Galois-theoretic features for 1-smooth pro-$p$ groups

Let $p$ be a prime. A pro-$p$ group $G$ is said to be 1-smooth if it can be endowed with a continuous representation $\theta\colon G\to\mathrm{GL}_1(\mathbb{Z}_p)$ such that every open subgroup $H$ of $G$, together with the restriction $\theta\vert_H$, satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-$p$ group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-$p$ Galois groups of fields, and that if a 1-smooth pro-$p$ group is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-$p$ Galois groups of fields. Finally we ask whether 1-smooth pro-$p$ groups satisfy a"Tits' alternative".


Introduction
Throughout the paper p will denote a prime number, and K a field containing a root of unity of order p. Let K(p) denote the compositum of all finite Galois p-extensions of K. The maximal pro-p Galois group of K, denoted by G K (p), is the Galois group Gal(K(p)/K), and it coincides with the maximal pro-p quotient of the absolute Galois group of K. Characterising maximal pro-p Galois groups of fields among pro-p groups is one of the most important -and challenging -problems in Galois theory. One of the obstructions for the realization of a pro-p group as maximal pro-p Galois group for some field K is given by the Artin-Scherier theorem: the only finite group realizable as G K (p) is the cyclic group of order 2 (cf. [1]).
The proof of the celebrated Bloch-Kato conjecture, completed by M. Rost and V. Voevodsky with Ch. Weibel's "patch" (cf. [12,27,29]) provided new tools to study absolute Galois groups of field and their maximal pro-p quotients (see, e.g., [2,3,17,21]). In particular, the now-called Norm Residue Theorem implies that the Z/p-cohomology algebra of a maximal pro-p Galois group G K (p) with Z/p a trivial G K (p)-module and endowed with the cup-product, is a quadratic algebra: i.e., all its elements of positive degree are combinations of products of elements of degree 1, and its defining relations are homogeneous relations of degree 2 (see § 2.3 below). For instance, from this property one may recover the Artin-Schreier obstruction (see, e.g., [17, § 2]).
More recently, a formal version of Hilbert 90 for pro-p groups was employed to find further results on the structure of maximal pro-p Galois groups (see [9,19,21]). A pair G = (G, θ) consisting of a pro-p group G endowed with a continuous representation θ : G → GL 1 (Z p ) is called a pro-p pair. For a pro-p pair G = (G, θ) let Z p (1) denote the continuous left G-module isomorphic to Z p as an abelian pro-p group, with G-action induced by θ (namely, g.v = θ(g) · v for every v ∈ Z p (1)). The pair G is called a Kummerian pro-p pair if the canonical map is surjective for every n ≥ 1. Moreover the pair G is said to be a 1-smooth pro-p pair if every closed subgroup H, endowed with the restriction θ| H , gives rise to a Kummerian pro-p pair (see Definition 2.1). By Kummer theory, the maximal pro-p Galois group G K (p) of a field K, together with the pro-p cyclotomic character θ K : G K (p) → GL 1 (Z p ) (induced by the action of G K (p) on the roots of unity of order a p-power lying in K(p)) gives rise to a 1-smooth pro-p pair G K (see Theorem 2.9).
In [5] -driven by the pursuit of an "explicit" proof of the Bloch-Kato conjecture as an alternative to the proof by Voevodsky -C. De Clerq and M. Florence introduced the 1-smoothness property, and formulated the so-called "Smoothness Conjecture": namely, that it is possible to deduce the surjectivity of the norm residue homomorphism (which is acknowledged to be the "hard part" of the Bloch-Kato conjecture) from the fact that G K (p) together with the pro-p cyclotomic character is a 1-smooth pro-p pair (see [5,Conj. 14.25] and [15, § 3.1.6], and Question 2.12 below).
In view of the Smoothness Conjecture, it is natural to ask which properties of maximal pro-p Galois groups of fields arise also for 1-smooth pro-p pairs. For example, the Artin-Scherier obstruction does: the only finite p-group which may complete into a 1-smooth pro-p pair is the cyclic group C 2 of order 2, together with the non-trivial representation θ : C 2 → {±1} ⊆ GL 1 (Z 2 ) (see Example 2.10 below).
A pro-p pair G = (G, θ) comes endowed with a distinguished closed subgroup: the θ-center Z(G) of G, defined by In [28], R. Ware proved the following result on maximal pro-p Galois groups of fields: if G K (p) is solvable, then it is locally uniformly powerful, i.e., G K (p) ≃ A ⋊ Z p , where A is a free abelian pro-p group, and the right-side factor acts by scalar multiplication by a unit of Z p (see § 3.1). We prove that the same property holds also for 1-smooth pro-p groups.
Theorem 1.2. Let G be a solvable torsion-free pro-p group, endowed with a representation θ : G → GL 1 (Z p ) such that G = (G, θ) is a 1-smooth pro-p pair. Then G is locally uniformly powerful.
This gives a complete description of solvable torsion-free pro-p groups which may be completed into a 1-smooth pro-p pair. Moreover, Theorem 1.2 settles the Smoothness Conjecture positively for the class of solvable pro-p groups.
is a 1-smooth pro-p pair with G solvable, then G is a Bloch-Kato pro-p group, i.e., the Z/p-cohomology algebra of every closed subgroup of G is quadratic. A solvable pro-p group does not contain a free non-abelian closed subgroup. For Bloch-Kato pro-p groups -and thus in particular for maximal pro-p Galois groups of fields containing a root of unity of order p -Ware proved the following Tits' alternative: either such a pro-p group contains a free non-abelian closed subgroup; or it is locally uniformly powerful (see [28,Cor. 1] and [17,Thm. B]). We conjecture that the same phenomenon occurs for 1-smooth pro-p groups. Conjecture 1.5. Let G be a torsion-free pro-p group which may be endowed with a representation θ : G → GL 1 (Z p ) such that G = (G, θ) is a 1-smooth pro-p pair. Then either G is locally uniformly powerful, or G contains a closed non-abelian free pro-p group.

Cyclotomic pro-p pairs
Henceforth, every subgroup of a pro-p group will be tacitly assumed to be closed, and the generatos of a subgroup will be intended in the topological sense.
In particular, for a pro-p group G and a positive integer n, G p n will denote the closed subgroup of G generated by the p n -th powers of all elements of G. Moreover, for two elements g, h ∈ G, we set . In particular, G ′ will denote the commutator subgroup [G, G] of G, and the Frattini subgroup G p · G ′ of G is denoted by Φ(G). Finally, d(G) will denote the minimal number of generatord of G, i.e., d(G) = dim(G/Φ(G)) as a Z/p-vector space.
2.1. Kummerian pro-p pairs. Let 1 + pZ p = {1 + pλ | λ ∈ Z p } ⊆ GL 1 (Z p ) denote the pro-p Sylow subgroup of the group of units of the ring of p-adic integers Z p . A pair G = (G, θ) consisting of a pro-p group G and a continuous homomorphism is called a cyclotomic pro-p pair, and the morphism θ is called an orientation of G (cf. [7, § 3] and [21]). A cyclotomic pro-p pair G = (G, θ) is said to be torsion-free if Im(θ) is torsion-free: this is the case if p is odd; or if p = 2 and Im(θ) ⊆ 1 + 4Z 2 . Observe that a cyclotomic pro-p pair G = (G, θ) may be torsion-free even if G has non-trivial torsion -e.g., if G is the cyclic group of order p and θ is constantly equal to 1. Given a cyclotomic pro-p pair G = (G, θ) one has the following constructions: for all a ∈ A, g ∈ G, and π the canonical projection A ⋊ G → G.
Given a cyclotomic pro-p pair G = (G, θ), the pro-p group G has two distinguished subgroups: (a) the subgroup Both Z(G) and K(G) are normal subgroups of G, and they are contained in Ker(θ). Moreover, Z(G) is abelian, while Thus, the quotient Ker(θ)/K(G) is abelian, and if G is torsion-free one has an isomorphism of pro-p pairs where the action is induced by θ, in the latter), and both pro-p groups are endowed with the orientation induced by θ (cf. Definition 2.1. Given a cyclotomic pro-p pair G = (G, θ), let Z p (1) denote the continuous G-module of rank 1 induced by θ, i.e., Z p (1) ≃ Z p as abelian pro-p groups, and g.λ = θ(g) · λ for every λ ∈ Z p (1). The pair G is said to be Kummerian if for every n ≥ 1 the map Observe that the action of G on Z p (1)/p is trivial, as Im(θ) ⊆ 1 + pZ p . We say that a pro-p group G may complete into a Kummerian, or 1-smooth, pro-p pair if there exists an orientation θ : G → 1 + pZ p such that the pair (G, θ) is Kummerian, or 1-smooth.
Kummerian pro-p pairs and 1-smooth pro-p pairs were introduced in [9] and in [5, § 14] respectively. In [21], if G = (G, θ) is a 1-smooth pro-p pair, the orientation θ is said to be 1-cyclotomic. Note that in [5, § 14.1], a pro-p pair is defined to be 1-smooth if the maps (2.4) are surjective for every open subgroup of G, yet by a limit argument this implies also that the maps (2.4) are surjective also for every closed subgroup of G (cf. [21,Cor. 3.2]).
cts denotes continuous cochain cohomology as introduced by J. Tate in [26].
One has the following group-theoretic characterization of Kummerian torsion-free pro-p pairs (cf.  Remark 2.4. Let G = (G, θ) be a cyclotomic pro-p pair with θ ≡ 1, i.e., θ is constantly equal to 1. Since K(G) = G ′ in this case, G is Kummerian if and only if the quotient G/G ′ is torsion-free. Hence, by Proposition 2.3, G is 1-smooth if and only if H/H ′ is torsion-free for every subgroup H ⊆ G. Pro-p groups with such property are called absolutely torsion-free, and they were introduced by T. Würfel in [30]. In particular, if G = (G, θ) is a 1-smooth pro-p pair (with θ non-trivial), then Res Ker(θ) (G) = (Ker(θ), 1) is again 1-smooth, and thus Ker(θ) is absolutely torsion-free. Hence, a pro-p group which may complete into a 1-smooth pro-p pair is an absolutely torsion-free-by-cyclic pro-p group.
(a) A cyclotomic pro-p pair (G, θ) with G a free pro-p group is 1-smooth for any orientation θ : (c) For p = 2 let G be the pro-p group with minimal presentation Then the pro-p pair (G, θ) is not Kummerian for any orientation θ : be the Heisenberg pro-p group. The pair (H, 1) is Kummerian, as H/H ′ ≃ Z 2 p , but H is not absolutely torsion-free. In particular, H can not complete into a 1-smooth pro-p pair (cf. [18,Ex. 5.4]). (e) The only 1-smooth pro-p pair (G, θ) with G a finite p-group is the cyclic group of order 2 G ≃ Z/2, endowed with the only non-trivial orientation θ : .
A torsion-free pro-p pair G = (G, θ) is said to be θ-abelian if the following equivalent conditions hold: Z p for some set I and some p-power q (possibly q = p ∞ = 0), and in this case Im(θ) = 1 + qZ p . In particular, a θ-abelian pro-p pair is also 1-smooth, as every open subgroup U of G is again isomorphic to Z I p ⋊ Z p , with action induced by θ| U , and therefore Res U (G) is θ| U -abelian.
Remark 2.7. From [9, Thm. 5.6], one may deduce also the following group-theoretic characterization of Kummerian pro-p pairs: a pro-p group G may complete into a Kummerian oriented pro-p group if, and only if, there exists an epimorphism of pro-p groups ϕ : G ։Ḡ such thatḠ has a minimal presentation (2.5), and Ker(ϕ) is contained in the Frattini subgroup of G (cf., e.g., [22,Prop. 3.11]). Remark 2.8. If G ≃ Z p , then the pair (G, θ) is θ-abelian, and thus also 1-smooth, for any orientation θ : G → 1 + pZ p .
On the other hand, if G = (G, θ) is a θ-abelian pro-p pair with d(G) ≥ 2, then θ is the only orientation which may complete G into a 1-smooth pro-p pair. Indeed, let G ′ = (G, θ ′ ) be a cyclotomic pro-p pair, with θ ′ : G → 1 + pZ p different to θ, and let {x 0 , x i , i ∈ I} be a minimal generating set of G as in the presentation (2.5) -thus, θ(x i ) = 1 for all i ∈ I, and θ(x 0 ) ∈ 1 + qZ p . Then for some i ∈ I one has θ ′ | H ≡ θ| H , with H the subgroup of G generated by the two elements x 0 and x i . In particular, one has θ([x 0 , , a contradiction as G is torsion-free by Remark 2.6. If θ ′ (x i ) = 1 then necessarily θ ′ (x 0 ) = θ(x 0 ), and thus 2.2. The Galois case. Let K be a field containing a root of 1 of order p, and let µ p ∞ denote the group of roots of 1 of order a p-power contained in the separable closure of K. Then µ p ∞ ⊆ K(p), and the action of the maximal pro-p Galois group G K (p) = Gal(K(p)/K) on µ p ∞ induces a continuous homomorphism θ K : G K (p) −→ 1 + pZ p -called the pro-p cyclotomic character of G K (p) -, as the group of the automorphisms of µ p ∞ which fix the roots of order p is isomorphic to 1 + pZ p (see, e.g., [8, p. 202] and [9, § 4]). In particular, if K contains a root of 1 of order p k for k ≥ 1, then Then by Kummer theory one has the following (see, e.g., [9,Thm. 4.2]). Theorem 2.9. Let K be a field containing a root of 1 of order p. Then G K = (G K (p), θ K ) is 1-smooth.
1-smooth pro-p pairs share the following properties with maximal pro-p Galois groups of fields.
(a) The only finite p-group which occurs as maximal pro-p Galois group for some field K is the cyclic group of order 2, and this follows from the pro-p version of the Artin-Schreier Theorem (cf. [1]). Likewise, the only finite p-group which may complete into a 1-smooth pro-p pair, is the cyclic group of order 2 (endowed with the only non-trivial orientation onto {±1}), as it follows from Example 2.5-(e) and Remark 2.6. (b) If x is an element of G K (2) for some field K and x has order 2, then x selfcentralizes (cf. [4,Prop. 2.3]). Likewise, if x is an element of a pro-2 group G which may complete into a 1-smooth pro-2 pair, then x self-centralizes (cf. [21, § 6.1]).

Bloch-Kato and the Smoothness Conjecture.
A non-negatively graded algebra A • = n≥0 A n over a field F, with A 0 = F, is called a quadratic algebra if it is 1-generated -i.e., every element is a combination of products of elements of degree 1 -, and its relations are generated by homogeneous relations of degree 2. One has the following definitions (cf. endowed with the cup-product, is a quadratic algebra over Z/p, then G is called a Bloch-Kato pro-p group. As the name suggests, a Bloch-Kato pro-p group is also weakly Bloch-Kato. By the Norm Residue Theorem, if K contains a root of unity of order p, then the maximal pro-p Galois group G K (p) is Bloch-Kato. The pro-p version of the "Smoothness Conjecture", formulated by De Clerq and Florence, states that being 1-smooth is a sufficient condition for a pro-p group to be weakly Bloch-Kato (cf. [5, Conj. 14.25]). Conjecture 2.12. Let G = (G, θ) be a 1-smooth pro-p pair. Then G is weakly Bloch-Kato.
In the case of G = G K for some field K containing a root of 1 of order p, using Milnor K-theory one may show that the weak Bloch-Kato condition implies that H • (G, Z/p)  [25] it is shown that if G = (G, θ) is a 1-smooth pro-p pair with G the pro-p completion of a rightangled Artin group induced by a simplicial graph Γ, then necessarily θ is trivial and Γ has the diagonal property -namely, G may be constructed starting from free pro-p groups by iterating the following two operations: free pro-p products, and direct products with Z p -, and thus G is Bloch-Kato (cf. [25, Thm. 1.2]).
Definition 3.1. A finitely generated pro-p group G is said to be powerful if one has G ′ ⊆ G p , and also G ′ ⊆ G 4 if p = 2. A powerful pro-p group which is also torsion-free and finitely generated is called a uniformly powerful pro-p group.
For the properties of powerful and uniformly powerful pro-p groups we refer to [6,Ch. 4].
A pro-p group whose finitely generated subgroups are uniformly powerful, is said to be locally uniformly powerful. As mentioned in the Introduction, a pro-p group G is locally uniformly powerful if, and only if, G has a minimal presentation (2.5) -i.e., G is locally powerful if, and only if, there exists an orientation θ : G → 1 + pZ p such that (G, θ) is a torsion-free θ-abelian pro-p pair (cf. [17,Thm. A] and [3,Prop. 3.5]).
Therefore, a locally uniformly powerful pro-p group G comes endowed authomatically with an orientation θ : G → 1 + pZ p such that G = (G, θ) is a 1-smooth pro-p pair. In fact, finitely generated locally uniformly powerful pro-p groups are precisely those uniformly powerful pro-p groups which may complete into a 1-smooth pro-p pair (cf. [18,Prop. 4

.3]).
Proposition 3.2. Let G = (G, θ) be a 1-smooth torsion-free pro-p pair. If G is locally powerful, then G is θ-abelian, and thus G is locally uniformly powerful.
It is well-known that the Z/p-cohomology algebra of a pro-p group G with minimal presentation (2.5) is the exterior Z/p-algebra -if p = 2 then n≥0 V is defined to be the quotient of the tensor algebra over Z/p generated by V by the two-sided ideal generated by the elements v ⊗ v, v ∈ V -, so that H • (G, Z/p) is quadratic. Moreover, every subgroup H ⊆ G is again locally uniformly powerful, and thus also H • (H, Z/p) is quadratic. Hence, a locally uniformly powerful pro-p group is Bloch-Kato.

3.2.
Normal abelian subgroups of maximal pro-p Galois groups. Let K be a field containing a root of 1 of order p (and also √ −1 if p = 2). In Galois theory one has the following result, due to A. Engler, J. Koenigsmann and J. Nogueira (cf. [11] and [10]). Theorem 3.3. Let K be a field containing a root of 1 of order p (and also √ −1 if p = 2), and suppose that the maximal pro-p Galois group G K (p) of K is not isomorphic to Z p . Then G K (p) contains a unique maximal abelian normal subgroup.
By [21,Thm. 7.7], such a maximal abelian normal subgroup coincides with the θ Kcenter Z(G K ) of the pro-p pair G K = (G K (p), θ K ) induced by the pro-p cyclotomic character θ K (cf. § 2.2). Moreover, the field K admits a p-Henselian valuation with residue characteristic not p and non-p-divisible value group, such that the residue field κ of such a valuation gives rise to the cyclotomic pro-p pair G κ isomorphic to G K /Z(G K ), and the induced short exact sequence of pro-p groups 3 and the splitting of (3.1) generalize to 1-smooth pro-p pairs whose underlying pro-p group is Bloch-Kato. Namely, if G = (G, θ) is a 1-smooth pro-p pair with G a Bloch-Kato pro-p group, then Z(G) is the unique maximal abelian normal subgroup of G, and it has a complement in G.

3.3.
Proof of Theorem 1.1. In order to prove Theorem 1.1 (and also Theorem 1.2 later on), we need the following result. Proof. Let H be the subgroup of G generated by y and [x, y]. Recall that by Remark 2.6, G (and hence also H) is torsion-free. If d(H) = 1 then H ≃ Z p , as H is torsion-free. Moreover, H is generated by y and x −1 yx, and thus xHx −1 ⊆ H. Therefore, x acts on H ≃ Z p by multiplication by 1 + pλ for some λ ∈ Z p . If λ = 0 then G is abelian, and thus G ≃ Z 2 p as it is absolutely torsionfree, and θ ≡ 1 by Remark 2.8. If λ = 0 then x acts non-trivially on the elements of H, and thus x ∩ H = {1} and G = H ⋊ x : by (2.5), (G, θ ′ ) is a θ ′ -abelian pro-p pair, with θ ′ : G → 1 + pZ p defined by θ ′ (x) = 1 + pλ and θ ′ (y) = 1. By Remark 2.8, one has θ ′ ≡ θ, and thus θ(x) = 1 + pλ and θ(y) = 1.
If d(H) = 2, then H is abelian by hypothesis, and torsion-free, and thus (H, θ ′ ) is θ ′ -abelian, with θ ′ ≡ 1 : H → 1 + pZ p trivial. By Remark 2.8, one has θ ′ = θ| H , and thus y, [x, y] ∈ Ker(θ). Now put z = [x, y] and t = y p , and let U be the open subgroup of G generated by x, z, t. Clearly, Res U (G) is again 1-smooth. By hypothesis one has z y = z, and hence commutator calculus yields ). Since t and z commute, from (3.2) one deduces Moreover, zt −λ/p ∈ Ker(θ| U ). Since Res U (G) is 1-smooth, by Proposition 2.3 the quotient Ker(θ| U )/K(Res U (G)) is a free abelian pro-p group, and therefore (3.3) implies that also zt −λ/p is an element of K(Res U (G)).
(i) The θ-center Z(G) is the unique maximal abelian normal subgroup of G.
(ii) The quotient G/Z(G) is a torsion-free pro-p group.
Proof. Recall that G is torsion-free by Remark 2.6. Since Z(G) is an abelian normal subgroup of G by definition, in order to prove (i) we need to show that if A is an abelian normal subgroup of G, then A ⊆ Z(G). First, we show that A ⊆ Ker(θ). If A ≃ Z p , let y be a generator of A. For every x ∈ G one has xyx −1 ∈ A, and thus xyx −1 = y λ , for some λ ∈ 1 + pZ p . Let H be the subgroup of G generated by x and y, for some x ∈ G such that d(H) = 2. Then the pair (H, θ ′ ) is θ ′ -abelian for some orientation θ ′ : H → 1 + pZ p such that y ∈ Ker(θ ′ ), as H has a presentation as in (2.5). Since both Res H (G) and (H, θ ′ ) are 1-smooth pro-p pairs, by Remark 2.8 one has θ ′ = θ| H , and thus A ⊆ Ker(θ).
If A ≃ Z p , then A is a free abelian pro-p group with d(A) ≥ 2, as G is torsionfree. Therefore, by Remark 2.4 the pro-p pair (A, 1) is 1-smooth. Since also Res A (G) is 1-smooth, Remark 2.8 implies that θ| A = 1, and hence A ⊆ Ker(θ). Now, for arbitrary elements x ∈ G and y ∈ A, put z = [x, y]. Since A is normal in G, one has z ∈ A, and since A is abelian, one has [z, y] = 1. Then Proposition 3.5 applied to the subgroup of G generated by {x, y} yields xyx −1 = x θ(x) , and this completes the proof of statement (i).
In order to prove statement (ii), suppose that y p ∈ Z(G) for some y ∈ G. Then y p ∈ Ker(θ), and since Im(θ) has no non-trivial torsion, also y lies in Ker(θ). Since G is torsion-free by Remark 2.6, y p = 1. Let H be the subgroup of G generated by y and x, for some x ∈ G such that d(H) ≥ 2. Since xy p x −1 = (y p ) θ(x) , commutator calculus yields (3.4) y Put z = [x, y], and let S be the subgroup of H generated by y, z. Clearly, Res S (G) is 1-smooth, and since y, z ∈ Ker(θ), one has θ| S = 1, and thus S/S ′ is a free abelian pro-p group by Remark 2.4. From (3.4) one deduces Since S/S ′ is torsion-free, (3.5) implies that z ≡ y 1−θ(x) −1 mod Φ(S), so that S is generated by y, and S ≃ Z p , as G is torsion-free. Therefore, S ′ = {1}, and (3.5) yields [x, y] = y 1−θ(x) −1 , and this completes the proof of statement (ii).
Remark 3.7. Let G be a pro-p group isomorphic to Z p , and let θ : G → 1 + pZ p be a non-trivial orientation. Then by Example 2.5-(a), G = (G, θ) is 1-smooth. Since G is abelian and θ(x) = 1 for every x ∈ G, x = 1, Z(G) = {1}, still every subgroup of G is normal and abelian.
In view of the splitting of (3.1) (and in view of Remark 3.4), it seems natural to ask the following question. If G = (G, θ) is a torsion-free pro-p pair, then either Ker(θ) = G, or Im(θ) ≃ Z p , hence in the former case one has G ≃ Ker(θ) ⋊ (G/ Ker(θ)), as the right-side factor is isomorphic to Z p , and thus p-projective (cf. [16, Ch. III, § 5]). Since Z(G) ⊆ Z(Ker(θ)) (and Z(G) = Z(G) if Ker(θ) = G), and since Ker(θ) is absolutely torsion-free if G is 1smooth, Question 3.8 is equivalent to the following question (of its own group-theoretic interest): if G is an absolutely torsion-free pro-p group, does G split as direct product One has the following partial answer (cf. [30,Prop. 5]): if G is absolutely torsion-free, and Z(G) is finitely generated, then Φ n (G) = Z(Φ n (G)) × H, for some n ≥ 1 and some subgroup H ⊆ Φ n (G) (here Φ n (G) denotes the iterated Frattini series of G, i.e., Φ 1 (G) = G and Φ n+1 (G) = Φ(Φ n (G)) for n ≥ 1).

Solvable pro-p groups
4.1. Solvable pro-p groups and maximal pro-p Galois groups. Recall that a (pro-p) group G is said to be meta-abelian if there is a short exact sequence such that both N andḠ are abelian; or, equivalently, if the commutator subgroup G ′ is abelian. Moreover, a pro-p group G is solvable if the derived series (G (n) ) n≥1 of G -i.e., G (1) = G and G (n+1) = [G (n) , G (n) ] -is finite, namely G (N +1) = {1} for some finite N .
Example 4.1. A non-abelian locally uniformly powerful pro-p group G is meta-abelian: if θ : G → 1+pZ p is the associated orientation, then G ′ ⊆ Ker(θ) p , and thus G ′ is abelian.
In Galois theory one has the following result by R. Ware (cf. [28,Thm. 3], see also [13] and [17,Thm. 4.6]). Theorem 4.2. Let K be a field containing a root of 1 of order p (and also √ −1 if p = 2). If the maximal pro-p Galois group G K (p) is solvable, then G K is θ K -abelian.
Even if there are several p-adic analytic pro-p groups which are solvable (e.g., finitely generated locally uniformly powerful pro-p groups), none of these two classes of pro-p groups contains the other one: e.g., (a) the wreath product Z p ≀ Z p ≃ Z Zp p ⋊ Z p is a meta-abelian pro-p group, but it is not p-adic analytic (cf. [23]); (b) if G is a pro-p-Sylow subgroup of SL 2 (Z p ), then G is a p-adic analytic pro-p group, but it is not solvable.
In addition, it is well-known that also for the class of pro-p completions of right-angled Artin pro-p groups one has a Tits' alternative: the pro-p completion of a right-angled Artin pro-p group contains a free non-abelian subgroup unless it is a free abelian pro-p group (i.e., unless the associated graph is complete) -and thus it is locally uniformly powerful.
In [18], it is shown that analytic pro-p groups which may complete into a 1-smooth pro-p pair are locally uniformly powerful. Therefore, after the results in [18] and [25], and Theorem 1.2, it is natural to ask whether a Tits' alternative, analogous to Theorem 4.6 (and its generalization to Bloch-Kato pro-p groups), holds also for all torsion-free 1smooth pro-p pairs. Question 4.7. Let G = (G, θ) be a torsion-free 1-smooth pro-p pair, and suppose that G is not θ-abelian. Does G contain a closed non-abelian free pro-p group?
In other words, we are asking whether there exists torsion-free 1-smooth pro-p pairs G = (G, θ) such that G is not analytic nor solvable, and yet it contains no free nonabelian subgroups. In view of Theorem 4.6 and of the Tits' alternative for Bloch-Kato pro-p groups [17,Thm. B], a positive answer to Question 4.7 would corroborate the Smoothness Conjecture.
Observe that -analogously to Quesion 3.8 -Question 4.7 is equivalent to asking whether an absolutely torsion-free pro-p group which is not abelian contains a closed non-abelian free subgroup. Indeed, by Proposition 3.5 (in fact, just by [18,Prop. 5.6]), if G = (G, θ) is a torsion-free 1-smooth pro-p pair and Ker(θ) is abelian, then G is θ-abelian.

Department of Mathematics and Applications, University of Milano Bicocca, 20125 Milan, Italy EU
Email address: claudio.quadrelli@unimib.it