The congruence subgroup problem for finitely generated nilpotent groups

David El-Chai Ben-Ezra; Alexander Lubotzky

doi:10.1515/jgth-2021-0039

Article Publicly Available

The congruence subgroup problem for finitely generated nilpotent groups

David El-Chai Ben-Ezra and Alexander Lubotzky

Published/Copyright: September 22, 2021

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From the journal Journal of Group Theory Volume 25 Issue 3

Abstract

The congruence subgroup problem for a finitely generated group Γ and for G ≤ Aut ⁢ ( Γ ) asks whether the map G ^ → Aut ⁢ ( Γ ^ ) is injective, or more generally, what its kernel C ⁢ ( G , Γ ) is. Here X ^ denotes the profinite completion of 𝑋. In the case G = Aut ⁢ ( Γ ) , we write C ⁢ ( Γ ) = C ⁢ ( Aut ⁢ ( Γ ) , Γ ) . Let Γ be a finitely generated group, Γ ¯ = Γ / [ Γ , Γ ] , and Γ * = Γ ¯ / tor ⁢ ( Γ ¯ ) ≅ Z ( d ) . Define

Aut * ⁢ ( Γ ) = Im ⁡ ( Aut ⁢ ( Γ ) → Aut ⁢ ( Γ * ) ) ≤ GL d ⁢ ( Z ) .

In this paper we show that, when Γ is nilpotent, there is a canonical isomorphism

C ⁢ ( Γ ) ≃ C ⁢ ( Aut * ⁢ ( Γ ) , Γ * ) .

In other words, C ⁢ ( Γ ) is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group Aut * ⁢ ( Γ ) . In particular, in the case where Γ = Ψ n , c is a finitely generated free nilpotent group of class 𝑐 on 𝑛 elements, we get that C ⁢ ( Ψ n , c ) = C ⁢ ( Z ( n ) ) = { e } whenever n ≥ 3 , and C ⁢ ( Ψ 2 , c ) = C ⁢ ( Z ( 2 ) ) = F ^ ω is the free profinite group on countable number of generators.

1 Introduction

Let G ≤ GL n ⁢ ( Z ) . The classical congruence subgroup problem (CSP) asks whether every finite index subgroup of 𝐺 contains a principal congruence subgroup, i.e. a subgroup of the form G ⁢ ( m ) = ker ⁡ ( G → GL n ⁢ ( Z / m ⁢ Z ) ) for some 0 ≠ m ∈ Z . Equivalently, it asks whether the natural map G ^ → GL n ⁢ ( Z ^ ) is injective, where G ^ and Z ^ are the profinite completions of the group 𝐺 and the ring ℤ, respectively. More generally, the CSP asks what is the kernel of this map. It is a classical 19th century result that, for G = GL n ⁢ ( Z ) , the answer is negative when n = 2 . Moreover (but not so classical, cf. [17, 16]), the kernel in this case is F ^ ω – the free profinite group on a countable number of generators. On the other hand, it was proved in the sixties by Mennicke [18] and Bass–Lazard–Serre [2] that, for n ≥ 3 , the map is injective, and the kernel is therefore trivial. This breakthrough led to a rich theory which studied the CSP for many other arithmetic groups. It has been solved for many arithmetic groups, but not yet for all. See [23, 24] for surveys.

By the observation GL n ⁢ ( Z ) ≅ Aut ⁢ ( Z ( n ) ) , the CSP can be generalized as follows. Let Γ be a group and G ≤ Aut ⁢ ( Γ ) . For a finite index characteristic subgroup M ≤ Γ , denote

G ⁢ ( M ) = ker ⁡ ( G → Aut ⁢ ( Γ / M ) ) .

Such a G ⁢ ( M ) is called a “principal congruence subgroup”, and a finite index subgroup of 𝐺 which contains G ⁢ ( M ) for some 𝑀 is called a “congruence subgroup”. The CSP for the pair ( G , Γ ) asks whether every finite index subgroup of 𝐺 is a congruence subgroup.

One can see that the CSP is equivalent to the question: is the congruence map G ^ = lim ← ⁡ G / U ↠ lim ← ⁡ G / G ⁢ ( M ) injective? Here, 𝑈 ranges over all finite index normal subgroups of 𝐺, and 𝑀 ranges over all finite index characteristic subgroups of Γ. When Γ is finitely generated, it has only finitely many subgroups of a given index 𝑚, and thus, the characteristic subgroups

M m = ⋂ { Δ ≤ Γ ∣ [ Γ : Δ ] = m }

are of finite index in Γ. Hence, one can write Γ = lim ← m ∈ N ⁡ Γ / M m and have^[1]

lim ← ⁡ G / G ⁢ ( M ) = lim ← m ∈ N ⁡ G / G ⁢ ( M m ) ≤ lim ← m ∈ N ⁡ Aut ⁢ ( Γ / M m ) = Aut ⁢ ( lim ← m ∈ N ⁡ ( Γ / M m ) ) = Aut ⁢ ( Γ ^ ) .

Therefore, when Γ is finitely generated, the CSP is equivalent to the question: is the congruence map G ^ → Aut ⁢ ( Γ ^ ) injective? More generally, the CSP asks what the kernel C ⁢ ( G , Γ ) of this map is. For G = Aut ⁢ ( Γ ) , we will also use the simpler notation C ⁢ ( Γ ) = C ⁢ ( Aut ⁢ ( Γ ) , Γ ) .

The classical CSP results mentioned above can therefore be reformulated as C ⁢ ( Z ( 2 ) ) = F ^ ω , while C ⁢ ( Z ( n ) ) = { e } for n ≥ 3 . Recently, it was proved that, when Γ = Φ n is the free metabelian group on 𝑛 generators, we have C ⁢ ( Φ 2 ) = F ^ ω , C ⁢ ( Φ 3 ) ⊇ F ^ ω , and for every n ≥ 4 , C ⁢ ( Φ n ) is abelian (see [4, 5, 6, 7]). So, while in the free abelian case there is a dichotomy between n = 2 and n ≥ 3 , in the free metabelian case, we have dichotomy between n = 2 , 3 and n ≥ 4 .

The goal of this paper is to show that, contrary to the above metabelian cases, when Γ is a finitely generated nilpotent group, the CSP for Γ is completely determined by the CSP for abelian groups. Let us put things more precisely. Let Γ be a finitely generated group, Γ ¯ = Γ / [ Γ , Γ ] and Γ * = Γ ¯ / tor ⁢ ( Γ ¯ ) , so Γ * ≅ Z ( d ) for some 𝑑. Define

Aut * ⁢ ( Γ ) = Im ⁡ ( Aut ⁢ ( Γ ) → Aut ⁢ ( Γ * ) ) ≤ GL d ⁢ ( Z ) .

When Γ is nilpotent, the group Aut * ⁢ ( Γ ) is known to be an arithmetic subgroup of GL d ⁢ ( Z ) , and every arithmetic subgroup of GL d ⁢ ( Z ) is obtained like that for some nilpotent group Γ ([3, 8, 9]).

The canonical map Aut ⁢ ( Γ ) → Aut * ⁢ ( Γ ) induces a map

C ⁢ ( Γ ) → C ⁢ ( Aut * ⁢ ( Γ ) , Γ * ) .

Here is the main theorem of the paper.

Theorem 1.1

Let Γ be a finitely generated nilpotent group. Then the canonical map C ⁢ ( Γ ) → C ⁢ ( Aut * ⁢ ( Γ ) , Γ * ) is an isomorphism.

So the CSP for nilpotent groups is completely reduced to the classic CSP. In particular, in the free cases, we have the following corollary.

Corollary 1.2

Let Γ = Ψ n , c be the free nilpotent group of class 𝑐 on 𝑛 elements. Then

C ⁢ ( Γ ) ≅ C ⁢ ( Aut * ⁢ ( Γ ) , Γ * ) = C ⁢ ( GL n ⁢ ( Z ) , Z ( n ) ) = C ⁢ ( Z ( n ) ) .

In particular,

for n = 2 , one has C ⁢ ( Ψ 2 , c ) ≅ C ⁢ ( Z ( 2 ) ) ≅ F ^ ω ;
for n ≥ 3 , one has C ⁢ ( Ψ n , c ) ≅ C ⁢ ( Z ( n ) ) ≅ { e } .

Remark 1.3

As mentioned above, every arithmetic subgroup 𝐷 of GL d ⁢ ( Z ) can appear as Aut * ⁢ ( Γ ) for a suitable nilpotent Γ. The possible congruence kernels for such arithmetic groups are not fully known as the classical CSP is not yet fully solved. But these include, besides the trivial groups and F ^ ω mentioned above, both finite cyclic groups (when 𝐷 is the restriction of scalars from suitable number fields) and infinite abelian groups of finite exponent (if 𝐷 is an arithmetic group of a non-simply connected group).

Here is the main line of the proof. For a finitely generated group Γ, consider the commutative exact diagram

when we define I ⁢ A * ⁢ ( Γ ) = ker ⁡ ( Aut ⁢ ( Γ ) → Aut ⁢ ( Γ * ) ) and Aut * ⁢ ( Γ ^ ) , I ⁢ A * ⁢ ( Γ ^ ) are defined to be the image and the kernel of the natural map

Aut ⁢ ( Γ ^ ) → Aut ⁢ ( Γ * ^ ) = GL d ⁢ ( Z ^ ) ,

respectively. This diagram gives rise to the commutative exact diagram (see [10, Lemma 2.1])

(1.1)

Note that C ⁢ ( Aut * ⁢ ( Γ ) , Γ * ) = ker ⁡ ( Aut * ⁢ ( Γ ) ^ → Aut * ⁢ ( Γ ^ ) ) . We prove the following theorem.

Theorem 1.4

Let Γ be a finitely generated nilpotent group.

For any G ≤ I ⁢ A * ⁢ ( Γ ) , the natural map G ^ → I ⁢ A * ⁢ ( Γ ^ ) is an embedding. In other words, C ⁢ ( G , Γ ) = { e } , so we have an affirmative solution to the CSP for any G ≤ I ⁢ A * ⁢ ( Γ ) .
The group I ⁢ A * ⁢ ( Γ ) is dense in I ⁢ A * ⁢ ( Γ ^ ) .

Note that, from the first part of Theorem 1.4, we obtain that in particular

C ⁢ ( I ⁢ A * ⁢ ( Γ ) , Γ ) = { e }

for any finitely generated nilpotent group Γ. This is not true in general (cf. [4, 5, 6, 7] for free metabelian groups). In some sense, the second part of Theorem 1.4 means that the map I ⁢ A * ⁢ ( Γ ) → I ⁢ A * ⁢ ( Γ ^ ) satisfies a “strong approximation” property. This is not true in general either (compare [15] for free groups). From the two parts of Theorem 1.4, we obtain the following corollary, which also implies Theorem 1.1.

Corollary 1.5

Let Γ be a finitely generated nilpotent group. Then

I ⁢ A * ⁢ ( Γ ) ^ ≅ I ⁢ A * ⁢ ( Γ ^ ) .

Corollary 1.5, together with chasing diagram (1.1), imply Theorem 1.1. Corollary 1.5 is a form of combination of congruence subgroup property as well as strong approximation for the group I ⁢ A * ⁢ ( Γ ) . Indeed, its proof boils down to these results for a suitable ℚ-unipotent group. But the reduction is slightly delicate: in § 3, it is shown that the proof of Corollary 1.5 (or Theorem 1.4) can be reduced to the case when Γ is torsion free. In § 2, we treat the torsion-free case, by reducing it first from Γ to Δ, when Δ is the “lattice hull” of Γ. This Δ contains Γ as a finite index subgroup, and it is contained in its Mal’cev completion 𝑅. It enjoys the property that log ⁡ ( Δ ) is a ℤ-lattice of the Lie algebra 𝐿 of 𝑅. This fact enables us to give I ⁢ A * ⁢ ( Δ ) the structure of the ℤ-points of a suitable unipotent group for which the Z ^ -points are exactly I ⁢ A * ⁢ ( Δ ^ ) . Hence, the classical CSP and strong approximation for this unipotent group imply Corollary 1.5.

In § 4, we sketch another proof to Corollary 1.5, which is more direct, in the case where Γ = Ψ n , c is a finitely generated free nilpotent group.

2 The case of torsion-free nilpotent groups

In this section, we are going to prove Theorem 1.4 in the case where Γ is torsion free. So let Γ be a finitely generated torsion-free nilpotent group. For our convenience, we will follow the approach presented in [25] and consider Γ as a subgroup of Tr 1 ⁢ ( n , Z ) , the group of n × n upper triangular matrices over ℤ with ones on the diagonal, for some 𝑛 (see [25, Chapter 5]). Recall the one-to-one correspondence given by the maps

log : Tr 1 ⁢ ( n , Q ) → Tr 0 ⁢ ( n , Q ) , exp : Tr 0 ⁢ ( n , Q ) → Tr 1 ⁢ ( n , Q ) ,

where Tr 0 ⁢ ( n , Q ) is the Lie algebra of n × n upper triangular matrices with zeros on the diagonal. Let 𝐿 be the Lie subalgebra of Tr 0 ⁢ ( n , Q ) spanned by log ⁡ ( Γ ) . The following is well known and can be found in [25, Chapter 6].

Theorem 2.1

There exists a unique (up to isomorphism) group 𝑅, called the radicable hull of Γ, or Mal’cev completion of Γ with the following properties.

Γ is a subgroup of 𝑅.
For every a ∈ R and m ∈ N , there exists b ∈ R such that b m = a .
For every a ∈ R , there exists m ∈ N such that a m ∈ Γ .
The group 𝑅 can be identified with exp ⁡ ( L ) ≤ Tr 1 ⁢ ( n , Q ) .

The connection between the group operation of 𝑅 and the Lie algebra operation of 𝐿 is given through the Baker–Campbell–Hausdorff (BCH) formula. One can use it in order to prove the following lemma [8, Lemma 2.1].

Lemma 2.2

Under the correspondence between 𝑅 and 𝐿, one has R ′ = exp ⁡ ( L ′ ) , where R ′ is the derived subgroup of 𝑅 and L ′ is the derived Lie subalgebra of 𝐿. This equality gives a natural group isomorphism between R / R ′ and the additive group L / L ′ .

One can use the BCH formula in order to prove that L ′ is the ℚ-span of log ⁡ ( Γ ′ ) , and hence R ′ can be identified with the radicable hull of Γ ′ . Denote

δ ⁢ ( Γ ) = ker ⁡ ( Γ → Γ * ≃ Z d ) .

Then, as any element of δ ⁢ ( Γ ) has some power in Γ ′ , we have log ⁡ ( δ ⁢ ( Γ ) ) ⊆ L ′ , and hence L ′ is also the ℚ-span of log ⁡ ( δ ⁢ ( Γ ) ) , and R ′ is also the radicable hull of δ ⁢ ( Γ ) . One gets from this the following lemma [8, Lemma 2.2].

Lemma 2.3

We have Γ ∩ R ′ = δ ⁢ ( Γ ) and dim Q ⁡ ( R / R ′ ) = rank Z ⁡ ( Γ / δ ⁢ ( Γ ) ) = d .

The following can be found in [25, Chapter 6].

Proposition 2.4

There exists a unique minimal intermediate subgroup Γ ≤ Δ ≤ R , called the lattice hull of Γ, such that its image log ⁡ ( Δ ) ⊆ L is a lattice. In other words, log ⁡ ( Δ ) is a free ℤ-module that spans 𝐿 over ℚ. One has [ Δ : Γ ] < ∞ .

Remark 2.5

Note that 𝑅 is also the radicable hull of Δ.

Given a set X ⊆ L , denote N Aut ⁢ ( L ) ⁢ ( X ) = { g ∈ Aut ⁢ ( L ) ∣ g ⁢ ( X ) = X } . The following can also be found in [25, Chapter 6].

Theorem 2.6

The correspondence between 𝑅 and 𝐿 induces an isomorphism

Aut ⁢ ( R ) ≃ Aut ⁢ ( L ) ≤ GL k ⁢ ( Q ) ,

where k = dim ⁡ ( L ) . Under this isomorphism, one can identify

Aut ⁢ ( Γ ) ≅ N Aut ⁢ ( L ) ⁢ ( log ⁡ ( Γ ) ) ≤ Aut ⁢ ( L ) , Aut ⁢ ( Δ ) ≅ N Aut ⁢ ( L ) ⁢ ( log ⁡ ( Δ ) ) ≤ Aut ⁢ ( L ) .

Moreover, we have Aut ⁢ ( Γ ) ≤ Aut ⁢ ( Δ ) ≤ Aut ⁢ ( R ) ≃ Aut ⁢ ( L ) . In other words, any automorphism of Γ can be uniquely extended to an automorphism of Δ, and any of the latter can be uniquely extended to an automorphism of 𝑅.

Theorem 2.6 and Lemma 2.3 imply the following proposition.

Proposition 2.7

Under the above notation, we can identify

I ⁢ A * ⁢ ( Γ ) = Aut ⁢ ( Γ ) ∩ ker ⁡ ( Aut ⁢ ( R ) → Aut ⁢ ( R / R ′ ) ) ≅ N Aut ⁢ ( L ) ⁢ ( log ⁡ ( Γ ) ) ∩ ker ⁡ ( Aut ⁢ ( L ) → Aut ⁢ ( L / L ′ ) ) ,

I ⁢ A * ⁢ ( Δ ) = Aut ⁢ ( Δ ) ∩ ker ⁡ ( Aut ⁢ ( R ) → Aut ⁢ ( R / R ′ ) ) ≅ N Aut ⁢ ( L ) ⁢ ( log ⁡ ( Δ ) ) ∩ ker ⁡ ( Aut ⁢ ( L ) → Aut ⁢ ( L / L ′ ) ) .

An immediate corollary of Proposition 2.7 and Theorem 2.6 is that I ⁢ A * ⁢ ( Γ ) is naturally embedded in I ⁢ A * ⁢ ( Δ ) . We would like now to show that the same property is valid also for the profinite completions of Γ and Δ.

Proposition 2.8

Any automorphism of Γ ^ can be uniquely extended to an automorphism of Δ ^ . In particular, I ⁢ A * ⁢ ( Γ ^ ) is naturally embedded as an open subgroup of I ⁢ A * ⁢ ( Δ ^ ) .

In order to prove Proposition 2.8, we are going to show that one can describe the relation between I ⁢ A * ⁢ ( Γ ^ ) and I ⁢ A * ⁢ ( Δ ^ ) in a very similar way to the description of the relation between I ⁢ A * ⁢ ( Γ ) and I ⁢ A * ⁢ ( Δ ) above. Before we do that, let us present an immediate consequence of Proposition 2.8.

Proposition 2.9

Let Γ be a finitely generated torsion free nilpotent group, and let Δ be the lattice hull of Γ. Let G ≤ I ⁢ A * ⁢ ( Γ ) ≤ I ⁢ A * ⁢ ( Δ ) .

If G ^ → I ⁢ A * ⁢ ( Δ ^ ) is injective, then G ^ → I ⁢ A * ⁢ ( Γ ^ ) is injective.
If I ⁢ A * ⁢ ( Δ ) is dense in I ⁢ A * ⁢ ( Δ ^ ) , then I ⁢ A * ⁢ ( Γ ) is dense in I ⁢ A * ⁢ ( Γ ^ ) .

Proof

The first statement is an immediate corollary of Proposition 2.8. Now, notice that, as Γ is open in Δ, we have Δ ∩ Γ ^ = Γ . Hence, by the identification in Proposition 2.7, one has I ⁢ A * ⁢ ( Δ ) ∩ I ⁢ A * ⁢ ( Γ ^ ) = I ⁢ A * ⁢ ( Γ ) . Thus, the second part also follows from Proposition 2.8. ∎

Proposition 2.9 shows us that, in order to prove Theorem 1.4 for a finitely generated torsion-free nilpotent group Γ, it is enough to show it for its lattice hull Δ. We turn now to describe the relation between Aut ⁢ ( Γ ^ ) and Aut ⁢ ( Δ ^ ) . The description is going to give more than just a proof to Proposition 2.8, and we are going to use it also to prepare for the rest of the section.

Let Γ p be the pro-𝑝 completion of Γ. As Γ is nilpotent, we have Γ ^ = ∏ p Γ p . In addition, as Γ is finitely generated and unipotent, it is arithmetic [25, Chapter 6]. Hence, by the affirmative solution to the congruence subgroup problem for arithmetic soluble groups (see [11, 21, 22]), we can view Γ ^ as the closure of Γ under the map

Γ ↪ Tr 1 ⁢ ( n , Z ) → Tr 1 ⁢ ( n , Z ^ ) = ∏ p Tr 1 ⁢ ( n , Z p ) .

As Tr 1 ⁢ ( n , Z p ) is a pro-𝑝 group, and Γ ^ = ∏ p Γ p , it follows that we can identify Γ p with the closure of Γ under the map

Γ ↪ Tr 1 ⁢ ( n , Z ) → ∏ p Tr 1 ⁢ ( n , Z p ) → Tr 1 ⁢ ( n , Z p ) .

Extending log : Tr 1 ⁢ ( n , Q p ) → Tr 0 ⁢ ( n , Q p ) and exp : Tr 0 ⁢ ( n , Q p ) → Tr 1 ⁢ ( n , Q p ) , log and exp are continuous with relation to the topology induced by Q p . We define L p to be the ℚ-span of log ⁡ ( Γ p ) and R p = exp ⁡ ( L p ) .

Lemma 2.10

The set L p is a Q p -Lie algebra.

Proof

The BCH clearly gives L p a ℚ-Lie algebra structure, just like it gives to 𝐿. We just need to explain why L p is closed under multiplication of scalars from Q p . By definition, it is enough to show that it is closed under multiplication of scalars from Z p . So let g ∈ Γ p and let m = lim i → ∞ ⁡ m i ∈ Z p for some m i ∈ Z . Let ρ g be the natural homomorphism ρ g : Z p → Γ p defined by sending the generator of Z p to 𝑔. Then, as log is continuous, we have

log ⁡ ( ρ g ⁢ ( m ) ) = log ⁡ ( ρ g ⁢ ( lim i → ∞ ⁡ m i ) ) = lim i → ∞ ⁡ log ⁡ ( ρ g ⁢ ( m i ) ) = lim i → ∞ ⁡ log ⁡ ( g m i ) = lim i → ∞ ⁡ ( m i ⋅ log ⁡ ( g ) ) = ( lim i → ∞ ⁡ m i ) ⋅ log ⁡ ( g ) = m ⋅ log ⁡ ( g ) .

It follows that the set log ⁡ ( Γ p ) is closed under multiplication by elements from Z p , and so is L p . ∎

The proof of the following is similar to the proof of the corresponding properties of the Mal’cev completion in Theorem 2.1.

Lemma 2.11

The set R p is a group containing Γ, 𝑅 and Γ p . Moreover, R p is a Mal’cev completion of Γ p in the sense that it satisfies the following properties.

For every a ∈ R p and m ∈ N , there exists b ∈ R p such that b m = a .
For every a ∈ R p , there exists m ∈ N such that a m ∈ Γ p .

Also, similarly to Lemma 2.2, one can use the BCH formula in order to show the following lemma.

Lemma 2.12

Under the correspondence between R p and L p , one has

R p ′ = exp ⁡ ( L p ′ ) .

This equality gives a natural group homomorphism between R p / R p ′ and the additive group L p / L p ′ .

One can use the BCH formula in order to prove that L p ′ is the ℚ-span of log ⁡ ( Γ p ′ ) , and hence R p ′ is a Mal’cev completion of Γ p ′ in the sense of Lemma 2.11. Denoting δ ⁢ ( Γ p ) = ker ⁡ ( Γ p → ( Γ * ) p ≃ Z p d ) , where d = rank Z ⁡ ( Γ * ) , we have also a similar property as in Lemma 2.3.

Lemma 2.13

One has Γ p ∩ R p ′ = δ ⁢ ( Γ p ) and

dim Q p ⁡ ( R p / R p ′ ) = rank Z p ⁡ ( Γ p / δ ⁢ ( Γ p ) ) = d .

Proof

We first prove that δ ⁢ ( Γ p ) = ker ⁡ ( Γ p → Γ ¯ p / tor ⁢ ( Γ ¯ p ) ) , where Γ ¯ p = Γ p / Γ p ′ . As ( Γ * ) p is a torsion-free abelian quotient of Γ p , it follows that

δ ⁢ ( Γ p ) ⊇ ker ⁡ ( Γ p → Γ ¯ p / tor ⁢ ( Γ ¯ p ) ) .

On the other hand, the map Γ → Γ p induces a map Γ * → Γ ¯ p / tor ⁢ ( Γ ¯ p ) . Now, as Γ p is a finitely generated pro-𝑝 group, Γ p ′ is closed in Γ p (see [13, Proposition 1.19]). Hence, Γ p → Γ ¯ p is a continuous homomorphism, and hence Γ p → Γ ¯ p / tor ⁢ ( Γ ¯ p ) is continuous as well. Thus, we can say that the image of Γ * under the map Γ * → Γ ¯ p / tor ⁢ ( Γ ¯ p ) is dense in Γ ¯ p / tor ⁢ ( Γ ¯ p ) . Hence, we have a surjective continuous homomorphism ( Γ * ) p → Γ ¯ p / tor ⁢ ( Γ ¯ p ) . It follows that

δ ⁢ ( Γ p ) ⊆ ker ⁡ ( Γ p → Γ ¯ p / tor ⁢ ( Γ ¯ p ) ) ,

so δ ⁢ ( Γ p ) = ker ⁡ ( Γ p → Γ ¯ p / tor ⁢ ( Γ ¯ p ) ) as required.

Now, as R p contains Γ p , we have Γ p / ( Γ p ∩ R p ′ ) ≤ R p / R p ′ . It follows that Γ p / ( Γ p ∩ R p ′ ) is a torsion-free abelian quotient of Γ p . Hence,

δ ⁢ ( Γ p ) = ker ⁡ ( Γ p → Γ ¯ p / tor ⁢ ( Γ ¯ p ) ) ⊆ Γ p ∩ R p ′ .

On the other hand, as every element of R p ′ has a power in Γ p ′ , it follows that we also have

Γ p ∩ R p ′ ⊆ ker ⁡ ( Γ p → Γ ¯ p / tor ⁢ ( Γ ¯ p ) ) = δ ⁢ ( Γ p ) ,

as required.

The equality Γ p ∩ R p ′ = δ ⁢ ( Γ p ) implies Γ p / δ ⁢ ( Γ p ) ≤ R p / R p ′ . Thus Γ p / δ ⁢ ( Γ p ) is a free Z p -module that spans the vector space R p / R p ′ over Q p . Hence,

dim Q p ⁡ ( R p / R p ′ ) = rank Z p ⁡ ( Γ p / δ ⁢ ( Γ p ) ) = d ,

as required. ∎

The following corollary will be needed later.

Corollary 2.14

We have dim Q ⁡ ( L ) = dim Q p ⁡ ( L p ) .

Proof

We saw that

dim Q p ⁡ ( L p / L p ′ ) = dim Q p ⁡ ( R p / R p ′ ) = rank Z p ⁡ ( Γ p / δ ⁢ ( Γ p ) ) = d = rank Z ⁡ ( Γ / δ ⁢ ( Γ ) ) = dim Q ⁡ ( R / R ′ ) = dim Q ⁡ ( L / L ′ ) .

By the fact that the commutator subgroup of a finitely generated nilpotent group is finitely generated, and L ′ is the ℚ-span of log ⁡ ( Γ ′ ) , one has

rank Z p ⁡ ( ( Γ ′ ) p / δ ⁢ ( ( Γ ′ ) p ) ) = rank Z ⁡ ( Γ ′ / δ ⁢ ( Γ ′ ) ) = dim Q ⁡ ( L ′ / L ′′ ) .

Using the CSP for arithmetic soluble groups, we can identify ( Γ ′ ) p with the closure of Γ ′ in Γ p ≤ Tr 1 ⁢ ( n , Z p ) . As explained in the proof of Lemma 2.13, the latter can be identified with ( Γ p ) ′ , and hence ( Γ ′ ) p = ( Γ p ) ′ . Hence, L p ′ , which is the ℚ-span of log ⁡ ( ( Γ p ) ′ ) , is actually the ℚ-span of log ⁡ ( ( Γ ′ ) p ) . It follows that

dim Q p ⁡ ( L p ′ / L p ′′ ) = rank Z p ⁡ ( ( Γ ′ ) p / δ ⁢ ( ( Γ ′ ) p ) ) = dim Q ⁡ ( L ′ / L ′′ ) .

Continuing like that, we obtain that dim Q ⁡ ( L ( i ) / L ( i + 1 ) ) = dim Q p ⁡ ( L p ( i ) / L p ( i + 1 ) ) for any 𝑖, where L ( i ) , L p ( i ) are the 𝑖-th derivatives of L , L p , respectively. Therefore,

dim Q ⁡ ( L ) = ∑ i dim Q ⁡ ( L ( i ) / L ( i + 1 ) ) = ∑ i dim Q p ⁡ ( L p ( i ) / L p ( i + 1 ) ) = dim Q p ⁡ ( L p )

as required. ∎

Remark 2.15

We presented a proof for Corollary 2.14, based on the previous line of discussion, but the knowledgeable reader can also deduce it by recalling that dim Q ⁡ ( L ) is equal to h ⁢ ( Γ ) – the Hirsch length of Γ, and h ⁢ ( Γ ) is equal to dim ⁡ ( Γ p ) – the dimension of the pro-𝑝 completion of Γ, and the latter is equal to dim Q p ⁡ ( L p ) as L p is the Lie algebra of Γ p .

Recall Δ, the lattice hull of Γ. For our convenience, without loss of generality, we can assume that Γ ≤ Δ ≤ Tr 1 ⁢ ( n , Z ) (see [25, Lemma 2 in Chapter 6]). Hence, Δ p can be identified with the closure of Δ in Tr 1 ⁢ ( n , Z p ) .

Lemma 2.16

One has log ⁡ ( Δ p ) = Z p ⁢ log ⁡ ( Δ ) ⊆ L p , where Z p ⁢ log ⁡ ( Δ ) is the Z p -lattice of L p spanned by log ⁡ ( Δ ) . In particular, log ⁡ ( Δ p ) is a Z p -lattice in L p and Γ p ≤ Δ p ≤ R p .

Proof

By an argument similar to Lemma 2.10, for any element m ∈ Z p and a ∈ log ⁡ ( Δ ) , we have m ⋅ a ∈ log ⁡ ( Δ p ) . Hence, we will get log ⁡ ( Δ p ) ⊇ Z p ⁢ log ⁡ ( Δ ) once we show that, for any g , h ∈ Δ p , there exists k ∈ Δ p such that

log ⁡ ( g ) + log ⁡ ( h ) = log ⁡ ( k ) .

By assumption, the group Δ is dense in Δ p , where the topology on Δ p coincides with the topology induced by Tr 1 ⁢ ( Q p ) . Let g i , h i , k i ∈ Δ be such that

lim i → ∞ ⁡ g i = g , lim i → ∞ ⁡ h i = h , log ⁡ ( k i ) = log ⁡ ( g i ) + log ⁡ ( h i ) .

We need to show that lim i → ∞ ⁡ k i = k ∈ Δ p exists and log ⁡ ( g ) + log ⁡ ( h ) = log ⁡ ( k ) . As exp and log are continuous, we have

k = lim i → ∞ ⁡ k i = lim i → ∞ ⁡ exp ⁡ ( log ⁡ ( g i ) + log ⁡ ( h i ) ) = exp ⁡ ( log ⁡ ( lim i → ∞ ⁡ g i ) + log ⁡ ( lim i → ∞ ⁡ h i ) ) = exp ⁡ ( log ⁡ ( g ) + log ⁡ ( h ) ) .

As Δ p is compact, we have k ∈ Δ p , where log ⁡ ( g ) + log ⁡ ( h ) = log ⁡ ( k ) .

For the opposite inclusion, Δ is dense in Δ p , so log ⁡ ( Δ ) is dense in log ⁡ ( Δ p ) . As Z p ⁢ log ⁡ ( Δ ) is closed in L p , it follows that log ⁡ ( Δ p ) ⊆ Z p ⁢ log ⁡ ( Δ ) , as required. ∎

Lemma 2.17

The group Δ p is a unique minimal subgroup Γ p ≤ Δ p ≤ R p such that its image in L p is a Z p -lattice.

Proof

Suppose that Λ p satisfies Γ p ≤ Λ p ≤ R p , log ⁡ ( Λ p ) is a Z p -lattice, and log ⁡ ( Δ p ) ⊈ log ⁡ ( Λ p ) . It follows that Γ ≤ Λ p ∩ Δ ≨ Δ , where

log ⁡ ( Λ p ∩ Δ ) = log ⁡ ( Λ p ) ∩ log ⁡ ( Δ )

is a lattice of 𝐿 that contains log ⁡ ( Γ ) . This is a contradiction to the minimality of Δ. ∎

Now, using Lemma 2.11 and the property Γ p ≤ Δ p ≤ R p (Lemma 2.16), and following the same steps for the proof of [25] to Theorem 2.6, we have the following theorem.

Theorem 2.18

There are natural isomorphisms

Aut ⁢ ( R p ) ≅ Aut ⁢ ( L p ) , Aut ⁢ ( Γ p ) ≅ N Aut ⁢ ( L p ) ⁢ ( log ⁡ ( Γ p ) ) ≤ Aut ⁢ ( L p ) , Aut ⁢ ( Δ p ) ≅ N Aut ⁢ ( L p ) ⁢ ( log ⁡ ( Δ p ) ) ≤ Aut ⁢ ( L p ) .

By Lemmas 2.12, 2.13 and Theorem 2.18, we obtain the next result.

Proposition 2.19

Denote I A * ( Γ p ) = ker ( Aut ( Γ p ) → Aut ( ( Γ * ) p ) . Then

I ⁢ A * ⁢ ( Γ p ) ≅ N Aut ⁢ ( L p ) ⁢ ( log ⁡ ( Γ p ) ) ∩ ker ⁡ ( Aut ⁢ ( L p ) → Aut ⁢ ( L p / L p ′ ) ) .

Similarly, as Γ p ≤ Δ p ≤ R p (Lemma 2.16), we also have

I ⁢ A * ⁢ ( Δ p ) ≅ N Aut ⁢ ( L p ) ⁢ ( log ⁡ ( Δ p ) ) ∩ ker ⁡ ( Aut ⁢ ( L p ) → Aut ⁢ ( L p / L p ′ ) ) .

We can now deduce Proposition 2.8.

Proof of Proposition 2.8

From Lemma 2.17 and Proposition 2.19, we get that, for any prime 𝑝, one has I ⁢ A * ⁢ ( Γ p ) ↪ I ⁢ A * ⁢ ( Δ p ) . Therefore,

I ⁢ A * ⁢ ( Γ ^ ) = ∏ p I ⁢ A * ⁢ ( Γ p ) ↪ ∏ p I ⁢ A * ⁢ ( Δ p ) = I ⁢ A * ⁢ ( Δ ^ ) .

Moreover, I ⁢ A * ⁢ ( Γ ^ ) = { α ∈ I ⁢ A * ⁢ ( Δ ^ ) ∣ α ⁢ ( Γ ^ ) = Γ ^ } ; hence it is open in I ⁢ A * ⁢ ( Δ ^ ) . ∎

As we mentioned previously, by Proposition 2.9, in order to prove Theorem 1.4 for Γ, it is enough to prove it for Δ. So, from now on, log ⁡ ( Δ ) is a lattice that spans 𝐿, and hence k = dim Q ⁡ ( L ) = rank Z ⁡ ( log ⁡ ( Δ ) ) . Our objective now is to construct a basis for log ⁡ ( Δ ) that, with relation to it, we will be able to view I ⁢ A * ⁢ ( Δ ) as the ℤ-points of a unipotent group scheme defined over ℤ, whose Z p -points are I ⁢ A * ⁢ ( Δ p ) . Once we show this, we will see that Theorem 1.4 (1) follows from the classical CSP for the arithmetic group I ⁢ A * ⁢ ( Δ ) , and Theorem 1.4 (2) is the classical strong-approximation theorem for this group.

Let g 1 , … , g d ∈ Δ be such that Δ * = Δ / δ ⁢ ( Δ ) ≃ Z d is generated by the images g ¯ 1 , … , g ¯ d . Then g ¯ 1 , … , g ¯ d also generate ( Δ * ) p = Δ p / δ ⁢ ( Δ p ) ≃ Z p d as a pro-𝑝 group. Hence, by Lemmas 2.3 and 2.13, l 1 = log ⁡ ( g 1 ) , … , l d = log ⁡ ( g d ) generate R / R ′ ≅ L / L ′ ≅ Q ( d ) over ℚ and generate R p / R p ′ ≅ L p / L p ′ ≅ Q p ( d ) over Q p . It follows that l 1 , … , l d are linearly independent over Q p (and over ℚ), and since 𝐿 and L p are nilpotent, they generate 𝐿 as a Lie algebra over ℚ and generate L p as a Lie algebra over Q p .

Lemma 2.20

Let L = γ 1 ⁢ ( L ) , L ′ = γ 2 ⁢ ( L ) , …, 0 = γ c ⁢ ( L ) be the lower central series of 𝐿. The set l 1 , … , l d can be completed to a basis

B = { l 1 , … , l d , l d + 1 , … , l k }

of the lattice log ⁡ ( Δ ) such that

𝐵 contains bases for γ j ⁢ ( L ) / γ j + 1 ⁢ ( L ) mod γ j + 1 ⁢ ( L ) for every 𝑗,
l j + 1 lies in the same term as l j in the lower central series of 𝐿, or deeper.

Proof

Now, l 1 , … , l d lie in γ 1 ⁢ ( L ) - γ 2 ⁢ ( L ) and provide a basis for

L / L ′ = γ 1 ⁢ ( L ) / γ 2 ⁢ ( L ) mod γ 2 ⁢ ( L ) .

Denote i 1 = d .

For every j ≥ 2 , the group ( log ⁡ ( Δ ) ∩ γ j ⁢ ( L ) ) / ( log ⁡ ( Δ ) ∩ γ j + 1 ⁢ ( L ) ) is a lattice in γ j ⁢ ( L ) / γ j + 1 ⁢ ( L ) , and hence

rank Z ⁡ ( ( log ⁡ ( Δ ) ∩ γ j ⁢ ( L ) ) / ( log ⁡ ( Δ ) ∩ γ j + 1 ⁢ ( L ) ) ) = dim Q ⁡ ( γ j ⁢ ( L ) / γ j + 1 ⁢ ( L ) ) .

Let l i j - 1 + 1 , … , l i j ∈ log ⁡ ( Δ ) ∩ γ j ⁢ ( L ) be a basis to

( log ⁡ ( Δ ) ∩ γ j ⁢ ( L ) ) / ( log ⁡ ( Δ ) ∩ γ j + 1 ⁢ ( L ) ) mod log ⁡ ( Δ ) ∩ γ j + 1 ⁢ ( L ) .

Then l i j - 1 + 1 , … , l i j gives a basis for γ j ⁢ ( L ) / γ j + 1 ⁢ ( L ) mod γ j + 1 ⁢ ( L ) , and they all lie in γ j ⁢ ( L ) - γ j + 1 ⁢ ( L ) .

This procedure gives us l 1 , … , l i c - 1 satisfying the two conditions in the lemma. Also, by the construction, l 1 , … , l i c - 1 span the abelian group log ⁡ ( Δ ) and provide a basis for 𝐿. It follows that i c - 1 = k and that l 1 , … , l k is the desired basis for log ⁡ ( Δ ) . ∎

Let 𝐵 be a basis for log ⁡ ( Δ ) as in the above lemma. Clearly, 𝐵 is also a basis for 𝐿 as a vector space over ℚ. As L p is generated as a Lie algebra by l 1 , … , l d over Q p , the set 𝐵 also spans L p over Q p . As dim Q ⁡ ( L ) = dim Q p ⁡ ( L p ) (Proposition 2.14), it follows that 𝐵 is also a basis for L p as a vector space over Q p . Hence, using the basis 𝐵, we can identify

Aut ⁢ ( L ) = GL k ⁢ ( Q ) ∩ Aut ⁢ ( L p ) ≤ GL k ⁢ ( Q p ) .

In addition, as log ⁡ ( Δ ) is a lattice and log ⁡ ( Δ p ) = Z p ⁢ log ⁡ ( Δ ) (Lemma 2.16), using the basis 𝐵, we can identify

Aut ⁢ ( Δ ) = N Aut ⁢ ( L ) ⁢ ( log ⁡ ( Δ ) ) = GL k ⁢ ( Z ) ∩ Aut ⁢ ( L ) , Aut ⁢ ( Δ p ) = N Aut ⁢ ( L p ) ⁢ ( log ⁡ ( Δ p ) ) = GL k ⁢ ( Z p ) ∩ Aut ⁢ ( L p ) .

Moreover, we can identify Aut ⁢ ( L ) and Aut ⁢ ( L p ) with the groups

Aut ⁢ ( L ) = { A ∈ GL k ⁢ ( Q ) ∣ [ A ⁢ x , A ⁢ y ] - A ⁢ [ x , y ] = 0 ⁢ for all ⁢ x , y ∈ B } , Aut ⁢ ( L p ) = { A ∈ GL k ⁢ ( Q p ) ∣ [ A ⁢ x , A ⁢ y ] - A ⁢ [ x , y ] = 0 ⁢ for all ⁢ x , y ∈ B } .

Now, notice that

σ ∈ ker ⁡ ( Aut ⁢ ( L ) → Aut ⁢ ( L / L ′ ) )

if and only if, for every i = 1 , … , d , we have σ ⁢ ( l i ) ∈ l i + L ′ . As l 1 , … , l d generate 𝐿 over ℚ, it follows that, for every σ ∈ ker ⁡ ( Aut ⁢ ( L ) → Aut ⁢ ( L / L ′ ) ) and every 𝑖 (not only for i = 1 , … , d ), we have σ ⁢ ( l i ) = l i + n i , where n i lies in a strictly deeper term in the lower central series of 𝐿 than the one that l i lies in. Now, denote the subgroup of Tr 1 ⁢ ( k , Q ) (resp. Tr 1 ⁢ ( k , Q p ) ) consisting of all the elements for which the upper left d × d -block is the identity matrix, by Tr 1 d ⁢ ( k , Q ) (resp. Tr 1 d ⁢ ( k , Q p ) ). By the construction in Lemma 2.20, with relation to 𝐵, we have

ker ⁡ ( Aut ⁢ ( L ) → Aut ⁢ ( L / L ′ ) ) = Aut ⁢ ( L ) ∩ Tr 1 d ⁢ ( k , Q ) ≤ GL k ⁢ ( Q p ) .

It follows that I ⁢ A * ⁢ ( Δ ) can be identified with the arithmetic group of ℤ-points of Aut ⁢ ( L ) ∩ Tr 1 d ⁢ ( k , Q ) . Similarly, we have

ker ⁡ ( Aut ⁢ ( L p ) → Aut ⁢ ( L p / L p ′ ) ) = Aut ⁢ ( L p ) ∩ Tr 1 d ⁢ ( k , Q p ) ≤ GL k ⁢ ( Q p ) ,

and so I ⁢ A * ⁢ ( Δ p ) can be identified with the Z p -points of Aut ⁢ ( L p ) ∩ Tr 1 d ⁢ ( k , Q p ) . By the description above, Aut ⁢ ( L ) ∩ Tr 1 d ⁢ ( k , Q ) and Aut ⁢ ( L p ) ∩ Tr 1 d ⁢ ( k , Q p ) are unipotent algebraic groups which are defined by the same equations over ℚ and Q p , respectively.

Now, I ⁢ A * ⁢ ( Δ ) is a unipotent subgroup of GL k ⁢ ( Z ) . Hence, by the congruence subgroup property for unipotent arithmetic groups, we have

and hence I ⁢ A * ⁢ ( Δ ) ^ ↪ I ⁢ A * ⁢ ( Δ ^ ) is injective. Now, as every finitely generated nilpotent group is subgroup separable (cf. [12]), the map G ^ → H ^ is injective whenever 𝐺 is a subgroup of a finitely generated nilpotent group 𝐻. Thus, for the finitely generated nilpotent group I ⁢ A * ⁢ ( Δ ) and for any G ≤ I ⁢ A * ⁢ ( Δ ) , we have G ^ ↪ I ⁢ A * ⁢ ( Δ ) ^ , and hence G ^ ↪ I ⁢ A * ⁢ ( Δ ^ ) . This gives the first part of Theorem 1.4 in the case of finitely generated torsion-free nilpotent groups.

For the second statement of Theorem 1.4, notice that, by the above description for I ⁢ A * ⁢ ( Δ ) and I ⁢ A * ⁢ ( Δ p ) , we obtain that I ⁢ A * ⁢ ( Δ ) is dense in I ⁢ A * ⁢ ( Δ p ) by the strong approximation property for arithmetic unipotent groups (see [20, Proposition 7.1, and the corollary afterward]). Therefore, as each of the groups I ⁢ A * ⁢ ( Δ p ) is a pro-𝑝 group, we obtain that I ⁢ A * ⁢ ( Δ ) is dense in

∏ p I ⁢ A * ⁢ ( Δ p ) = I ⁢ A * ⁢ ( Δ ^ ) .

This completes the proof of Theorem 1.4 for finitely generated torsion-free nilpotent groups.

3 The general case

The aim of this section is to prove Theorem 1.4 given its validity for finitely generated torsion-free nilpotent group. Along the section, Γ is a finitely generated nilpotent group, and Δ = Γ / tor ⁢ ( Γ ) . As a finitely generated nilpotent group, Γ is residually finite. Hence, we can think on Γ as a subgroup of Γ ^ . One has tor ⁢ ( Γ ^ ) = tor ⁢ ( Γ ) (see [14, Lemma 2.3], [26, Corollary 7.5]). In other words, tor ⁢ ( Γ ) is a normal subgroup of Γ ^ , and

Δ ^ = Γ / tor ⁢ ( Γ ) ^ = Γ ^ / tor ⁢ ( Γ ) = Γ ^ / tor ⁢ ( Γ ^ )

is torsion free. In addition, we get that tor ⁢ ( Γ ) is characteristic, not only as a subgroup of Γ, but also as a subgroup of Γ ^ . Therefore, we have a map

I ⁢ A * ⁢ ( Γ ^ ) → I ⁢ A * ⁢ ( Δ ^ ) ,

and we obtain the commutative diagram

Denote K ~ = ker ⁡ ( I ⁢ A * ⁢ ( Γ ) → I ⁢ A * ⁢ ( Δ ) ) . Let x 1 , … , x n be a generating set for Γ. Then every α ∈ K ~ can be described by x i ↦ x i ⁢ a i for some a i ∈ tor ⁢ ( Γ ) . As tor ⁢ ( Γ ) is finite, K ~ is also finite.

Now, let G ≤ I ⁢ A * ⁢ ( Γ ) , and denote its image in I ⁢ A * ⁢ ( Δ ) by 𝐻. Then

K = ker ⁡ ( G → H ) ≤ K ~

is finite, and hence we obtain the following commutative exact diagram:

Note that, as G ≤ I ⁢ A * ⁢ ( Γ ) ↪ I ⁢ A * ⁢ ( Γ ^ ) , the map K → I ⁢ A * ⁢ ( Γ ^ ) is injective. Note also that, as 𝐾 is finite, we have K = K ^ . Hence, moving to the profinite completion of the upper row, we get the commutative exact diagram

It follows that K ^ → I ⁢ A * ⁢ ( Γ ^ ) is injective, and by § 2, also H ^ → I ⁢ A * ⁢ ( Δ ^ ) is injective. Hence, by diagram chasing, G ^ → I ⁢ A * ⁢ ( Γ ^ ) is also injective. This proves the first assertion of Theorem 1.4 in the general case.

We move now to prove the second assertion. As Γ is residually finite and tor ⁢ ( Γ ) is finite, there exists t ∈ N such that tor ⁢ ( Γ ) ∩ Γ t = { e } , where Γ t is the normal subgroup

Γ t = ⟨ g t ∣ g ∈ Γ ⟩ ⁢ ⊲ ⁢ Γ .

Fix this 𝑡. It follows that, whenever t ∣ m , we have tor ⁢ ( Γ ) ∩ Γ m = { e } , and therefore tor ⁢ ( Γ ) is naturally embedded in Γ / Γ m for such 𝑚. As Γ is nilpotent and finitely generated, Γ / Γ m and Δ / Δ m are finite for any 𝑚 (see [26, Corollary 3.3]), and hence one has

Γ ^ = lim ← m ∈ N ⁡ Γ / Γ m = lim ← t ∣ m ⁡ Γ / Γ m , Δ ^ = lim ← m ∈ N ⁡ Δ / Δ m = lim ← t ∣ m ⁡ Δ / Δ m .

Now, let α ^ ∈ I ⁢ A * ⁢ ( Γ ^ ) , and write

α ^ = ( α m ) t ∣ m ∈ I ⁢ A * ⁢ ( Γ ^ ) ≤ lim ← t ∣ m ⁡ Aut ⁢ ( Γ / Γ m ) ,

where α m ∈ Aut ⁢ ( Γ / Γ m ) . In order to prove the second part of Theorem 1.4, i.e. that I ⁢ A * ⁢ ( Γ ) is dense in I ⁢ A * ⁢ ( Γ ^ ) , it is enough to show the following result.

Proposition 3.1

Let 𝑚 be such that t ∣ m , and let α ^ , α m be as above. Then there exists α ∈ I ⁢ A * ⁢ ( Γ ) such that α → α m through the map I ⁢ A * ⁢ ( Γ ) → Aut ⁢ ( Γ / Γ m ) .

So fix 𝑚 such that t ∣ m , and let β ^ ∈ I ⁢ A * ⁢ ( Δ ^ ) be the image of α ^ under the map I ⁢ A * ⁢ ( Γ ^ ) → I ⁢ A * ⁢ ( Δ ^ ) . Write

β ^ = ( β m ) t ∣ m ∈ I ⁢ A * ⁢ ( Δ ^ ) ≤ lim ← t ∣ m ⁡ Aut ⁢ ( Δ / Δ m ) ,

where β m ∈ Aut ⁢ ( Δ / Δ m ) is the image of α m ∈ Aut ⁢ ( Γ / Γ m ) under the map

Aut ⁢ ( Γ / Γ m ) → Aut ⁢ ( Δ / Δ m ) .

By assumption, I ⁢ A * ⁢ ( Δ ) is dense in I ⁢ A * ⁢ ( Δ ^ ) , and hence there exists β ∈ I ⁢ A * ⁢ ( Δ ) such that β → β m . So we have the diagram

We want to use α m and 𝛽 in order to construct α ∈ I ⁢ A * ⁢ ( Γ ) such that α → α m .

Let us recall the following notion: let P 1 , P 2 and 𝑄 be groups with epimorphisms π i : P i ↠ Q , i = 1 , 2 . The group

U = P 1 × Q P 2 = { ( x , y ) ∈ P 1 × P 2 ∣ π 1 ⁢ ( x ) = π 2 ⁢ ( y ) }

is called the fiber product of P 1 , P 2 along 𝑄 (or the pullback). It is easy to check that, as π i are surjective, we get the following commutative diagram of surjective maps:

such that P 1 ≃ U / ( U ∩ ( { e } × P 2 ) ) and P 2 ≃ U / ( U ∩ ( P 1 × { e } ) ) . Regarding our context, as we have the commutative surjective diagram

we get a natural map Γ ⁢ → ρ ⁢ Δ × Δ / Δ m Γ / Γ m . Now, as we assume that t ∣ m , we have

ker ⁡ ( Γ → Γ / Γ m ) ∩ ker ⁡ ( Γ → Δ ) = Γ m ∩ tor ⁢ ( Γ ) = { e } ,

and hence the map 𝜌 is injective. In addition,

ker ⁡ ( Γ → Δ / Δ m ) = Γ m ⋅ tor ⁢ ( Γ ) = ker ⁡ ( Γ → Γ / Γ m ) ⋅ ker ⁡ ( Γ → Δ ) ,

and hence one can check that it follows that 𝜌 is also surjective. Therefore, we can identify Γ with Δ × Δ / Δ m Γ / Γ m , the fiber product of Δ and Γ / Γ m along Δ / Δ m . The following lemma is elementary.

Lemma 3.2

With notation as above, let σ i ∈ Aut ⁢ ( P i ) for i = 1 , 2 be automorphisms preserving ker ⁡ ( π i ) and inducing σ ¯ i ∈ Aut ⁢ ( Q ) . If σ ¯ 1 = σ ¯ 2 , then there exists σ ∈ Aut ⁢ ( U ) such that 𝜎 preserves U ∩ ( P 1 × { e } ) and U ∩ ( { e } × P 2 ) and induces σ 1 on P 1 and σ 2 on P 2 .

Proof

For ( x , y ) ∈ U , define σ ⁢ ( x , y ) = ( σ 1 ⁢ ( x ) , σ 2 ⁢ ( y ) ) . We claim that we have ( σ 1 ⁢ ( x ) , σ 2 ⁢ ( y ) ) ∈ U . Indeed,

π 1 ⁢ ( σ 1 ⁢ ( x ) ) = σ ¯ 1 ⁢ ( π 1 ⁢ ( x ) ) = σ ¯ 1 ⁢ ( π 2 ⁢ ( y ) ) = σ ¯ 2 ⁢ ( π 2 ⁢ ( y ) ) = π 2 ⁢ ( σ 2 ⁢ ( y ) ) .

Showing that 𝜎 is a bijective homomorphism follows from the same properties for σ 1 , σ 2 . The other properties of 𝜎 follow straightforwardly from the definition. ∎

Applying Lemma 3.2 to α m ∈ Aut ⁢ ( Γ / Γ m ) , β ∈ I ⁢ A * ⁢ ( Δ ) , we obtain α ∈ Aut ⁢ ( Γ ) whose projection is α m through Aut ⁢ ( Γ ) → Aut ⁢ ( Γ / Γ m ) . Hence, in order to finish the proof of Theorem 1.4, it remains to show that the 𝛼 we thus obtained is not only in Aut ⁢ ( Γ ) but also in α ∈ I ⁢ A * ⁢ ( Γ ) .

Lemma 3.3

Recall that Γ * = Γ ¯ / tor ⁢ ( Γ ¯ ) , where Γ ¯ = Γ / Γ ′ , and denote

Δ * = Δ ¯ / tor ⁢ ( Δ ¯ ) , where ⁢ Δ ¯ = Δ / Δ ′ .

Then we have a canonical isomorphism Γ * ≃ Δ * .

Proof

We have a natural projection

Δ = Γ / tor ⁢ ( Γ ) ↠ Γ ¯ / tor ⁢ ( Γ ¯ ) = Γ *

which gives rise to a map Δ ¯ = Δ / Δ ′ ↠ Γ * and to a map Δ * = Δ ¯ / tor ⁢ ( Δ ¯ ) ↠ Γ * . Obviously, this map is the inverse of the natural map Γ * ↠ Δ * . So Γ * ≃ Δ * as required. ∎

From this lemma, we get that the preimage of I ⁢ A * ⁢ ( Δ ) under Aut ⁢ ( Γ ) → Aut ⁢ ( Δ ) is I ⁢ A * ⁢ ( Γ ) . Thus, as 𝛼 is a preimage of 𝛽 through the map Aut ⁢ ( Γ ) → Aut ⁢ ( Δ ) and β ∈ I ⁢ A * ⁢ ( Δ ) , it follows that indeed α ∈ I ⁢ A * ⁢ ( Γ ) , as required.

4 The free cases

In this section, we sketch a more straightforward proof to Corollary 1.5 in the special case where Ψ c is a free nilpotent group on 𝑛 elements, a proof which does not refer either to the CSP or to the strong approximation for unipotent groups. Note that, as Ψ c * = Ψ c / Ψ c ′ = Z ( n ) and Ψ ^ c * = Ψ c / Ψ c ′ ^ = Z ^ ( n ) , in this case, we have I ⁢ A ⁢ ( Ψ c ) = I ⁢ A * ⁢ ( Ψ c ) and I ⁢ A ⁢ ( Ψ ^ c ) = I ⁢ A * ⁢ ( Ψ ^ c ) . Let us formulate the assertion.

Theorem 4.1

Let Ψ c be the free nilpotent group on 𝑛 elements. For every c ∈ N , the map I ⁢ A ⁢ ( Ψ c ) ^ → I ⁢ A ⁢ ( Ψ ^ c ) is an isomorphism.

Proof

We will prove this by induction on 𝑐. For c = 1 , 2 , the result is trivial, as Ψ 1 = { e } and Ψ 2 = Z ( n ) , so

I ⁢ A ⁢ ( Ψ 1 ) ^ = I ⁢ A ⁢ ( Ψ ^ 1 ) = I ⁢ A ⁢ ( Ψ 2 ) ^ = I ⁢ A ⁢ ( Ψ ^ 2 ) = { e } .

For the induction step, we will use the (easy) facts that, for every 𝑐, the natural map Aut ⁢ ( Ψ c + 1 ) → Aut ⁢ ( Ψ c ) is surjective (see [1]), and also the natural map Aut ⁢ ( Ψ ^ c + 1 ) → Aut ⁢ ( Ψ ^ c ) is surjective (see [15, Section 5.2]). So let c ≥ 2 . Denote

A ⁢ ( Ψ c + 1 ) = ker ⁡ ( I ⁢ A ⁢ ( Ψ c + 1 ) ↠ I ⁢ A ⁢ ( Ψ c ) ) , A ⁢ ( Ψ ^ c + 1 ) = ker ⁡ ( I ⁢ A ⁢ ( Ψ ^ c + 1 ) ↠ I ⁢ A ⁢ ( Ψ ^ c ) ) .

We have the commutative exact diagram

which gives rise to the commutative exact diagram

By the induction hypothesis and diagram chasing, it is enough to prove that the map A ⁢ ( Ψ c + 1 ) ^ → A ⁢ ( Ψ ^ c + 1 ) is an isomorphism.

Let x 1 , … , x n be a set of free generators for Ψ c + 1 . Define

Z ⁢ ( Ψ c + 1 ) ( n ) = Z ⁢ ( Ψ c + 1 ) × ⋯ × Z ⁢ ( Ψ c + 1 ) ⏟ n ,

where Z ⁢ ( Ψ c + 1 ) is the center of Ψ c + 1 . Observe now that A ⁢ ( Ψ c + 1 ) can be viewed as a subgroup of Z ⁢ ( Ψ c + 1 ) ( n ) in the following way: for α ∈ A ⁢ ( Ψ c + 1 ) , one can describe it by its action on the generators of Ψ c + 1 and write α ⁢ ( x i ) = x i ⁢ u i for some u i ∈ Z ⁢ ( Ψ c + 1 ) . We claim that the map α ↦ ( u 1 , … , u n ) ∈ Z ⁢ ( Ψ c + 1 ) ( n ) is a natural injective homomorphism

A ⁢ ( Ψ c + 1 ) → Z ⁢ ( Ψ c + 1 ) ( n ) .

Indeed, let α , β ∈ A ⁢ ( Ψ c + 1 ) be defined by α ⁢ ( x i ) = x i ⁢ u i and β ⁢ ( x i ) = x i ⁢ v i for some u i , v i ∈ Z ⁢ ( Ψ c + 1 ) . Now, as u i ∈ Z ⁢ ( Ψ c + 1 ) ⊆ Ψ c + 1 ′ , it can be written as a product of commutators of words on the generators x i . As u i , v i ∈ Z ⁢ ( Ψ c + 1 ) , it is easy to see that ( β ∘ α ) ⁢ ( x i ) = x i ⁢ v i ⁢ u i . This proves the claim. In [1], it is proven that the above map A ⁢ ( Ψ c + 1 ) → Z ⁢ ( Ψ c + 1 ) ( n ) is also surjective, and hence it is an isomorphism. In other words, for every choice of elements u 1 , … , u n ∈ Z ⁢ ( Ψ c + 1 ) , one can define an element α ∈ A ⁢ ( Ψ c + 1 ) by defining α ⁢ ( x i ) = x i ⁢ u i for the generators x 1 , … , x n of Ψ c + 1 . This is indeed an automorphism as { x i ⁢ u i } i = 1 n generate Ψ c + 1 , which is a free nilpotent group.

Using a similar approach, one can prove that

A ⁢ ( Ψ ^ c + 1 ) ≅ Z ⁢ ( Ψ ^ c + 1 ) ( n )

(see also [15]), and thus

A ⁢ ( Ψ c + 1 ) ^ ≅ Z ⁢ ( Ψ c + 1 ) ^ ( n ) ≅ Z ⁢ ( Ψ ^ c + 1 ) ( n ) ≅ A ⁢ ( Ψ ^ c + 1 ) ,

as required. ∎

Funding source: National Science Foundation

Award Identifier / Grant number: 1502651

Award Identifier / Grant number: DMS-1700165

Funding source: H2020 European Research Council

Award Identifier / Grant number: 692854

Funding statement: During the period of the research, the first author was supported by the Rudin foundation and, not concurrently, by NSF research training grant (RTG) # 1502651. The second author is indebted for support from the NSF (Grant No. DMS-1700165) and the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (Grant No. 692854).

Communicated by: Benjamin Klopsch

References

[1] S. Andreadakis, On the automorphisms of free groups and free nilpotent groups, Proc. Lond. Math. Soc. (3) 15 (1965), 239–268. 10.1112/plms/s3-15.1.239Search in Google Scholar

[2] H. Bass, M. Lazard and J.-P. Serre, Sous-groupes d’indice fini dans S ⁢ L ⁢ ( n , Z ) , Bull. Amer. Math. Soc. 70 (1964), 385–392. 10.1090/S0002-9904-1964-11107-1Search in Google Scholar

[3] G. Baumslag, Automorphism groups of nilpotent groups, Amer. J. Math. 91 (1969), 1003–1011. 10.2307/2373314Search in Google Scholar

[4] D. E.-C. Ben-Ezra, The congruence subgroup problem for the free metabelian group on two generators, Groups Geom. Dyn. 10 (2016), no. 2, 583–599. 10.4171/GGD/357Search in Google Scholar

[5] D. E.-C. Ben-Ezra, The IA-congruence kernel of high rank free metabelian groups, Ann. K-Theory 4 (2019), no. 3, 383–438. 10.2140/akt.2019.4.383Search in Google Scholar

[6] D. E.-C. Ben-Ezra, The congruence subgroup problem for the free metabelian group on n ≥ 4 generators, Groups Geom. Dyn. 14 (2020), no. 3, 729–764. 10.4171/GGD/561Search in Google Scholar

[7] D. E.-C. Ben-Ezra and A. Lubotzky, The congruence subgroup problem for low rank free and free metabelian groups, J. Algebra 500 (2018), 171–192. 10.1016/j.jalgebra.2017.01.001Search in Google Scholar

[8] R. M. Bryant and J. R. J. Groves, Algebraic groups of automorphisms of nilpotent groups and Lie algebras, J. Lond. Math. Soc. (2) 33 (1986), no. 3, 453–466. 10.1112/jlms/s2-33.3.453Search in Google Scholar

[9] R. M. Bryant and A. Papistas, Automorphism groups of nilpotent groups, Bull. London Math. Soc. 21 (1989), no. 5, 459–462. 10.1112/blms/21.5.459Search in Google Scholar

[10] K.-U. Bux, M. V. Ershov and A. S. Rapinchuk, The congruence subgroup property for Aut ⁡ F 2 : A group-theoretic proof of Asada’s theorem, Groups Geom. Dyn. 5 (2011), no. 2, 327–353. 10.4171/GGD/130Search in Google Scholar

[11] J. S. Chahal, Solution of the congruence subgroup problem for solvable algebraic groups, Nagoya Math. J. 79 (1980), 141–144. 10.1017/S0027763000018985Search in Google Scholar

[12] J. Deré and M. Pengitore, Effective subgroup separability of finitely generated nilpotent groups, J. Algebra 506 (2018), 489–508. 10.1016/j.jalgebra.2018.03.035Search in Google Scholar

[13] J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic Pro-𝑝 Groups, Cambridge University, Cambridge, 2003. Search in Google Scholar

[14] P. H. Kropholler and J. S. Wilson, Torsion in profinite completions, J. Pure Appl. Algebra 88 (1993), no. 1–3, 143–154. 10.1016/0022-4049(93)90018-OSearch in Google Scholar

[15] A. Lubotzky, Combinatorial group theory for pro-𝑝-groups, J. Pure Appl. Algebra 25 (1982), no. 3, 311–325. 10.1016/0022-4049(82)90086-XSearch in Google Scholar

[16] A. Lubotzky, Free quotients and the congruence kernel of SL 2 , J. Algebra 77 (1982), no. 2, 411–418. 10.1016/0021-8693(82)90263-0Search in Google Scholar

[17] O. V. Mel’nikov, Congruence kernel of the group SL 2 ⁢ ( Z ) , Dokl. Akad. Nauk SSSR 228 (1976), no. 5, 1034–1036. Search in Google Scholar

[18] J. L. Mennicke, Finite factor groups of the unimodular group, Ann. of Math. (2) 81 (1965), 31–37. 10.2307/1970380Search in Google Scholar

[19] N. Nikolov and D. Segal, Finite index subgroups in profinite groups, C. R. Math. Acad. Sci. Paris 337 (2003), no. 5, 303–308. 10.1016/S1631-073X(03)00349-2Search in Google Scholar

[20] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, Boston, 1993. Search in Google Scholar

[21] V. P. Platonov, The congruence problem for solvable integral groups, Dokl. Akad. Nauk BSSR 15 (1971), 869–872. Search in Google Scholar

[22] V. P. Platonov and A. A. Šaromet, The congruence subgroup problem for linear groups over arithmetic rings (in Russian), Dokl. Akad. Nauk BSSR 16 (1972), 393–396. Search in Google Scholar

[23] M. S. Raghunathan, The congruence subgroup problem, Proc. Indian Acad. Sci. Math. Sci. 114 (2004), no. 4, 299–308. 10.1007/BF02829437Search in Google Scholar

[24] A. S. Rapinchuk, The congruence subgroup problem, Algebra, 𝐾-Theory, Groups, and Education (New York 1997), Contemp. Math. 243, American Mathematical Society, Providence (1999), 175–188. 10.1090/conm/243/03693Search in Google Scholar

[25] D. Segal, Polycyclic Groups, Cambridge Tracts in Math. 82, Cambridge University, Cambridge, 2005. Search in Google Scholar

[26] R. B. Warfield, Jr., Nilpotent groups, Lecture Notes in Math. 513, Springer, Berlin, 1976. 10.1007/BFb0080152Search in Google Scholar

Received: 2021-03-09

Revised: 2021-07-30

Published Online: 2021-09-22

Published in Print: 2022-05-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2021-0039