MILNOR PROPERTY IN FINITELY GENERATED SOLUBLE GROUPS.

FRANÇOISE POINT

Abstract. We give a characterization of the finitely generated nilpotent-by-finite groups among the class of finitely generated soluble groups.

1. Introduction.

In [9], we considered a family of identities which we called Milnor identities after a property singled out by J. Milnor in [6]. These identities generalize monoidal (positive) identities, Engel or more generally m-identities

Let us recall the definition of those identities. Let w(x, y) be a word in x, y and assume that it is homogeneous in x and y. This means that the total degrees of both x and y in w are 0. So, for instance we can write w(x, y) as a product of conjugates of x by powers of y, namely w(x, y) = ∏ y^{n_{i}}.x^{m_{i}}.y^{−n_{i}}. To the above homogeneous word, we found convenient to associate the following rational function by replacing the factor y^{n_{i}}.x^{m_{i}}.y^{−n_{i}} by m_{i}.Y^{n_{i}}; and so we obtain: ∑ m_{i}.Y^{n_{i}} = Y^{ℓ}.P (Y), where ℓ ∈ ℤ and P (Y) a polynomial with non zero constant term whenever w(x, y) ∉< x^{y^{k}}; k ∈ ℤ >′, where ” ′ ” means that we take the derived subgroup. An identity of the form w(x, y) = 1 is called a Milnor identity (see [9]) if it implies in the variety of all groups an identity of the form y^{n_{0}}.x^{m_{0}}.y^{−n_{0}} · · · .y^{n_{r}}.x^{m_{r}}.y^{−n_{r}}.v(x, y) = 1, where n_{i}, m_{i} ∈ ℤ, n_{0} < ⋯ < n_{r}, r ∈ ω, gcd(m_{i}; 1 ≤ i ≤ r) = 1 and v(x, y) ∈< x^{y^{k}}; k ∈ ℤ >′.

The definition we gave in [9] is syntactic, but G. Endimioni noted that we have the following nice characterization in terms of varieties:

Proposition 1.1. (See [2]) Let A be the variety of abelian groups and A^{p} be the variety of abelian groups of exponent p. Then, w(x, y) = 1 is a Milnor identity iff the variety of groups satisfying w = 1 does not include any varieties of the form A^{p}A, with p a prime number. □

In the class of finitely generated elementary amenable groups, to satisfy a monoidal identity implies the property of being nilpotent-by-finite (see [9]). Moreover, thanks to E. Zel’manov solution of the restricted Burnside problem, one can show that the nilpotency class and the exponent are bounded in terms of the identity (see [1]). However, it is now well-known that outside that class there are finitely generated groups satisfying a monoidal identity which are not soluble-by-periodic (see [8]).

Date: June 20, 2002.

Senior Research Associate at the ”Fonds National de la Recherche Scientifique”.

Here, we will consider a weaker property namely that any subgroup generated by two elements satisfies a Milnor identity with a bound on the absolute value of the coefficients of the corresponding family of polynomials. We called that property locally N-Milnor and we will show that a finitely generated soluble group locally N-Milnor is nilpotent-by-finite, using the description of the just non-polycyclic groups ([14]). Conversely note that any nilpotent-by-finite group is locally N-Milnor. In [3], G. Endimioni gave another characterization of nilpotent-by-finite groups in the class of finitely generated soluble groups (see Remark 3 below).

2. Bounded Milnor property.

Notation 2.1. We will use the following definition for the commutator: [a, b] := a.b.a^{−1}.b^{−1} and we will denote by a_{i} the conjugate b^{i}.a.b^{−i}. Further, [a,_{n} b] is defined by induction as [[a,_{n−1} b], b]; note that the polynomial associated to that word is equal to (1 − Y)^{n}.

If H denotes a group, then H′ is the derived subgroup i.e. the subgroup of H generated by the commutators of elements of H.

Definition 2.1. Let H_{a,b}=< a^{b^{z}} : z ∈ ℤ > and H′_{a,b} its derived subgroup. A group G is locally N-Milnor if for all a, b in G there is n ∈ ℕ and m_{1}, ..., m_{n} ∈ ℤ such that

(⋆) ba^{m_{1}}b^{−1}...b^{n}a^{m_{n}}b^{−n}.u = 1,

where u belongs to H′_{a,b}, with 1. gcd(m_{1}, ..., m_{n})=1 and

2. |m_{i}| ≤ N for all i ∈ {1, ⋯, n}.

Remark 2.2. If H_{a,b} is finitely generated, then for instance we have bab^{−1}= b^{2}a^{m_{1}}b^{−2}...b^{n}a^{m_{n}}b^{−n}.u with u in H′_{a,b} and the exponents of b strictly increasing. So if G is polycyclic the first condition above is always satisfied. But what might happen is that when considering H_{a,b^{n}}, the maximum in absolute value of the exponents of a in the corresponding expression (⋆) may increase with n.

Remark 2.3. Let G be a group with a Fitting subgroup F which is torsion-free abelian and which has a free abelian subgroup F_{0} of finite rank such that F/F_{0} is a π-group for π a finite set of prime numbers. If a ∈ F, let ord_{F_{0}}(a) be the order of a in the quotient F/F_{0}. Let r be the rank of F_{0}, so given any a_{i}, if n = lcm(ord_{F_{0}}(a_{i}); 1 ≤ i ≤ r+1), there exist m_{1}, ⋯, m_{r} such that a_{1}^{n.m_{1}}. · · · .a_{r+1}^{n.m_{r+1}} = 1 and w.l.o.g. we may assume that gcd(m_{1}, ..., m_{r+1})=1. This relation then entails that a_{1}^{m_{1}}. · · · .a_{r+1}^{m_{r+1}} = 1 since the Fitting subgroup F is torsion-free.

Remark 2.4. Let us recall G. Endimioni’s characterization. Let a, b belong to G and consider a word of the form w(a, b).[a,_{n} b]^{e_{0}}. · · · .[a,_{n+m} b]^{e_{m}} = 1, with e_{i} ∈ ℤ, e_{0}, e_{m} ≠ 0 and w(a, b) ∈< a^{<a,b>} >′. Note that the polynomial P (Y) associated to that word is of the form: (1 − Y)^{n}.(e_{0} + e_{1}.(1 − Y) + ⋯ + e_{m}.(1 − Y)^{m}). (Note that the gcd of the coefficients of P (Y) is equal to the gcd(e_{i}; 0 ≤ i ≤ m).) Let Φ_{∗}(a, b) (respectively Φ^{∗}(a, b)) the least integer |e_{0}| (respectively |e_{m}|) such that a, b satisfy a relation of this form and put Φ_{∗}(a, b)=Φ^{∗}(a, b)=∞ if it is not the case.

G. Endimioni showed that a finitely generated soluble group is polycyclic iff for any a, b we have Φ^{∗}(a, b) = 1 and that a polycyclic group G is nilpotent if for any a, b ∈ G, the sequence (Φ_{∗}(a, b^{k}))_{k>0} is bounded.

First, let us show that any nilpotent-by-finite group is locally N-Milnor.

Proposition 2.1. Let G be a group with a normal nilpotent subgroup G_{0} of nilpotency class c and of index n. Then, G is locally N-Milnor.

Proof: First, a nilpotent-by-finite group satisfies a monoidal identity. This has been proved independently in two papers ([7], [5]). Second, any group satisfying a monoidal identity is (locally) 1-Milnor (see [11] Lemma 4.9 and [9] Lemma 6).

Since the proof is elementary, we will recall it below. W.l.o.g. we may assume that the monoidal identity is of the form w_{1}(x, y) = w_{2}(x, y), where w_{1}, w_{2} are two (positive) words in {x, y} beginning and ending with different letters and of the same length (if it is not the case and if both words are non-trivial, replace w_{1} by w_{1}.w_{2} and w_{2} by w_{2}.w_{1}; otherwise if the identity has the form w(x, y) = 1, replace it by w(x, y) = w(y, x)).

Now in order to transform the identity into the equality between two distinct products of conjugates of x by strictly increasing positive powers of y, one makes the substitution x → x.y. Namely, G ⊧ ∀x, y w_{1}(x, y) = w_{2}(x, y) is equivalent to G ⊧ ∀x, y w_{1}(x.y, y) = w_{2}(x.y, y), in turn equivalent to G ⊧ ∀x, y x.y^{i_{1}}.x.y^{−i_{1}}. · · · .y^{i_{n}}.x.y^{−i_{n}} = y^{j_{1}}.x.y^{−j_{1}}. · · · .y^{j_{m}}.x.y^{−j_{m}}, 1 ≤ i_{1} < i_{2} < ⋯ < i_{n} < d, 1 ≤ j_{1} < j_{2} < ⋯ < j_{m} < d where d is equal to the length of the word w_{1}(x, y) with the convention that whenever n or m are zero the sequence is empty. So in Definition 2.1, in the corresponding expression (⋆), the exponents m_{i} of x belong to {−1, 1} and the word u is trivial. □

Theorem 2.2. Let G be a finitely generated soluble group, locally N-Milnor. Then G is nilpotent-by-finite.

The proof of the theorem will follow from the two propositions below. But, let us first state a corollary.

Corollary 2.3. Let G be a finitely generated linear group and suppose that G is locally N-Milnor. Then G is nilpotent-by-finite.

Proof: We use Tits alternative for linear groups. Let F_{2} be the free group on two generators. Since F′_{2}/F′′_{2} is free abelian of rank ω, a locally N-Milnor group cannot contain F_{2}. So, by J. Tits result (see [12]), G is soluble-by-finite and so by the preceding Proposition, G is nilpotent-by-finite. □

Proposition 2.4. Let G be a finitely generated soluble group, locally N-Milnor. Then G is polycyclic.

Proof: We argue by contradiction and assume that G is not polycyclic. Then by [14], G has a quotient G/N that is just non-polycyclic (jnp). This means that G/N is not polycyclic but every proper quotient of G/N is polycyclic. As the property of being locally N-Milnor is invariant under taking quotient, we can assume that G is a finitely generated jnp locally N-Milnor group. We will make use of the description of the jnp groups, first undertaken by J.R.J. Groves and then by D. Robinson and J. Wilson. We summarize in a lemma those properties that we need on the Fitting subgroup of such group (see [14] (2.2), (2.3) and [4] Lemma 5.2).

Lemma 2.5. Let G be a jnp group and let F be its Fitting subgroup, then F is a non-trivial abelian normal subgroup of G and either:

1. F is an elementary abelian p-group for some prime number p.

2. F is a torsion free abelian group with a free abelian subgroup F_{0} of finite rank

such that F/F_{0} is a π-group for some finite set π of prime numbers.

□

As G is jnp, F cannot be finitely generated. But, in both cases, we will get a contradiction by showing that the property of being locally N-Milnor forces F to be finitely generated.

In order to prove that F is finitely generated, it suffices to show that

H(a, b) =< a^{b^{i}}; i ∈ ℤ >

is finitely generated for all a ∈ F and all b ∈ G (see Lemmas 1, 2 and 3 in [6])). Indeed, as G/F is polycyclic, it is finitely presented and thus F =< c_{1}, c_{2}, ⋯, c_{r} >^{G} for some finite number of elements c_{1}, ⋯, c_{r} of F. Let F = H_{0} ⊴ H_{1} ⊴ ⋯ ⊴ H_{s} = G with each of the successive quotients being cyclic, namely H_{i+1}/H_{i}=< b_{i+1}H_{i} > for some b_{i+1} ∈ G with 0 ≤ i ≤ s − 1. Then, for 1 ≤ j ≤ r,

< c_{j} >^{G}=< c_{j}^{b_{1}^{n_{1}}.··· .b_{s}^{ns}} : n_{1}, ⋯, n_{s} ∈ ℤ >,

and we apply the fact that each H(c_{j}, b_{1}) is finitely generated, 1 ≤ j ≤ r; then we apply again to each generator say d_{k} of the group spanned by all H(c_{j}, b_{1}), 1 ≤ j ≤ r, that H(d_{k}, b_{2}) is finitely generated and we continue inductively.

1. In the case where F is an elementary abelian p-group, this is a direct consequence of the fact that G is locally N-Milnor thanks to condition 1 in Definition 2.1. Indeed, for any a ∈ F and b ∈ G, we have, using Notation 2.1,

a_{0}^{m_{0}}. · · · .a_{s}^{m_{s}} = 1,

with p |̸m_{0}.m_{s}. So, H(a, b) is generated by a_{1}, ⋯, a_{s}.

2. In the case where F is torsion-free, the proof that F is finitely generated is combinatorially more involved. We will show in next lemma that H(a, b) is finitely generated by embedding it in the torsion-free finitely generated abelian group F_{0}. The map will be the n^{th}-power map, for some n, which is in this case injective since F is torsion-free and a morphism since F is abelian. This follows from the next lemma.

Let F_{p} be the set of elements of F of order not divisible by p in the quotient F/F_{0}. Note that F_{p} is a subgroup of F and ⋂_{p∈π} F_{p} ⊆ F_{0}.

Lemma 2.6. Let p ∈ π and let a ∈ F and b ∈ G. Then H(a, b)^{p^{α}} ⊆ F_{p}, for some positive integer α.

Proof:

Let a ∈ F. We denote by |a| the order of the element aF_{p} in the quotient F/F_{p} and we will write a ≡ b to mean that ab^{−1} ∈ F_{p}. Recall that a_{i} denotes the conjugate b^{i}.a.b^{−i}.

We will prove the lemma by showing that the sequence |a_{i}|, |a_{i+1}|, ⋯ with increasing indices as well as the sequence |a_{i}|, |a_{i−1}|, ⋯ with decreasing indices are bounded. We prove only the first claim as the proof of the latter is similar. We argue by contradiction by assuming that the sequence of orders modulo F_{p} with increasing indices is unbounded (∗).

By replacing a by a conjugate if necessary we can assume that a ∉ F_{p}. Since G is locally N-Milnor, there exist integers m_{0}, ⋯, m_{s} with m_{0}.m_{s} ≠ 0, gcd(m_{0}, ⋯, m_{s}) = 1 such that

a_{0}^{m_{0}}. · · · .a_{s}^{m_{s}} = 1 (1)

with |m_{i}| < N, 0 ≤ i ≤ s and s > 0.

Let ℓ be the largest integer in {0, ⋯, s} such that m_{ℓ} is prime to p. By assumption (∗) on the orders, we have that ℓ < s. Let (k_{i})_{i∈ω} be a strictly increasing sequence of natural numbers defined inductively as follows. Let k_{0} be the smallest index j in {0, ⋯, ℓ} such that |a_{k_{0}}| = max{|a_{j}| : 0 ≤ j ≤ ℓ} and for i ≥ 0, let k_{i} be the smallest integer strictly greater than k_{i−1} such that |a_{k_{i}}| > |a_{k_{i−1}}|. (The choice of k_{0} imposes that k_{1} > ℓ.) In other words, the sequence of the k_{i}’s is the sequence of indices of the a_{i}’s where the order strictly increases.

The next claim will prove that the distance between two consecutive indices of elements in this sequence is less than or equal to s. Claim: We have that k_{i+1} − k_{i} ≤ s, for i ≥ 1. Proof of the Claim: We conjugate the relation (1) by b^{k_{i}−ℓ} (note that this is a positive power of b since k_{1} > ℓ) and we get:

a_{k_{i}−ℓ}^{m_{0}}. · · · .a_{k_{i}}^{m_{ℓ}}. · · · .a_{k_{i}+s−ℓ}^{m_{s}} = 1. (2)

Let p^{n+1} be the order of a_{k_{i}}, n ≥ 1, then raising (2) to the power p^{n}, and calculating modulo F_{p} we get:

a_{k_{i}}^{p^{n}.m_{ℓ}}. · · · .a_{k_{i}+s−ℓ}^{p^{n}.m_{s}} ≡ 1.

Since m_{ℓ} is prime to p, we have that a_{k_{i}}^{p^{n}.m_{ℓ}} ∉ F_{p}. Therefore,

a_{k_{i}+1}^{p^{n}.m_{ℓ+1}}. · · · .a_{k_{i}+s−ℓ}^{p^{n}.m_{s}} ∉ F_{p}.

By choice of ℓ, all the integers m_{ℓ+1}, ⋯, m_{s} are divisible by p. Let j = min{1, ⋯, s − ℓ} such that a_{k_{i}+j}^{p^{n}.m_{ℓ+j}} ∉ F_{p}, then |a_{k_{i}+j}| > p^{n+1} = |a_{k_{i}}|. So, we get that k_{i} < k_{i+1} ≤ k_{i} + j ≤ k_{i} + s. □

It follows from the Claim that if k_{1} ≤ j < k_{i} for i ≥ 2 and k_{i} − j ≥ r.s, then |a_{k_{i}}| ≥ p^{r}.max{|a_{t}| : 0 ≤ t ≤ j}.

Now, choose a natural number f satisfying

p^{f} ≤ N < p^{f+1}.

Let k = (f + 1).s. We now apply the local N-Milnor property to the pair a and b^{k} and obtain the following relation:

a_{0}^{n_{0}}.a_{k}^{n_{1}} · · · .a_{k.t}^{n_{t}} = 1, (3)

with n_{0}, ⋯, n_{t} ∈ ℤ, n_{0}.n_{t} ≠ 0 and |n_{i}| < N, 0 ≤ i ≤ t.

Conjugating relation (3) by a suitable power of b, we may assume that k.t is equal to some k_{i}, i ≥ 1.

Now, in one hand, raising (3) to power p^{w} := max{|a_{j}| : 0 ≤ j ≤ k.(t − 1)}, we get that a_{k.t}^{n_{t}.p^{w}} ≡ 1 and so:

|a_{k.t}| ≤ p^{w}.p^{f},

and on the other hand applying the Claim, that

|a_{k.t}| > p^{w}.p^{f}.

This contradiction finishes the proof of the lemma and of Proposition 2.4. □

Proposition 2.7. Let G be a polycyclic group, locally N-Milnor. Then G is nilpotentby-finite.

Proof: Here we will use the proof of J. Tits of the result of J.A. Wolf that a polycyclic group of polynomial growth is nilpotent-by-finite (see A2 Lemma 2 in [13]).

Let G_{0} be a subgroup of G of finite index which is poly-infinite cyclic ([10]). By induction on the Hirsh number of G, G_{0} has a nilpotent-by-finite subgroup G′_{0} such that G_{0}/G′_{0} is isomorphic to ℤ. Let t be an element of G_{0} − G′_{0} generating the quotient. Denote by G_{1} the Fitting subgroup of G_{0}. We will show that for some k, t^{k} centralizes G_{1} and so the subgroup < G_{1}, t^{k} > is nilpotent and has finite index in G. We may refine the lower central series of G_{1} into a central series which is normalized by t and where each infinite quotient is a finitely generated free abelian group. Consider such infinite quotient, say of the form ℤ^{m}, for some m, and set V := ℤ^{m} ⊗ ℂ. Extend the action of t on V by linearity and represent this action by a matrix in coefficients in ℂ. It is conjugated to a matrix of the form

( \begin{matrix}λ & 1 & ¯0 & ⋯ \\ 0 & λ & 1 & ¯0 \\ ⋯ & & ⋯ \\ ¯0 & ⋯ & λ & 1 \\ ¯0 & ⋯ & & λ \end{matrix} )

with λ in ℂ. Since G is locally N-Milnor, for every integral power t^{z} of t there are polynomials P_{z}(x) belonging to ℤ[x] with coefficients bounded by N, such that P_{z}(λ^{z}) = 0 (⋆), z ∈ ℤ. Using the same reasoning as in [13] A2, Lemma 3, this implies that |λ|=1. For sake of completeness, let us indicate the proof below.

First, just working with t itself, let us prove that |λ| < N. Consider all the expressions of the form e_{0} + e_{1}.λ + ... + e_{i}.λ^{i}, e_{j} in {0, ..., N − 1}. We will prove that if |λ| ≥ N, then these are all distinct, which contradicts (⋆). By the way of contradiction, suppose that ∑_{i=0}^{m} e′_{i}.λ^{i} = ∑_{j=0}^{m} e′_{j}.λ^{j}, for some positive m, where 0 ≤ e′_{i}, e′_{j} ≤ N − 1 and (e′_{i}) ≠ (e′_{j}). We then get an expression of the form:

∑_{k<n} ±f_{k}.λ^{k} = f_{n}.λ^{n}, where 1 ≤ f_{k} ≤ N − 1, 0 ≤ k ≤ n ≤ m. So, |f_{n}.λ^{n}|= | ∑_{i<n} ±f′_{i}.λ^{i}| ≤ ∑_{i<n} |f′_{i}|.|λ^{i}| < |λ|^{n} with 1 ≤ f′_{i} ≤ N − 1, which is absurd.

Second, repeating the above argument with every power t^{z} of t, z ∈ ℤ, we get that |λ|=1.

By [13] A2, Lemma 3 (i), the orbits of {t^{z} : z ∈ ℤ} in V have compact closure and so in ℤ^{m}, these orbits are finite. So, there exists ℓ such that t^{ℓ} acts as the identity on ℤ^{m}. Proceeding in the same way on each infinite quotient, we finally get a natural number k such that t^{k} centralizes G_{1}, therefore, G is nilpotent-by-finite. □

Remark 2.5. Note that in the above Lemma, we only used condition 2 of being locally N-Milnor.

Acknowledgments. We would like to thank the referee for taking the time to read (and understand) the first version of this paper. We are indebted to him for entirely rewriting the proof of Proposition 2.4.

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Françoise Point, Institut de Mathématique et Informatique, Université de Mons-Hainaut, Le Pentagone, 6, avenue du Champ de Mars, B-7000 Mons, Belgium

E-mail address: point@logique.jussieu.fr