Polynomiality properties of group extensions with torsion-free abelian kernel

by Manfred Hartl

Institut de Recherche Mathématique Avancée, ULP et CNRS, 7 rue René-Descartes, 67084 Strasbourg Cedex, France

In studying torsion-free nilpotent groups of class 2 it is a key fact that each central group extension ZZ^{n} ⟶^{>} E ⟶ ZZ^{m} is representable by a bilinear cocycle. So for investigating nilpotent groups of higher class it is natural to ask for a generalization of this fact, namely when ZZ^{m} is replaced by some torsion-free nilpotent group G and ZZ^{n} by some torsion-free abelian group B with nilpotent G-action. In this paper we study a suitable notion of (bi)polynomial cocycles (in the strong sense of polynomiality introduced by Passi) and prove the desired representability theorem (4.3). This was known before only for central extensions with divisible kernel (Passi-Sucheta-Tahara 1987) and with kernel ZZ if G is abelian (Igodt 1984), nilpotent of class 2 (Dekimpe 1993) or if the quotients of the lower central series of G are torsion-free (Passi 1990).

Our representability result implies a convergence theorem for an approximation of H^{2}(G, B) by polynomial cohomology groups (4.4). Since the latter ones are well accessible to computation one obtains a formula for H^{2}(G, B) in terms of a presentation of G which can be evaluated by integral matrix calculus (6.4). More precisely, the convergence theorem amounts to an identification of H^{2}(G, B) with the cohomology of a cochain complex consisting of finitely generated free ZZ - modules if the groups G and B are finitely generated. Moreover, this complex can be read off directly from a given free presentation of G. As an application, we establish an explicit bijection between T -groups and ZZ -torsion-free nilpotent Lie rings both finitely generated and nilpotent of class ≤ 3 (7.2).

In case a given group extension is representable by a polynomial cocycle we also determine the minimal degree of polynomiality for which this holds. Indeed, dropping the assumption that G is torsion-free nilpotent, we give an intrinsic characterization of all group extensions with torsion-free abelian kernel which are representable by a polynomial cocycle of degree ≤ n, provided that the cokernel is finitely generated and acts nilpotently on the kernel (4.2). Thus a polynomiality theory for group extensions as asked for by Passi is now achieved in case the kernel is torsion-free. A motivation for this – apart from calculating H^{2}(G, B) explicitely – comes from a question posed by J. Milnor in 1977, namely whether all finitely generated, torsion-free virtually polycyclic groups arise as fundamental groups of compact, complete affinely flat manifolds. Even the case of torsion-free nilpotent groups is interesting but far from being understood, notably since counterexamples of this type have recently been discovered, see [4] and [6]. A connection of Milnor’s question for nilpotent groups with polynomial constructions was first indicated in the work of Igodt and Lee [17], [18], [20]. Recently a very close connection of this type was established in [12], and in [13] an obstruction theory is developed for the problem in terms of polynomial cohomology which is based on the results of this paper.

supported by the German Israel Science Foundation, project no. 6507, by the Sonderforschungsbereich 343, Bielefeld, and the Human Capital and Mobility Programme of the European Union

On the other hand, polynomiality properties of extensions with torsion kernel are closely related to dimension subgroups, as was observed by Passi in case of central extensions. An extension of this idea to the non-central case is provided by theorem 3.4 below. This might be of interest in connection with the recent discovery of Gupta and Kuz’min [7] that the quotient group D_{n}(G)/γ_{n}(G) is an abelian normal but in general non-central subgroup of G/γ_{n}(G).

Moreover, we obtain a functorial equivalence between extensions of torsion-free nilpotent groups and of ZZ -torsion-free nilpotent modules (5.1). This improves a general result of Reiner-Roggenkamp in the special situation of torsion-free nilpotent groups. As applications, it yields an inductive cohomological description of automorphism groups of torsion-free nilpotent groups [9] and a generalization of the classical Dold-Kan equivalence – between simplicial abelian groups and chain complexes of abelian groups – to simplicial groups of class 2 [8], [14]. Applications in localization theory are also to be expected, namely to questions concerning the genus of torsion-free nilpotent groups.

All above-mentioned results are based on the more technical work of the first section. There also generalizations of theorems of Witt and Quillen are obtained concerning certain Lie algebras associated with groups; these results might be of independent interest.

Contents

1 The relative polynomial construction 3

2 Polynomial 2-cocycles 15 3 Extensions of groups and modules 18

4 Representability of group extensions by polynomial cocycles 22

5 Abelian models of torsion-free nilpotent groups 23

6 The second cohomology of torsion-free nilpotent groups 27

7 Torsion-free nilpotent groups of class 3 31

1 The relative polynomial construction

The polynomial construction P_{n,R}(E) for groups E and commutative coefficient rings R is due to classical work of Passi [23]. A relative version, P_{n,R}(E, B) for B ⊲ E, however, appears only implicitely in several contexts, for example in Passi’s studies of the dimension subgroup problem [24], [27], in Roquette’s proof of the Golod-Šafarevič inequality [33] or in Kuz’min’s study of a varietal version of the classical shifting isomorphism [19].

In this section we study the absolute and the relative polynomial construction modulo torsion in their own right. Moreover, the torsion subgroup of P_{n,Z Z}(E, B) is identified in terms of the groups B ⊲ E in favourable situations.

Let G be a group. We define commutators in G by (a, b) = aba^{−1}b^{−1}. Let R be a commutative ring and let Δ_{R}(G) denote the augmentation ideal of the group algebra R(G). A left R(G)-module B is called nilpotent of class ≤ k if Δ_{R}^{k}(G) · B = 0. For example, let B be an abelian normal subgroup of a group E and let the action of G = E/B on B be induced by conjugation in E. Then the ZZ(G)-module B is nilpotent of class ≤ k if and only if all elements (a_{1}, (…, (a_{k}, b) …) vanish for a_{i} ∈ E, b ∈ B. As usual, we shall write ‘G -module’ instead of ‘ZZ(G)-module’.

1.1 Definitions: Let E be a group, R as above and B ⊲ E a normal subgroup. We define the quotient ring

P_{n,R}(E, B) = Δ_{R}(E)/ (Δ_{R}(B)Δ_{R}(E) + Δ_{R}^{n+1}(E)) .

Thus P_{n,R}(E) = P_{n,R}(E, {1}) is the polynomial construction of Passi. The canonical quotient map from Δ_{R}(E) to P_{n,R}(E) or to P_{n,R}(E, B) is denoted by ρ. Moreover, for an abelian group A we write τA = torsion of A, τ̄A = A/τA and τ̄ : A→ →τ̄A . We define the quotient rings

Pn^{√} (E, B) = τ̄ P_{n,Z Z}(E, B) , Pn^{√} (E) = Pn^{√} (E, {1}) .

Via left multiplication, P_{n,R}(E, B), Pn^{√} (E, B) and Pn^{√} (E/B) are nilpotent left (E/B)-modules of class ≤ n. Moreover, the maps

p_{n,R}: E → P_{n,R}(E, B) , pn^{√} : E → Pn^{√} (E, B)

sending a ∈ E to the coset of a−1 are (E → → E/B)-derivations; as such, they are universal with respect to nilpotent R(E/B)-modules of class ≤ n or to nilpotent ZZ -torsion-free (E/B)-modules of class ≤ n, respectively. A canonical isomorphism P_{n,R}(E, B) ≅ R(E/B) ⊗_{Z Z(E)} P_{n,Z Z}(E) of left R(E/B)-modules is induced by this property of p_{n,R}. Finally, suitable group homomorphisms induce ring homomorphisms on the constructions introduced above in the obvious way.

Now let

\underline{E}: B ^{≻}→^{i} E → →^{π}G (1)

be a group extension with abelian kernel B. Then it is an elementary fact that the sequence

R ⊗_{Z Z} B ⟶^{p_{n,R}i} P_{n,R}(E, B) ⟶^{P_{n,R}(π)} P_{n,R}(G) → → 0 (2)

is an exact sequence of R(G)-linear homomorphisms, where the G-action on R⊗_{Z Z} B is given by R-linear extension of the G-action on B induced by conjugation in E. Moreover, the map p_{n,R}i here denotes the R-linear extension of the map p_{n,R}i defined above.

The main goal of this section is to study the effect of replacing the terms P_{n,R}(−) in (2) by the corresponding terms Pn^{√} (−).

In order to explain the results first we postpone all proofs to the end of the section.

Let E = γ_{1}(E) ⊃ γ_{2}(E) ⊃ … denote the lower central series of a group E. For U < E let \sqrt{U} = \{a ∈ E | ∃m ∈ ZZ : a^{m} ∈ U\}, the isolator of U, which is a subgroup if U contains γ_{q}(E) for some q. A group is torsion-free nilpotent of class ≤ c iff \sqrt{γ_{c+1}(G)} = \{1\}. We write T_{c}, T for the class (or category) of all torsion-free nilpotent groups of class ≤ c or of arbitrary class, respectively.

For a subset H ⊂ E and u ∈ ZZ let H^{u} = {a^{u} | a ∈ H}. The map i in (1) will frequently be regarded as an inclusion and consequently suppressed from notation.

Theorem 1.2 Let B ^{≻}→^{i} E → →^{π}G be a group extension with torsion-free abelian kernel B, and let n ≥ 1. Then the sequence of G-linear homomorphisms

0 ⟶ B ⟶^{p_{n}^{√} i} Pn^{√} (E, B) ⟶^{P_{n}^{√} (π)} Pn^{√} (G) ⟶ 0 (3)

is exact if and only if the following three relations (A) – (C) hold in E:

(A) B ∩ γ_{n+1}(E) = {1}

(B) \sqrt{B γ_{n+1}(E)} ⊂ B \sqrt{γ_{n+1}(E)} (C) Let 1 ≤ j ≤ n, u ≥ 2, r ≥ 1, and choose elements b ∈ B∩((E − B \sqrt{γ_{j}(E)})^{u}γ_{j}(E))

and a_{q} ∈ \sqrt{B γ_{s_{q}}(E)} , 1 ≤ q ≤ r, such that s_{1} + … + s_{r} + j > n. Then

(a_{1}, (… (a_{r}, b) …) ∈ B^{u} .

Here relation (B) can be replaced by the relation

(B)’ B ∩ ((E − B \sqrt{γ_{n+1}(E)})^{u}γ_{n+1}(E)) ⊂ B^{u} for u ≥ 2.

Moreover, if the properties (A) – (C) are satisfied then the torsion subgroup of P_{n,Z Z}(E, B) is generated by the set p_{n,Z Z} \sqrt{γ_{n+1}(E)} together with the elements

p_{n,Z Z}(a_{1}) ⋯ p_{n,Z Z}(a_{r+1}) − p_{n,Z Z} ((a_{1}, (… (a_{r}, b) …)^{\frac{1}{u}} ) , (4)

where r ≥ 1, a_{q} ∈ \sqrt{B γ_{s_{q}}(E)} , 1 ≤ q ≤ r + 1, such that s_{1} + … + s_{r+1} > n, and a_{r+1}^{u} = bc for some u ≥ 2, b ∈ B, c ∈ γ_{s_{r+1}}(E). □

The element (a_{1}, (… (a_{r}, b) …) in the last part will be proved to be a u-th power in B and to be uniquely determined by u and the elements a_{q}.

Remark 1.3 Relations (A) and (B) are equivalent to saying that the restriction π: \sqrt{γ_{n+1}(E)} → \sqrt{γ_{n+1}(G)} is an isomorphism. In particular, the extension \underline{E} is

induced by the extension B ⟶^{>}^{i} E/\sqrt{γ_{n+1}(E)} ⟶^{π} G/\sqrt{γ_{n+1}(G)} of torsion-free nilpotent groups, and the G-action on B induced by conjugaton in E is nilpotent of class ≤ n.

Also note that relation (C) is trivially satisfied if B is divisible or is central in E.

Corollary 1.4 Let B ^{≻}→^{i} E → →^{π}G be a group extension with torsion-free abelian kernel. Then sequence (3) is exact provided there is a number c ≤ n such that the following three properties are satisfied:

(a) π: \sqrt{γ_{n+1}(E)} → \sqrt{γ_{n+1}(G)} is isomorphic

(b) π: \sqrt{γ_{j}(E)} → \sqrt{γ_{j}(G)} is surjective for c < j ≤ n

(c) n ≥ c + k − 1 where k is the nilpotency class of the G-action on B, see the remark above. □

The following two corollaries are immediate applications of 1.4.

Corollary 1.5 Let B ^{≻}→^{i} E → →^{π}G be a group extension with abelian kernel where E and G are torsion-free nilpotent. Let c, e, k be the nilpotency classes of G, E and of the G-action on B induced by conjugation in E, respectively. Then for n ≥ max{e, c + k − 1} sequence (3) is exact. □

Corollary 1.6 Let E be a T_{e}-group, B an abelian normal subgroup. Then \sqrt{B} is abelian. Let c, k be the nilpotency classes of G = E/\sqrt{B} and of the G-action on \sqrt{B} induced by conjugation in E. Then for n ≥ max\{e, c + k − 1\} the sequence

0 ⟶ \sqrt{B} ⟶ Pn^{√} (E , \sqrt{B}) ⟶ Pn^{√} (G) ⟶ 0

of G-modules is exact. □

Note that this in particular applies when B = γ_{f}(E) where f is the smallest integer ≥ \frac{e+1}{2} , or when B = γ_{2}(E) if E is metabelian. In both cases one has n = e.

Now we consider the structure of the ‘absolute’ polynomial construction Pn^{√} (G). The crucial result here is the following integral analogue of Quillen’s description of the ring Gr(Q(G)) [30].

The filtration \{\sqrt{γ_{i}(G)}\} of G is an N-series, see [27, IV.1.3], whence a graded Lie ring L^{√} (G) is defined by

L^{√} (G) = ∑_{i≥1} \sqrt{γ_{i}(G)}/\sqrt{γ_{i+1}(G)} ,

[ a\sqrt{γ_{i+1}(G)} , b\sqrt{γ_{j+1}(G)} ] = (a, b)\sqrt{γ_{i+j+1}(G)} .

The enveloping algebra UL^{√} (G) carries a canonical grading which is induced by the grading of L^{√} (G). On the other hand, the N-series \{\sqrt{γ_{i}(G)}\} induces a descending filtration (Δn_{√} (G))_{n} of ZZ(G) by both-sided ideals, where

Δn_{√} (G) = ∑_{\begin{matrix}s≥1 \\ j_{1}+...+j_{s}≥n \end{matrix}} Δ_{Z Z}(\sqrt{γ_{j_{1}}(G)}) ⋯ Δ_{Z Z}(\sqrt{γ_{j_{s}}(G)}) ,

see [27, III.1.5]. Let Gr^{√} (ZZ(G)) and Gr^{τ}(ZZ(G)) denote the associated graded rings of ZZ(G) with respect to the filtrations \{Δn_{√} (G)\} and \{ ^{+} \sqrt{Δ_{Z Z}^{n}(G)} \}, respectively.

Theorem 1.7 Let G be a group. Then for all n ≥ 1 we have an equality

Δn_{√} (G) = ^{+} \sqrt{Δ_{Z Z}^{n}(G)} , (5)

and there is a natural isomorphism

θ_{Z Z} : U L^{√} (G) ⟶^{≅} Gr^{√} (ZZ(G)) (6)

of additively torsion-free graded rings. Here the map θ_{Z Z} is induced by the maps

\sqrt{γ_{i}(G)} → Δi_{√} (G), a ↦ a − 1. □

The isomorphism U L^{√} (G) ≅ Gr^{τ}(ZZ(G)) obtained by combining (5) and (6) was also proved by B. Hartley using different methods [16].

Corollary 1.8 Let G be a group and n ≥ 1. Then the following statements hold:

(i) There is a natural ring isomorphism

Pn^{√} (G) ≅ Δ_{Z Z}(G)/Δn+1_{√}(G) .

Thus by 1.7 there is a natural isomorphism

Gr^{√} ( Pn^{√} (G)) ≅ UL^{√} (G)/ ∑_{i>n} U_{i}L^{√} (G)

of graded rings induced by θ_{Z Z}^{−1}. Whence for finitely generated groups G there exists a (nonnatural) isomorphism of finitely generated free abelian groups

Pn^{√} (G) ≅ ⨁_{i=1}^{n} U_{i}L^{√} (G)

(since the groups L^{√} (G) and consequently U_{i}L^{√} (G) are free abelian in this case). In particular, the rank of Pn^{√} (G) is combinatorially determined by the numbers r_{i} = rank (\sqrt{γ_{i}(G)}/\sqrt{γ_{i+1}(G)}), 1 ≤ i ≤ n, according to the Birkhoff-Witt-theorem.

(ii) The following sequence consists of G-linear homomorphisms and is exact:

0 ⟶ \sqrt{γ_{n}(G)} / \sqrt{γ_{n+1}(G)} ⟶^{p_{n}^{√}} Pn^{√} (G) ⟶^{P_{n}^{√} (π)} Pn^{√} (G/\sqrt{γ_{n}(G)}) ⟶ 0 ,

with π: G ⟶ G/ \sqrt{γ_{n}(G)} .

(iii) The divisibility in Pn^{√} (G) of elements in pn^{√} (γ_{n}(G)) is given by the iden-

tity

+ \sqrt{pn^{√} (γ_{n}(G))} = pn^{√} (\sqrt{γ_{n}(G)}) .

Here the symbol ^{+} √ indicates that the corresponding isolator is taken with respect to the additive structure of Pn^{√} (G). If K_{n} denotes the subgroup generated by the n-fold ring commutators of Pn^{√} (G) we have the following isomorphism and identities in Pn^{√} (G):

\sqrt{γ_{n}(G)}/\sqrt{γ_{n+1}(G)} ≅^{p_{n}^{√}} pn^{√} \sqrt{γ_{n}(G)} = ^{+} \sqrt{pn^{√} (γ_{n}(G))} = ^{+} \sqrt{K_{n}} .

□

Part (iii) may be viewed as a generalization of Witt’s theorem that the commutator Lie ring L(G) = ⨁_{i≥1} γ_{i}(G)/γ_{i+1}(G) of a free group G is free. In fact, let T denote the noncommutative polynomial ring generated by a basis of G and let T̄ be its augmentation ideal. Then Witt’s theorem says that L(G) is isomorphic to the sub-Lie algebra of T generated by homogenous elements of degree 1. This is a special case of 1.8 (iii) since we have \sqrt{γ_{n}(G)} = γ_{n}(G), Pn^{√} (G) ≅ T̄/T̄^{n+1} (see 6.2 below) and ^{+} \sqrt{K_{n}} = K_{n}. As an application we derive the following

Theorem 1.9 Let G be a group. Then the graded Lie ring L^{√} (G) is determined by the group ring ZZ(G) up to isomorphism. In fact, let L^{τ}(G) be the sub-Lie ring of Gr^{τ}(Δ_{Z Z}(G)) which is the additive isolator of the sub-Lie ring generated by Gr_{1}^{τ}(Δ_{Z Z}(G)) . Then the inclusion G ↪ Δ_{Z Z}(G) , a ↦ a − 1, induces a natural isomorphism of Lie rings

L^{√} (G) ≅ L^{τ}(G) . □

This result is an integral analogue of Quillen’s result in [30] that the Lie algebra L^{√} (G) ⊗ Q is determined by the ring Q(G).

Now we turn to the proofs. We start by considering the situation for rational coefficients. The ‘descent to ZZ ’ then turns out to be much simpler than in comparable situations in the literature; e.g., lattice subgroups do not play a distinguished role. Also Mal’cev basises, orderings and weighting arguments as commonly used in similar contexts do not occur.

1.10 First note that we have for all groups the elementary relations Q ⊗ (γ_{i}(G)/γ_{i+1}(G)) ≅ Q ⊗ (\sqrt{γ_{i}(G)}/\sqrt{γ_{i+1}(G)}), Δ_{Q} (\sqrt{γ_{i}(G)}) ⊂ Δ_{Q}^{i} (G) . In

particular, the maps \sqrt{γ_{i}(G)} → Δ_{Q}^{i} (G), a ↦ a − 1, induce a natural homomorphism

θ_{Q} : U(Q ⊗ L^{√} (G)) → Gr(Q(G)) (7)

of graded algebras which is actually an isomorphism for all groups G [30].

Lemma 1.11 Let L be a graded Lie ring which is torsion-free as an abelian group. Then the enveloping algebra UL over ZZ is also torsion-free as an abelian group.

Proof: The Lie ring L is the colimit of its sub-Lie rings L_{α} which are locally (i.e. in each degree) finitely generated as abelian groups. Since the enveloping algebra as a left adjoint functor preserves colimits we get

(8) UL ≅ colim_{α}UL_{α} = ⨆_{α} UL_{α} / ∼

where x ∼ y , x ∈ UL_{α}, y ∈ UL_{β}, if there is an inclusion i_{αβ}: L_{α} ⊂ L_{β} in L such that U(i_{αβ})(x) = y .

Since L is supposed to be additively torsion-free each L_{α} is locally and hence globally a free ZZ -module. The Birkhoff-Witt-theorem then provides a natural isomorphism Gr(UL_{α}) ≅S(L_{α}) between the associated graded ring of UL_{α} (with respect to the ascending filtration by word length) and the symmetric ZZ -Algebra over L_{α}. It follows that UL_{α} is additively free abelian and whence torsion-free, and that the homomorphisms Gr(U(i_{αβ})) induced by the inclusions i_{αβ}: L_{α} ⊂ L_{β} are injective. Thus also the maps U(i_{αβ}) are injective which implies the assertion via (8). □

1.12 Proof of 1.7 and 1.8: For an abelian group A let ν or ν(A): A ≅ ZZ ⊗ A → Q ⊗ A be the canonical map. Using 1.10 and (7) we obtain the following factorization of ν(UL^{√} (G)):

UL^{√} (G) ⟶^{θ_{Z Z}} Gr^{√} (ZZ(G)) ⟶^{ν̄} Gr(Q(G)) θ_{Q}^{−1} ; ≅ U(Q ⊗ L^{√} (G)) ≅ Q ⊗ UL^{√} (G)

Here ν̄ is induced by ν: ZZ(G) → Q(G). Since L^{√} (G) and thus by 1.11 also UL^{√} (G) are torsion-free as abelian groups we successively obtain

(9) injectivity of ν(UL^{√} (G)) and whence

(10) injectivity of θ_{Z Z}, i.e. (6)

(11) injectivity of ν̄.

By 1.10 we have a factorization

ν: P_{n,Z Z}(G) ⟶^{ω} Δ_{Z Z}(G)/Δn+1_{√}(G) ⟶^{ν_{2}} P_{n,Q} (G) ≅ Q ⊗ P_{n,Z Z}(G) ,

where ω is the canonical quotient map. But

(12) Gr^{√} (ν_{2}) = ν̄ | ⨁_{i≤n} Gri^{√} (ZZ(G))

where the associated graded map Gr^{√} (ν_{2}) is taken with respect to the filtration

\{\frac{Δ^{k} √ (G)}{Δ{n+1 \atop √}(G)}\} of Δ_{Z Z}(G)/Δn+1_{√}(G) and with respect to the augmentation filtration

of P_{n,Q} (G). Now (11) entrains injectivity of ν_{2}, which implies

(13) τ P_{n,Z Z}(G) =Ker(ν) =Ker(ω) = Δn+1_{√}(G) / Δ_{Z Z}^{n+1}(G)

and thus relation (5) and part (i) of 1.8. Now consider part (ii). In order to see that the map \sqrt{γ_{n}(G)}/\sqrt{γ_{n+1}(G)} → Pn^{√} (G) induced by pn^{√} is welldefined and injective consider its factorization

\sqrt{γ_{n}(G)}/\sqrt{γ_{n+1}(G)} ≅ Ln^{√} (G) ⟶^{κ_{n}} U_{n}L^{√} (G) ≅^{θ_{n,Z Z}} Grn^{√} (ZZ(G)) ↪ Pn^{√} (G) ,

where κ is the canonical map, θ_{n,Z Z} is isomorphic by 1.7 and the last factor is given by (i). Injectivity of κ_{n} is derived from the commutative diagram

L^{√} (G)  ⟶^{κ}  UL^{√} (G)
↓^{ν}  ↓^{ν}
Q ⊗ L^{√} (G) ⟶^{κ} U(Q ⊗ L^{√} (G)

and by using injectivity of the bottom map κ due to the Birkhoff-Witt theorem. Moreover, the second factor of the composition

Gr^{√} ( Pn^{√} (G)/pn^{√} \sqrt{γ_{n}(G)}) ≅ Gr^{√} ( Pn^{√} (G))/pn^{√} Ln^{√} (G) → Gr^{√} ( Pn^{√} (G/\sqrt{γ_{n}(G)}))

is isomorphic by (i), so the surjection Pn^{√} (G)/pn^{√} \sqrt{γ_{n}(G)} → → Pn^{√} (G/\sqrt{γ_{n}(G)}) is isomorphic, too. Thus part (ii) is proved.

In order to derive (iii) from (ii) we calculate

Pn^{√} (π) ^{+} \sqrt{pn^{√} (γ_{n}(G))} ⊂ ^{+} \sqrt{pn^{√} (γ_{n} (G/\sqrt{γ_{n}(G)}))} = ^{+} \sqrt{0} = 0

Now (ii) implies

+ \sqrt{pn^{√} (γ_{n}(G))} ⊂ pn^{√} \sqrt{γ_{n}(G)} .

The converse inclusion follows from the linearity of pn^{√} |\sqrt{γ_{n}(G)} . □

Proof of 1.9: The ring structure of Δ_{Z Z}(G) is determined by the ring structure of ZZ(G) [29] p. 664, whence also the Lie ring L^{τ}(G). The asserted isomorphism with L^{√} (G) is provided by 1.8 (iii) as an isomorphism of abelian groups.

Now let a ∈ \sqrt{γ_{k}(G)} und b ∈ \sqrt{γ_{l}(G)} . Then (a, b) ∈ \sqrt{γ_{k+l}(G)} , and we have

pk+l^{√}(a, b) = [pk+l^{√}(a) , pk+l^{√}(b)] + [pk+l^{√}(a) , pk+l^{√}(b)] pk+l^{√}(a^{−1}b^{−1}) (14)

by the relation (a, b) − 1 = [a − 1 , b − 1]a^{−1}b^{−1} in ZZ(G). But the last term in (14) lies in the image of Δk+l+1_{√}(G) in Pk+l^{√}(G) and thus vanishes by 1.8 (i). This proves the theorem. □

Now we turn to the relative construction Pn^{√} (E, B). The rational key result here is

Lemma 1.13 [19] Let E be a torsion-free nilpotent group of class ≤ n and let B be an abelian normal subgroup. Then the map

p_{n,Q} : E → P_{n,Q} (E, B)

is injective. □

Together with (2) we obtain

Theorem 1.14 Let \underline{E} = ( B ^{≻}→^{i} E → →^{π}G) be a group extension with abelian kernel B. Then the following sequence of G-linear homomorphisms is exact:

0 ⟶ Q ⊗ (B / B ∩ γ_{n+1}(E)) ⟶^{Q⊗p_{n,Z Z}i} P_{n,Q} (E, B) ⟶^{P_{n,Q} (π)} P_{n,Q} (G) ⟶0

The ‘descent to ZZ ’ now is done by means of the following

Lemma 1.15 Let E be a group, B ⊲ E and n ≥ 1. Then the isolator subgroup of pn^{√} (B) in Pn^{√} (E, B) is generated by the set pn^{√} \sqrt{B γ_{n+1}(E)} together with the

elements \frac{1}{u} pn^{√} (a_{1}, (… (a_{r}, b) …) where u ≥ 2, b ∈ B ∩((E − B \sqrt{γ_{j}(E)})^{u}γ_{j}(E)) ,

1 ≤ j ≤ n and a_{q} ∈ \sqrt{B γ_{s_{q}}(E)} , 1 ≤ q ≤ r, such that s_{1} + … + s_{r} + j > n.

Moreover, the set pn^{√} \sqrt{B γ_{n+1}(E)} consists of the elements of pn^{√} (B) together

with the elements \frac{1}{u} pn^{√} (b), u ≥ 2 and b ∈ B ∩ ((E − B \sqrt{γ_{n+1}(E)})^{u}γ_{n+1}(E)).

Proof: Let π : E → →E/B = G . By the identity τ P_{n,Z Z}(G) = Δn+1_{√}(G)/Δ_{Z Z}^{n+1}(G) proved in (13) and by the relation

π^{−1} \sqrt{γ_{j}(G)} = \sqrt{B γ_{j}(E)} (15)

we have

(16)  \sqrt{p_{n}i(B)} = \sqrt{p_{n,Z Z}i(B)} = P_{n,Z Z}(π)^{−1}τ P_{n,Z Z}(G)
= ∑_{I} Δi_{√} (G) · p_{n}\sqrt{B γ_{j}(E)}

whence also

\sqrt{pn^{√} i(B)} = ∑_{I} Δi_{√} (G) · pn^{√} \sqrt{B γ_{j}(E)} (17)

with I = {(i, j) | i ≥ 0, 1 ≤ j ≤ n+1, i+j > n}, and where we use the G-action on Pn^{√} (E, B) and set Δo_{√} (G) = ZZ(G). Indeed, it suffices to take j ≤ n+1 since

for j > n + 1 one has Δi_{√} (G) · pn^{√} \sqrt{B γ_{j}(E)} ⊂ Δo_{√} (G) · pn^{√} \sqrt{B γ_{n+1}(E)} .

Let (i, j) ∈ I, x ∈ Δi_{√} (G) and a ∈ \sqrt{B γ_{j}(E)} with a^{u} = bc, b ∈ B,

c ∈ γ_{j}(E) and u ≥ 1 (without restriction). Since xpn^{√} (a) ∈ \sqrt{pn^{√} i(B)} we have for

every k > 1 the relation xpn^{√} (a)^{k} ∈ \sqrt{ρΔ_{Z Z}(B)} ρΔ_{Z Z}(E) ⊂ \sqrt{ρ(Δ_{Z Z}(B) Δ_{Z Z}(E))} =

\sqrt{0} = 0. Thus

u xpn^{√} (a) = x((pn^{√} (a) + 1)^{u} − 1) = xpn^{√} (a^{u}) = xpn^{√} (bc)
= xpn^{√} (b) + xpn^{√} (c) + xpn^{√} (b) pn^{√} (c) .

But pn^{√} (b) pn^{√} (c) ⊂ ρ(Δ_{Z Z}(B) Δ_{Z Z}(E)) = 0, and applying the inclusion

ν : Pn^{√} (E, B) ⟶^{>} P_{n,Q} (E, B) we see that

ν(xpn^{√} (c)) ⊂ Δ_{Q}^{i} (G) · Δ_{Q}^{j} (E) = Δ_{Q}^{i+j−1}(G) · Δ_{Q} (E) = 0

since the G-action on P_{n,Q} (E, B) is nilpotent of class n ≤ i + j − 1. Thus

xpn^{√} (a) = \frac{1}{u}xpn^{√} (b) = \frac{1}{u}pn^{√} (x · b) (18)

Together with (15) and (16) this proves the first part of the assertion, except from the fact that for u > 1 it suffices to take the elements a ∈ \sqrt{B γ_{j}(E)} outside

B \sqrt{γ_{j}(E)} . For if a = b′w, b′ ∈ B, w ∈ \sqrt{γ_{j}(E)} , then xpn^{√} (a) = xpn^{√} (b′) for

the same reasons as above, using the relation Δ_{Q} (\sqrt{γ_{j}(E)}) ⊂ Δ_{Q}^{j} (E). Whence by (18)

\frac{1}{u}pn^{√} (x · b) = xpn^{√} (a) = xpn^{√} (b′) = pn^{√} (x · b′) ⊂ pn^{√} (B) .

The ‘moreover’ part of the assertion is the case i = 0 and j = n + 1. □

Proof of theorem 1.2: By 1.10 we have a factorization

p_{n,Q} : E → → E/\sqrt{γ_{n+1}(E)} ⟶^{p_{n}^{√}} Pn^{√} (E, B) ⟶^{>}^{ν} P_{n,Q} (E, B) .

(19) Thus by Kuz’min’s lemma 1.13 the map pn^{√} : E/\sqrt{γ_{n+1}(E)} → Pn^{√} (E, B)

is injective. In particular, Ker(pn^{√} i) = B ∩ \sqrt{γ_{n+1}(E)} .

(20) Using that B is torsion-free we see that the map pn^{√} i: B → Pn^{√} (E, B) is injective if and only if relation (A) holds.

Moreover, exactness of sequence (2) implies KerPn^{√} (π) = ^{+} \sqrt{pn^{√} i(B)}. Now suppose relation (A) holds. Then it follows from 1.15 and (20) that sequence (3) is exact iff the relations (C) and pn^{√} \sqrt{B γ_{n+1}(E)} ⊂ pn^{√} (B) hold. By (19) the latter relation is equivalent with relation (B). Moreover, the ‘moreover’ part of 1.15 implies that relation (B) can be replaced by relation (B)’.

Now suppose that the relations (A) – (C) hold. To determine the torsion subgroup τ P_{n,Z Z}(E, B) consider the short exact sequence

τ P_{n,Z Z}(E, B) ↪ ^{+} \sqrt{p_{n,Z Z}i(B)} ⟶^{τ̄} ^{+} \sqrt{pn,Z Z^{√}i(B)} .

By the first part of the theorem it is split by means of the composition

s: ^{+} \sqrt{pn^{√} i(B)} = pn,Z Z^{√}i(B) ≅^{p_{n}^{√} i} B ⟶^{p_{n,Z Z}i} ^{+} \sqrt{p_{n,Z Z}i(B)} .

Thus if ^{+} \sqrt{p_{n,Z Z}i(B)} is generated by elements \{y_{k}\} then τ P_{n,Z Z}(E, B) is generated by the elements y_{k} − sτ̄(y_{k}). Using (17), (18) and relation (C) we find that τ P_{n,Z Z}(E, B) is generated by the set (1 − sτ̄) p_{n,Z Z} \sqrt{B γ_{n+1}(E)} together with the

elements described in (4). By relation (B) and by τ̄ p_{n,Z Z} \sqrt{γ_{n+1}(E)} = 0 we have

p_{n,Z Z} \sqrt{γ_{n+1}(E)} = (1 − sτ̄)p_{n,Z Z} \sqrt{γ_{n+1}(E)}
⊂ (1 − sτ̄) p_{n,Z Z} \sqrt{B γ_{n+1}(E)}
= (1 − sτ̄)p_{n,Z Z}(B\sqrt{γ_{n+1}(E)})
⊂ (1 − sτ̄)(p_{n,Z Z}i(B) + p_{n,Z Z} \sqrt{γ_{n+1}(E)})
= p_{n,Z Z} \sqrt{γ_{n+1}(E)} ,

whence (1 − sτ̄)p_{n,Z Z} \sqrt{B γ_{n+1}(E)} = p_{n,Z Z} \sqrt{γ_{n+1}(E)} . This completes the proof. □

Proof of 1.4: By assumption (a) and 1.3 it remains to check that relation (C) in theorem 1.2 holds. For j > c the set B ∩ ((E − B \sqrt{γ_{j}(E)})^{u}γ_{j}(E)) is empty;

for if a^{u}d ∈ B with a ∉ B \sqrt{γ_{j}(E)} and d ∈ γ_{j}(E) then a ∈ \sqrt{B γ_{j}(E)} which

equals B \sqrt{γ_{j}(E)} by (b). So let j ≤ c, u ≥ 2, b ∈ B ∩ ((E − B \sqrt{γ_{j}(E)})^{u}γ_{j}(E)),

a_{q} ∈ \sqrt{B γ_{s_{q}}(E)} , 1 ≤ q ≤ r, such that s_{1} + … s_{r} + j > n. Using the G-action on B we write

(a_{1}, … (a_{r}, b) …) = (π(a_{1}) − 1) ⋯ (π(a_{r}) − 1) · b
∈ Δ_{Z Z} (\sqrt{γ_{s_{1}}(G)}) ⋯ Δ_{Z Z} (\sqrt{γ_{s_{r}}(G)}) · B

Applying the injection ν : B ⟶^{>} Q ⊗ B we get

ν(a_{1}, … (a_{r}, b) …) ∈ Δ_{Q} (\sqrt{γ_{s_{1}}(G)}) ⋯ Δ_{Q} (\sqrt{γ_{s_{r}}(G)}) · (Q ⊗ B)
⊂ Δ_{Q}^{s_{1}}(G) ⋯ Δ_{Q}^{s_{r}}(G) · (Q ⊗ B)
= 0

since s_{1} + … + s_{r} > n − j ≥ n − c ≥ k − 1 by (c). Thus (a_{1}, … (a_{r}, b) …) = {1} ∈ B^{u} as required. □

Proof of 1.6: We only have to show that \sqrt{B} is abelian; then 1.5 applies. Consider the extension

B ↪ \sqrt{B} → → T

with T = \sqrt{B}/B. This group is a nilpotent torsion group, whence locally finite. Since \sqrt{B} is nilpotent the conjugation action of T on B is nilpotent, too, of class r + 1, say. Then the homomorphism

μ_{B}: P_{r}(T) ⊗ B → B , p_{r}(a) ⊗ b = (a, b)

is welldefined. But P_{r}(T) is a torsion group since

Gr(P_{r}(T)) = ∑_{i=1}^{r} Δ_{Z Z}^{i}(T)/Δ_{Z Z}^{i+1}(T) ,

and since the canonical homomorphism

(T^{ab})^{⊗i} → Δ_{Z Z}^{i}(T)/Δ_{Z Z}^{i+1}(T) , a_{1}γ_{2}(T) ⊗ … ⊗ a_{i}γ_{2}(T) ↦ (a_{1} − 1) ⋯ (a_{i} − 1)

is welldefined and surjective. Hence μ_{B} is trivial since B is torsion-free. Thus B is central in \sqrt{B}. Now the assertion follows from the fact that \sqrt{B} is torsion-free via the following lemma.

Lemma 1.16 Let B ⟶^{>} Q ⟶^{π} T be a central group extension such that T is locally finite. Then γ_{2}(Q) is also locally finite.

Proof: Let q : F → →Q be a surjection with F being a free group. Let R = Ker(πq). Then we have a commutative diagram of homorphisms with exact top row

\frac{R ∩ (F, F)}{(F, R)} ↪ \frac{(F, F)}{(F, R)} ⟶^{πq} γ_{2}(T)
↓q̃  ∥
γ_{2}(Q) ⟶^{π} γ_{2}(T)

where q̃ induced by q is surjective. Now by Hopf’s formula the left hand term equals H_{2}(T) = ⋃ { Im(H_{2}(U) → H_{2}(T)) | U < T, U finitely generated }. But each U is finite by assumption whence so is H_{2}(U). It follows that H_{2}(T) is locally finite, and since γ_{2}(T) is, too, the same holds for (F, F)/(F, R) (here we use that subgroups of finite index in finitely generated groups are finitely generated). Thus also γ_{2}(Q) is locally finite, as asserted. □

2 Polynomial 2-cocycles

Let G be a group, B a G-module. Recall that a function d: G^{r} = G×…×G → B is called an r-cochain and that it is called normalized if d(a_{1}, …, a_{r}) = 0 whenever one of the a_{i} is 1. A 2-cochain d is a 2-cocycle if the relation

a d(b, c) − d(ab, c) + d(a, bc) − d(a, b) = 0 (21)

holds for all a, b, c ∈ G. Moreover, d is the coboundary of a 1-cochain f if the relation d(a, b) = af(b) − f(ab) + f(a) (22)

holds for all a, b ∈ G.

To an r-cochain d assign the homomorphism

d̄: Δ_{Z Z}(G)^{⊗r} = Δ_{Z Z}(G) ⊗ … ⊗ Δ_{Z Z}(G) ⊂ ZZ(G)^{⊗r} ≅ ZZ(G^{r})⟶^{d}B (23)

of abelian groups where d is extended linearly to ZZ(G^{r}). Then it is easily checked that a normalized 2-cochain d is a 2-cocycle if and only if the relation

xd̄(y ⊗ z) = d̄(xy ⊗ z) − d̄(x ⊗ yz) (24)

holds for all x, y, z ∈ Δ_{Z Z}(G) , and that d is the coboundary of a normalized 1-cochain f if and only if the relation

d̄(x ⊗ y) = xf̄(y) − f̄(xy) (25)

holds for all x, y ∈ Δ_{Z Z}(G) .

Definition 2.1 A normalized 2-cocycle d: G × G → B is called a polynomial 2-cocycle of degree ≤ (m, n) if for every a ∈ G, the maps d(a, −), d(−, a) : G → B are polynomial of degree ≤ n and ≤ m respectively. Equivalently, the relation

d̄(x ⊗ Δ_{Z Z}^{n+1}(G)) = 0 = d̄(Δ_{Z Z}^{m+1}(G) ⊗ x)

holds for all x ∈ Δ_{Z Z}(G) . In other words, d̄ factors through group homomorphisms

Δ_{Z Z}(G) ⊗ Δ_{Z Z}(G) ⟶^{ρ⊗ρ} P_{m,Z Z}(G) ⊗ P_{n,Z Z}(G) ⟶^{d̃} B .

Lemma 2.2 Let B be a nilpotent G-module of class ≤ k (i.e. Δ_{Z Z}^{k}(G) · B = 0) and let d: G × G → B be a polynomial 2-cocycle of degree ≤ (m, n). Then the following statements are true:

(i) The map d̄ satisfies

d̄(Δ_{Z Z}^{i}(G) ⊗ Δ_{Z Z}^{j}(G)) = 0 if i ≥ k and i + j > min{m + 1, n + k}.

In particular, d is a polynomial 2-cocycle of degree ≤ (n + k − 1, n), and if the G-action on B is trivial then d is of degree ≤ (min{m, n},min{m, n}).

(ii) Suppose that d is the coboundary of a normalized 1-cochain f : G → B. Then f is a polynomial map of degree ≤ min{n + k, max{k, m + 1}}.

Proof: Set formally Δ_{Z Z}^{0}(G) = ZZ(G). By (24) one has for i ≥ k

d̄(Δ_{Z Z}^{i}(G) ⊗ Δ_{Z Z}^{j}(G)) = d̄(Δ_{Z Z}^{k}(G)Δ_{Z Z}^{i−k}(G) ⊗ Δ_{Z Z}^{j}(G))
= d̄(Δ_{Z Z}^{k}(G) ⊗ Δ_{Z Z}^{j+i−k}(G))
= d̄(Δ_{Z Z}^{i+j−1}(G) ⊗ Δ_{Z Z}(G))

The last term vanishes if i + j − 1 > m and the penultimate one if i + j − k > n.

Now suppose that d is the coboundary of f as in (ii). By (25) we have for r > k

f̄(Δ_{Z Z}^{r}(G)) = f̄(Δ_{Z Z}^{k}(G) Δ_{Z Z}^{r−k} (G)) = − d̄(Δ_{Z Z}^{k}(G) ⊗ Δ_{Z Z}^{r−k} (G))

so assertion (ii) follows from (i). □

Definition 2.3 Let G be a group and B be a G-module. Let P_{n}H^{2}(G, B) be the subset of H^{2}(G, B) consisting of those elements which are representable by polynomial cocycles of degree ≤ (n, n+1). Clearly P_{n}H^{2}(G, B) is a subgroup of H^{2}(G, B) and we have a natural filtration

0 = P_{o}H^{2}(G, B) ⊂ P_{1}H^{2}(G, B) ⊂ P_{2}H^{2}(G, B) ⊂ P_{n}H^{2}(G, B) ⊂ …

of H^{2}(G, B).

By 2.2(i) this filtration generalizes the filtration defined by Passi [27] p. 63 for trivial G-modules B and studied by several authors [26], [34], [28], [17], [5], [35]. It is, however, still an open problem (even in the central case) to give an intrinsic characterization of the extension classes lying in P_{n}H^{2}(G, B). In case B is an additively torsion-free nilpotent module this problem is solved in theorem 4.2 below.

In order to gain a better description of polynomial cocycles we introduce the following construction.

Definition 2.4 Let A be an algebra with augmentation ϵ over a commutative ring R, Ā = Ker(ϵ). Let I ⊂ Ā be an ideal and M a left A-module such that IM = 0. Then a left A/I -module (Ā/I) ˜⊗ M is defined by taking (Ā/I) ˜⊗M = (Ā/I)⊗_{R} M as an R-module and by imposing the A/I-multiplication

a(b ⊗ x) = ab ⊗ x − (a − ϵ(a) · 1) ⊗ bx ,

a ∈ A/I, b ∈ Ā/I, x ∈ M.

Let G be a group and let (B_{i}(G), δ_{i}) denote the normalized bar resolution of ZZ. Then it is easily verified that a G-linear isomorphism

B_{2}(G)/Im(δ_{3}) ≅ Δ_{Z Z}(G) ˜⊗Δ_{Z Z}(G)

is induced by sending the element [a|b] to (a − 1) ⊗ (b − 1) for a, b ∈ G.

From this observation or from (24) we obtain

Lemma 2.5 A normalized 2-cochain d is a 2-cocycle if and only if the homomorphism d̄: Δ_{Z Z}(G) ˜⊗ Δ_{Z Z}(G) → B (23) is G-linear. Consequently, the correspondence d ↦ d̃ in 2.1 is a 1-1 correspondence between polynomial 2-cocycles of degree ≤ (n, n + 1) and G-linear homomorphisms P_{n,Z Z}(G) ˜⊗ P_{n+1,Z Z}(G) → B.

We remark that the construction ˜⊗ above can be used in a similar way to define polynomial 2-cohomology of nilpotent Lie algebras, which serves as a basic tool for studying their affine structures [13].

Remark 2.6 If in 2.4 A = R(G) is a group algebra and I = 0 then there is an A-linear isomorphism

Ā ˜⊗ M ⟶^{≅} Ā ⊗_{R} M

with the diagonal representation of G on Ā ⊗_{R} M, given by (a − 1) ⊗ x ↦ (a − 1) ⊗ ax, with inverse map (a − 1) ⊗ x ↦ (a − 1) ⊗ a^{−1}x, a ∈ G, x ∈ M. For I ≠ 0, however, Ā ˜⊗ M is not in general isomorphic to the diagonal representation. For example, when G is a free group, R = ZZ, I = Δ_{Z Z}^{3}(G) and M = P_{3,Z Z}(G), then P_{2,Z Z}(G) ˜⊗ P_{3,Z Z}(G) is a nilpotent G-module of class ≤ 3 which is not true for the diagonal action on P_{2,Z Z}(G) ⊗_{Z Z} P_{3,Z Z}(G).

Now let B be a nilpotent G-module of class ≤ n + 1. Then the group P_{n}H^{2}(G, B) can be entirely described in terms of polynomial cocycles and polynomial coboundaries, as follows.

Define a homomorphism

γ: Hom_{Z Z}(P_{n+1,Z Z}(G), B) → Hom_{G}(P_{n,Z Z}(G) ˜⊗ P_{n+1,Z Z}(G), B)

by γ(f)(x⊗y) = xf(y)−fμ(x⊗y) where μ: P_{n,Z Z}(G) ⊗ P_{n+1,Z Z}(G) → P_{n+1,Z Z}(G) is the homomorphism induced by multiplication.

Proposition 2.7 Let G be a group, B a nilpotent G-module of class ≤ k and let n ≥ k − 1. Then there is a natural isomorphism

P_{n}H^{2}(G, B) ≅^{ψ} Hom_{G}(P_{n,Z Z}(G) ˜⊗ P_{n+1,Z Z}(G) , B)/γ Hom_{Z Z}(P_{n+1,Z Z}(G) , B) .

Using the bar resolution of G the map ψ is given by ψ^{−1}(g)[a|b] = g(p_{n}(a) ⊗ p_{n+1,Z Z}(b)) for g ∈ Hom_{G}(P_{n,Z Z}(G) ˜⊗ P_{n+1,Z Z}(G), B) and a, b ∈ G. In particular, if B is a trivial module then

P_{n}H^{2}(G, B) ≅ Hom_{G}(P_{n,Z Z}(G) ⊗_{G} P_{n,Z Z}(G) , B)/μ^{∗} Hom_{Z Z}(P_{n+1,Z Z}(G) , B) .

□

This follows immediately from 2.5, 2.2(ii) and (25).

We consider P_{n}H^{2}(G, B) as a polynomial approximation of H^{2}(G, B) which has the advantage to be well computable from a finite presentation of G via 2.7, cf. sections 6 and 7. For this reason we now turn to a study of the problem of convergence by first relating it to other polynomiality properties of group extensions.

3 Extensions of groups and modules

Let G be a group, B a G-module and I ⊂ Δ_{Z Z}(G) a left ideal in ZZ(G). Let

\underline{M}: B ⟶^{>}^{j} M ⟶^{q} Δ_{Z Z}(G)/I

be an extension of left G-modules where G acts on Δ_{Z Z}(G)/I by left multiplication.

3.1 Then a group extension

P ull(G, \underline{M}) = (B ⟶^{>}^{j′} E_{o} ⟶^{pr_{2}} G)

is defined by

E_{o} = { (x, a) ∈ M × G | q(x) = a − 1 + I } ,
(x, a) · (y, b) = (x + a · y , ab)

and j′x = (jx, 1) . Moreover, the map pr_{1} : E_{o} → M is a pr_{2}-derivation.

3.2 One easily checks that the G-action on B induced by conjugation in E coincides with the given one.

3.3 The extension class of P ull(G, \underline{M}) in H^{2}(G, B) is also obtained by applying the Bockstein operator

β\underline{M} : H^{1}(G, Δ_{Z Z}(G)/I) → H^{2}(G, B)

associated with \underline{M} to the canonical derivation a ↦ a − 1 + I.

Alternatively, it is obtained by applying the composite homomorphism

Ext_{G}^{1}(Δ_{Z Z}(G)/I, B) ⟶^{ρ^{∗}}Ext_{G}^{1}(Δ_{Z Z}(G), B) ≅^{σ} H^{2}(G, B)

to the extension class of \underline{M} ; here ρ : Δ_{Z Z}(G) → → Δ_{Z Z}(G)/I is the canonical quotient map and σ is the classical shifting isomorphism.

We say that a group extension \underline{E}: B ^{≻}→^{i} E → →^{π}G is induced by a module extension \underline{M}: B ⟶^{>}^{j} M ⟶^{q} Δ_{Z Z}(G)/I if it is congruent to the extension P ull(G, \underline{M}).

For example, in studying extensions in the variety N_{n} of nilpotent groups of class ≤ n one considers the case I = Δ_{Z Z}^{n+1}(G) . For torsion-free groups B Kuz’min proved in [19] that each extension \underline{E} in N_{n} is induced by an extension B ⟶^{>} M ⟶ P_{n,Z Z}(G) of modules over ZZ_{(n)}(G): = ZZ(G)/Δ_{Z Z}^{n}(G) . For non-torsion-free groups B this is no longer true in general, even for n = 2; the situation is much more complicated since it involves, among other phenomena [21], the classical dimension subgroup problem. For n = 2, at least, the situation is completely cleared up in [11].

For studying the filtration P_{n}H^{2}(G, B) of H^{2}(G, B) for torsion-free B, however, it is more appropriate to consider I = ^{+} \sqrt{Δ_{Z Z}^{n+1}(G)}, the additive isolator. This becomes apparent in the following theorem which compares several polynomiality properties of group extensions to each other.

Theorem 3.4 Let \underline{E}: B ^{≻}→^{i} E → →^{π}G be a group extension with abelian kernel B, n ≥ 1. Then the following properties of \underline{E} are equivalent:

(i) The G-action on B induced by conjugation in E is nilpotent of class ≤ n, and \underline{E} is representable by a polynomial 2-cocycle of degree ≤ (n − 1, n), i.e. the extension class of \underline{E} lies in filtration P_{n−1}H^{2}(G, B) of H^{2}(G, B).

(ii) \underline{E} is induced by an extension \underline{M}: B ⟶^{>}^{j} M ⟶^{q} P_{n,Z Z}(G) of ZZ_{(n)}(G)- modules which is split over ZZ.

(iii) The homomorphism p_{n,Z Z}i: B → P_{n,Z Z}(E, B) of groups is split injective.

Moreover, if the group B is torsion-free then these properties are equivalent to the following ones:

(iv) \underline{E} is induced by an extension \underline{N}: B ⟶^{>}^{ι} N ⟶^{κ} Pn^{√} (G) of ZZ_{(n)}(G)- modules which is split over ZZ.

(v) The sequence 0 → B ⟶^{p_{n}^{√} i} Pn^{√} (E, B) ⟶^{P_{n}^{√} (π)} Pn^{√} (G) → 0 of ZZ_{(n)}(G)-

modules is exact and split over ZZ.

For central extensions the equivalence of (i) and (iii) was discovered by Passi [25], [27] p. 64.

Proof of 3.4. Suppose that the G-action on B induced by conjugation in E is nilpotent of degree ≤ n and let \underline{M}: B ⟶^{>}^{j} M ⟶^{q} P_{n,Z Z}(G) be an extension of abelian groups which is split by means of a retraction r: M → B and dual section s: P_{n,Z Z}(G) → M, sq = id − jr. Consider a homomorphism

W : P_{n−1,Z Z}(G) ⊗ M → M

which fits into a commutative diagram

P_{n−1,Z Z}(G) ⊗ B ⟶^{1⊗j} P_{n−1,Z Z}(G) ⊗ M ⟶^{1⊗q} P_{n−1,Z Z}(G) ⊗ P_{n,Z Z}(G)
↓^{μ_{B}}  ↓^{W}  ↓^{μ}
B  ⟶^{j}  M  ⟶^{q}  P_{n,Z Z}(G)

where μ_{B} is induced by the G-action map Δ_{Z Z}(G) ⊗ B → B (which is supposed to satisfy Δ_{Z Z}^{n}(G)B = 0). Then one readily deduces from 2.5 that the following properties are equivalent:

(a) W defines a G-action on M by a · m = m + W (p_{n−1,Z Z}(a) ⊗ m) which makes \underline{M} into an extension of ZZ_{(n)}(G)-modules;

(b) the component

W′: P_{n−1,Z Z}(G) ⊗ P_{n,Z Z}(G) ⟶^{1⊗s} P_{n−1,Z Z}(G) ⊗ M ⟶^{W} M ⟶^{r} B

of W defines a polynomial 2-cocycle d_{W} of degree ≤ (n − 1, n) by the composition

d_{w}: G × G ⟶^{p_{n−1,Z Z}⊗p_{n,Z Z}} P_{n−1,Z Z}(G) ⊗ P_{n,Z Z}(G) ⟶^{W′} B .

Moreover, if (a) holds then the group extension induced by \underline{M} is represented just by the 2-cocycle d_{W}. This follows from 3.3 and the following calculation involving (25) for a, b ∈ G:

β\underline{M}(p_{n,Z Z})(a, b) = (p_{n,Z Z}i)^{−1}(asp_{n,Z Z}(b) − sp_{n,Z Z}(ab) + sp_{n,Z Z}(a)))
= (p_{n,Z Z}i)^{−1}(asp_{n,Z Z}(b) − s(ap_{n,Z Z}(b)))
= (p_{n,Z Z}i)^{−1}(id − sP_{n,Z Z}(π))(asp_{n,Z Z}(b))
= r(asp_{n,Z Z}(b))
= r((a − 1)sp_{n,Z Z}(b))
= W′(p_{n−1,Z Z}(a) ⊗ p_{n,Z Z}(b)) .

So if \underline{E} is representable by a polynomial 2-cocycle of degree ≤ (n − 1, n) and d̄: P_{n−1,Z Z}(G) ⊗ P_{n,Z Z}(G) → B is the corresponding homomorphism then \underline{E} is induced by the extension of ZZ_{(n)}(G)-modules defined by taking M = B ⊕ P_{n,Z Z}(G) and

W = ( \begin{matrix}μ_{B} & d̄ \\ 0 & μ \end{matrix}) .

This proves the implication (i) ⇒ (ii). Now suppose that the homomorphism p_{n,Z Z}i: B → P_{n,Z Z}(E, B) is split injective. Then the sequence

\underline{M}: 0 → B ⟶^{p_{n,Z Z}i} P_{n,Z Z}(E, B) ⟶^{P_{n,Z Z}(π)} P_{n,Z Z}(G) → 0

is an extension of ZZ_{(n)}(G)-modules which is split over ZZ. In particular, the G-action on B is nilpotent of degree ≤ n since the action on P_{n,Z Z}(E, B) is. Moreover, the map (p_{n,Z Z}, π) : E → P_{n,Z Z}(E, B) × G induces a congruence \underline{E} → P ull(G, \underline{M}). So if the homomorphism W : P_{n−1,Z Z}(G) ⊗ P_{n,Z Z}(E, B) → P_{n,Z Z}(E, B) is given by the G-action on P_{n,Z Z}(E, B) then the polynomial 2-cocycle d_{W} is a cocycle for \underline{E}. This proves (iii) ⇒ (i). To prove the implication (ii) ⇒ (iii) let \underline{E} be induced by an extension \underline{M} as in (ii), i.e. there is an isomorphism α: E → E_{o} such that αi = j′ and pr_{2}α = π. Then the map pr_{1}α: E → M is a π-derivation which admits a factorization pr_{1}α: E ⟶^{p_{n,Z Z}} P_{n,Z Z}(E, B) ⟶^{ᾱ} M through a G-linear homomorphism ᾱ by the universal property of p_{n,Z Z}, see 1.1. We obtain a commutative diagram of G-linear homomorphisms

B ⟶^{>}^{j}  M  ⟶^{q} P_{n,Z Z}(G)
∥  ↑ᾱ  ∥
B ⟶^{p_{n,Z Z}i} P_{n,Z Z}(E, B) ⟶^{P_{n,Z Z}(π)} P_{n,Z Z}(G)

since ᾱp_{n,Z Z}i(b) = pr_{1}αi(b) = pr_{1}j′(b) = j(b) and qᾱp_{n,Z Z}(a) = qpr_{1}α(a) = p_{n,Z Z}pr_{2}α(a) = p_{n,Z Z}π(a) = P_{n,Z Z}(π)p_{n,Z Z}(a) for b ∈ B, a ∈ G. It follows that the map p_{n,Z Z}i: B → P_{n,Z Z}(E, B) is injective and consequently ᾱ is isomorphic since both rows are exact. So if r: M → B is a retraction of j then rᾱ is a retraction of p_{n,Z Z}i, as required.

For the remainder of the proof suppose that B is torsion-free. If (ii) holds then q maps τM isomorphicly onto τP_{n,Z Z}(G), whence the sequence \underline{N} : B ⟶^{>}^{τ̄j} τ̄M ⟶^{τ̄(q)} Pn^{√} (G) is an extension of ZZ_{(n)}(G)-modules which is split over ZZ. It induces \underline{E} since the quotient map τ̄ : M → →τ̄M induces a congruence \underline{E} ≅ P ull(G, \underline{M})⟶^{≅} P ull(G, \underline{N}). This proves (ii) ⇒ (iv). To prove (iv) ⇒ (v) suppose that (iv) holds. Similarly as in the proof of the implication (ii) ⇒ (iii) we obtain a commutative diagram of G-linear homomorphisms

B ⟶^{>}^{ι}  N  ⟶^{κ} Pn^{√} (G)
∥  ↑ᾱ  ↑τ̄
B ⟶^{p_{n,Z Z}i} P_{n,Z Z}(E, B) ⟶^{P_{n,Z Z}(π)} P_{n,Z Z}(G)

By Ker(ᾱ) ∩ p_{n,Z Z}i(B) = 0 and by exactness of the bottom row the subgroup Ker(ᾱ) is mapped injectively into Ker(τ̄) = τP_{n,Z Z}(G) by P_{n,Z Z}(π), whence lies in τP_{n,Z Z}(E, B). On the other hand, N is torsion-free, whence Ker(ᾱ) = τP_{n,Z Z}(E, B) and there is an injective map ᾱ′: Pn^{√} (E, B) → N induced by ᾱ;

(26) for later use, we note that ᾱ′ is actually a G-isomorphism.

Thus the map pn^{√} i is injective and KerPn^{√} (π) = Ker(κᾱ′) = p_{n,Z Z}i(B). It follows that the sequence in (v) is exact, whence an extension of ZZ_{(n)}(G)-modules isomorphic to \underline{N} via ᾱ′ and, in particular, split over ZZ.

Finally, if (v) holds and r: Pn^{√} (E, B) → B is a retraction of pn^{√} i then the composition rτ̄: P_{n,Z Z}(E, B) → B is a retraction of p_{n,Z Z}i, i.e. (iii) holds and the proof is complete. □

4 Representability of group extensions by polynomial cocycles

4.1 In this section let

\underline{E}: B ^{≻}→^{i} E → →^{π}G

be a group extension such that

• B is torsion-free abelian

• the G-action on B induced by conjugation in E is nilpotent of class ≤ k

• either the group B is divisible or G is finitely generated.

Furthermore, let n ≥ k.

We are going to give a necessary and sufficient condition for representability of \underline{E} by a polynomial cocycle. For this, recall 3.4. First of all, if G is finitely generated so is the group P_{n,Z Z}(G), whence Pn^{√} (G) is finitely generated free abelian and the splitting condition in (v) is redundant; the same conclusion holds if B is a divisible and whence injective abelian group. But exactness of the sequence in (v) was characterized intrinsically in 1.2, so all results there can be interpreted in view of 3.4, as follows.

Theorem 4.2 Under the assumptions of 4.1, \underline{E} is representable by a polynomial cocycle of degree ≤ (n − 1, n) if and only if the three conditions (A) - (C) in 1.2 are satisfied.

In particular, this holds provided there is a number c ≤ n such that the properties (a) - (c) in 1.4 are satisfied. □

Corollary 4.3 Let G in 4.1 be torsion-free nilpotent and let c, e be the nilpotency classes of G and E, respectively. Then \underline{E} is representable by a polynomial cocycle of degree ≤ (n − 1, n) with n = e if B is divisible and with n = max{e, c + k − 1} if G is finitely generated. □

Thus we obtain our desired

Convergence Theorem 4.4 Let G be a torsion-free nilpotent group of class ≤ c and B a ZZ -torsion-free nilpotent G-module of class ≤ k such that either the group B is divisible or G is finitely generated. Then

P_{c+k−1}H^{2}(G, B) = H^{2}(G, B) . □

Special cases of this result are obtained in the literature, where the module structure of B is always supposed to be trivial and either B is divisable [28] or B = ZZ but in addition G is abelian [17], nilpotent of class 2 [5], has torsion-free lower central quotients (Passi 1991, see [35]), or is an arbitrary finitely generated T -group [8], [9].

The following application of 1.6 may also be worthwile to be written out.

Corollary 4.5 Let E be a finitely generated T_{e}-group, B an abelian normal subgroup. Then \sqrt{B} is abelian, cf. 1.6. Let c, k be the nilpotency classes of G = E/\sqrt{B} and of the G-action on \sqrt{B} induced by conjugation in E. Then the extension \sqrt{B} ↪ E → → G is representable by a polynomial cocycle of degree ≤ (n − 1, n) with n = max{e, c + k − 1}. □

The corollary in particular applies when B = γ_{f}(E) where f is the smallest integer ≥ \frac{e+1}{2} , or when B = γ_{2}(E) if E is metabelian. In both cases one has n = e.

5 Abelian models of torsion-free nilpotent groups

It is a classical result [32] that for a group G the category of group extensions B ⟶^{>} E ⟶ G with abelian kernel is equivalent to the category of G-module extensions B ⟶^{>} M ⟶ Δ_{Z Z}(G) . We here show that for extensions of torsion-free nilpotent groups G and E an analogous ‘model theorem’ holds using ZZ -torsion-free nilpotent modules and replacing Δ_{Z Z}(G) by the polynomial quotient Pn^{√} (G) . From this result we obtain inductive models for torsion-free nilpotent groups in terms of nilpotent rings. This leads to an inductive description of automorphism groups of T -groups in [8], [10]. Moreover, it amounts to a generalization of the Dold-Kan theorem for simplicial abelian groups to simplicial groups of class 2 [8], [14].

Fix integers c, e, k ≥ 1 such that c, k ≤ e ≤ c + k. Consider the category G_{c,e,k} of group extensions \underline{E} = ( B ^{≻}→^{i} E → →^{π}G) where G and E are T -groups of class ≤ c and ≤ e, respectively, and where the G-action on B induced by conjugation in E is nilpotent of class ≤ k. Morphisms from \underline{E}_{1} to \underline{E}_{2} in G_{c,e,k} are tripels (f, F, f′) of homomorphisms which make the following diagram commutative:

B_{1} ⟶^{>}^{i_{1}} E_{1} ⟶^{π_{1}} G_{1}
↓^{f′}  ↓^{F}  ↓^{f}
B_{2} ⟶^{>}^{i_{2}} E_{2} ⟶^{π_{2}} G_{2}

On the other hand, a (c,e,k)-model is a pair (G, \underline{M}) where G is a T_{c}-group and where \underline{M} = (B ⟶^{>}^{j} M ⟶^{q} Pn^{√} (G)) is an extension of ZZ -torsion-free left G-modules with the following properties:

(i) n = max{e, c + k − 1}

(ii) the G-action on M is nilpotent of class ≤ n

(iii) the G-action on B is nilpotent of class ≤ k

(iv) i(B) ∩ Δ_{Z Z}^{e}(G)M = 0.

We assign to a group extension \underline{E} = ( B ^{≻}→^{i} E → →^{π}G) in G_{c,e,k} the (c, e, k)- model

P (\underline{E}) = (G, B ⟶^{>}^{p_{n}^{√} i} Pn^{√} (E, B) ⟶^{P_{n}^{√} (π)} Pn^{√} (G)) .

The term on the right has the required properties of a (c, e, k)-model by 1.5 and 1.13. Conversely, a (c, e, k)-model (G, \underline{M}) gives rise to the group extension P ull(G, \underline{M}), see 3.1. We have to check that E_{o} is nilpotent of class ≤ e: One has γ_{e+1}(E_{o}) ⊂ j′B since e ≥ c; but iγ_{e+1}(E_{o}) = pr_{1}γ_{e+1}(E_{o}) ⊂ Δ_{Z Z}^{e}(G)M by the G-isomorphism M ≅ Pn^{√} (E_{o}, j′B) in (26) and by the relation γ_{e+1}(E_{o}) ⊂ Δ_{Z Z}^{e+1}(E_{o}) .

We also want to compare the automorphism groups of group extensions in G_{c,e,k} with those of their (c, e, k)-models. For this we define the category M_{c,e,k} of (c, e, k)-models by taking as morphisms from (G_{1}, \underline{M}_{1}) to (G_{2}, \underline{M}_{2}) all tripels (f, F, f′) with the following properties:

• f : G_{1} → G_{2} and f′: B_{1} → B_{2} are homomorphisms of groups;

• F : M_{1} → M_{2} is an f -equivariant homomorphism, i.e. F (ax) = f(a) F (x) for x ∈ M_{1}, a ∈ G_{1};

• the following diagram is commutative:

B_{1} ⟶^{>}^{i_{1}} M_{1} ⟶^{π_{1}} Pn^{√} (G_{1})
↓^{f′}  ↓^{F}  ↓Pn^{√} (f)
B_{2} ⟶^{>}^{i_{2}} M_{2} ⟶^{π_{2}} Pn^{√} (G_{2})

The constructions P and P ull above extend to functors

G_{c,e,k} ⟶^{P} ; ⟵_{P ull} M_{c,e,k}

by defining P (f, F, f′) = (f, Pn^{√} (F), f′) and P ull(f, F, f′) = (f, F × f, f′).

Theorem 5.1 A group extension \underline{E} ∈ G_{c,e,k} is uniquely determined by its (c,e,k)-model up to congruence and conversely. Furthermore, the automorphism group of \underline{E} is mapped isomorphicly onto the automorphism group of its (c, e, k)- model by the functor P. Actually, the functors P and P ull are mutually inverse equivalences of categories.

Proof: Natural transformations P ∘ P ull ⟶ id and id ⟶ P ull ∘ P are readily established by the universal properties of pn^{√} and of the pullback used in the definition of P ull, resp. These are isomorphisms by the five-lemma. □

Remark. In comparison with the general model theorem on arbitrary group extensions in [32] our special result for T -groups has several advantages:

• For finitely generated groups E the models P (\underline{E}) consist of finitely generated free abelian groups.

• Pn^{√} (G) is easily computable by abelian generators and relations if G is given by a finite presentation, cf. 6.2 below. This is due to the fact that for free groups F the ring ZZ(F)/Δ_{Z Z}^{n}(F) is nothing more than a truncated noncommuative polynomial ring, i.e. it is free in the category of rings of index n + 1, while the ring ZZ(G) itself is not free but has a much more complicated structure.

• The modules involved in our models are nilpotent which is well adapted to inductive constructions.

From 5.1 we obtain the following inductive abelian models for torsion-free nilpotent groups and their homomorphisms.

Let S be a ring (without unit) of index n + 1, i.e. S^{n+1} = 0. Then K_{n}(S) denotes the additive subgroup of S generated by the n-fold ring commutators of S and ^{+} \sqrt{K_{n}(S)} denotes its isolator taken with respect to the additive structure of S.

We define the category R_{n} of models for T_{n}-groups as follows: Objects are pairs \underline{S} = (G, π : S → → Pn^{√} (G) ) where G is a T_{n−1}-group, S is a ring of index n + 1 whose additive group is torsion-free and where π is a surjective ring homomorphism such that Ker(π) = ^{+} \sqrt{K_{n}(S)}. Morphisms from \underline{S}_{1} to \underline{S}_{2} in R_{n} are pairs (f, F) with f ∈ Hom(G_{1}, G_{2}) and where F : S_{1} → S_{2} is a ring homomorphism such

that Pn^{√} (f) π_{1} = π_{2} F.

Now we assign to a T_{n}-group E the model

R_{n}(E) = (E/\sqrt{γ_{n}(E)} , Pn^{√} (π) : Pn^{√} (E) → →Pn^{√} (E/\sqrt{γ_{n}(E)}) )

where π : E → →E/\sqrt{γ_{n}(E)} is the canonical quotient map. This is in fact an object in R_{n} by 1.8 (ii) and (iii). Conversely, an arbitrary object \underline{S} = (G, π : S → → Pn^{√} (G) ) in R_{n} gives rise to a group

G_{n}(\underline{S}) = \{(x, a) ∈ S × G | π(x) = pn^{√} (a) \} with (x, a)(y, b) = (x + y + xy, ab) .

As P and P ull above, the constructions R_{n} and G_{n} can be extended to morphisms in a canonical way to yield functors

T_{n} ⟶^{R_{n}} ; ⟵_{G_{n}} R_{n} .

Theorem 5.2 A T_{n}-group E is determined by its model R_{n}(E) up to isomorphism, and its automorphism group is canonically isomorphic to that of R_{n}(E). Actually, the constructions R_{n} and G_{n} are mutually inverse equivalences between the category of T_{n}-groups and the category R_{n}.

In some sense, this result is a substitute of the Mal’cev correspondence between rational nilpotent groups and rational nilpotent Lie algebras in the nondivisible case. It might be useful for the construction of ‘small’ models for simplicial T - groups and thus for integral homotopy types of spaces, in analogy with Quillen’s use of the Mal’cev correspondence as a key ingredient in his modelization of rational homotopy types by rational differential graded Lie algebras. As a first step in this direction, we derive from 5.2 a Dold-Kan theorem for simplicial groups of class 2 [8], [14]; the ‘small models’ obtained in this way actually extend (in the simply connected case) the ‘quadratic modules’ introduced by Baues [3]. Applications to group cohomology with respect to the variety of nilpotent groups of class n are also to be expected.

Theorem 5.2 is an immediate consequence of 5.1 using the following

Remark 5.3 An extension of G-modules B ⟶^{>} M ⟶^{q} Pn^{√} (G) can be viewed as a singular extension of rings without unit with trivial right ZZ_{(n)}(G) -module structure on B, and conversely. In fact, the module M can be endowed with the multiplication

M ⊗ M ⟶^{q⊗M} Pn^{√} (G) ⊗ M ⟶^{μ} M

where μ is induced by the given G-action on M, μ(pn^{√} (a) ⊗ m) = (a − 1)m for a ∈ G, m ∈ M. The converse is clear.

6 The second cohomology of torsion-free nilpotent groups

Let G be a group. Then ZZ(n)^{√}(G) = τ̄(ZZ_{(n)}(G)) with ZZ_{(n)}(G) = ZZ(G)/Δ_{Z Z}^{n}(G) is an augmented ring. Now let B be a ZZ -torsion-free left G-module which is nilpotent of class ≤ n. Consider B as a bimodule over ZZ(n)^{√}(G) the right module structure of which is defined via the augmentation map of ZZ(n)^{√}(G). Let

H_{n}^{2}(ZZ(n)^{√}(G) , B) ⊂ H_{H}^{2}(ZZ(n)^{√}(G) , B)

be the subgroup of the Hochschild cohomology of ZZ(n)^{√}(G) consisting of congruence classes of singular ring extensions B ⟶^{>} S ⟶^{π} ZZ(n)^{√}(G) for which S̄ = Ker(ϵ π) satisfies S̄^{n+1} = 0.

Theorem 6.1 Let G be a T_{c}-group and let B be a ZZ -torsion-free nilpotent G-module of class ≤ k. Then there are natural isomorphisms

(1) H^{2}(G, B) ≅ Ext_{Z Z_{(c+k)}(G)}^{1}(Pc+k^{√}(G) , B)

(2) H^{2}(G, B) ≅ H_{c+k}^{2}(ZZ(c+k+1)^{√}(G) , B)

(3) H^{2}(G, B) ≅ Hom_{G}(P_{c+k−1}(G) ˜⊗ P_{c+k}(G), B)/γ Hom_{Z Z}(P_{c+k}(G), B)

cf. 2.7, where for (2) and (3) we suppose in addition that either the group B is divisible or G is finitely generated. □

Assertions (1) and (3) are immediate consequences of 5.1 and of 4.4 and 2.7, respectively. Assertion (2) follows from (1) by 5.3 and the same arguments as quoted after 4.1.

If G is given by a free presentation then part (3) of the theorem amounts to a method of computing H^{2}(G, B) by using matrix calculus, as follows.

Let G = ⟨X | R⟩ be a free presentation of a group where X = {x_{i}} is some set and R = {r_{j}} is some subset of the free group F = ⟨X⟩ generated by X. Write π : F → →G = F/N where N is the normal subgroup generated by R. Let T be the noncommutative polynomial ring in the variables x_{i} with unit and augmentation map ZZ ⟶^{η} T ⟶^{ϵ} ZZ, respectively. For n ≥ 1 let T̄ = Ker(ϵ), T_{n} = T/T̄^{n+1}, T̄_{n} = T̄/T̄^{n+1}. For m > n let ρ : T_{m} → →T_{n} be the canonical ring quotient and ν : T_{n} ⟶^{>} T_{m} the additive inclusion homomorphism defined by use of the basises for T and T_{i} consisting of monomials in the x_{i}.

Now fix n. Write r_{j} = x_{j1}^{±1} ⋯ x_{js_{j}}^{±1} in F and let

r̄_{j} = (1 + x_{j1})^{±1} ⋯ (1 + x_{js_{j}})^{±1} − 1 in T̄_{n}

with (1 + x)^{−1} = 1 + ∑_{μ=1}^{n}(−1)^{μ}x^{μ} , and let R̄ = span{r̄_{j}}. Let S ⊂ T̄_{n} be a subgroup spanned by elements generating T_{n}R̄T_{n} as a two-sided ideal so that T_{n}R̄T_{n} = T_{n}ST_{n} (e.g., one can take S = R̄).

Lemma 6.2 The surjection m = m_{G} : T̄_{n} → →P_{n,Z Z}(G) defined by m(x_{t_{1}} ⋯ x_{t_{m}}) = p_{n,Z Z}π(x_{t_{1}}) ⋯ p_{n,Z Z}π(x_{t_{m}}) induces an isomorphism of rings

m̄: T̄_{n}/T_{n}ST_{n} ≅ P_{n,Z Z}(G) .

Proof: For R = ∅ an inverse isomorphism of m_{F} : T̄_{n} → →P_{n,Z Z}(F) is readily established by the universal property of the ring ZZ(F). For arbitrary R one has

Ker(ZZ(π) : ZZ(F) → ZZ(G)) = ZZ(F)Δ_{Z Z}(N)ZZ(F) = ZZ(F)Δ_{Z Z}(R)ZZ(F)

by the identities ab−1 = a(b−1)+(a−1), a^{−1} −1 = −a^{−1}(a−1) and a^{−1}ba−1 = a^{−1}[b−1, a−1]+(b−1). Hence Ker(m̄_{G}) = m_{F}^{−1}Ker(P_{n,Z Z}(π)) = T_{n}R̄T_{n} = T_{n}ST_{n}.

□

Now consider the T_{n−1}-module T̄_{n−1} ˜⊗ T̄_{n} defined in 2.4 with A = T and I = T̄^{n}. One defines a T_{n−1}-linear homomorphism

ϕ: T_{n−1} ⊗ (T̄_{n−1} ⊗ T̄_{1}) → T̄_{n−1} ˜⊗ T̄_{n}

on the extended T_{n−1}-module on the left by extension of the ZZ -linear homomorphism id ⊗ ν: T̄_{n−1} ⊗ T̄_{1} → T̄_{n−1} ⊗ T̄_{n}. Then one easily checks the following

Lemma 6.3 ϕ is an isomorphism with inverse map

χ: T̄_{n−1} ⊗ T̄_{n} ≅ ZZ ⊗ T̄_{n−1} ⊗ (T̄_{1} ⊕ T̄_{n}^{2}) ⟶^{(η⊗1⊗1,χ_{2})} T_{n−1} ⊗ T̄_{n−1} ⊗ T̄_{1} ,

where χ_{2}(1 ⊗ y ⊗ ν(z)x_{i}) = 1 ⊗ yz ⊗ x_{i} − y ⊗ z ⊗ x_{i} , y, z ∈ T̄_{n−1} .

Now let B be a nilpotent G-module of class ≤ n. Define a sequence of group homomorphisms

Hom_{Z Z}(T̄_{n}/T_{n}ST_{n} , B) ⟶^{δ_{1}} Hom_{Z Z}(T̄_{n}^{2}/T_{n}ST̄_{n} , B) ⟶^{δ_{2}} Hom_{Z Z}(T̄_{n−1} ⊗ S, B) ,

δ_{1}(f)(x_{i_{1}} ⋯ x_{i_{t}} + T_{n}ST̄_{n}) = (π(x_{i_{1}}) − 1) ⋯ (π(x_{i_{t−1}}) − 1)f(x_{i_{t}}) − f(x_{i_{1}} ⋯ x_{i_{t}})

T̄_{n−1} ⊗ S ↪ T̄_{n−1} ⊗ T̄_{n}  ⟶^{χ} T_{n−1} ⊗ (T̄_{n−1} ⊗ T̄_{1})
↓^{δ_{2}(g)}  ↓^{m⊗μ}
B  ⟵^{μ_{B}} ZZ_{(n)}(G) ⊗ B ⟵^{1⊗g} ZZ_{(n)}(G) ⊗ T̄_{n}^{2}/T_{n}ST̄_{n}

where μ_{B}: ZZ_{(n)}(G) ⊗ B → B is given by the G-action on B and μ: T̄_{n−1} ⊗ T̄_{1} → T̄_{n}^{2}/T_{n}ST̄_{n} is defined by μ(y ⊗ x_{i}) = ν(y)x_{i} + T_{n}ST̄_{n} , y ∈ T̄_{n−1}.

Proposition 6.4 Let G = ⟨X | R⟩ and B be a nilpotent G-module of class k ≤ n. Then there is an isomorphism

P_{n−1}H^{2}(G, B) ≅ Ker(δ_{2})/Im(δ_{1})

such that a 2-cocycle d: G × G → B corresponding to g ∈ Ker(δ_{2}) is given by

d (x_{p_{1}}^{m_{1}} ⋯ x_{p_{r}}^{m_{r}} , x_{q_{1}}^{n_{1}} ⋯ x_{q_{s}}^{n_{s}}) =

∑_{I_{1}} \binom{m_{1}}{μ_{1}} ⋯ \binom{m_{r}}{μ_{r}}\binom{n_{1}}{ν_{1}} ⋯ \binom{n_{s}}{ν_{s}} g (x_{p_{1}}^{μ_{1}} ⋯ x_{p_{r}}^{μ_{r}}x_{q_{1}}^{ν_{1}} ⋯ x_{q_{s}}^{ν_{s}}) +

− ∑_{I_{2}} \binom{m_{1}}{μ_{1}} ⋯ \binom{m_{r}}{μ_{r}}\binom{n_{1}}{ν_{1}} ⋯ \binom{n_{s}}{ν_{s}} (π(x_{p_{1}}) − 1)^{μ_{1}} ⋯ (π(x_{p_{r}}) − 1)^{μ_{r}} g (x_{q_{1}}^{ν_{1}} ⋯ x_{q_{s}}^{ν_{s}})

with

I_{1} = {(μ_{1}, …, μ_{r}, ν_{1}, …, ν_{s}) | 0 ≤ μ_{i}, ν_{j}; 1 ≤ ∑ μ_{i}, ∑ ν_{j}; ∑ μ_{i} + ∑ ν_{j} ≤ n }

I_{2} = {(μ_{1}, …, μ_{r}, ν_{1}, …, ν_{s}) | 0 ≤ μ_{i}, ν_{j}; 1 ≤ ∑ μ_{i} ≤ k − 1 ; 1 < ∑ ν_{j} ≤ n }

As usual, \binom{s}{μ} = \frac{s(s−1)···(s−μ+1)}{μ!} for all s ∈ ZZ − \{0\}, and \binom{s}{0}x^{0} =^{def} 1.

Note that \binom{s}{μ} is an integer valued rational polynomial of degree μ in s for fixed μ. Hence the coefficients in the sum expansion of d above are integer valued rational polynomials in the variables m_{1}, …, m_{r}, n_{1}, …, n_{s}, of total degree ≤ n − 1 in the variables m_{1}, …, m_{r}, of total degree ≤ n in the variables n_{1}, …, n_{s}, and of total degree ≤ n + k − 1 in all variables altogether.

Proof: First note that there are T_{n−1}-linear isomorphisms

T̄_{n−1} ˜⊗ T̄_{n}/ (T_{n−1}ρ(S)T_{n−1} ⊗ νT̄_{1}) ⟶^{χ̄} T_{n−1} ⊗ (T̄_{n−1}/(T_{n−1}ρ(S)T_{n−1}) ⊗ T̄_{1})
↓^{1⊗μ̄}
T_{n−1} ⊗ (T̄_{n}^{2}/T_{n}ST̄_{n})

induced by χ and μ. So in view of 2.7 and 6.3 it remains to show that the subgroup

(m ⊗ μ)χ(T_{n−1}ρ(S)T_{n−1} ⊗ T̄_{n}^{2} + T̄_{n−1} ⊗ T_{n}ST_{n})

of ZZ_{(n)}(G) ⊗ T̄_{n}^{2}/T_{n}ST̄_{n} lies in the G-submodule generated by the subgroup (m ⊗ μ)χ(T̄_{n−1} ⊗ S). Let σ : T̄_{n}^{2} → →T̄_{n}^{2}/T_{n}ST̄_{n} . Then we calculate

(m ⊗ μ)χ(T_{n−1}ρ(S)T_{n−1} ⊗ T̄_{n}^{2}) ⊂ 1 ⊗ σ(ν(T_{n−1})ST̄_{n}^{2}) + m(T_{n−1}ρ(S)T_{n−1}) ⊗ σ(T̄_{n}^{2})
= 0
(m ⊗ μ)χ(T̄_{n−1} ⊗ (T_{n}S)T̄_{n}) ⊂ 1 ⊗ σ(T̄_{n}ST̄_{n}) + ZZ_{(n)}(G) ⊗ σ(T_{n}ST̄_{n})
= 0 .

Finally, let x ∈ T̄_{n−1}, y ∈ T̄_{n}, s ∈ S. Then

(m ⊗ μ)χ(x ⊗ ys) = (m ⊗ μ)χ(xy ⊗ s − x(y ⊗ s))
⊂ ZZ_{(n)}(G) · (m ⊗ μ)χ(T̄_{n−1} ⊗ S)

by T_{n−1}-linearity of χ and m ⊗ μ.

Thus the isomorphism H^{2}(G, B) ≅ Ker(δ_{2})/ Im(δ_{1}) is proved. The explicit form of the cocycle in the assertion is straightforward from the definition of the isomorphism m̄ in 6.2 and from the binomial formula, used to evaluate the term x^{m} − 1 = ((x − 1) + 1)^{m} − 1. □

Example: Consider the Heisenberg groups G_{q} = ⟨a, b, c | (a, b) = c^{q}, (a, c) = (b, c) = 1⟩, q ∈ ZZ. Here 6.1 and a computation by the procedure indicated above as well as a related convergence result for a polynomial cofiltration of τ̄H_{2}(G) (cf. [8, 2.3.20]) give isomorphisms

τ̄H_{2}(G) ≅ ZZ · τ̄p̄_{3}{(a, c)} ⊕ ZZ · τ̄p̄_{3}{(b, c)}
H^{2}(G, ZZ) ≅ ZZ/qZZ ⊕ ZZ ⊕ ZZ ,

where the tupel (u, w, z) is represented by the polynomial cocycle

D(u, w, z)(a^{a_{1}}b^{b_{1}}c^{c_{1}} , a^{a_{2}}b^{b_{2}}c^{c_{2}}) =

u b_{1}a_{2} + w(qb_{1}\binom{a_{2}}{2} − c_{1}a_{2}) + z(qb_{1}a_{2}b_{2} + q\binom{b_{1}}{2}a_{2} − c_{1}b_{2}) ,

a_{i}, b_{i}, c_{i} ∈ ZZ. Moreover, the Kronecker product

< , >: H^{2}(G, ZZ) × τ̄H_{2}(G)⟶ZZ

is given by

< (u, w, z) , k{(a, c)} + l{(b, c)} > = wk + zl .

An application of the formula in 6.4 for non-trivial coefficient modules is given in the last section.

7 Torsion-free nilpotent groups of class 3

Based on the cohomological results of the previous section a 1-1- correspondence between finitely generated T_{3}-groups and finitely generated ZZ -torsion-free nilpotent Lie rings of class ≤ 3 is established. Thus the construction of these groups reduces to free abelian groups and homomorphisms.

Let A be a free abelian group with basis {x_{i}}, and let S(A) be the symmetric ZZ -algebra over A which is isomorphic to the polynomial ring ZZ[{x_{i}}]. Viewing A also as an abelian Lie ring one has ring isomorphisms

ZZ_{(n)}(A) ≅^{m̄} S(A)/\overline{S(A)}^{n} ≅ U(A)/\overline{U(A)}^{n}

see 6.2, whence both nilpotent group and Lie ring representations of A of class ≤ n are given by modules over S(A)/\overline{S(A)}^{n}, and the term ‘nilpotent A-module’ may be understood in either sense. We write ax, a ∈ A, x ∈ M, for the group action and a^{∘}x = (a − 1)x for the corresponding Lie action.

Let B be a nilpotent A-module. Consider the following part of the dual of the Koszul complex where we replace E_{2}(A) by Λ^{2}A = A ⊗ A/ span{x ⊗ x | x ∈ A}.

Hom_{Z Z}(A, B) ⟶^{d_{2}^{∗}} Hom_{Z Z}(Λ^{2}A , B) ⟶^{d_{3}^{∗}} Hom_{Z Z}(E_{3}(A), B)

where one defines for x, y, z ∈ A, f ∈ Hom_{Z Z}(A, B) and g ∈ Hom_{Z Z}(Λ^{2}A , B)

d_{2}^{∗}(f)(x ∧ y) = x^{∘}f(y) − y^{∘}f(x) ,
d_{3}^{∗}(g)(x ∧ y ∧ z) = x^{∘}g(y ∧ z) + y^{∘}g(z ∧ x) + z^{∘}g(x ∧ y) .

It is an elementary fact that there is a bijection between

H_{Lie}^{2}(A, B) =^{def} Ker(d_{3}^{∗})/Im(d_{2}^{∗})

and the set of congruence classes of Lie ring extensions of A by B. Indeed, for g ∈ Ker(d_{3}^{∗}) a Lie ring L_{g} is defined by

L_{g} = B ⊕ A , [(b_{1}, a_{1}) , (b_{2}, a_{2})] = a_{1}^{∘}b_{2} − a_{2}^{∘}b_{1} + g(a_{1} ∧ a_{2}) ,

and B ⟶^{>}^{(1,0)^{t}} L_{g} ⟶^{(0,1)} A is the corresponding extension.

Theorem 7.1 Let A be a free abelian group with basis {x_{i}}, and let B be a nilpotent A-module of class ≤ 2. Then there is an isomorphism

Φ : P_{2}H^{2}(A, B) ⟶^{≅} H_{Lie}^{2}(A, B)

defined as follows: For a group extension \underline{E}: B ⟶^{>}^{ι} E ⟶^{π} A choose x̄_{i} ∈ E with π(x̄_{i}) = x_{i} and define g\underline{E} ∈ Hom(Λ^{2}A, B) by g\underline{E} (x_{i} ∧ x_{j}) = ι^{−1}(x̄_{i}x̄_{j}x̄_{i}^{−1}x̄_{j}^{−1}). Then Φ\{\underline{E} \} = \{g\underline{E} \}. Conversely, if h: Λ^{2}A → B is a Lie-cocycle, then

Φ^{−1}(h)(∑_{i} m_{i}x_{i} , ∑_{k} n_{k}x_{k}) = ∑_{i>k} m_{i}n_{k}h(x_{i} ∧ x_{k}) +
+ ∑_{i>k} \binom{m_{i}}{2}n_{k}x_{i}^{∘}h(x_{i} ∧ x_{k}) + ∑_{i>k} m_{i}\binom{n_{k}}{2}x_{k}^{∘}h(x_{i} ∧ x_{k}) +
+ ∑_{i<j>k} m_{i}m_{j}n_{k}x_{i}^{∘}h(x_{j} ∧ x_{k}) + ∑_{i>l>k} m_{i}n_{k}n_{l}x_{k}^{∘}h(x_{i} ∧ x_{l})
□

Remark: We note that the cocycle Φ^{−1}(h) which here is obtained by evaluating the formula in 6.4, equals the factor set produced by commutator calculus when requiring that ι^{−1}(x̄_{i}x̄_{j}x̄_{i}^{−1}x̄_{j}^{−1}) = h(x_{i} ∧ x_{j}). In view of this fact, the formula for P_{n}H^{2}(G, B) in 6.4 here has a twofold function: on the one hand, it manages the commutator calculus of computing the factor set for arbitrary linear combinations of basis elements x_{i} in terms of the ‘generating’ commutators x̄_{i}x̄_{j}x̄_{i}^{−1}x̄_{j}^{−1}; on the other hand, it exhibits the condition on these generating commutators which ensures that the factor set produced by that commutator calculus satisfies the cocycle property. So it seems worthwhile to review the formula in 6.4 with regard to this aspect also in the general case, notably for standard commutator presentations.

Together with 4.4 we obtain

Corollary 7.2 Let A be a free abelian group, B a torsion-free nilpotent Amodule of class ≤ 2. Then there is an isomorphism

H^{2}(A, B) ≅ H_{Lie}^{2}(A, B) .

In particular, there is a (nonnatural) bijection between T -groups and ZZ -torsion-free nilpotent Lie rings both finitely generated and nilpotent of class ≤ 3.

Indeed, a group or Lie ring of this type can be canonically identified with a group or Lie ring extension

\sqrt{γ_{2}(G)} ↪ G ⟶ τ̄G^{ab} , \sqrt{[L, L]} ↪ L ⟶ τ̄L^{ab} .

We finally remark that this result generalizes to all metabelian T -groups, see [15].

Proof of 7.1: We apply 6.4 for the canonical presentation of A, X = {x_{i}}, R = {x_{i}x_{j}x_{i}^{−1}x_{j}^{−1}}. Then we can take S = {[x_{i}, x_{j}] = x_{i}x_{j} − x_{j}x_{i}}. Choose a total ordering of X.

According to the Birkhoff-Witt theorem one has a decomposition

T̄_{3}^{2} = Λ^{2}A ⊕ S^{2}A ⊕ L_{3}A ⊕ Λ^{2}A ⊗ A ⊕ S^{3}A

using the basises

{ [x_{i}, x_{j}] | x_{i} > x_{j} }  of Λ^{2}A
{ x_{i_{1}} ⋯ x_{i_{m}} | x_{i_{1}} ≤ … ≤ x_{i_{m}} } of S^{m}A
{ [x_{i}, [x_{j}, x_{k}]] | x_{i} ≤ x_{j} > x_{k} } of L_{3}A .

Consider a group homomorphism

g: T̄_{3}^{2}/T_{3}ST̄_{3} = Λ^{2}A ⊕ S^{2}A ⊕ L_{3}A ⊕ S^{3}A ⟶ B .

Then for x ∈ T̄_{2}, y, z ∈ X we have

δ_{2}(g)(x ⊗ [y, z]) = 0 if x ∈ T̄_{2}^{2}
δ_{2}(g)(x ⊗ [y, z]) = g(x(yz − zy)) − x^{∘}g(yz − zy)) if x ∈ X
= g[x, [y, z]] − x^{∘}g[y, z]

since x[y, z] ≡ [x, [y, z]] mod Λ^{2}A ⊗ A.

Thus we find that δ_{2}(g) = 0 if and only if g[x, [y, z]] = x^{∘}g[y, z] for x, y, z ∈ X and (g|Λ^{2}A) ∈ Ker(d_{3}^{∗}) , observing that the group L_{3}A is isomorphic to A ⊗ Λ^{2}A modulo the Jacobi identity. Thus we obtain a commutative diagram with horizontal isomorphisms

Ker(d_{3}^{∗}) ⊕ Hom_{Z Z}(S^{2}A ⊕ S^{3}A, B)  ⟶^{≅}  Ker(δ_{2})
↑^{Δ}  ↑^{δ_{1}}
Hom_{Z Z}(A, B) ⊕ Hom_{Z Z}(S^{2}A ⊕ S^{3}A, B) ⟶^{≅} Hom_{Z Z}(T̄_{3}/T_{3}ST_{3}, B)

with Δ = ( \begin{matrix}d_{2}^{∗} & 0 \\ Δ_{2} & −id \end{matrix} ), Δ_{2}(f)(x_{i}x_{j}, x_{k}x_{l}x_{m}) = x_{i}^{∘}f(x_{j}). Thus

Ker(d_{3}^{∗})/Im(d_{2}^{∗}) ≅ coker(Δ) ≅ Ker(δ_{2})/Im(δ_{1}) ≅ P_{2}H^{2}(A, B)

by 6.4. That the inverse of this composite isomorphism is just Φ follows easily by revealing the definitions and observing the following fact: If d is any cocycle representing an extension \underline{E} : B ⟶^{>} E ⟶ A then

g\underline{E} (x_{i} ∧ x_{j}) = ι^{−1}(x̄_{i}x̄_{j}x̄_{i}^{−1}x̄_{j}^{−1}) = d(x_{i}, x_{j}) − d(x_{j}, x_{i}) .
□

Acknowledgments

To the supervisors of my thesis, Professors H. J. Baues and F. Grunewald, I am indebted for many useful discussions and support.

I am grateful to Prof. Yu. V. Kuz’min for making me aware of his paper [19] which enabled me to generalize former work in my thesis to non-central extensions.

Moreover, I would like to express my gratitude to the German-Israel science foundation, the Sonderforschungsbereich 343 in Bielefeld and the Human Capital and Mobility programme of the European Union for awarding successive researchs positions to me which enable me to persue freely my work programme. Moreover, I am very grateful to Professor H. Abels at Bielefeld and Professors J.-L. Loday and C. Kassel at Strasbourg for creating a friendly and stimulating atmosphere and for many interesting discussions.

Finally, I am grateful to the Katholieke Universiteit Leuven for awarding a three months junior fellowship to me, thus providing the oppurtunity to give a seminar on the work presented in this paper. In particular, I am indebted to Prof. P. Igodt, Dr. K. Dekimpe and W. Malfait for pointing out many weak points in a former preprint version and for making my stay at Campus Kortrijk very enjoyable in any sense.

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