Georgian Mathematical Journal Volume 9 (2002), Number 4, 703–708

A MAGNUS–WITT TYPE ISOMORPHISM FOR NON-FREE GROUPS

G. ELLIS

Dedicated to Professor Hvedri Inassaridze

Abstract. We use the theory of nonabelian derived functors to prove that certain Baer invariants of a group G are torsion when G has torsion second integral homology. We use this result to show that if such a group has torsion-free abelianisation then the Lie algebra formed from the quotients of the lower central series of G is isomorphic to the free Lie algebra on G_{ab}. We end the paper with some related remarks about precrossed modules and partial Lie algebras.

2000 Mathematics Subject Classification: Primary: 20E14, 20F40; Secondary: 20J06, 18G50. Key words and phrases: Baer invariants, nonabelian derived functors, precrossed module, partial Lie albebra, Peiffer commutator.

1. Introduction

The lower central series of a group G is defined by setting γ_{1}G = G and γ_{n+1}G = [γ_{n}G, G] for n ⩾ 1. The quotients of this series form a Lie algebra (over Z)

ΓG = ⨁_{n⩾1}γ_{n}G/γ_{n+1}G,

where the Lie bracket is induced by the commutator maps γ_{m}G × γ_{n}G → γ_{m+n}G, (x, y) ↦ [x, y] = xyx^{−1}y^{−1}.

Also associated to the group G is the free Lie algebra L(G_{ab}) on the abelian group G_{ab} = G/γ_{2}G. This is defined, up to isomorphism, by the following universal property: there is an additive homomorphism ι: G_{ab} → L(G_{ab}) such that, for any other Lie algebra M and additive homomorphism α: G_{ab} → M, there exists a unique Lie homomorphism \overline{α}: L(G_{ab}) → M making the triangle

\xymatrix{G_{ab} \ar[r]^{ι} \ar[dr]^{α} & L(G_{ab}) \ar[d]^{\overline{α}} \\  & M}

commute. The existence of L(G_{ab}) follows from general considerations, and the universal property implies that ι must be injective.

The canonical inclusion ϕ: G_{ab} → ΓG induces a (surjective) homomorphism \overline{ϕ}: L(G_{ab}) ↠ ΓG. A result of Magnus [10] and Witt [13] asserts that \overline{ϕ} is an isomorphism when G is a free group. In this paper we shall prove the following.

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Theorem 1. If the second integral homology H_{2}(G, Z) is a torsion group then ker \overline{ϕ} lies in the torsion subgroup of the additive group of L(G_{ab}). In particular \overline{ϕ}: L(G_{ab}) ↠ ΓG is an isomorphism under the additional assumption that G_{ab} is torsion free.

A variant of this theorem was proved by Strebel [12]. His proof involves the Poincaré-Birkhoff-Witt theorem and techniques from classical homological algebra. Although Theorem 1 could be deduced from Strebel’s work, we shall give an alternative proof involving the theory of nonabelian derived functors developed by Hvedri Inassaridze [6] and others. We shall derive the theorem from the following result on the Baer invariants

M^{(n)}(G) = \frac{R ∩ γ_{n+1}F}{γ_{n+1}(R, F)} , n ⩾ 1,

which are defined in terms of a free presentation G ≅ F/R. Here F is a free group with normal subgroup R, γ_{1}(R, F) = R and γ_{n+1}(R, F) = [γ_{n}(R, F), F] for n ⩾ 1. The invariants M^{(n)}(G) are abelian groups and are independent of the choice of free presentation (cf. [9]). Note that Hopf’s formula states M^{(1)}(G) ≅ H_{2}(G, Z).

Theorem 2. If M^{(1)}(G) is a torsion group then M^{(n)}(G) is a torsion group for all n ⩾ 1.

We end the paper with some related remarks about the partial Lie algebra formed from the lower Peiffer central series of a precrossed module.

2. The Proof of Theorem 1

For all n ⩾ 1 a normal inclusion N ⩽ G gives rise to a natural five term exact sequence (cf. [8])

M^{(n)}(G) → M^{(n)}(G/N) → N/γ_{n+1}(N, G) → G/γ_{n+1}G → G/Nγ_{n+1}G → 1 .

On taking N = γ_{2}G we obtain the exact sequence

M^{(n)}(G) ⟶^{λ} M^{(n)}(G_{ab}) → γ_{n+1}G/γ_{n+2}G → 0 .  (1)

It is explained in [3] that there is a natural isomorphism of abelian groups

L(G_{ab}) ≅ ⨁_{n⩾1}M^{(n)}(G_{ab}) .  (2)

When G = F is a free group this isomorphism is precisely the Magnus-Witt isomorphism. The isomorphism is obtained for arbitrary G by describing L((F/R)_{ab}) as a quotient of L(F_{ab}). We can combine (1) and (2) to obtain an exact sequence

⨁_{n⩾1}M^{(n)}(G) ⟶^{⊕λ} L(G_{ab}) ⟶^{\overline{ϕ}} ΓG → 1.

The first assertion of Theorem 1 follows from this sequence and Theorem 2. The second assertion then follows from the following claim applied to A = G_{ab}.

Claim 1. If A is a torsion-free abelian group, then the additive group of the free Lie algebra L(A) is torsion-free.

To prove the claim suppose that A is a torsion-free abelian group. Let T (A) = ⊕_{n⩾1}(⊗^{n}A) denote the tensor algebra on A. Thus T (A) is an associative algebra (over Z) formed by taking the direct sum of iterated tensor products of the abelian group A. Since the tensor product of torsion-free groups is torsion-free, the abelian group underlying T (A) is torsion-free. The Poincaré-Birkhoff-Witt theorem for Z-modules [5] implies that the free Lie algebra L(A) imbeds into T (A). This proves the claim.

3. The Proof of Theorem 2

Recall that a free simplicial resolution of G consists of a simplicial group F_{∗} = {F_{m}}_{m⩾0} with π_{0}(F_{∗}) ≅ G, π_{m}(F_{∗}) = 0 for m ⩾ 1, and with the F_{m} free groups whose bases are preserved by the boundary and degeneracy morphisms d_{i}^{m}: F_{m} → F_{m−1}, s_{i}^{m}: F_{m} → F_{m+1}, 0 ⩽ i ⩽ m. In this context the homotopy groups of F_{∗} can be defined as π_{m}(F_{∗}) = ker(∂_{m})/im(∂_{m+1}) where ∂_{m}: ∩_{i=1}^{m} ker d_{i}^{m} ⟶ ∩_{i=1}^{m−1} ker d_{i}^{m−1} is the restriction of d_{0}^{m}: F_{m} → F_{m−1}. In other words, π_{m}(F_{∗}) is the mth homology group of the Moore complex

N(F_{∗}) : ⋯ N(F_{∗})_{m} ⟶^{∂_{m}} N(F_{∗})_{m−1} → ⋯ → N(F_{∗})_{0} → 1

with N(F_{∗})_{m} = ∩_{i=1}^{m} ker d_{i}^{m}.

Given a functor T: (Groups) → (Groups), we define its left derived functors as

L_{m}^{T}(G) = π_{m}(T (F_{∗})), m ⩾ 0.

The groups L_{m}^{T}(G) are independent of the choice of free simplicial resolution. For more details see for instance [6].

We shall consider the derived functors of the functor

τ_{n}: (Groups) → (Groups), G ↦ G/γ_{n+1}G

for n ⩾ 1. In [4] it was observed that there are natural isomorphisms

L_{0}^{τ_{n}}(G) ≅ τ_{n}G = G/γ_{n+1}G ,
L_{1}^{τ_{n}}(G) ≅ M^{(n)}(G) .

Let F_{∗} be a free simplicial resolution of G. The short exact sequence of simplicial groups

1 → γ_{n+1}(F_{∗}) → F_{∗} → τ_{n}(F_{∗}) → 1

gives rise to a long exact sequence of homotopy groups part of which is

π_{1}(F_{∗}) → π_{1}(τ_{n}(F_{∗})) → π_{0}(γ_{n+1}(F_{∗})) → π_{0}(F_{∗}) → π_{0}(τ_{n}(F_{∗})) → 1 .

Using the various isomorphisms we can rewrite this as an exact sequence

0 → M^{(n)}(G) → π_{0}(γ_{n+1}(F_{∗})) → G → τ_{n}(G) → 1

from which we obtain the natural isomorphism

M^{(n)}(G) ≅ π_{0}(γ_{n+1}(F_{∗}))  (3)

The short exact sequence of simplicial groups

1 → γ_{n+2}(F_{∗}) → γ_{n+1}(F_{∗}) → γ_{n+1}(F_{∗})/γ_{n+2}(F_{∗}) → 1

and (3) yield the exact sequence

π_{1}(γ_{n+1}(F_{∗})/γ_{n+2}(F_{∗})) → M^{(n+1)}(G) → M^{(n)}(G) .  (4)

Theorem 2 follows by induction from (4) and the following. Claim 2. If M^{(1)}(G) is a torsion group, then so too is π_{1}(γ_{n+1}(F_{∗})/γ_{n+2}(F_{∗})) for n ⩾ 1.

To prove the claim we introduce the functors

Γ_{n}: (Groups) → (Groups), G ↦ γ_{n}G/γ_{n+1}G, n ⩾ 1.

Thus, as an abelian group, ΓG is equal to ⨁_{n⩾1} Γ_{n}G. Define L_{n}(G_{ab}) to be the preimage of Γ_{n}G under the canonical surjection L(G_{ab}) ↠ ΓG. The result of Magnus and Witt gives a natural isomorphism L_{n}(F_{ab}) ≅ Γ_{n}F for any free group F.

An abelian group is torsion if and only if it is trivial when tensored by the rationals. Tensoring the Lie algebra L(F_{ab}) by the rationals yields a Lie algebra L(F_{ab}) ⊗ Q over Q. Let L(F_{ab} ⊗ Q) denote the free lie algebra (over Q) on the vector space F_{ab} ⊗ Q. By verifying the appropriate universal property we see that there is a natural isomorphism L(F_{ab}) ⊗ Q ≅ L(F_{ab} ⊗ Q). Let L_{n}(F_{ab} ⊗ Q) denote the vector subspace of L(F_{ab} ⊗ Q) generated by the canonical image of L_{n}(F_{ab}). So L_{n}(F_{ab} ⊗ Q) ≅ L_{n}(F_{ab}) ⊗ Q.

We have

π_{1}(γ_{n+1}(F_{∗})/γ_{n+2}(F_{∗})) ⊗ Q = π_{1}( Γ_{n+1}(F_{∗}) ) ⊗ Q
≅ π_{1}( L_{n+1}((F_{∗})_{ab}) ) ⊗ Q
= H_{1}( N( L_{n+1}((F_{∗})_{ab})) ) ) ⊗ Q
≅ H_{1}( N( L_{n+1}((F_{∗})_{ab}) ) ⊗ Q ) (5)
≅ H_{1}( N( L_{n+1}((F_{∗})_{ab}) ⊗ Q ) ) (6)
= π_{1}( L_{n+1}((F_{∗})_{ab}) ⊗ Q )
≅ π_{1}( L_{n+1}((F_{∗})_{ab} ⊗ Q) ).

Isomorphism (5) follows from the Universal Coefficient Theorem for chain complexes of free abelian groups (see for instance [11]). Isomorphism (6) follows from the Dold-Kan theorem [7] (which expresses a simplicial abelain group A_{∗} in terms of its Moore complex N(A_{∗})) and the fact that tensor products commute with direct sums.

By the Universal Coefficient Theorem, π_{1}((F_{∗})_{ab} ⊗ Q) ≅ π_{1}((F_{∗})_{ab}) ⊗ Q ≅ M^{(1)}(G)⊗Q. Suppose that M^{(1)} is a torsion group. Then M^{(1)}(G)⊗Q = 0 and in low dimensions (⩽ 2) the simplicial abelian group (F_{∗})_{ab} ⊗ Q coincides with a free simplicial resolution of the vector space G_{ab} ⊗ Q. Since vector spaces are ‘free objects’, the simplicial group (F_{∗})_{ab} ⊗ Q admits a contracting homotopy in low dimensions. Therefore π_{1}(L_{n+1}((F_{∗})_{ab} ⊗ Q)) = 0. This proves the claim.

4. Peiffer Commutators and Partial Lie Algebras

The isomorphism of Magnus and Witt has been generalised by H. J. Baues and D. Conduché [1]. In the generalisation free groups are replaced by ‘free precrossed modules’ and Lie algebras are replaced by “partial Lie algebras”. We suspect that Theorem 1 might also admit a generalisation in this direction.

Recall that a precrossed module is a homomorphism of groups ∂: M → P with an action of P on M satisfying ∂(^{p}m) = p(∂m)p^{−1} for all m ∈ M, p ∈ P. An element in M of the form

mm′m^{−1}(^{∂m}m′)^{−1}

with m, m′ ∈ M is called a Peiffer commutator and denoted by ⟨m, m′⟩. A precrossed module is called a crossed module if all Peiffer commutators are trivial. For any subgroup N in M let us denote by ⟨N, M⟩ the subgroup of M generated by those Peiffer commutators with either m ∈ N or m′ ∈ N. The lower Peiffer central series is then defined by setting M_{1} = M and M_{n+1} = ⟨M_{n}, M⟩ for n ⩾ 1. Each term M_{n} is a normal subgroup of M closed under the action of P. For n ⩾ 2 the quotient group M_{n}/M_{n+1} is abelian. It is natural to set C = (M/M_{2})_{ab} and form the direct sum

ΓM = C ⊕ ⨁_{n⩾2}M_{n}/M_{n+1} .

The Peiffer commutator map M_{m} × M_{n} → M_{m+n}, (m, m′) ↦ ⟨m, m′⟩ induces a bilinear mapping [−, −]: ΓM × ΓM → ΓM. The structure (ΓM, [−, −]) is not in general a Lie algebra. It is however the motivating example of what is called a ‘partial Lie algebra’ in [1]. The main result in [1] asserts that ΓM is a free partial Lie algebra if ∂: M → P is a free precrossed module with P a free group. The Magnus-Witt isomorphism corresponds to the case when P is the trivial group (for in this case the lower Peiffer central series of M coincides with the lower central series of M considered as a group).

Integral homology groups H_{m}(M)_{P} for a precrossed module were introduced in [2] for dimensions m = 1, 2. Theorem 1 in [2] with the main result in [1] immediately imply the following.

Proposition 1. Let ∂: M → P be a precrossed module such that: 1) P is a free group; 2) the induced map ∂: M/M_{2} → P is a free crossed module; 3) H_{2}(M)_{P} is trivial. Then ΓM is a free partial Lie algebra.

This proposition with P equal to the trivial group is a special case of Theorem 1 (since in this case: hypothesis (2) asserts that M_{ab} is a free abelian group; H_{2}(M)_{1} is the second integral homology of the group M; and ΓM is a Lie algebra). It would be interesting to know whether Proposition 1 can be generalized in the direction of Theorem 1 when P is not trivial.

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(Received 17.08.2002)

Author’s address: Mathematics Department National University of Ireland Galway Ireland E-mail: 0002319S@bodkin.nuigalway.ie