Strong indecomposability of the outer automorphism groups of nonabelian free profinite groups

Fact 1

The center of F m is trivial.

Fact 2

The center of Out ⁢ ( F m ) is trivial.

Theorem 1.1

Let Σ ⊆ P be a subset such that either | Σ | = 1 or | P ∖ Σ | is finite. Let G ⊆ Out ⁢ ( Π Z Q ̄ Σ ) be a closed subgroup such that

• 𝐺 contains an open subgroup of ρ Z Σ ⁢ ( G K ) ,

• there exists a prime number l ∈ Σ such that the image of 𝐺 via the natural morphism

  ϕ l : Out ⁢ ( Π Z Q ̄ Σ ) ↠ Out ⁢ ( Π Z Q ̄ { l } ) ,

  which is necessarily surjective by [14, Proposition 4.5.4 (b)], is slim, i.e., the center of every open subgroup of ϕ l ⁢ ( G ) is trivial.

Then 𝐺 is strongly indecomposable, i.e., no open subgroup of 𝐺 has a nontrivial direct product decomposition.

Corollary 1.2

Let m ≥ 2 be an integer and Σ ⊆ P a subset such that either | Σ | = 1 or | P ∖ Σ | is finite. Then Aut ⁢ ( F m Σ ) and Out ⁢ ( F m Σ ) are strongly indecomposable.

Definition 3.1

Let 𝐺 be a profinite group and H ⊆ G a closed subgroup of 𝐺.

1. We shall write Z G ⁢ ( H ) for the centralizer of 𝐻 in 𝐺, i.e., the closed subgroup { g ∈ G ∣ g ⁢ h ⁢ g − 1 = h ⁢ for any ⁢ h ∈ H } , and Z ⁢ ( G ) ⁢ = def ⁢ Z G ⁢ ( G ) .

2. We shall say that 𝐺 is slim if Z G ⁢ ( U ) = { 1 } for every open subgroup 𝑈 of 𝐺, or, equivalently, Z ⁢ ( U ) = { 1 } for every open subgroup 𝑈 of 𝐺.

3. We shall say that 𝐺 is decomposable if there exist nontrivial normal closed subgroups H 1 ⊆ G and H 2 ⊆ G such that G = H 1 × H 2 . We shall say that 𝐺 is indecomposable if 𝐺 is not decomposable. We shall say that 𝐺 is strongly indecomposable if every open subgroup of 𝐺 is indecomposable.

Remark 3.2

Let 𝐺 be a slim profinite group. From [12, §0] and [8, Lemma 1.6], we recall the following well known facts.

1. Every finite normal closed subgroup of 𝐺 is trivial.

2. Let H ⊆ G be an open subgroup and α ∈ Aut ⁢ ( G ) . Suppose that 𝛼 induces the identity automorphism on 𝐻. Then 𝛼 is the identity automorphism on 𝐺.

Lemma 3.3

Let 𝐺 be a profinite group and { G i } i ∈ I a directed subset of the set of characteristic open subgroups of 𝐺, where j ≥ i ⇔ G j ⊆ G i , such that

Write ϕ i : Out ⁢ ( G ) → Out ⁢ ( G / G i ) for the natural homomorphism. Then

Proof

We consider σ ∈ ⋂ i ∈ I Ker ⁢ ( ϕ i ) ⊆ Out ⁢ ( G ) and a lifting σ ̃ ∈ Aut ⁢ ( G ) . For each i ∈ I , write σ ̃ i ∈ Aut ⁢ ( G / G i ) for the automorphism induced by σ ̃ . Then, since σ ∈ Ker ⁢ ( ϕ i ) , the automorphism σ ̃ i is inner. Let γ i ∈ G / G i be an element which induces σ ̃ i via conjugation. Write C i ⁢ = def ⁢ γ i ⋅ Z ⁢ ( G / G i ) ⊆ G / G i . Here, we note that if i 1 ≥ i 2 ( i 1 , i 2 ∈ I ), then the natural surjection G / G i 1 ↠ G / G i 2 induces a map C i 1 → C i 2 . Observe that, since C i ( i ∈ I ) is a finite nonempty set, the inverse limit lim ← i ∈ I ⁡ C i is nonempty; see [14, Corollary 1.1.6]. Then σ ̃ is the inner automorphism given by any element γ ∈ lim ← i ∈ I C i ( ⊆ lim ← i ∈ I G / G i = G ) . This completes the proof of Lemma 3.3. ∎

Lemma 3.4

Let 𝐺 be a topologically finitely generated profinite group and S ⊆ P a finite subset. Then the natural homomorphism

is injective.

Proof

Since 𝐺 is topologically finitely generated, there exists a directed subset { G i } i ∈ I of the set of characteristic open subgroups of 𝐺, where j ≥ i ⇔ G j ⊆ G i , such that ⋂ i ∈ I G i = { 1 } ; see [14, Proposition 2.5.1 (b)]. Fix such a family. For each i ∈ I , let p i ∈ P ∖ S be such that p i does not divide the order of the finite group G / G i . Then the natural surjection G ↠ G / G i factors through the natural surjection G ↠ G ( p i ) ′ . Thus, Lemma 3.4 follows from Lemma 3.3. ∎

Definition 3.5

Let 𝑘 be a field, k ̄ an algebraic closure of 𝑘, and 𝑋 a smooth curve (i.e., a one-dimensional, smooth, separated, finite-type, geometrically connected scheme) over 𝑘. Write X ̄ k ̄ for the smooth compactification of X k ̄ over k ̄ . Then we shall refer to any element of the underlying set of X ̄ k ̄ ∖ X k ̄ as a cusp of X k ̄ . We shall say that 𝑋 is a smooth curve of type ( g , r ) over 𝑘 if the genus of X k ̄ is 𝑔, and the cardinality of the set of cusps of X k ̄ is 𝑟. If 𝑋 is a smooth curve of type ( g , r ) over 𝑘, and 2 ⁢ g − 2 + r > 0 , then we shall say that 𝑋 is a hyperbolic curve over 𝑘.

Definition 3.6

Let 𝑘 be an algebraically closed field, 𝑍 a hyperbolic curve over 𝑘, 𝑄 a profinite group, and q : Π Z ↠ Q an epimorphism (in the category of profinite groups).

1. Write Z ̄ for the smooth compactification of 𝑍 over 𝑘, 𝐾 for the function field of Z ̄ , K ̃ for the maximal Galois extension of 𝐾 that is unramified over 𝑍 within a fixed separable closure K sep , Z ̄ ̃ for the normalization of Z ̄ in K ̃ , and S ̃ for the inverse image of S ⁢ = def ⁢ Z ̄ ∖ Z in Z ̄ ̃ . Note that we have a natural action of Π Z ≅ Gal ⁢ ( K ̃ / K ) on Z ̄ ̃ . Then, for each point z ∈ S , we shall refer to the stabilizer subgroup ⊆ Π Z of any point in S ̃ lying over 𝑧 as a cuspidal inertia subgroup of Π Z associated to 𝑧. We shall refer to the image under 𝑞 of any such cuspidal inertia subgroup of Π Z as a cuspidal inertia subgroup of 𝑄 associated to 𝑧.

2. In the set-up of (i), we shall write

   Out C ⁢ ( Q ) ⊆ Out ⁢ ( Q ) (respectively,  Out | C | ⁢ ( Q ) ⊆ Out ⁢ ( Q ) )

   for the subgroup of outer automorphisms of 𝑄 that induce permutations (respectively, the identity permutation) on the set of the conjugacy classes of cuspidal inertia subgroups of 𝑄.

Definition 3.7

Let 𝐺 be a profinite group, Π a topologically finitely generated profinite group such that Z ⁢ ( Π ) = { 1 } , and G → Out ⁢ ( Π ) a continuous homomorphism. Then we shall write

Lemma 3.8

Let 𝑘 be an algebraically closed field, 𝑍 a hyperbolic curve over 𝑘, 𝑄 a profinite group, and q : Π Z ↠ Q an epimorphism. Suppose that 𝑄 is topologically finitely generated and slim. Let J ⊆ Out C ⁢ ( Q ) be a closed subgroup and V ⊆ Q an open subgroup. In particular, q − 1 ⁢ ( V ) ⊆ Π Z may be naturally identified with the étale fundamental group of a hyperbolic curve over 𝑘. Write

for the natural surjections. Then, for any sufficiently small open subgroup M ⊆ J , there exist an outer action of 𝑀 on 𝑉 and an open injection V ⁢ ⋊ out ⁢ M ↪ Q ⁢ ⋊ out ⁢ J such that

1. the outer action of 𝑀 preserves and induces permutations on the set of the conjugacy classes of cuspidal inertia subgroups of 𝑉,

2. the outer action of 𝑀 on 𝑉 extends uniquely (by the slimness of 𝑄) to a 𝑉-outer action on 𝑄 that is compatible with the outer action of 𝐽 ( ⊇ M ) on 𝑄,

3. the injection V ⁢ ⋊ out ⁢ M ↪ Q ⁢ ⋊ out ⁢ J is the injection determined by the inclusions V ⊆ Q and M ⊆ J , together with the 𝑉-outer actions on 𝑉 and 𝑄.

Proof

Write n ∈ Z ≥ 1 for the index of 𝑉 in 𝑄. Write

for the subgroup of Aut ⁢ ( Q ) consisting of elements that induce automorphisms of 𝑉, and moreover, induce permutations on the set of the conjugacy classes of cuspidal inertia subgroups of 𝑉. First, we note that, since 𝑉 is an open subgroup of 𝑄, Inn V ⁢ ( Q ) is an open subgroup of Inn ⁢ ( Q ) . In particular, there exists an open subgroup M 1 ⊆ Aut ⁢ ( Q ) such that M 1 ∩ Inn ⁢ ( Q ) = Inn V ⁢ ( Q ) . Next, we consider the set

Here, observe that, since 𝑄 is topologically finitely generated, Q n is a finite set; see [14, Proposition 2.5.1 (a)]. Thus, if we write Sym ⁢ ( Q n ) for the group of permutations on Q n , then the kernel of the natural homomorphism Aut ⁢ ( Q ) → Sym ⁢ ( Q n ) is an open subgroup of Aut ⁢ ( Q ) . Write

In particular, it is easy to see from the relevant definitions that

is an open subgroup of ϕ Q − 1 ⁢ ( J ) satisfying the following conditions:

1. M Aut ∩ Inn ⁢ ( Q ) ⊆ Inn V ⁢ ( Q ) ,

2. M Aut ⊆ Aut [ V ] ⁢ ( Q ) .

Write

Then we have a commutative diagram of profinite groups

where the horizontal arrows in the third line are surjective; see the definitions of M V , 𝑀, M V , Aut . Now we verify the following assertion.

Indeed, we note that, in light of our assumption that 𝑄 is slim, the natural morphism Aut [ V ] ⁢ ( Q ) → Aut ⁢ ( V ) is injective; see Remark 3.2 (ii). Then it follows from the commutativity of the diagram

that Aut [ V ] ⁢ ( Q ) / Inn V ⁢ ( Q ) → Out ⁢ ( V ) is injective, hence that M V , Aut → M V is bijective. On the other hand, it is easy to see from the above condition (i) that M V , Aut → M is injective. This completes the proof of Claim 3.8.A.

In light of Claim 3.8.A, we obtain a natural outer action

Then, since the natural morphism V ⁢ ⋊ out ⁢ M → Aut ⁢ ( V ) factors through Aut [ V ] ⁢ ( Q ) , we have a natural open injection V ⁢ ⋊ out ⁢ M ↪ Q ⁢ ⋊ out ⁢ J . Finally, it is easy to see from the relevant definitions that the morphisms M ↪ Out ⁢ ( V ) and V ⁢ ⋊ out ⁢ M ↪ Q ⁢ ⋊ out ⁢ J satisfy conditions (a), (b), (c). This completes the proof of Lemma 3.8. ∎

Definition 4.1

Let 𝑘 be a field, k ̄ an algebraic closure of 𝑘, and 𝑍 a hyperbolic curve over 𝑘. Then we have an exact sequence of profinite groups

We shall write ρ Z : G k → Out ⁢ ( Π Z k ̄ ) for the outer representation determined by the above exact sequence. Let Σ ⊆ P be a nonempty subset. Then we shall write

for the outer representation induced by ρ Z , and

Let 𝑝 be a prime number. If Σ = { p } , then we shall also write ρ Z p ⁢ = def ⁢ ρ Z Σ .

Lemma 4.2

Let 𝑙 be a prime number and Σ ⊆ P a subset such that l ∈ Σ . Let n ∈ Z ≥ 1 be a Σ-integer, K ⊆ Q ̄ a number field, and Z ⊆ P K 1 ∖ { 0 , 1 , ∞ } an open subscheme obtained by forming the complement of a finite subset of 𝐾-rational points of P K 1 ∖ { 0 , 1 , ∞ } . In particular, 𝑍 is a hyperbolic curve of genus 0 over 𝐾. Write ( P Q ̄ 1 ⊇ ) Y Q ̄ → Z Q ̄ ( ⊆ P Q ̄ 1 ) for the finite étale Galois covering of Z Q ̄ of degree 𝑛 determined by t ↦ t n , and

Then the following hold.

1. We have a natural homomorphism

   Out | C | ⁢ ( Π Z Q ̄ Σ ) → Out | C | ⁢ ( Q ) .

2. Write 𝜌 for the composite of natural homomorphisms

   G K → ρ Z Σ Out | C | ⁢ ( Π Z Q ̄ Σ ) → Out | C | ⁢ ( Q ) ;

   recall our assumption that all cusps of 𝑍 are 𝐾-rational. Then

   Z Out | C | ⁢ ( Q ) ⁢ ( Im ⁢ ( ρ ) ) = { 1 } .

3. Let G ⊆ Out | C | ⁢ ( Q ) be a closed subgroup such that 𝐺 contains an open subgroup of Im ⁢ ( ρ ) . Then 𝐺 is slim.

Proof

We begin by observing that the normal open subgroup

(determined by the finite étale Galois covering Y Q ̄ → Z Q ̄ ) may be characterized as the normal open subgroup topologically generated by the cuspidal inertia subgroups of Π Z Q ̄ Σ that is not associated to the cusps 0, ∞, and the unique closed subgroups of the cuspidal inertia subgroups of Π Z Q ̄ Σ associated to the cusps 0, ∞, of index 𝑛. Then it is easy to see from the relevant definitions that any element in Out | C | ⁢ ( Π Z Q ̄ Σ ) induces an element in Out | C | ⁢ ( Q ) . This completes the proof of assertion (i).

Next, we verify assertion (ii). Let σ ∈ Z Out | C | ⁢ ( Q ) ⁢ ( Im ⁢ ( ρ ) ) . Then it follows from the above observation that any lifting in Aut ⁢ ( Q ) of 𝜎 induces an automorphism of Π Y Q ̄ l . Let σ ̃ ∈ Aut ⁢ ( Q ) be a lifting of 𝜎 such that the automorphism

induced by σ ̃ preserves the Π Y Q ̄ l -conjugacy class of cuspidal inertia subgroups of Π Y Q ̄ l associated to the cusp 1. Here, we note that, since σ ̃ preserves the 𝑄-conjugacy class of cuspidal inertia subgroups of 𝑄 associated to the cusp 0 (respectively, ∞), and the finite étale Galois covering Y Q ̄ → Z Q ̄ is totally ramified over the cusp 0 (respectively, ∞), it follows that σ ̃ | Π Y Q ̄ l preserves the Π Y Q ̄ l -conjugacy class of cuspidal inertia subgroups of Π Y Q ̄ l associated to the cusp 0 (respectively, ∞). Write

for the outer automorphism determined by

Observe that, since the outer action of G K , together with σ Y , on Π Y Q ̄ l preserves the Π Y Q ̄ l -conjugacy class of cuspidal inertia subgroups of Π Y Q ̄ l associated to the cusp 1, it follows from our assumption that

that σ Y commutes with the outer action of G K on Π Y Q ̄ l . Then it follows from the Grothendieck Conjecture [9, Theorem A] that σ Y arises from a unique isomorphism f : Y Q ̄ ⁢ → ∼ ⁢ Y Q ̄ of schemes over Q ̄ . Note that, since σ ̃ | Π Y Q ̄ l induces the identity permutation on the set of the Π Y Q ̄ l -conjugacy classes of cuspidal inertia subgroups of Π Y Q ̄ l associated to the cusps 0, 1, ∞, it follows that 𝑓 induces the identity permutation on the subset { 0 , 1 , ∞ } ⊆ P Q ̄ 1 . In particular, we conclude that 𝑓 is the identity automorphism, hence that σ Y is the identity outer automorphism. Recall that the automorphism

is the restriction of σ ̃ ∈ Aut ⁢ ( Q ) . Thus, since 𝑄 is slim by [12, Proposition 1.4], it follows from Remark 3.2 (ii) that σ ̃ is an inner automorphism, hence that 𝜎 is the identity outer automorphism. This completes the proof of assertion (ii).

Finally, we verify assertion (iii). Since every open subgroup of 𝐺 contains an open subgroup of Im ⁢ ( ρ ) , to verify (iii), it suffices to show that Z ⁢ ( G ) = { 1 } . But this follows from assertion (ii). This completes the proof of assertion (iii), hence of Lemma 4.2. ∎

Definition 4.3

Let 𝑙 be a prime number. We shall say that a profinite group 𝐺 is almost Z l if there exists an open subgroup H ⊆ G such that 𝐻 is isomorphic to Z l .

Lemma 4.4

Let 𝑝 be a prime number, Σ ⊆ P a nonempty subset such that p ∉ Σ , and 𝑘 a finite field of characteristic 𝑝. In the notation of Definition 4.1, suppose that 𝑍 is a hyperbolic curve of genus 0 over 𝑘 such that all cusps of 𝑍 are 𝑘-rational. Write ρ ⁢ = def ⁢ ρ Z Σ . Then the following hold.

1. Suppose that | P ∖ Σ | is finite. Then the natural homomorphism

   Aut ⁢ ( Z k ̄ ) → Out ⁢ ( Π Z k ̄ Σ )

   determines an isomorphism

   Aut ⁢ ( Z k ̄ ) ⁢ → ∼ ⁢ Z Out ⁢ ( Π Z k ̄ Σ ) ⁢ ( ρ ⁢ ( G k ) ) .

2. Suppose that either | Σ | = 1 or | P ∖ Σ | is finite. Then, if we write

   χ Σ : Out | C | ⁢ ( Π Z k ̄ Σ ) → ( Z ̂ Σ ) ×

   for the pro-Σ cyclotomic character, which is obtained by considering the actions on the cuspidal inertia subgroups of Π Z k ̄ Σ , then the natural composite

   Z Out | C | ⁢ ( Π Z k ̄ Σ ) ⁢ ( ρ ⁢ ( G k ) ) ⊆ Out | C | ⁢ ( Π Z k ̄ Σ ) → χ Σ ( Z ̂ Σ ) ×

   is injective.

3. Let 𝑙 be a prime number ≠ p . Then Z Out ⁢ ( Π Z k ̄ l ) ⁢ ( ρ ⁢ ( G k ) ) is almost Z l .

Proof

First, we verify assertion (i). Write Out G k ⁢ ( Π Z [ Σ ] ) for the group of Π Z k ̄ Σ -outer automorphisms of Π Z [ Σ ] that lie over G k ; see Definition 4.1. Then, since Π Z k ̄ Σ is center-free by [12, Proposition 1.4], it is well known that the natural homomorphism

is an isomorphism; see [17, Lemma 7.1]. On the other hand, since G k is abelian, it follows from [15, Theorem D], together with the definition of Out G k ⁢ ( Π Z [ Σ ] ) , that

where Aut ⁢ ( Z k ̄ / Z ) ⊆ Aut ⁢ ( Z k ̄ ) denotes the subgroup consisting of automorphisms of Z k ̄ that induce automorphisms of 𝑍 compatible with the natural morphism Z k ̄ → Z .

Next, we verify the following assertion.

Indeed, let α ∈ Aut ⁢ ( Z k ̄ ) and σ ∈ G k ( ↪ Aut ⁢ ( Z k ̄ ) ) . Then, since G k is abelian, it follows that

Next, we note that 𝛾 induces the identity permutation on the set of cusps of Z k ̄ . Thus, we conclude that γ = 1 , hence that 𝛼 induces a unique automorphism in Aut ⁢ ( Z ) compatible with the natural morphism Z k ̄ → Z . This completes the proof of Claim 4.4.A.

Thus, by applying Claim 4.4.A, we obtain a natural isomorphism

This completes the proof of assertion (i).

Next, we verify assertion (ii). If | Σ | = 1 , then the desired conclusion follows from the latter half of the proof of [13, Proposition 2.2.4]. Thus, we may assume without loss of generality that | P ∖ Σ | is finite. Write Aut | C | ⁢ ( Z k ̄ ) ⊆ Aut ⁢ ( Z k ̄ ) for the subgroup of automorphisms of Z k ̄ that induce the identity permutation on the set of cusps of Z k ̄ . Then 𝜙 induces a composite

Observe that this composite factors as the composite of the natural injection

with the pro-Σ cyclotomic character G F p → ( Z ̂ Σ ) × . Thus, since this character is injective by [2, Théorème 1], we conclude that the natural composite

is injective. This completes the proof of assertion (ii).

Finally, we verify assertion (iii). We begin by observing that, since the image of the 𝑙-adic cyclotomic character G k → Z l × is infinite, it follows from [10, Corollary 2.7 (i)] that

Moreover, we observe that, since

it follows that

Thus, to verify assertion (iii), it suffices to show that

On the other hand, since ρ ⁢ ( G k ) is an infinite abelian group by [8, Lemma 4.2 (iv)], we conclude from assertion (ii) that

as desired. This completes the proof of assertion (iii), hence of Lemma 4.4. ∎

Remark 4.5

It is natural to pose the following question.

However, at the time of writing, the authors do not know whether the answer to this question is affirmative or not.

Lemma 4.6

Let 𝑙 be a prime number and K ⊆ Q ̄ a number field. In the notation of Definition 4.1, suppose that k = K , and 𝑍 is a hyperbolic curve over 𝐾. Then every open subgroup U ⊆ Im ⁢ ( ρ Z l ) is nonabelian.

Proof

Recall that, since the image of the 𝑙-adic cyclotomic character G K → Z l × is infinite, it follows that Im ⁢ ( ρ Z l ) is infinite [8, Lemma 4.2 (iv)]. In particular, 𝑈 is infinite. Write K ′ ⊆ Q ̄ for the finite extension of 𝐾 determined by 𝑈. Suppose that 𝑈 is abelian. Then, since

the centralizer

is infinite. However, since Aut K ′ ⁢ ( Z K ′ ) is finite, this contradicts the Grothendieck Conjecture for hyperbolic curves over number fields [9, Theorem A]. Thus, we conclude that 𝑈 is nonabelian. This completes the proof of Lemma 4.6. ∎

Lemma 5.1

Let Σ ⊆ P be a nonempty subset and K ⊆ Q ̄ a number field. In the notation of Definition 4.1, suppose that k = K , and 𝑍 is a hyperbolic curve of genus 0 over 𝐾. Let

be a closed subgroup such that

• 𝐺 contains an open subgroup of ρ Z Σ ⁢ ( G K ) ,

• there exists a prime number l ∈ Σ such that the image of 𝐺 via the natural morphism

  ϕ l : Out ⁢ ( Π Z Q ̄ Σ ) ↠ Out ⁢ ( Π Z Q ̄ l ) ,

  which is necessarily surjective by [14, Proposition 4.5.4 (b)], is slim,

• there exist normal closed subgroups G 1 ⊆ G , G 2 ⊆ G such that G = G 1 × G 2 .

Then either

1. ϕ l ⁢ ( G 1 ) = { 1 } and G 1 ⊆ Out | C | ⁢ ( Π Z Q ̄ Σ ) , or

2. ϕ l ⁢ ( G 2 ) = { 1 } and G 2 ⊆ Out | C | ⁢ ( Π Z Q ̄ Σ ) .

Proof

First, by replacing 𝐾 by a finite extension of 𝐾, we may assume without loss of generality that ρ Z Σ ⁢ ( G K ) ⊆ G , and all cusps of 𝑍 are 𝐾-rational. Let 𝔭 be a maximal ideal of the ring of integers of 𝐾 such that

• the characteristic of the residue field at 𝔭 is not equal to 𝑙, and

• 𝑍 has good reduction at 𝔭.

Let F ∈ G K be a lifting of the Frobenius element at 𝔭. We shall write,

• for each i = 1 , 2 , pr i : G ↠ G i for the natural projection,

• J ⊆ G K for the closed subgroup topologically generated by 𝐹, where we note that 𝐽 is isomorphic to Z ̂ ,

• I ⁢ = def ⁢ ρ Z Σ ⁢ ( J ) ⊆ G ,

• I 1 ⁢ = def ⁢ pr 1 ⁢ ( I ) × { 1 } ⊆ G 1 × G 2 = G , I 2 ⁢ = def ⁢ { 1 } × pr 2 ⁢ ( I ) ⊆ G 1 × G 2 = G .

Here, we note that, since 𝐼 is abelian, it follows that

hence that

Moreover, we note that, since 𝑍 has good reduction at 𝔭, it follows from the theory of specialization isomorphism, together with Lemma 4.4 (iii), and [8, Lemma 4.2 (iv)] that

• Z Out ⁢ ( Π Z Q ̄ l ) ⁢ ( ϕ l ⁢ ( I ) ) is almost Z l , and

• ϕ l ⁢ ( I ) is infinite.

In particular, either ϕ l ⁢ ( I 1 ) is infinite, or ϕ l ⁢ ( I 2 ) is infinite. We may assume without loss of generality that ϕ l ⁢ ( I 2 ) is infinite. Observe that, since Z Out ⁢ ( Π Z Q ̄ l ) ⁢ ( ϕ l ⁢ ( I ) ) is almost Z l , it follows that ϕ l ⁢ ( I 2 ) ∩ ϕ l ⁢ ( I ) ⊆ ϕ l ⁢ ( I ) is an open subgroup. Then, since G 1 ⊆ Z G ⁢ ( I 2 ) , there exists an open subgroup I † ⊆ I such that

Now suppose that ϕ l ⁢ ( G 1 ) is infinite. Then, since

and moreover, by Lemma 4.4 (iii),

it follows that ϕ l ⁢ ( G 1 ) ∩ ϕ l ⁢ ( I ) ⊆ ϕ l ⁢ ( I ) is an open subgroup. On the other hand, since G 2 ⊆ Z G ⁢ ( G 1 ) , there exists an open subgroup I ‡ ⊆ I † ( ⊆ I ) such that

In summary, we have

Then, since Z Out ⁢ ( Π Z Q ̄ l ) ⁢ ( ϕ l ⁢ ( I ‡ ) ) is almost Z l by Lemma 4.4 (iii), it follows that there exists an open subgroup U ⊆ ρ Z l ⁢ ( G K ) such that 𝑈 is abelian, a contradiction; see Lemma 4.6. Thus, we conclude that ϕ l ⁢ ( G 1 ) is finite. Therefore, since ϕ l ⁢ ( G 1 ) ⊆ ϕ l ⁢ ( G ) is a finite normal subgroup, it follows from our assumption that ϕ l ⁢ ( G ) is slim that ϕ l ⁢ ( G 1 ) = { 1 } ; see Remark 3.2 (i).

Finally, we verify the inclusion G 1 ⊆ Out | C | ⁢ ( Π Z Q ̄ Σ ) . Write

for the 𝑙-adic cyclotomic character, which is obtained by considering the actions on the cuspidal inertia subgroups of Π Z Q ̄ Σ . Then, since χ l ⁢ ( I ) is infinite, it follows from [10, Corollary 2.7 (i)] that

In particular, since ϕ l ⁢ ( I 1 ) ⊆ ϕ l ⁢ ( G 1 ) = { 1 } , we have I 1 ⊆ Out | C | ⁢ ( Π Z Q ̄ Σ ) ; see [10, Proposition 1.2 (i)]. Thus, we obtain an inclusion

where we write

Here, observe that, since χ l factors through the restriction of ϕ l on Out | C | ⁢ ( Π Z Q ̄ Σ ) , it follows that

hence that χ l ⁢ ( I 2 | C | ) is infinite. Therefore, we conclude from [10, Corollary 2.7 (i)] that

In particular, since ϕ l ⁢ ( G 1 ) = { 1 } , we have G 1 ⊆ Out | C | ⁢ ( Π Z Q ̄ Σ ) ; see [10, Proposition 1.2 (i)]. This completes the proof of Lemma 5.1. ∎

Remark 5.2

In the notation of Lemma 5.1, suppose that

Then the second assumption on 𝐺 (concerning the slimness of ϕ l ⁢ ( G ) ) follows automatically from the first assumption on 𝐺. Indeed, to verify the slimness of ϕ l ⁢ ( G ) , by replacing 𝐾 by a finite extension of 𝐾, we may assume without loss of generality that 𝑍 is an open subscheme of P K 1 ∖ { 0 , 1 , ∞ } obtained by forming the complement of a finite subset of 𝐾-rational points of P K 1 ∖ { 0 , 1 , ∞ } . Then the slimness of ϕ l ⁢ ( G ) follows from Lemma 4.2 (iii).