L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Yves CORNULIER

On the Cantor-Bendixson rank of metabelian groups Tome 61, n^o2 (2011), p. 593-618.

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ON THE CANTOR-BENDIXSON RANK OF METABELIAN GROUPS

by Yves CORNULIER

Abstract. — We study the Cantor-Bendixson rank of metabelian and virtually metabelian groups in the space of marked groups, and in particular, we exhibit a sequence (Gn) of 2-generated, finitely presented, virtually metabelian groups of Cantor-Bendixson rankωⁿ.

Résumé. — On étudie le rang de Cantor-Bendixson des groupes métabéliens ou virtuellement métabéliens dans l’espace des groupes marqués, et on exhibe notam- ment une suite (Gn) de groupes virtuellement métabéliens de présentation finie à deux générateurs, de rang de Cantor-Bendixson égal àωⁿ.

1. Introduction

LetGbe a discrete group. Under pointwise convergence, the setN(G) of normal subgroups is a Hausdorff compact, totally disconnected space. This topology, sometimes referred to as the Chabauty topology, was studied in many papers, including [6, 12, 7, 8, 11]. IfF_d denotes the non-abelian free group ondgenerators, we can viewN(Fd) as the set Gd ofmarked groups ondgenerators, through the identification N7→F_d/N.

As a topological space, the identification of Gd seems to be a difficult problem. We focus here on the Cantor-Bendixson rank, which is defined as follows. IfX is a topological space, we define its derived subspaceX⁽¹⁾ as the subset of accumulation points inX. Iterating over ordinals

X⁽⁰⁾=X, X^(α+1)=X^(α)(1), X^(λ)= \

β<λ

X^(β)for limitλ,

Keywords: Metabelian groups, space of marked groups, Cantor-Bendixson analysis, Bieri-Strebel invariant, lattice of subgroups.

Math. classification:20E15, 13E05, 20F05, 20F16, 57M07.

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we have a non-increasing familyX^(α)of closed subsets. Ifx∈X, we write CBX(x) = sup{α|x∈X^(α)}

if this supremum exists, in which case it is a maximum. Otherwise we say that x is in the condensation part (or perfect kernel) of X and we write CBX(x) =C, where the symbolCis not an ordinal. If CBX(x)6=Cfor all x∈X, i.e. if X^(α) is empty for some ordinal, we say that X is scattered.

IfG is a group, we define its (intrinsic) Cantor-Bendixson rankcb(G) as CB_N(G)({1}).

GroupsGwith cb(G) = 0, which include finite groups and simple groups, are calledfinitely discriminableand were considered in [11]. However, most groups, like infinite residually finite groups, are not finitely discriminable.

Let us begin by a very simple example (contained in Proposition 4.1).

Proposition 1.1. — Let G be a finitely generated nilpotent group.

Then cb(G) =h(G), the Hirsch length ofG.

For instance, we have cb(Z^k) =k, which was already mentioned, without proof, in [11, Section 6]. The reader can check it as an warm-up exercise;

precisely the statement to prove by induction is that if A is a finitely generated abelian group virtually isomorphic to Z^k, then it has Cantor- Bendixson rank cb(A) =k.

So far all known examples either satisfied cb(G)< ωor cb(G) =C. Our main result is to leap fromω toω^ω.

Theorem 1.2. — Fix any d> 2. Then Gd contains points of Cantor- Bendixson rank equal to any ordinal α < ω^ω. More precisely, for every α < ω^ω, there exists a finitely presented, 2-generated metabelian-by-(finite cyclic) groupH with cb(H) =α.

The second statement implies the first as for every finitely presented d-generated group H, the space Gd contains N(H) as a clopen subset.

Note that it is known that, on the other hand, as a particular case of [18, Theorem 3], every non-elementary hyperbolic groupGsatisfies cb(G) =C (another proof is given by [7] when G is torsion-free) and in particular cb(Fd) =C.

A pleasant class of groups, for which the study of the Cantor-Bendixson rank can be carried out, is the class of groups satisfying max-n, i.e. in which there is no infinite increasing sequence of normal subgroups.

Proposition 1.3. — LetGbe a group satisfying max-n. Then the space N(G)is scattered. If moreover every quotient ofGis residually finite, then

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we have cb(G) = sup{cb(H) + 1}, where H ranges over quotients groups ofGwith infinite kernel.

An important class of groups with max-n is the class of finitely generated, virtually abelian-by-polycyclic groups [14], and these groups are residually finite (as well as their quotients) by a result of Roseblade [19]. In particular, this includes finitely generated, virtually metabelian groups, for which however residual finiteness is much easier to obtain [15].

The gist of Theorem 1.2 is the study of the Cantor-Bendixson rank of finitely generated metabelian groups. However in this case (metabelian in- stead of virtually metabelian) we have a bound on the exponent of the Cantor-Bendixson rank in terms of the number of generators. Recall the (standard) wreath productHoGrefers to the semidirect productH^(G)oG.

Theorem 1.4. — LetGbe a finitely generated metabelian group.

(1) Fixd>0. Suppose that Gsits inside an exact sequence 1→M →G→Q→1,

whereM is abelian andQis abelian ofQ-rank 6d. Then cb(G)< ω^d+1.

Moreover, this bound is sharp, as the wreath productZ^koZ^dsatisfies cb(Z^koZ^d) =ω^d·k

for allk>1.

(2) IfGisd-generated andd>2, then cb(G)6ω^d·(d−1),

with equality if and only ifGis isomorphic to the free metabelian group ondgenerators.

(3) If G is d-generated and the above exact sequence is split, then cb(G)< ω^d, and this bound is sharp ifd>2.

We actually give a precise computation of cb(G) for any finitely generated metabelian group. For a general finitely generated metabelian groupG, we proceed as follows. If N is a normal subgroup ofG, define the G-Hirsch lengthhG(N) as the supremum of lengthskof chains of normal subgroups ofGcontained inN

N0⊂N1⊂ · · · ⊂Nk, Ni/Ni−1 infinite∀i.

TheHirsch radical ofGis the largest normal subgroup Hir(G) =N of G such thathG(N)<∞. This is well-defined, sinceGsatisfies max-n.

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In Section 3, we recall the notion of reduced length of modules intro- duced in [10]; this is an ordinal-valued length characterized, whenA is a finitely generated commutative ring, by the property, for finitely generated A-modulesM

`⁰(M) = sup{`⁰(M/N) + 1 :N infiniteA-submodule of M}.

Notably, ifM has Krull dimensiond>0, then ω^d−16`⁰(M)< ω^d

(withω⁻¹= 0). LetW(M) denote the largestA-submodule ofM of Krull dimension61.

Theorem 1.5. — LetGbe a finitely generated metabelian group in an extension

1→M →G→Q→1,

withM, Qabelian, and viewM as aZ[Q]-module. Then (1) We have cb(G) =`⁰(M/W(M)) +h_G(Hir(G)).

(2) In particular, ifdis the Krull dimension of theZ[Q]-moduleM, we have

`⁰(M)6cb(G)< `⁰(M) +ω.

(3) IfP is a prime ideal inZ[Q], ifZ[Q]/P has Krull dimensiond>2, M is isomorphic to a torsion-free Z[Q]/P-module of rank r, and the action ofQonM is faithful, then cb(G) =ω^d−1·r.

To prove the finite presentability of the virtually metabelian groups in Theorem 1.2, we apply a general criterion due to Bieri and Strebel [4]. This criterion is explained in Section 6. Besides, it is important that the groups in Theorem 1.2 are finitely presented: indeed, ifGis a group withdgiven generators, thenN(G) is always closed inGd, but is open if and only ifG is finitely presented (see [11, Lemma 1.3]). Otherwise we can define cbê(G) as the Cantor-Bendixson rank ofGas an element ofGd. By [11, Lemma 1], this does not depend on the choice of a finite generating family of G. If G is finitely presented then cbê(G) = cb(G). Groups with cbê(G) = 0, called isolated groups, are finitely presented and are the main subject of the paper [11]. On the other hand, we have the following result, which was asserted without proof in [11, Section 6] in the case ofZoZ. We give here a proof in Section 8.

Proposition 1.6. — LetH, Gbe finitely generated groups, withH 6=

{1} andGinfinite. Then

cb^e(HoG) =C.

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Question 1.7. — Is it true that any infinitely presented, finitely generated metabelian group satisfies cb^e(G) =C?

With L. Guyot, we obtain a positive answer in the special case of abelian- by-cyclic finitely generated groups, for instance forZ[1/6]o2/3Z.

Question 1.8. — The bounds given in Theorem 1.4 are not optimal when the metabelian group is assumed finitely presented. What are then the optimal bounds?

Let X be a topological space. The least α such thatX^(α) =X^(α+1) is called theCantor-Bendixson rank ofX and is denoted by CB(X). This is always well-defined, since the non-increasing chain (X^(α)) always stabilizes.

For instance, CB(X) = 0 if and only if X is perfect. In general, CB(X) is the supremum of CB_X(x) + 1, where x ranges over the points not in the condensation part of X. By Theorem 1.2, for every d > 2, we have CB(G_d) > ω^ω. If G is a group, and if cb(G) 6= C, then CB(N(G)) >

cb(G) + 1. This is an equality under the assumptions of Proposition 1.3, but not in general: for instance, in [11, Proof of Theorem 5.3], an isolated group G was given with a normal subgroup K such that G/K is free of rank two. Then sinceGis isolated, cb(G) = 0, but it follows from Theorem 1.2 that CB(N(G))>ω^ω, because sinceG/K is finitely presented, N(G) containsN(G/K) as clopen subset.

Last but not least, we can ask

Question 1.9. — For26d <∞, do we have CB(Gd)> ω^ω?

I do not have any clue how to construct a finitely generated group G with cb(G) = ω^ω, although it does probably exist. Since Gd is compact metrizable, CB(G_d) is a countable ordinal. It would be surprising if its value would depend ond>2.

Outline.The groups referred to in Theorem 1.2 are constructed in Section 2, at the end of which we indicate why they satisfy the claimed property;

this relies on results of Sections 3, 5, and 6. The assertions in Proposition 1.3 are particular cases of Lemma 3.6, Proposition 3.14 and Corollary 3.15.

The inequality in Theorem 1.4(1) follows from Theorem 1.5(2), and the example in Theorem 1.4(1) is obtained in Section 4. Theorem 1.4(2),(3) are obtained in Section 7. Theorem 1.5 is proved in Paragraph 3.4.

Section 2 essentially extends the introduction. Sections 4, 5, and Para- graph 7.2 partly rely on Section 3. At this notable exception, the different sections can be read independently.

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Acknowledgments. I thank Alexander Kechris, Simon Thomas, Robert Young, and especially Luc Guyot and Pierre de la Harpe for valuable dis- cussions and suggestions.

2. Examples of finitely presented virtually metabelian groups

2.1. Construction

Let us describe the groups we need to obtain the second assertion of Theorem 1.2. We postpone all the proofs to Paragraph 2.2. The easiest (and most natural) construction provides a 4-generated group; we then explain how to reduce to 3, and then 2 generators.

Fix the integerd>1, anddformal variables (xi), which for convenience we view as indexed byZ/dZ. Consider the ringA_d=Z[(x_i), s⁻¹] wheres= Q

i(xi−x²_i). ThenZ^2d=Z^Z/dZ×Z^Z/dZacts by multiplication onAd, where, if the canonical basis is denoted by ((ei),(fi)),ei acts by multiplication by x_i andf_i by multiplication by 1−x_i.

The semidirect productHd=AdoZ^2d has a faithful representation by square 2-matrices overA_d

(0, ei)7→

xi 0 0 1

; (0, fi)7→

1−xi 0

0 1

; u= (1,0)7→

1 1 0 1

,

whose image is the set of all matrices of the form Q

ixⁿ_iⁱ(1−xi)^mⁱ P

0 1

,((n_i),(m_i))∈Z^2d, P ∈A_d.

Proposition 2.1. — The (2d+ 1)-generated metabelian group H_d is finitely presented, and

cb(H_d) =ω^d.

The finite presentability of H_d is obtained from the computation of the Bieri-Strebel geometric invariant carried out in Section 6.

The computation of the Cantor-Bendixson rank essentially relies on the computation of the Cantor-Bendixson rank of the ringA_d (i.e. of the ideal {0} in the set of ideals ofAd), which was proved in [10] to be equal toω^d. Next, we can form the semidirect productH_doZ/dZ, whereZ/dZ(whose canonical generator we denote by σ) permutes shifts the variables. This group is virtually metabelian, and is generated by{u, e₁, f₁, σ}. As it con- tainsHd as a subgroup of finite index, it is finitely presented as well.

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Proposition 2.2. — The4-generated virtually metabelian groupΓd= HdoZ/dZis finitely presented and satisfies

cb(Γd) =ω^d.

Now the proof relies on the study of the space of Z/dZ-invariant ideals in A_d. This fits in the context of modules endowed with an action of a finite group, and was not considered in [10], so we prove the necessary preliminaries in Section 5.

To pass from 4 to 3 generators, assume that d is odd and replace σ by the generator γ of Z/2dZ which acts on Ad by ring automorphisms, mappingx_i to 1−x_i+1for alli∈Z/dZ. In particularγ^dsendsx_i to 1−x_i and γ^d+1 = σ. So the group Γ⁰_d generated by {u, e1, γ} contains Γd as a subgroup of index 2. For similar reasons, it has Cantor-Bendixson rankω^d. Finally, to get a 2-generated group, we consider the subgroup Λd generated by{ue₁, γ}. Denote by Λ⁰_d the normal subgroup generated byue₁, so that Λd= Λ⁰_dohγi.

Proposition 2.3. — The metabelian groupΛ⁰_dis finitely presented, as well asΛd. It lies in an extension

1→M →Λ⁰_d →Q→1,

withQ 'Z^2d, freely generated by the images of all ue_i and uf_i, and M is isomorphic as aZ[Q]-module to the kernel of the ring homomorphism of Z[Q]ontoZ[1/2]mapping all generators to1/2. Moreover, we have

cb(Λd) =cb(Λ⁰_d) =ω^d.

2.2. Proofs

The finite presentability of H_d is a consequence of Corollary 6.6, using Bieri-Strebel’s characterization of finitely presented metabelian groups (Theorem 6.1). So its overgroups of finite index Γd and Γ⁰_d are also finitely presented. Finally Lemma 6.7 implies that Λ⁰_d is finitely presented, as well as its overgroup of finite index Λd.

All the groupsG=H_d,Γ_d,Γ⁰_d,Λ_d,Λ⁰_d arise in an extension 1→M →G→R→1,

whereR containsZ^2d as a subgroup of finite index, and M is an ideal in A_d. ForG=H_d,Γ_d,Γ⁰_d,M =A_d; in the two last cases,M 6={0}because [ue1, uf1] 6= 1. As the Krull dimension of Ad is d+ 1, by Corollary 5.3,

`⁰_Q(M) = ω^d in all cases. Since M satisfies the hypotheses of Corollary 3.18, we obtain`⁰(G) =`⁰_Q(M) =ω^d.

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3. Length and Cantor-Bendixson rank

3.1. Length of noetherian modules

LetAbe a ring (not necessarily commutative). Recall that anA-module M hasfinite lengthif there is an upper bound on the lengthdof increasing chains

0 =M0⊂M1⊂ · · · ⊂Md=M

ofA-submodules of M, and the least bound is called thelengthof M. By a theorem of Jordan and Hölder, modules of finite length can be characterized as modules that are simultaneously noetherian (every increasing chain of submodules stabilizes) and artinian (every decreasing chain stabilizes). However in general, most noetherian modules have infinite length and it is natural to expect a notion of ordinal length. This was first done by Bass [2] (in a commutative setting), using well-ordered decreasing chains of submodules. Then Gulliksen [13] provided the inductive definition which follows, slightly less intuitive at first sight, but more handy to deal with.

Definition 3.1. — Define inductively, for every noetherianA-module, itsordinal length `(M) =`_A(M)as

`(M) = sup{`(M/N) + 1 :N nonzeroA-submodule ofM}.

This has to be viewed as an inductive definition: the starting point is

`({0}) = sup(∅) = 0. In more generality, if X is a noetherian partially ordered set, we can define an ordinal-valued function on X by `(x) = sup{`(y) + 1 : y > x}. The uniqueness of ` follows from noetherianity of X. For the existence, set M_x ={y :y >x}, consider the setV of u∈X such that there exists a function ` satisfying the inductive condition on M_u. IfV 6=X, then its complement contains a maximal elementu. So for any x > u, the number `(x) is uniquely defined. Therefore the inductive definition shows that` can be defined on Mu, contradicting that u /∈ V. Here, the partially ordered set is the set of quotients of the moduleM.

If α is a non-zero ordinal, there exists a unique ordinal β such that ω^β 6α < ω^β+1, and we writeβ = deg(α).

Definition 3.2. — Let M be a noetherian A-module. The Krull di- mensionof M is defined as the ordinaldeg(`(M))if M 6={0}, and−1 if M ={0}.

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The more usual notion of Krull dimension, used in the non-commutative setting, is called the “deviation of the poset of submodules ofM” (see [17, Chap. 6]). We do not need this definition, but it is important to mention that it is equivalent to the one given here [13, Theorem 2.3]. Moreover, whenAis commutative, it coincides [17, Chap. 6.4] with the very classical notion of Krull dimension defined in terms of chains of prime ideals, defined inductively as

dim(M) = sup{dim(A/P)+1 :P non-minimal prime ideal of A/Ann(M)}.

LetA again be arbitrary (not necessarily commutative) and letM be a noetherianA-module. If `(M) =ω^α for some ordinalα, we say thatM is critical, orα-critical. This means that the Krull dimension ofM isα, but the Krull dimension of any proper quotient ofM is< α. Acritical series forM is a composition series

0 =M₀⊂M₁⊂ · · · ⊂Mk =M, where eachMi/Mi−1isαi-critical and

α₁6· · ·6αk.

A critical series always exists forM [17, 6.2.20], and in practice is easy to write down. It is a particular case of [13, Theorem 2.1] that we then have

`(M) =ω^α^k+· · ·+ω^α¹;

ifnα is the number ofisuch that αi=α, then the family of non-negative integers (n_α) is finitely supported, and we can rewrite this formula as

`(M) =X

α

ω^α·n_α (sum in reverse order).

In particular, we have

Proposition 3.3. — Let A be a commutative noetherian ring, P a prime ideal such thatA/P has Krull dimensionα>1. LetM be a torsion- freeA/P-module of rankr. Then`_A(M) =ω^d·r.

Proof. — Let (M_i) be a critical series as above. We haveα_i6αfor alli.

SinceM is torsion-free overA/P, so isM1, so we haveα1=α. Therefore αi=dfor alli; sinceMi/M_i−1is a subquotient ofM, this implies that it is torsion-free; since it is critical, it is torsion-free of rank one. In particular, k=r. The formula above gives`(M) =ω^d·r.

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3.2. General lengths

LetH be a semigroup. AnH-group is by definition a groupGendowed with an action ofH by group endomorphisms.

Example 3.4. — If A is a ring and M is an A-module, then M is an A-group, whereA is viewed as a multiplicative semigroup.

An H-group satisfies H-max-n if every non-decreasing sequence of H- stable normal subgroups ofGstabilizes. WhenGis anH-group andN an H-stable normal subgroup ofG, the groupN has natural action of bothH andG, hence of the semidirect product GoH, which we denote byGH for short.

Let G be an H-group satisfying H-max-n. We can define by induction thelength ofGas

`H(G) = sup(`H(G/N) + 1),

where G/N ranges over all proper H-quotients of G. Here of course we assume sup(∅) = 0, which gives`_H({1}) = 0.

Lemma 3.5. — For any ordinal β 6 `(M), there exists a H-quotient G/N ofGsuch that`H(G/N) =β.

Proof. — Letβ be the smallest counterexample, andγ6αthe smallest ordinal > β such that `H(G/N) = γ for some H-quotient G/N. Then the definition of `H(G/N) leads to the existence of N⁰ such that β 6

`_H(G/N⁰)< γ, contradicting the definition of γ.

Let N_H(G) denote the set of H-stable normal subgroups of G. Let cbH(G) denote the Cantor-Bendixson rank of{1} inNH(G).

Lemma 3.6. — LetGbe anH-group satisfyingH-max-n. ThenNH(G) is scattered and cb_H(G)6`_H(G).

Proof. — This is a straightforward induction on`_H(G).

However this result is not optimal in general (see Proposition 3.14).

If α, β are ordinals, their natural sum is defined inductively as follows [21, XIV.28]

α⊕β = max sup

γ<α

((γ⊕β) + 1),sup

γ<β

((α⊕γ) + 1)

! ,

with sup∅ = 0. Recall that any ordinal αhas a unique Cantor form [21, XIV.19]

α=X

γ

ω^γ·nγ,

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where the sum is indexed by γ ranging over the ordinals in decreasing order, and (nγ) is a finitely supported family of non-negative integers. If β=Pω^γ·n⁰_γ is also in Cantor form, then their natural sum is

α⊕β =X

γ

ω^γ·(nγ+n⁰_γ).

The following lemma generalizes Theorem 2.1 and Proposition 2.11 in [13].

Lemma 3.7. — Let G be an H-group with H-max-n, in an exact sequence ofH-groups

1→M →G→Q→1.

Then

`H(Q) +`GH(M) 6 `H(G) 6 `H(Q)⊕`GH(M).

WhenG=M×Qas anH-group,

`H(G) =`H(Q)⊕`GH(M).

Proof. — All facts are directly obtained by induction on `H(G), in a similar fashion. For instance, let us check that`_H(G)6`_H(Q)⊕`_GH(M).

IfN is anH-invariant normal subgroup ofG, then ifN∩M is non-trivial, by induction

`_H(G/(N∩M))6`_H(Q)⊕`_GH(M/(N∩M))< `_H(Q)⊕`_GH(M) by definition of the natural sum. IfN∩M ={1} and the projectionp(N) ofN onQis non-trivial, then by induction

`_H(G/N)6`_H(Q/p(N))⊕`_GH(M)< `_H(Q)⊕`_GH(M)

again by definition of the natural sum. In all cases, if N is non-trivial, we get`H(G/N)< `H(Q)⊕`GH(M), so passing to the supremum we get

`_H(G)6`_H(Q)⊕`_GH(M).

3.3. Reduced length

There is a well-defined notion of left Euclidean division for ordinals. In particular, ifαis an ordinal, it is easy to check that there is a unique ordinal α⁰ such thatα=ω·α⁰+rwithr < ω. For instance, 1⁰= 0, (ωⁿ⁺¹)⁰=ωⁿ forn < ω, and (ω^α)⁰ =ω^α forα>ω.

Definition 3.8. — Let G be a H-group satisfying H-max-n. Define

`⁰_H(G)as (`_H(M))⁰.

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Proposition 3.9. — We have

`⁰_H(G) = sup{`⁰_H(G/N) + 1},

whereG/N ranges over allH-quotients ofGwith`_GH(N)>ω.

Proof. — Define`⁰⁰_H(G) as above by the inductive formula

`⁰⁰_H(G) = sup{`⁰⁰_H(G/N) + 1 :`_GH(N)>ω}.

Let us prove by induction onα=`H(G) that`⁰_H(G) =`⁰⁰_H(G). LetN be a normalH-subgroup ofGwith `_GH(N)>ω. By Lemma 3.7, we have

`H(G/N) +`GH(N)6`H(G).

So

ω·(`⁰_H(G/N) + 1)6ω·`⁰_H(G);

since we can “simplify” by ω on the left, this gives, using the induction hypothesis

`⁰⁰_H(G/N) + 16`⁰_H(G);

taking the supremum overN, we get`⁰⁰_H(G)6`⁰_H(G).

Conversely, takeα < `⁰_H(G). Soω·α+ω6`_H(G). By Lemma 3.5, there exists a normal H-subgroup N of G with `H(G/N) = ω·α. In particu- lar,`⁰_H(G/N) = α, so by induction hypothesis,`⁰⁰_H(G/N) =α. If we had

`GH(N)< ωthen Lemma 3.7 would imply`H(G)< ω·α+ω, a contradic- tion. Therefore`⁰⁰_H(G)> α. Since this holds for anyα < `⁰_H(G), we deduce

`⁰_H(G)6`⁰⁰_H(G).

Lemma 3.10. — Let Gbe an H-group with H-max-n, in an exact sequence ofH-groups

1→M →G→Q→1.

Then

`⁰_H(Q) +`⁰_GH(M) 6 `⁰_H(G) 6 `⁰_H(Q)⊕`⁰_GH(M).

WhenG=M×Qas anH-group,

`⁰_H(G) =`⁰_H(Q)⊕`⁰_GH(M).

Proof. — Since the sum and natural sum commute with α 7→ α⁰, this

immediately follows from Lemma 3.7.

Lemma 3.11. — Let G be an H-group satisfying H-max-n. Then for every H-stable normal subgroup F of G with `GH(F) < ω, we have

`⁰_H(G/F) =`⁰_H(G).

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Proof. — By definition of`⁰_H, we have`⁰_GH(F) = 0. So this follows readily

from Lemma 3.10.

Lemma 3.12. — Let G be an H-group satisfying H-max-n. Suppose thatGis residually finite as anH-group. Then

`_H(G)< ω⇔`⁰_H(G) = 0⇔Gis finite.

Proof. — The left-hand equivalence is true by definition of`⁰_H, without assuming residual finiteness. Clearly G finite implies `H(G) < ∞. Con- versely ifG is infinite, it has a decreasing sequence (Mn) ofH-stable (finite index) normal subgroups, the sequence `H(G/Mn) is increasing and

`H(M)>ω.

Definition 3.13. — Using Lemma 3.7, we can define, for every H- groupGwithH-max-n,E_H(G), [resp.W_H(G)], as its unique largest normal H-invariant subgroupNwith`H(N)< ω[resp.`⁰_H(N)< ω, i.e.`H(N)< ω²].

It follows from Lemmas 3.7 and 3.10 that EH(G/EH(G)) = {1} and WH(G/WH(G)) ={1}.

Proposition 3.14. — LetGbe anH-group satisfyingH-max-n. Sup- pose thatGis residually finite as anH-group, as well as all itsH-quotients.

Then cbH(G) =`⁰_H(G).

Proof. — Consider a counterexample G with `H(G) = α and assume that the proposition is proved for everyH-group with`H < α.

By Lemma 3.12, M = E_H(G) is finite. Therefore if N is close enough to {1} in NH(G), we have N ∩M = {1}; if N 6= {1} this forces N to be infinite. In this case, by induction cb_H(G/N) = `⁰_H(G/N) < `⁰_H(G).

Accordingly, cbH(G)6`⁰_H(G/N) + 1, so cbH(G)6`⁰_H(G).

Conversely, consider any ordinal β < αand a finite subsetI ofG− {1}.

Then Ghas a proper quotient G/N with `_GH(N)> ω (so N is infinite) and `⁰_H(G/N) = β, and has an H-stable normal finite index subgroup L withL∩I=∅; necessarilyN∩F 6={1}. AsL/(L∩N) is finite, we have

`⁰_GH(L/(L∩N))< ω. So by Lemma 3.11,`⁰_H(G/(L∩N)) =`⁰_H(G/L) =β and cb_H(G/(L∩N)) = β again by induction. Thus every neighbourhood of{1}inNH(G) contains an element of Cantor-Bendixsonβ. As this holds for everyβ < α, we get cbH(G)>α=`⁰_H(G).

Examples with `⁰_H(G) < cbH(G) < `H(G) were obtained in [10, Lemma 15].

Corollary 3.15. — Under the same assumptions, CB(N_H(G)) = cbH(G) + 1.

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Proof. — Set α= cbH(G). Then by Proposition 3.14, N(G)^(α) is contained in the set of finite normalH-invariant subgroups ofG(it is actually equal in view of Lemma 3.11) and contains{1}. The assumptionH-max-n then implies that the non-empty setN(G)^(α) is finite, soN(G)^(α+1)=∅

and CB(N(G)) =α+ 1.

Lemma 3.16. — Let G be an H-group with H-max-n and N an H- stable normal subgroup of G contained in EH(G) (resp. WH(G)). Then

`H(G) =`H(G/N) +`GH(N)(resp.`⁰_H(G) =`⁰_H(G/N) +`⁰_GH(N)). More- over,`(G/E_H(G))and `⁰(G/W_H(G))are not successor ordinals.

Proof. — The first statement is a particular case of Lemmas 3.7 and 3.10, since whenα, βare ordinals withβ finite,α⊕β =α+β.

If`_H(G) =α+ 1, then by definition of`_H, for some non-trivialH-stable normal subgroupN, we have`H(G/N) =α. From the left-hand inequality in Lemma 3.7 we deduce `_H(N)61, so`_H(N) = 1, soN ⊂E_H(G) and EH(G) is non-trivial. Similarly if `⁰_H(G) is a successor ordinal,WH(G) is

non-trivial. This proves the second statement.

Suppose that we have an extension of H-groups 1→M →G→Q→1, for which we want to compute`⁰_H(G).

Proposition 3.17. — LetGbe anH-group withH-max-n lying in an extension

1→M →G→Q→1 with`⁰_H(Q)<∞. Then

`⁰_H(G) =`⁰_GH(M/WGH(M)) +`⁰_GH(WH(G)).

Proof. — By Lemma 3.16,`⁰_H(G) =`⁰_H(G/W_H(G))+`⁰_GH(W_H(G)). Now WH(G)∩M =WGH(M), we have an extension

1→M/WGH(M)→G/WH(G)→Q⁰ →1, for some quotientQ⁰ ofQ.

We consider two cases.

• M/WGH(M) ={1}. ThenG=WH(G) and the lemma holds.

• M/W_GH(M)6={1}. Then by Lemma 3.16,`⁰_GH(M/W_GH(M)) is a limit ordinal, so as`⁰_H(Q⁰)< ω, we get

`⁰_H(Q⁰) +`⁰_GH(M/WGH(M)) =`⁰_GH(M/WGH(M)).

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By Lemma 3.10

`⁰_GH(M/WGH(M))6`⁰(G/WH(G))6`⁰_GH(M/WGH(M)) +`H(Q⁰);

and by Lemma 3.16,`⁰(G/W_H(G)) is a limit ordinal and`⁰_H(Q⁰)<

ω, so we thus get `⁰(G/WH(G)) =`⁰_GH(M/WGH(M)).

Corollary 3.18. — Under the same hypotheses, ifM contains its own centralizer inGandWGH(M) ={1}, then

`⁰_H(G) =`⁰_GH(M).

3.4. Proof of Theorem 1.5

Lemma 3.19. — LetGbe anH-group with max-n. Then if finite,`⁰_H(G) is the supremum of lengthsk of chains of GH-subgroups1 =N0 ⊂ · · · ⊂ N_k=Gwith`⁰_H(N_i/N_i−1)>1for alli.

Proof. — If we have such a chain, then it follows from Lemma 3.7 that

`⁰_H(G)>k. The converse is a trivial induction on`⁰_H(G).

Let us now prove Theorem 1.5. It follows from the discussion in Para- graph 3.1 that ifM is a noetherian module over a commutative ring, then dim(M)61 (Krull dimension) if and only if`⁰(M)< ω. Therefore the def- initions ofW(M) given in the introduction and in Paragraph 3.3 coincide.

Besides, we have

Suppose thatGis residually finite as well as its quotients andNis a normal subgroup. By Lemma 3.19, if`⁰_G(N)< ω, then it coincides withhG(N) as defined in the introduction. Similarly,W(G) coincides with Hir(G), also defined in the introduction. Also, cb(G) =`⁰(G) by Proposition 3.14.

Given these remarks, we see that (1) of the theorem appears as a particular case of Proposition 3.17.

For (2), the left-hand inequality is clear since`⁰_G(M)6`⁰_G(G) = cb(G).

Finally (3) is a consequence of (1). Indeed, W(M) ={0} and `⁰(M) = ω^d−1·rby Proposition 3.3; moreover Hir(G) ={1}. Indeed, sinceW(M) = {1} (M being now written multiplicatively), Hir(G)∩M ={1}. In particular, Hir(G) centralizesM. Since the action of QonM is faithful, this implies that the projection of Hir(G) on Qis trivial, hence Hir(G)⊂M, hence Hir(G) ={1}.

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4. Some examples

Proposition 4.1. — IfGis a virtually polycyclic group, then`⁰(G)6 h(G), the Hirsch length ofG. The equality`⁰(G) =h(G)holds if and only ifGis supersolvable (e.g.Gis nilpotent).

Proof. — Indeed, by Lemma 3.12, if finite,`⁰(G) is the greatest number of infinite subfactors in anormal series ofG. On the other hand,h(G) is the greatest number of subfactors in asubnormalseries of G. To say that Gis supersolvable just means that there exists a normal series in which all infinite subfactors are cyclic, whence the equality. IfG is virtually polycyclic, there exists a normal series with exactly `⁰(G) infinite subfactors, all torsion-free (at the cost of adding some finite subfactors in the normal series). IfGis not supersolvable, then one of these infinite subfactors has to have rank at least two, soh(G)> `⁰(G).

Example 4.2. — IfG=Z^koF withF finite, then`⁰(G) is the number of irreducible representations in which Q^k decomposes under the action ofF.

Proposition 4.3. — For alld>0, n>1,

`⁰(Z^koZ^d) =ω^d·k.

Proof. — The group G = Z^k oZ^d can be written as Z[Q]^k oQ with Q=Z^d. AsZ[Q] is a domain of Krull dimensiond+1, we have`⁰_Q(Z[Q]^k) = ω^d ·k by Proposition 3.3. Now, if d > 1, Corollary 3.18 applies to give

`⁰(G) =ω^d·k. The cased= 0 is a particular case of Proposition 4.1.

Denote byCmthe cyclic group of ordermand byδ(m) the total number factors in a prime decomposition ofm(e.g.δ(18) = 3).

Proposition 4.4. — For alld>1,

`⁰(CmoZ^d+1) =ω^d·δ(m); `⁰(CmoZ) =δ(m) + 1.

Proof. — The group G = CmoZ^d+1 can be written as Z/mZ[Q]oQ.

The Z[Q]-module Z/mZ[Q] can be written as an iterated extension of δ(m) modules, each of the form Z/pZ[Q]. As the latter is a domain of Krull dimension d+ 1, it has `⁰ = ω^d. Now Lemma 3.10 implies that

`⁰(Z/mZ[Q]) =ω^d·δ(d).

IfQ=Z, then in the principal ideal domainZ/pZ[Q], every nonzero ideal has finite index, so `⁰_Q(Z/pZ[Q]) = 1. So Z/mZ[Q] has a normal series of length δ(m) in which each subfactor has `⁰_Q = 1, so `⁰_Q(Z/mZ[Q]) = δ(m) by Lemma 3.10. Again by Lemma 3.10, we deduce that `⁰(G) =

δ(m) + 1.

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5. Actions of finite groups

Let A be a ring and G a group acting on A by ring automorphisms.

We call a GA-module an A-module endowed with a G-action by group automorphisms, such that, for all g ∈ G, a ∈ A and m ∈ M, we have g(am) = (ga)(gm).

A GA-submodule is the same as aG-invariant A-submodule. In partic- ular, ifG is finite, a module is finitely generated, resp. Noetherian as an A-module if and only if it so as aGA-module.

Assume now that M is a Noetherian GA-module and that G is finite.

We consider the length and reduced length as defined in Section 3, withH the underlying multiplicative semigroup ofA. As we do this all along this section, we drop the indexAon`⁰.

So the definitions of Section 3 read as

`⁰_G(M) = sup{`⁰_G(M/N)|N non-ArtinianGA-submodule of M} and

`⁰(M) = sup{`⁰(M/N)|N non-ArtinianA-submodule ofM}, as it was defined in [10]. Clearly,`⁰_G(M)6`⁰(M).

Suppose thatAis Noetherian, and, to simplify the exposition, that it has finite Krull dimension. Let M be a finitely generated A-module of Krull dimensiond>1. We have (see Section 3)

`(M) =ω^d·`d(M) +o(ω^d)

whereo(ω^d) denotes some ordinal< ω^dand and`d(M) is a positive integer.

Similarly define`G,d(M) so that`(M) =ω^d·`G,d(M) +o(ω^d). Note thata priori`d(M) is a non-negative integer.

Proposition 5.1. — Suppose thatA is Noetherian of finite Krull dimensiond, andG is finite of order n. Let M be a finitely generated GA- module, of Krull dimension6d(as anA-module). Then

`_G,d(M)6`_d(M)6n`_G,d(M).

Proof. — The left-hand inequality is an obvious consequence of`G(M)6

`(M). We prove the right-hand inequality by induction on`(M) +d. First, if`d(M) = 0 this is trivial. So we suppose`d(M)>1.

Since M has Krull dimension d, there exists an associated prime ideal of coheightd, i.e. a A-submoduleN ofM with`(N) =ω^d. LetN⁰ be the GA-submodule generated byN: it is generated by thegN forg ∈G and therefore is, as anA-module, a quotient ofNⁿ. So `d(N⁰)6n`d(N) = n.

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By [10, Lemma 7],`_d is additive on exact sequences of modules of Krull dimension6d. If`(N⁰)< `(M), the inequality to proves holds for bothN⁰ andM/N⁰, so by additivity holds forM.

So suppose `(M)6`(N⁰). Then`d(M)6`d(N⁰)6n. So we just have to prove that `G,d(M) > 1, or equivalently that `G(M) > ω^d. If d = 0 this is obvious. Otherwise, for any integerm,M has a submoduleL with ω^d−1·mn 6 `(L) < ω^d. Replacing L by T

g∈GgL if necessary, we can suppose that L is G-invariant. By induction hypothesis, `_G,d−1(M/L) >

m. So `G(M) > ω^d−1·m. Since this holds for any m, we deduce that

`_G(M)>ω^d.

Corollary 5.2. — Under the same hypotheses, if `⁰(M) =ω^d, then

`⁰_G(M) =ω^d.

Corollary 5.3. — LetGbe a finite group, Abe a finitely generated domain of Krull dimension d > 1 with a G-action, and I a non-zero G- invariant ideal inA. Then`⁰_G(I) =ω^d−1.

Proof. — In view of Corollary 5.2, it is enough to check that `⁰(I) = ω^d−1, which is a particular case of Proposition 3.3.

The following lemma is well-known.

Lemma 5.4. — LetAbe a Noetherian ring andM a finitely generated A-module. ThenM is residually Artinian as anA-module.

Proof. — Pick a non-zero element x0 in M. Let W be a maximal A- submodule ofM not containingx0. We claim thatM/W is Artinian.

We can suppose thatW ={0}, i.e. thatx₀is contained in every non-zero A-submodule ofM and we have to prove that M is Artinian.

If M is non-Artinian, then it has an associate idealP such thatA/P is not a field. So M contains an A-submodule N isomorphic to A/Q. Pick a non-zero non-invertible elementain A/P. By a standard application of Artin-Rees’ lemma, we haveT

n>0aⁿN ={0}. By the assumption onx0, we get aⁿN = {0} for some n, i.e. aⁿ(A/P) = 0, and therefore as P is

prime, we obtaina∈P, a contradiction.

Lemma 5.5. — Let G be a finite group, A be a Noetherian ring with aG-action, andM a finitely generatedGA-module. ThenM is residually Artinian as aGA-module.

Proof. — Pick a non-zero elementx₀inM. Then there exists by Lemma 5.4 anA-submoduleN ofM such thatx0∈/N andM/N is Artinian.

If N⁰ =T

g∈GgN, then M/N⁰ embeds intoQ

g∈GM/gN, so is Artinian as well. Moreover,N⁰ is aGA-submodule and x0∈/N⁰.

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Proposition 5.6. — LetGbe a finite group,Abe a finitely generated ring with a G-action, and M a finitely generated GA-module. Then the Cantor-Bendixson rank ofM as a GA-module, i.e. the Cantor-Bendixson rank of{0}in the set ofGA-submodules ofM, is`⁰_G(M).

Proof. — Any Artinian finitely generated A-module is finite: this is a classical consequence of the Nullstellensatz (see for instance [10, Lemma 13]). Therefore, using Lemma 5.5,M is residually finite as aGA-module.

So the proposition appears as a particular case of Proposition 3.14.

6. The Bieri-Strebel invariant and tensor products In all this section, we consider a finitely generated metabelian group G, inside an extension

1→M →G→Q→1,

withQabelian andM abelian. SoM is a finitely generatedZ[Q]-module.

Ifv∈Hom(Q,R), we setQv={q∈Q|v(q)>0}. We set

Γ(M) ={v∈Hom(Q,R)|M is not a finitely generatedQv-module} ∪ {0}.

This is a closed subset [4, Proposition 2.2] of the vector space Hom(Q,R), and is further studied in [5]. We write Γ^±(M) = Γ(M)∩(−Γ(M)).

Theorem 6.1 (Bieri-Strebel [4]). — The finitely generated metabelian groupGis finitely presented if and only if

Γ^±(M) ={0}.

Lemma 6.2. — Fixv∈Hom(Q,R)− {0}. LetV be theQ_v-submodule generated by some finite generating subset of theQ-moduleM. Then we have the equivalences

• v /∈Γ(M);

• qV =V for everyq∈Q;

• qV ⊂V for someqwithv(q)<0.

Proof. — Let us first check that the two first assertions are equivalent.

Suppose thatqV 6=V for someq∈Q. Replacing qbyq⁻¹if necessary we can suppose thatv(q)>0. SoqV ⊂V, and we deduce that the sequence (q⁻ⁿV) of Q_v-submodules of M is strictly increasing, so that M is not noetherian, hence not finitely generated as aQv-module, i.e.v∈Γ(V).

Conversely the assumption qV =V for allq∈Qclearly implies that V is aQ-module, henceV =M, so M is a finitely generatedQv-module.

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The second assertion clearly implies the third, and the converse holds because the set ofq satisfyingqV ⊂V is a subsemigroup, and clearlyQis generated as a subsemigroup byQ_v∪ {q}wheneverq /∈Q_v. SoqV ⊂V for allq∈Q, and multiplying byq⁻¹ we getV ⊂q⁻¹V for allq∈Q, so asQ is closed under inversion,qV =V for allq∈Q.

Suppose that Q = Q1×Q2, and let Ai be the ring generated by Qi. Suppose M = M₁⊗_ZM₂, where M_i is a finitely generated A_i-module, and M is naturally viewed as a Q-module. We have the identification Hom(Q,R) = Hom(Q₁,R)×Hom(Q₂,R).

Lemma 6.3. — We have the inclusion

Γ(M)⊂Γ(M₁)×Γ(M₂)

Proof. — Suppose that (v1, v2)∈/Γ(M1)×Γ(M2), sayv1∈/Γ(M1). Con- siderVi⊂Mias in Lemma 6.2. So there existsq₁∈Q₁withv₁(q₁)<0 and q1V1 =V1. Sov(q1,1) = v1(q1)<0 and (q1,1)(V1⊗V2)⊂(q1V1⊗V2) = V1⊗V2. So the Qv-submodule V generated by V1 ⊗V2 is (q1,1)-stable,

hence it is aQ-submodule, sov /∈Γ(M).

Corollary 6.4. — IfΓ^±(M1) ={0}andΓ^±(M2) ={0}thenΓ^±(M) = {0}.

Here is a classical example (below the coefficient ring Zcan be replaced byZ/kZ). WriteR+= [0,∞).

Lemma 6.5. — Suppose thatA=M =Z[u,(u+u²)⁻¹], Q=Z² acts by(m, n)·P(u) =P(u)u^m(1 +u)ⁿ. ThenΓ(M) = R₊(1,0)∪R₊(0,1)∪ R+(−1,−1). In particular,Γ^±(M) ={0}.

Proof. — First observe thatA =M has a ring automorphism of order three given by u 7→ −(1 +u)/u 7→ −1/(u+ 1) 7→ u. This implies that Γ(M) is invariant under the matrix

−1 1

−1 0

of order three, which rotates (0,1) 7→(1,0) 7→(−1,−1). So it is enough to check that (0,1) belongs to Γ(M), but not (a, b) ifa, b >0.

Ifv(m, n) =n, thenZ[Qv] consists exactly ofZ[u, u⁻¹]. So (0,1)∈Γ(M).

Considerv(m, n) =am+bnwitha, b >0. LetV be theZ[Q_v]-submodule ofZ[Q] generated by 1. This is clearly a ring, so we just have to check that it containsu, u⁻¹,(1 +u)⁻¹. Sincea>0,u∈V. ThereforeZ[u]⊂V. Since a >0, we know thatv(n,−1)>0 for largen, soV contains (−u)ⁿ/(1 +u) for largen, which can be written as (1−(1+u))ⁿ/(1+u) =P₁(u)+1/(1+u) withP1(u)∈Z[u]. So 1/(1 +u)∈V. As b >0,V contains (1 +u)ⁿ/ufor

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largen. Since we can write (1 +u)ⁿ/u=P₂(u) + 1/uwith P₂(u)∈Z[u],

we deduce thatV containsu⁻¹.

AsM is the tensor product ofkcopies ofZ[u,(u²+u)⁻¹], from Corollary 6.4 we get

Corollary 6.6. — IfM =A=Z[u₁, . . . , u_k, s⁻¹]where s=

k

Y

i=1

(u²_i + ui), thenΓ^±(M) ={0}.

We will also need the following easy consequence of Theorem 6.1.

Lemma 6.7. — Let G be a finitely presented metabelian group in an exact sequence

1→M →G→Q→1

withM andQabelian. Let H be a subgroup of Gwhose projection onQ is surjective (i.e.HM =G). ThenH is finitely presented as well.

Proof. — By assumption we have an exact sequence 1→M ∩H →H →Q→1.

SoM∩His aQ-submodule ofM, hence is finitely generated as aQ-module, soH is finitely generated. Next, we see that Γ(M∩H)⊂Γ(H) (this uses the fact that the ringsZQv implied in the definition of the Bieri-Strebel invariant, are noetherian, as localizations of polynomial rings, although they may be infinitely generated). So Theorem 6.1 implies thatH is finitely

presented.

7. Free and split metabelian groups

7.1. Free metabelian groups

Let FMd =hx1, . . . , xdidenote the free metabelian group ondgenerators. Consider the extension

1→M →FMd→Q→1 withQ'Z^d andM = [FM_d,FM_d].

Proposition 7.1. — As aZ[Q]-module,M is torsion-free of rankd−1.