On automorphism groups of torsion-free nilpotent groups

by Manfred Hartl

We describe an inductive computation of the automorphism group of torsionfree nilpotent groups (= T -groups) in terms of iterated group extensions. All terms involved are computable by use of linear algebra in case G is given by a finite presentation.

In particular we consider the automorphism groups of free nilpotent groups which are of particular interest in algebraic K-theory; for nilpotency class 2 they are related to the exotic element of K_{3}(ZZ) as was pointed out by Baues-Dreckmann [2].

Let G be a free nilpotent group of class n, i.e. G = F/γ_{n+1}(F) for some free group F, where γ_{i}(F) is given by the lower central series of F. For the free abelian group A = G^{ab} = G/γ_{2}(G) let T (A) = ZZ ⊕ ⨁_{i>1} A^{⊗i} be the tensor ring over A, T̄ its augmentation ideal, T̄_{n} = T̄/T̄^{n+1} the quotient ring, and let L_{n}(A) be the subgroup of the free Lie-ring over A which is generated by the commutators of length n. Then we have the following two exact sequences of abelian groups for n ≥ 2:

L_{n}(A) ⟶^{>}^{l_{n}} T̄_{n}^{2} ⟶^{μ} T̄_{n}^{2}/L_{n}(A) ↪^{j} T̄_{n}/L_{n}(A) ⟶^{ρ} A

Here l_{n} is the restriction of the canonical inclusion for the universal envelopping algebra UL_{∗}(A) ≅ T (A) .

Note that G_{n} =^{def} G/γ_{n}(G) is a free nilpotent group of class n – 1. The sequences above are sequences of left Aut(G_{n}) - modules: Let f ∈ Aut(G_{n}). Then f acts on A ≅ G_{n}^{ab} and on L_{n}(A) by the induced map f_{♯} = f^{ab} and f_{♯} = L_{n}(f^{ab}) respectively. Next choose a basis {x_{i}} of G_{n} and write

f(x_{i}) = x_{i_{1}}^{±1} ⋯ x_{i_{s}}^{±1} γ_{n}(G) .

Note that T̄_{n} is isomorphic with the truncated (non commutative) polynomial ring with generators {x_{i}}. Ring isomorphisms f_{♯} on T̄_{n} and T̄_{n}^{2} , respectively, are defined by sending the generator x_{i} to the coset of the element

(1 + x_{i_{1}})^{±1} ⋯ (1 + x_{i_{s}})^{±1} − 1

with (1 + x)^{−1} = 1 + ∑_{ν=1}^{n}(−1)^{ν}x^{ν}. The isomorphisms f_{♯} now yield the action of f on T̄_{n}^{2}, T̄_{n}^{2}/L_{n}(A) and T̄_{n}/L_{n}(A).

The exact sequences above induce the following short exact sequences with Hom = Hom_{Z Z}.

I:Hom(A, L_{n}(A)) ⟶^{>}^{ρ^{∗}} Hom(T̄_{n}/L_{n}(A) , L_{n}(A)) ⟶^{j^{∗}} Hom(T̄_{n}^{2}/L_{n}(A) , L_{n}(A))

II: Hom(T̄_{n}^{2}/L_{n}(A) , L_{n}(A)) ⟶^{>}^{l_{n∗}} Hom(T̄_{n}^{2}/L_{n}(A) , T̄_{n}^{2})

⟶^{μ_{∗}} Hom(T̄_{n}^{2}/L_{n}(A) , T̄_{n}^{2}/L_{n}(A))

These are actually sequences of left Aut(G_{n})-modules by defining f · α(x) = f_{♯}α(f_{♯}^{−1}x) where α is an element in one of the Hom-modules in I or II.

Theorem A: Let G be a free nilpotent group of class n with A = G^{ab}. Then the classifying cohomology class of the group extension

0 ⟶ Hom(A , L_{n}(A)) ⟶^{1^{+}} Aut(G) ⟶^{Π} Aut(G/γ_{n}(G)) ⟶ 1

coincides with the element

β_{I}β_{II}(ϵ) ∈ H^{2}(Aut(G/γ_{n}(G)) , Hom(A , L_{n}(A))

Here β_{I} and β_{II} are the Bockstein operators associated with the sequences I and II respectively and ϵ is the 0-dimensional cohomology class given by the identity of T̄_{n}^{2}/L_{n}(A). .

The homomorphism Π in the theorem is reduction mod γ_{n}(G) and the homomorphism 1^{+} is defined for x ∈ G by 1^{+}(α)(x) = x · y where y = i α(x γ_{2}(G)) ∈ G by the isomorphism of Witt i: L_{n}(A) ≅ γ_{n}(G).

Next we consider the more general case of a finitely generated torsion-free nilpotent group G of class n.

Let \sqrt{γ_{n}(G)} = \{a ∈ G | a^{k} ∈ γ_{n}(G) for some k\} be the isolator of γ_{n}(G) so

that G_{n} =^{def} G/\sqrt{γ_{n}(G)} is a T -group of class n – 1. Now the group Aut(G) can be computed from the group Aut(G_{n}) in a similar way as in Theorem A. Here, however, an additional obstruction operator arises, and the polynomial ring T̄_{n} above has to be replaced by the following more general construction: Define the ring

Pn^{√} (G) = (Δ_{Z Z}(G)/Δ_{Z Z}^{n+1}(G))/ZZ − torsion

where Δ_{Z Z}(G) denotes the augmentation ideal of the group ring ZZ(G), and let

μ_{n} = μ_{n}(G) : Pn−1^{√}(G_{n}) ⊗_{G_{n}} Pn−1^{√}(G_{n}) → Pn^{√} (G)

be the canonical factorization for the multiplication map of Pn^{√} (G). All these terms can be computed from the truncated polynomial ring T̄_{n} above in case G is given by a free presentation R ↪ F ⟶ G with A = F^{ab}.

Let K_{n} denote the isolator in Pn^{√} (G) of the additive subgroup of Pn^{√} (G) which is generated by the n-fold ring commutators of Pn^{√} (G).

Proposition ([3], 1.3.5(iii)): Let G be a group. Then for n ≥ 1 there is a natural isomorphism

k_{n}: \sqrt{γ_{n}(G)}/\sqrt{γ_{n+1}(G)} ≅ K_{n}

induced by the map a ↦ {a − 1} ∈ Pn^{√} (G) . This may be viewed as a generalization of Witt’s isomorphism γ_{n}(G) ≅ L_{n}(A) used in theorem A to arbitrary groups since in case G is free we have isomorphisms

Pn^{√} (G) ≅ T̄_{n} and K_{n} ≅ L_{n}(A) ⟶^{>}^{l_{n}} T̄_{n}.

Now for f ∈ Aut(G_{n}) the induced isomorphism Pn−1^{√}(f)⊗Pn−1^{√}(f) of Pn−1^{√}(G_{n})⊗_{G_{n}} Pn−1^{√}(G_{n}) restricts to an isomorphism f_{μ} of the subgroup M = μ_{n}(G)^{−1}K_{n} since one has the identity M = μ_{n}(G)^{−1}K_{n} = Ker μ_{n}(G_{n}). Let

cAut(G_{n}) ⊂ Aut(G)

be the subgroup of all automorphisms f for which there exists f_{♯} ∈ Aut(K_{n}) such that

f_{♯} (μ_{n}|M) = (μ_{n}|M) f_{μ} .

The map f_{♯} is uniquely determined since the span of the n-fold ring commutators of Pn^{√} (G) is contained in K_{n} ∩ Pn^{√} (G)^{2} = K_{n}∩ Im μ_{n} and has finite index in K_{n} where K_{n} is torsion-free. Hence for f ∈ cAut(G_{n}) one obtains a unique automorphism of

Λ_{n} =^{def} K_{n} + Pn^{√} (G)^{2} ,

also denoted by f_{♯}, satisfying

f_{♯}μ_{n} = μ_{n}Pn−1^{√}(f) ⊗ Pn−1^{√}(f) .

Thus K_{n} and Λ_{n} are left cAut(G_{n})-modules by use of f ↦ f_{♯}.

Lemma ([3], 2.3.24(1)): One has the following short exact sequence of homomorphisms between abelian groups:

Pn^{√} (G_{n})^{2} ↪^{j} Pn^{√} (G_{n}) ⟶^{ρ} G_{n}^{ab} ,

where ρ sends the coset of a − 1, a ∈ G, to the element aγ_{2}(G).

Therefore the following sequences are exact sequences of cAut(G_{n})-bimodules.

I′ : Hom(G_{n}^{ab}, K_{n}) ⟶^{>}^{ρ^{∗}} Hom( Pn^{√} (G_{n}), K_{n}) ⟶^{j^{∗}} Hom( Pn^{√} (G_{n})^{2}, K_{n})

⟶^{E^{∗}} Ext_{Z Z}^{1}(G_{n}^{ab}, K_{n})

II′ : Hom( Pn^{√} (G_{n})^{2} , K_{n}) ⟶^{>}^{ι_{∗}} Hom( Pn^{√} (G_{n})^{2} , Λ_{n}) ⟶^{μ_{n}(G_{n})_{∗}}

Hom( Pn^{√} (G_{n})^{2} , Pn^{√} (G_{n})^{2})

with ι: K_{n} ↪ Λ_{n}. As above, these sequences are sequences of left cAut(G_{n})- modules by setting f · α = f_{∗}f^{−1∗}α.

Now a derivation

O: cAut(G_{n}) → Hom( Pn^{√} (G_{n})^{2}, K_{n})

with respect to the left cAut(G_{n})-action is given as follows: Since Pn^{√} (G) is a finitely generated free abelian group one can choose a ZZ -linear splitting t of the surjection

Pn^{√} (π) : Pn^{√} (G) → → Pn^{√} (G_{n})

where π : G→ →G_{n} is the canonical quotient map. Here t carries Pn^{√} (G_{n})^{2} to Λ_{n}, so one gets the restriction

l: Pn^{√} (G_{n})^{2} → Λ_{n} of t.

Thus we obtain for f ∈ cAut(G_{n}) the welldefined homomorphism

O(f) = ι^{−1}(f_{♯}^{∘}l^{∘}( Pn^{√} (f^{−1})| Pn^{√} (G_{n})^{2}) − l)

Hence the operator O satisfies O(f) = ι^{−1}(f · l − l) so that O is a derivation. Theorem A is a special case of the next result.

Theorem B: Let G be a finitely generated T -group of class n.

(i) The following sequence is exact:

1 ⟶ Hom(G_{n}^{ab}, K_{n}) ⟶^{1^{+}} Aut(G)⟶^{Π} cAut(G_{n}) ⟶^{E^{∗}O}Ext_{Z Z}^{1}(G_{n}^{ab}, Ln^{√} (G))

Here the homomorphism Π in the theorem is reduction mod \sqrt{γ_{n}(G)} and the homomorphism 1^{+} is defined for x ∈ G_{n} by 1^{+}(α)(x) = x · y where y = k_{n}^{−1} α(x γ_{2}(G_{n})) ∈ G by the isomorphism k_{n}: \sqrt{γ_{n}(G)} ≅ K_{n} in the proposition above. The derivation E^{∗}O is given by E in I′. Moreover, the action of Ker E^{∗}O on

Hom(G_{n}^{ab}, K_{n}) ≅^{k_{n}^{−1}_{∗}} Hom (G_{n}^{ab}, \sqrt{γ_{n}(G)})

defined by conjugation in Aut(G) coincides with the restriction of the left cAut(G_{n})-action defined above.

(ii) The classifying cohomology class of the group extension

Hom(G_{n}^{ab}, K_{n}) ⟶^{>}^{1^{+}} Aut(G) ⟶^{Π} Ker(E^{∗}O)

obtained from (i) coincides with the element

β′_{I}{O|Ker(E^{∗}O)} ∈ H^{2}(Ker(E^{∗}O) , Hom(G_{n}^{ab}, K_{n})) ,

where

β′_{I} : H^{1}(Ker(E^{∗}O) , KerE^{∗}) → H^{2}(Ker(E^{∗}O) , Hom(G_{n}^{ab}, K_{n}))

is the Bockstein operator associated with the extension I′ of Ker(E^{∗}O)- modules.

(iii) Let (G_{n})^{ab} be torsion-free (this, in particular, holds if G^{ab} is torsion-free, i.e. \sqrt{γ_{2}(G)} = γ_{2}(G)). Then the Ext-term in (i) vanishes so that one has the group extension

Hom(G_{n}^{ab}, K_{n}) ⟶^{>}^{1^{+}} Aut(G) ⟶^{Π} cAut(G_{n}) .

The classifying cohomology class of this extension coincides with the element

β′_{I}β′_{II}(ϵ) ∈ H^{2}(cAut(G_{n}) , Hom(G_{n}^{ab}, K_{n})) .

Here β′_{II} is the Bockstein operator associated with extension II′ of Ker E^{∗}Ōmodules defined above and ϵ is the 0-dimensional cohomolgy class given by the identity of Pn^{√} (G_{n})^{2}.

Theorem A and B are corollaries of more general results on the category T_{n} of T -groups of class ≤ n. They are based on a functorial equivalence between extensions of torsion-free nilpotent groups (finitely generated) and certain extensions of additively torsion-free nilpotent modules (additively finitely generated), established in [3], [4]. One also needs the notions of ‘linear extension of categories’, ‘exact sequences for functors’ and the ‘cohomology of categories’ as established by Baues [1]. Using these new concepts we proved results on the categories T_{n} which are almost literally of the same nature as theorem A and B. This in fact shows that not only automorphism groups but also all diagrams in T_{n} can be computed by results as above.

The proofs are part of the author’s doctoral thesis, Bonn 1991, written under supervision of Prof. H.J. Baues. There also automorphism groups of nilpotent groups with dimension property are described in a similar way.

References

[1] Baues, H.J., Algebraic Homotopy, Cambridge Studies in Advanced Mathematics 15, Cambridge University Press (1988).

[2] Baues, H.J. und Dreckmann, W., The cohomology of homotopy categories and the general linear group, K-Theory (1989) , 307-338.

[3] Hartl, Manfred, Abelsche Modelle nilpotenter Gruppen, Dissertation Bonn (1991).

[4] Hartl, Manfred, On polynomiality properties of group extensions, to appear in J. Algebra.