group G(u, \sqrt{C}) which satisfies the separation condition (S) due to the case 1. Now by corollary 7 G(u, A_{0}) is an ω-residually G(u, \sqrt{C}) group. Hence, by lemma 3 G(u, A_{0}) satisfies the condition (S). The theorem follows. □

Combining theorems 3, 4 and lemma 5 one can obtain the following very general result.

Theorem 5 (Global approximations for arbitrary extensions) Let G be a torsion-free hyperbolic group and A a torsion-free abelian group. Then G(u, A) is an ω-residually G(u, Q) group.

Proof. Let A_{Q} be a Q-completion of A, i.e. A_{Q} = A ⨂ Q. It is known that the canonical map a ⟶ a ⨂ 1 is a monomorphism from A to A_{Q}. Therefore, G(u, A) is embeddable in G(u, A_{Q}). Hence it is enough to prove that the latter group is an ω-residually G(u, Q) group. The centralizer C_{G}(u) is the infinite cyclic group (because G is torsionfree hyperbolic), therefore the isolator \sqrt{C}(u) in A_{Q} is isomorphic to Q. So, \sqrt{C}(u) is a direct summand in A_{Q}), say, A_{Q} = \sqrt{C}(u) × B for some divisible subgroup B. The subgroup B has a structure of a vector space over Q , hence, it is a direct sum of copies of Q. It implies that B is an ω-residually Q group. Now we are in a position to apply the theorem 3, and it follows that G(u, A_{Q}) is a strongly ω-residually G(u, Q) group. □

Note, that in general the group G(u, Q) is not a subgroup of G(u, A), therefore the separating homomorphisms are not retracts.

A lot of interesting examples of global approximations of extensions of centralizers G(u, A) come from this theorem. We would like to mention the following, which is a crucial one for extensions of centralizers by fields of characteristic 0.

Corollary 8 (Global approximations for divisible extensions). Let G be a torsion-free hyperbolic group and A a torsion-free divisible abelian group. Then G(u, A) is strongly ω-residually G(u, Q) group.

Now we are ready to formulate an important and somewhat surprising result about finitely generated subgroups of extensions of centralizers.

Corollary 9 (Local approximations). Let G be a torsion-free hyperbolic group and G(u, A) be an arbitrary A-extension of a centralizer C(u) by a torsion-free abelian group A. Then any finitely generated subgroup H of G(u, A) is an ω-residually ⟨G, t | u = t^{m}⟩ group, where ⟨G, t | u = t^{m}⟩ is a torsion-free hyperbolic group which is obtained from G by free extracting an m-root from u, m depends on H.

Proof. Let H be a finitely generated subgroup of G(u, A). The whole group G(u, A) and hence the subgroup H is ω-residually G(u, Q). But any finitely generated subgroup in the latter group is contained in a subgroup of the type ⟨G, t | u = t^{m}⟩, which is hyperbolic (see [BF] or [KM]). □

5 Extensions of centralizers of G as a source of ω-residually G groups

In this section we will show that starting from a CSA-group G satisfying the separation condition (S), for example, starting from a torsionfree hyperbolic G, one can constract different types of ω-residually G groups. It is worthwhile to mention among such groups free extensions of centralizers of G, free products of copies of G, some special types of free products with amalgamation and HNN-extensions - all those groups that are embeddable in free extensions of centralizers of G. The idea of this type of examples was discovered by G. Baumslag [GB4] and has proved to be very fruitful. From other hand, it turned out that all known examples of finitely generated ω-residually G groups (at least, when G is torsion-free hyperbolic) are embeddable in a group which is constructed from G using a finite number of extensions of centralizers. All such groups are contained in the Z[x]-completion G^{Z[x]} of the group G (see the next section). It leads to the following main conjecture about ω-residually G groups.

Conjecture 1 Let G be a torsion-free hyperbolic group. Then every finitely generated ω-residually G group is embeddable in the group G^{Z[x]}.

Propostion 4 Let G be a nonabelian CSA-group. Then G ∗ G is embeddable in any free rank 1 extension G(u, t) of a centraizer of G.

Proof. Let G(u, t) = ⟨G, t | [C(u), t] = 1⟩ be a free rank 1 extension of the centralizer C(u) of an element u ∈ G. The group G is not abelian, therefore there exists v ∈ G such that [u, v] ≠ 1. From CSAcondition we have C(u)^{v} ∩ C(u) = 1. Let z = (tvt)^{2} ∈ G(u, t) and G^{z} be a conjugate of G by z in G(u, t). We claim that the subgroup H =< G, G^{z} > in G(u, t) generated by G and G^{z} is isomorphic to G ∗ G.

Let

x = g_{0}(z^{−1}h_{1}z)g_{1}(z^{−1}h_{2}z)g_{2} … (z^{−1}h_{n}z)g_{n}  (2)

be an element in H such that: h_{i} ≠ 1, i = 1, …, n and g_{j} ≠ 1, j = 1, …, n − 1 in the case n ≠ 0; and g_{1} ≠ 1 in the case n = 0. Using the method of normal forms for free products to prove that H ≅ G ∗ G one needs only to prove that x ≠ 1 in the HNN-extension G(u, t). To get a contradiction, let us suppose that x = 1. If n ≠ 0 then the word ( 2) contains a pinch. Let us consider a “typical” subword of the word ( 2):

(z^{−1}h_{i}z)g_{i}(z^{−1}h_{i+1}z)  (3)

There are only two possible types of pinches in ( 3), namely, h_{i} ∈ C(u) and g_{i+1} ∈ C(u). In those cases one has t^{−1}h_{i}t = h_{i} and t^{−1}g_{i+1}t = g_{i+1}. Then “making a pinch” one can rewrite the word ( 3) according to the reductions in HNN-extensions. In the rewritten word new pinches can occur only in subwords of the type v^{−1}h_{i}v and vg_{i+1}v^{−1}. But by the choice of v and the conditions h_{i} ∈ C(u), g_{i+1} ∈ C(u) one can deduce that v^{−1}h_{i}v ∉ C(u) and vg_{i+1}v^{−1} ∉ C(u). It means that there are no more pinches in ( 3). This arguments show one that pinches in ( 3) and hence in ( 2) do not affect each other and, therefore, after all possible reductions in ( 2) the remaining word represents a nontrivial element in G(u, t) in the case when n ≠ 0. But if n = 0 then x = g_{0} ≠ 1 by the supposition. The contradiction proves the proposition. □

As a corollary of the proposition above and theorem 3 one can obtain the following result which was mentioned already in section 2.

Theorem 6 Let G be a CSA-group satisfying the separation condition (S) and containing an element of infinite order, H_{1} and H_{2} be arbitrary groups. Then the following statements are equivalent to one another:

1) H_{1} and H_{2} are ω-residually G groups;

2) H_{1} ∗ H_{2} is an ω-residually G group.

Proof. 2) implies 1) because the property “to be an ω-residually G group” is inherited by subgroups. Let us prove that 1) implies 2). Using normal forms of elements in a free product and universal property of free products one can prove that H_{1} ∗ H_{2} is an ω-residually G ∗ G group. To finish the prove one needs only to apply the proposition above embed G ∗ G into some extension of centralizer G(u, t) of rank 1 and then apply the theorem 3 to ω-approximate G(u, t), and hence G ∗ G, in G. □

For free groups the theorem 6 was proved by B. Baumslag in [BB], but in his case a free product of two free groups is obviuosly a free group, so he did not need proposition 4.

Corollary 10 Let G be a torsion-free hyperbolic group, H_{1} and H_{2} be arbitrary groups. Then the following statements are equivalent to one another:

1) H_{1} and H_{2} are ω-residually G groups;

2) H_{1} ∗ H_{2} is an ω-residually G group.

Now we will describe another very fruitfull method to construct ω-residually G groups. Let G be a group, u ∈ G and ϕ ∈ Aut(G) be an automorphism of G. By G_{ϕ,u} = G ∗_{C(u)=C(u)^{ϕ}} G we denote the free product of G by G with the amalgamation of C(u) and C(u)^{ϕ} with respect to ϕ. This group is called a double of G with respect to ϕ and u. The following result was mentioned in [GB4].

Propostion 5 Let G(u, t) be a free rank 1 extension of a centralizer C(u) of G. Then the group G ∗_{C(u)=C(u)^{ϕ}} G is embeddable in G(u, t) for any automorphism ϕ ∈ Aut(G).

Sketch of the proof. Let H = ⟨G, G^{t}⟩ be a subgroup in G(u, t) generated by G and t-conjugate G^{t}. Using normal forms in HNN-extensions one can prove that H ≃ G_{ϕ,u}. □

And again we have the corresponding corollary for torsion-free hyperbolic groups.

Corollary 11 Let G be a torsion-free hyperbolic group, u ∈ G and ϕ ∈ Aut(G) be an automorhism of G. Then the group G_{ϕ,u} is an ω-residually G group.

Using subgroups of the double G_{ϕ,u} one can constract ω-residually G groups with very complicated presentations.

Namely, let S be a subgroup of G containing C(u) and S = ⟨X | R⟩ a presentation of S. Let g(x) be a word representing an element g ∈ G in terms of the generators X of S. Then the subgroup of the double G_{ϕ,u} which is generated by G and S^{ϕ} is ω-residually G and has the representation ⟨G, S^{ϕ} | g = g^{ϕ}(x)⟩_{g∈C(u)}. This method to construct ωresidually G groups shows one that the extesion of a centralizer G(u, t) contains subgroups which look (on the first glance) very different from extensions of centralizers, free products or doubles.

6 Approximation theorems for exponential groups

In this section we apply the results about extensions of centralizers to exponential groups. The key point here is that a tensor completion of a CSA-group can be constructed using extensions of centralizers. Therefore, all theorems about approximations of extensions of centralizers from section 3 are valid (in proper reformulation) for tensor completions of CSA-groups.

We will assume that the notions of an A-group and A-completion are known (see [MR2] for details). R.Lyndon [L] consedered a Z[x]- completions of a free group F, G.Baumslag [ GB1] investigated a Qcompletion of F. It turned out that this were the basic examples of the most principal types of completions.

Let A be an torsion-free abelian group with a fixed subgroup C ≤ A and G be a C-group. According to the definition of A-completion G^{A} of the C-group G we have an action of A on G^{A} and this action extends the already existed action of C on G. Moreover, the action of A on G^{A} is unique up to an automorphism of A identical on C.

In particular, any group G is a Z-group, therefore in an A-completi-on of G we identify the standard action of integers Z on G with the action of some fixed cyclic subgroup C =< c >≤ A by the rule g^{n} = g^{nc} for all g ∈ G, n ∈ Z (here c is a fixed generator of C).

Definition 11 The A-completion G^{A} of a C-group G is termed to be of Lyndon’s type if the subgroup C ≤ A is an isolated subgroup of A.

Definition 12 An A-completion of a C-group G is termed to be of Baumslag’s type if \sqrt{C} = A.

Let G^{A} be an A-completion of a CSA C-group G by a torsion-free abelian group A with respect to an inclusion C ≤ A. Then:

G^{\sqrt{C}} is an \sqrt{C}-completion of G and it is of Baumslag’s type; G^{\sqrt{C}} is a \sqrt{C}-subgroup of G^{A} and G^{A} is an A-completion of G^{\sqrt{C}}

which is of the Lyndon’s type .

Definition 13 The sequence of completions

G ≤ G^{\sqrt{C}} ≤ G^{A}

is called the canonical Baumslag-Lyndon decomposition (shortly, BLdecomposition).

Below we list some properties of A-completions from the paper [MR2]. Let G be a nonabelian CSA group and A a torsion-free abelian group. Then:

1). The A-completion G^{A} can be constructed as union of infinite chain of A-extensions of centralizers, starting from the group G:

G = G_{0} ≤ G_{1} ≤ … ≤ G_{α} ≤ G_{α+1} ≤ … ≤ G_{δ} = G^{A},

where for each ordinal α < δ the group G_{α+1} is an A-extension of some centralizer in G_{α} and G_{λ} is a union of the all preceeding terms of this chain for a limit ordinal λ.

2). Let C ≤ B be subgroups of A and G be a C-group Then the canonnical map G^{B} :⟶ G^{A} is a monomorphism.

3). If H is a finitely generated subgroup of G^{A} then H is contained in a subgroup of the type G^{A_{0}}, where A_{0} is a finitely generated subgroup of A.

Now we will formulate a simple lemma which allows one to constract a separating retracts for direct limits.

Lemma 6 Let

G = G_{0} ≤ G_{1} ≤ … ≤ G_{α} ≤ G_{α+1} ≤ … ≤ G_{δ} = _{α<δ}^{⋃} G_{α}

be a chain of groups such that G_{α+1} is strongly ω-residually G_{α} for any α < δ. Then G_{δ} is ω-residually G_{α} for any α < δ.

Using the lemma above one can easily reformulate the theorems from Section 3 to the case of completions of CSA-groups.

We start with the main technical theorem.

Theorem 7 Let C ≤ A be torsion-free abelian groups and A is strongly ω-residually C, let G be a torsion-free CSA C-group satisfying the separation condition (S). Then the tensor A-completion G^{A} is a strongly ω-residually G.

As corollaries of the main theorem we formulate several results about approximations of A-completions of Lyndon’s type.

Corollary 12 (Approximations of Lyndon’s type) Let G be a nonabelian CSA-group satisfying the separation condition (S), A be a torsion-free Abelian group and G^{A} be a completion of Lyndon’s type. Then the following statements are true:

1. (Global) If A is a free abelian, then G^{A} is strongly ω-residually G;

2. (Local) Any finitely generated subgroup of G^{A} is strongly ωresidually G.

Proof. 1) follows from the fact that any free abelian group is ωresidually Z. Let’s prove 2). Any finitely generated subgroup H of G^{A} is contained in a subgroup G^{B} for a suitable finitely generated subgroup B of A. In this case B is a free Abelian group, hence by theorem 3 G^{B} is strongly ω-residually G, so is H. □.

Corollary 13 Let G be a torsion-free hyperbolic group, A be a torsionfree Abelian group and G^{A} a completion of Lyndon’s type. Then:

1. If A is a free abelian, then G^{A} is strongly ω-residually G;

2. Any finitely generated subgroup of G^{A} is strongly ω-residually G.

Now we can formulate the corresponding results for completions of the general type.

Theorem 8 Let G be a torsion-free hyperbolic group and A be a torsionfree abelian group. Then the completion G^{A} satisfy the separation condition (S).

Theorem 9 (Global approximations for arbitrary completions) Let G be a torsion-free hyperbolic group and A a torsion-free abelian group. Then the completion G^{A} is an ω-residually G^{Q} group.

Corollary 14 (Global approximations for divisible completions) Let G be a torsion-free hyperbolic group and A a torsion-free divisible abelian group. Then the completion G^{A} is a strongly ω-residually G^{Q} group.

Corollary 15 (Local approximations, general type) Let G be a torsionfree hyperbolic group and A a torsion-free abelian group. Then any finitely generated subgroup of the A-completion G^{A} is an ω-residually {G_{i} | i ∈ I} , where G_{i} is a torsion free hyperbolic group which is obtained from G by a finite sequence of freely extracting roots.

Proof. According to Global approximations for arbitrary completions the group G^{A} is an ω-residually G^{Q} group. In [KM] it was proved that any finitely generated subgroup of G^{Q} is contained in a subgroup which is obtained from G by a finite sequence of free extracting roots and, moreover, the latter subgroup is torsion-free hyperbolic. □

Finally, we will show the relationship between the approximations of the group A and approximations of the completion G^{A}.

Let A, A_{i}, i ∈ I, be abelian groups, Φ_{i} ⊂ Hom(A, A_{i}) and Φ = ⋃ Φ_{i}. We will say Φ is additively closed iff Φ_{i} is a subgroup of Hom(A, A_{i}). If A is an ω-residually {A_{i} | i ∈ I} group and Φ is the separating family of homomorphisms, then, taking the subgroups generated by Φ_{i}, one can allways assume that Φ is additively closed. For a group G a homomorphism ϕ : A ⟶ A^{ϕ} induces a homomorphism f_{ϕ} : G^{A} ⟶ G^{A^{ϕ}} of the corresponding completions by the rule: f_{ϕ} : g^{a} ⟶ g^{ϕ(a)}.

Propostion 6 Let A_{i}, i ∈ I be torsion-free Abelian groups and a group A is ω-residually {A_{i} | i ∈ I}. Suppose the corresponding separating family of homomorphisms Φ is additively closed. If G is a CSA-group satisfying the separation condition (S), then G^{A} is an ω-residually {G^{A^{ϕ}} | ϕ ∈ Φ} and a separating family of homomorphisms can be chosen as follows {f_{ϕ} : G^{A} ⟶ G^{A^{ϕ}}|ϕ ∈ Φ}.

Proof. Let us denote F = {f_{ϕ} : G^{A} ⟶ G^{A^{ϕ}}|ϕ ∈ Φ} and choose arbitrary nontrivial elements h_{1}, …, h_{n} ∈ G^{A}. This elements are contained in a subgroup H ≤ G^{A} which is obtained from G by finitely many extensions of centralizers. By induction on the number of such extensions we can assume that H contains a subgroup H_{0}, which is ω-separated in {G^{A^{ϕ}} | ϕ ∈ Φ} by F, and H is an extension of the centralizer of an element u ∈ H_{0}. In this case any element h ∈ H can be represented in a reduced form h = g_{0}a_{1}g_{2} … a_{m}g_{m}, where [g_{i}, u] ≠ 1 for any i = 1, … m − 1. Therefore, the image of h under f = f_{ϕ} ∈ F will be equal to h^{f} = g_{0}^{f}a_{1}^{f}g_{2}^{f} … a_{m}^{f}g_{m}^{f}. We can choose the homomorphism f such that [g_{i}^{f}, u^{f}] ≠ 1, it implies that elements g_{0}^{f}, …, g_{m}^{f} and u^{f}, …, u^{f} satisfy the separation condition (CS). The family F is additively closed, hence we can assume also, that a_{1}^{f}, …, a_{m}^{f} are big enough powers in the group A^{ϕ}. According to theorem 8 the group G^{A^{ϕ}} satisfies the separation condition (S), consequently h^{ϕ} ≠ 1. It is clear that one can choose the same ϕ for all elements h_{1}, …, h_{n}. □

7 Applications

7.1 Equationally Noetherian groups and Hilbert’s basis theorem

To go further we need some terminology about equations over groups. Let G be a group, F = F (X) the free group with a basis X = {x_{1}, x_{2}, …}, G[X] = G ∗ F free product of G and F.

An element w from G[X] is called an equation over group G. We write this as w = 1. As an element of the free product w can be written as a product of some elements x_{1}, …, x_{n} from X ∪X^{−1} (which are called variables) and elements g_{1}, …, g_{m} from G (constants). To emphasize this we will write, sometimes, w(x_{1}, …, x_{n}, g_{1}, …, g_{m}) = 1 or, simply, w(x, g) = 1. A system of equations over group G is an arbitrary set of equations W = {w_{i} = 1 | i ∈ I} (notation: W = 1). A system W = 1 is termed to be coefficient-free if it does not contain constants from G.

A solution of a system W (x_{1}, …, x_{n}) = 1 over a group G is a tuple of elements a_{1}, …, a_{n} ∈ G such that after replacement of each x_{i} by a_{i} in every equation w(x, g) = 1 one gets a trivial element in a group G. On other hand, a solution of the system W = 1 over G can be described as a retract ϕ : G[X] ⟶ G such that ϕ(W) = 1. It is clear that these definitions are equivalent, we will use both of them. By S(W) we denote the set of all solutions in G of the system W = 1. Two systems W_{1} = 1 and W_{2} = 1 are equivalent over G if S(W_{1}) = S(W_{2}).

Now we ready to formulate the Noethereian property for equations which is, in fact, an analogue of Hilbert’s Basis theorem for groups.

Definition 14 A group G is Equationally Noetherian if any system of equations over G is equivalent to some finite part of it.

The first result in this area was obtained by Guba in [Gub], where he proved that free groups are equationally Noetherian.

Propostion 7 The following statements are true:

1) The general linear group GL(n, R) over an arbitrary noetherian ring R is equationally Noetherian;

2) Any subgroup of an equationally Noetherian group is equationally Noetherian.

Proof. First of all, let us notice that the group GL(n, R) is a subgroup of the group SL(n + 1, R). Therefore, it is enough to prove the theorem only in the case SL(n, R). Any system of equations over group SL(n, R) corresponds to some system of polynomials over the ring R. Indeed, to construct the system one needs only to replace each variable x in an equation w = 1 by the corresponding n × n matrix (x_{kl}), where x_{kl} are new variables, to multiply all matrices in w and to equalize the entries (polynomials!) in the equation w = 1. According to the classical Hilbert’s basis theorem the polynomial system is equivalent to some finite part of it. Going backward, one can choose a finite set of corresponding group equations which have induced this finite polynomial part. It is not hard to see that this finite subsystem of group equations is equivalent to the whole original system. It proves item 1).

The second claim is quite obvious. □

This result is contained in [S].

Corollary 16 Any linear group is equationally Noetherian.

In particular, the following groups are linear over a Noetherian commutative ring and hence they are equationally Noetherian: free groups , polycyclic groups [Aus], holomorphs of polycyclic groups [Mer], finitely generated metabelian groups [R2].

Coefficient-free systems of equations play important role in an investigation of universal theories of groups. It turned out that for finitely generated groups the property “to be equationally Noetherian” depends only on coefficient-free systems of equations.

Propostion 8 Let G be a finitely generated group. Then if any coefficient-free system of equations over G is equivalent to some finite part of it, then the group G is equationally Noetherian.

Proof.Suppose any system of coefficient-free equations over G is equivalent to some finite part of it. Any system of equations over G can be written as W (x, a) = 1 where x is a finite tuple of variables and a is a finite generaing set for G, so we have only finitely many variables and finitely many constants. Let us replace the generators from a by new variables, say y. The system W (x, y) = 1 is equivalent over G to some finite part W_{0}(x, y) = 1. One can consider now only those solutions of the system W_{0}(x, y) = 1 for which y = a. It will be exactly the solution set for W_{0}(x, a) = 1. But this solutions also satisfy the whole system W (x, y) = 1, therefore they are solutions of the system W (x, a) = 1. Hence this systems are equivalent.□

Propostion 9 Let N be the class of all equationally Noetherian groups. Then

1) N is closed under subgroups, direct products and ultrapowers;

2) N is universally axiomatizable, in paricular, it is closed under ∀- equivalence i.e. if a group G is an equationally Noetherian group and H ≡_{∀} G then H is an equationally Noetherian group too;

3) N is closed under approximations, i.e. if G is an equationally Noetherian group and H is residually G then H is an equationally Noetherian group.

The proof of 1) is straightforward and we omit it. 2) follows from 1). Let us prove 3). Let G ∈ N and H is a residually G group. Any coefficient-free system W (x_{1}, …, x_{n}) = 1 is equivalent over G to some finite part W_{0}(x_{1}, …, x_{n}) = 1 of it. Assume, to the contrary, that W = 1 is not equivalent to W_{0} = 1 over H. It means, that there are some elements r_{1}, …, r_{n} ∈ H such that W_{0}(r_{1}, …, r_{n}) = 1 in H but w(r_{1}, …, r_{n}) ≠ 1 in H for some w ∈ W. By the conditions of the proposition one can find a homomorphism f : H ⟶ G with the property f(w(r_{1}, …, r_{n})) ≠ 1 in G. It contradicts the choice of the subsystem W_{0}, because W_{0}(f(r_{1}), …, f(r_{n})) = 1 in G. □

We will say a group G is locally ω-residually H if any finitely generated subgroup of G is locally ω-residually H.

Theorem 10 (A criterion for ∀-equivalence). The following is true:

1) if G ≡_{∀} H and at least one of them is an equationally Noetherian group, then G is locally ω-residually H and H is locally ω-residually G;

2) if G is locally ω-residually H and H is locally ω-residually G, then G ≡_{∀} H.

Proof. Let G ≡_{∀} H and one of them is equationally Noetherian. By previuos proposition both of G, H are equationally Noetherian groups. Let G_{0} be a finitely generated subgroup of G and

< x_{1}, …, x_{n}|r_{1}, r_{2}, …, >

be a finitely generated presentation of G_{0}. The system of equations {r_{i}(x_{1}, …, x_{n}) = 1|i = 1, 2, …} is equivalent over H to some finite part of it, say, {r_{i}(x_{1}, …, x_{n}) = 1|i ≤ m}. Let

u_{1}(x_{1}, …, x_{n}), …, u_{k}(x_{1}, …, x_{n})

be arbitrary elements from the group G_{0}. We have to find a homomorphism from G_{0} to H which separates the given elements

u_{1}(x_{1}, …, x_{n}), …, u_{k}(x_{1}, …, x_{n}).

The formula

Φ = ∃x_{1} … x_{n}(m ; ⋀ ; 1 r_{i}(x_{1}, …, x_{n}) = 1) k ; ⋀ ; 1 u_{j}(x_{1}, …, x_{n}) ≠ 1)

holds in G therefore it holds in H. Consequently, there exist some elements h_{1}, …, h_{n} ∈ H such that r_{1}(h_{1}, …, h_{n}) = 1, …, r_{m}(h_{1}, …, h_{n}) = 1, but u(h_{1}, …, h_{n}) ≠ 1, …, u_{k}(h_{1}, …, h_{n}) ≠ 1. From the choice of the subsystem r_{i}, i ≤ m we have all equalities r_{i}(h_{1}, …, h_{n}) = 1, i = 1, 2, …, . Hence, the map x_{i} ⟶ h_{i}, i = 1, …, n, can be extended to a homomorphism f_{0} : G_{0} ⟶ H which separates the elements u_{1}(x_{1}, …, x_{n}), …, u_{k}(x_{1}, …, x_{n}). It proves that G is locally ω-residually H. Due to the symmetry for G and H one can conclude that H is also locally ω-residually G.

Now, let us suppose that a group G is locally ω-residually H. We claim that any ∃-sentence which holds in G is true in H. From general predicate calculus we know that one can consider only formulae of the type

Φ = ∃x_{1}, …, x_{n}( s ; ⋀ ; 1 u_{i}(x_{1}, …, x_{n}) = 1 t ; ⋀ ; 1 v_{j}(x_{1}, …, x_{n}) ≠ 1)

Let elements g_{1}, …, g_{n} ∈ G satisfy this sentence in G. Denote by G_{0} the subgroup < g_{1}, …, g_{n} >. By the conditions of the theorem there exists a homomorphism f : G_{0} ⟶ H which separates the elements v_{1}(g_{1}, …, g_{n}), …, v_{t}(g_{1}, …, g_{n}) in H. It implies that the images f(g_{1}), …, f(g_{n}) satisfy the same equalities u_{i}(f(g_{1}), …, f(g_{n})) = 1, i = 1, …, s, and inequalities v_{i}(f(g_{1}), …, f(g_{n})) ≠ 1, i = 1, …, t, in H. Therefore, the sentence Φ holds in H. It shows, according to the symmetry of G and H, that G ≡_{∃} H, and hence G ≡_{∀} H. □

Corollary 17 Let G, H be groups and at least one of them be a linear group. Then G ≡_{∀} H iff G is locally ω-residually H and H is locally ω-residually G.

The proof follows from the theorem above and the fact that linear groups are equationally Neotherian. □

In the case when one of the groups G or H is a nonabelian free group the corollary above was proved in [R].

A subgroup G is called existentially closed in a group H, if any existential sentence with constants from G holds in the whole group H if and only if it holds in the subgroup G.

Corollary 18 Let G ≤ H be groups and H is locally strongly ωresidually G. Then G is existentially closed in H, in particular, G ≡_{∀} H.

Proof is exactly the same as in the second part of the theorem 10.

Theorem 11 Let G be a torsion-free CSA-group satisfying the separation condition (S) (see section 3), A be a torsion-free abelian group and G^{A} be a tensor A-completion of Lyndon’s type. Then G is existentially closed in G^{A} and G^{A} ≡_{∀} G.

Proof. Due to the corollary 12 the group G^{A} is locally strongly ωresidually G. The conclusion comes from the corollary above. □

Corollary 19 Let G be a torsion free hyperbolic group, A be a torsion free abelian group and G^{A} be a tensor A-completion of G of the Lyndon’s type. Then G is existentially closed in G^{A} and G^{A} ≡_{∀} G.

Corollary 20 Let G be a torsion-free hyperbolic group and A be an abelian Q-group. Then G is existentially closed in G^{A} and G^{A} ≡_{∀} G^{Q}.

Proof follows from the corollary 14 and corollary 18.□

Theorem 12 Let F be a nonabelian free group and A be an abelian torsion-free group. Then the tensor completion F^{A} of Lyndon’s type has a decidable universal theory.

Proof. From the theorem 11 the group F^{A} has the same universal theory as the group F. The universal theory of F is decidable [Mak]. □

8 Problems

Problem 1 Does any group acting freely on some Λ-tree satisfy the separation condition (S)?

Problem 2 Is any finitely generated group acting freely on some Λtree an ω-residually torsion-free hyperbolic group?

Problem 3 Describe the one- relator groups which are ω-residually free.

Problem 4 What kinds of free products with amalgamation along the cyclic subgroups preserve the property to be a residually free group?

The second block of problems conserns the equationally Noetherian groups.

Problem 5 Is an arbitrary hyperbolic group equationally Noetherian?

Problem 6 Is an arbitrary group acting freely on a Λ-tree equationally Noetherian?

Problem 7 Is a free product of equationally Noetherian groups equationally Noetherian?

Problem 8 Is a free pro-p-group equationally Noetherian?

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Partially supported by RFBR grant N 95-01-00513.