2.1 Kummerian pro-p pairs

Let $1+p\mathbb {Z}_p=\{1+p\lambda \mid \lambda \in \mathbb {Z}_p\}\subseteq \mathrm {GL}_1(\mathbb {Z}_p)$ denote the pro-p Sylow subgroup of the group of units of the ring of p-adic integers $\mathbb {Z}_p$ . A pair $\mathcal {G}=(G,\theta )$ consisting of a pro-p group G and a continuous homomorphism

$$ \begin{align*}\theta\colon G\longrightarrow1+p\mathbb{Z}_p\end{align*} $$

is called a cyclotomic pro- p pair, and the morphism $\theta $ is called an orientation of G (cf. [Reference Efrat7, Section 3] and [Reference Quadrelli and Weigel21]).

A cyclotomic pro-p pair $\mathcal {G}=(G,\theta )$ is said to be torsion-free if $\operatorname {\mathrm {Im}}(\theta )$ is torsion-free: this is the case if p is odd; or if $p=2$ and $\operatorname {\mathrm {Im}}(\theta )\subseteq 1+4\mathbb {Z}_2$ . Observe that a cyclotomic pro-p pair $\mathcal {G}=(G,\theta )$ may be torsion-free even if G has nontrivial torsion—e.g., if G is the cyclic group of order p and $\theta $ is constantly equal to 1. Given a cyclotomic pro-p pair $\mathcal {G}=(G,\theta )$ one has the following constructions:

1. (a) if H is a subgroup of G, $\operatorname {\mathrm {Res}}_H(\mathcal {G})=(H,\theta \vert _H)$ ;

2. (b) if N is a normal subgroup of G contained in $\operatorname {\mathrm {Ker}}(\theta )$ , then $\theta $ induces an orientation $\bar \theta \colon G/N\to 1+p\mathbb {Z}_p$ , and we set $\mathcal {G}/N=(G/N,\bar \theta )$ ;

3. (c) if A is an abelian pro-p group, we set $A\rtimes \mathcal {G}=(A\rtimes G,\theta \circ \pi )$ , with $a^g=a^{\theta (g)^{-1}}$ for all $a\in A$ , $g\in G$ , and $\pi $ the canonical projection $A\rtimes G\to G$ .

Given a cyclotomic pro-p pair $\mathcal {G}=(G,\theta )$ , the pro-p group G has two distinguished subgroups:

1. (a) the subgroup

   (2.1) $$ \begin{align} K(\mathcal{G})=\left\langle\left. h^{-\theta(g)}\cdot h^{g^{-1}}\right|g\in G,h\in\operatorname{\mathrm{Ker}}(\theta)\right\rangle \end{align} $$
   introduced in [Reference Efrat and Quadrelli9, Section 3];

2. (b) the $\theta $ -center

   (2.2) $$ \begin{align} Z(\mathcal{G})=\left\langle h\in\operatorname{\mathrm{Ker}}(\theta)\left|ghg^{-1}=h^{\theta(g)}\;\forall\:g\in G\right.\right\rangle \end{align} $$
   introduced in [Reference Quadrelli17, Section 1].

Both $Z(\mathcal {G})$ and $K(\mathcal {G})$ are normal subgroups of G, and they are contained in $\operatorname {\mathrm {Ker}}(\theta )$ . Moreover, $Z(\mathcal {G})$ is abelian, while

$$ \begin{align*} K(\mathcal{G})\supseteq\operatorname{\mathrm{Ker}}(\theta)',\qquad\text{and} \qquad K(\mathcal{G})\subseteq\Phi(G). \end{align*} $$

Thus, the quotient $\operatorname {\mathrm {Ker}}(\theta )/K(\mathcal {G})$ is abelian, and if $\mathcal {G}$ is torsion-free one has an isomorphism of pro-p pairs

(2.3) $$ \begin{align} \mathcal{G}/K(\mathcal{G})\simeq(\operatorname{\mathrm{Ker}}(\theta)/K(\mathcal{G}))\rtimes (\mathcal{G}/\operatorname{\mathrm{Ker}}(\theta)), \end{align} $$

namely, $G/K(\mathcal {G})\simeq (\operatorname {\mathrm {Ker}}(\theta )/K(\mathcal {G}))\rtimes (G/\operatorname {\mathrm {Ker}}(\theta ))$ (where the action is induced by $\theta $ , in the latter), and both pro-p groups are endowed with the orientation induced by $\theta $ (cf. [Reference Quadrelli18, Equation 2.6]).

Definition 2.1 Given a cyclotomic pro-p pair $\mathcal {G}=(G,\theta )$ , let $\mathbb {Z}_p(1)$ denote the continuous G-module of rank 1 induced by $\theta $ , i.e., $\mathbb {Z}_p(1)\simeq \mathbb {Z}_p$ as abelian pro-p groups, and $g.\lambda =\theta (g)\cdot \lambda $ for every $\lambda \in \mathbb {Z}_p(1)$ . The pair $\mathcal {G}$ is said to be Kummerian if for every $n\geq 1$ the map

(2.4) $$ \begin{align} H^1(G,\mathbb{Z}_p(1)/p^n)\longrightarrow H^1(G,\mathbb{Z}_p(1)/p), \end{align} $$

induced by the epimorphism of G-modules $\mathbb {Z}_p(1)/p^n\to \mathbb {Z}_p(1)/p$ , is surjective. Moreover, $\mathcal {G}$ is 1-smooth if $\operatorname {\mathrm {Res}}_H(\mathcal {G})$ is Kummerian for every subgroup $H\subseteq G$ .

Observe that the action of G on $\mathbb {Z}_p(1)/p$ is trivial, as $\operatorname {\mathrm {Im}}(\theta )\subseteq 1+p\mathbb {Z}_p$ . We say that a pro-p group G may complete into a Kummerian, or 1-smooth, pro-p pair if there exists an orientation $\theta \colon G\to 1+p\mathbb {Z}_p$ such that the pair $(G,\theta )$ is Kummerian, or 1-smooth.

Kummerian pro-p pairs and 1-smooth pro-p pairs were introduced in [Reference Efrat and Quadrelli9] and in [Reference De Clercq and Florence5, Section 14] respectively. In [Reference Quadrelli and Weigel21], if $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair, the orientation $\theta $ is said to be 1-cyclotomic. Note that in [Reference De Clercq and Florence5, Section 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. [Reference Quadrelli and Weigel21, Corollary 3.2]).

Remark 2.1 Let $\mathcal {G}=(G,\theta )$ be a cyclotomic pro-p pair. Then $\mathcal {G}$ is Kummerian if, and only if, the map

$$ \begin{align*} H^1_{\mathrm{cts}}(G,\mathbb{Z}_p(1))\longrightarrow H^1(G,\mathbb{Z}_p(1)/p), \end{align*} $$

induced by the epimorphism of continuous left G-modules $\mathbb {Z}_p(1)\twoheadrightarrow \mathbb {Z}_p(1)/p$ , is surjective (cf. [Reference Quadrelli and Weigel21, Proposition 2.1])—here $H_{\mathrm {cts}}^*$ denotes continuous cochain cohomology as introduced by Tate in [Reference Tate26].

One has the following group-theoretic characterization of Kummerian torsion-free pro-p pairs (cf. [Reference Efrat and Quadrelli9, Theorems 5.6 and 7.1] and [Reference Quadrelli20, Theorem 1.2]).

A torsion-free pro-p pair $\mathcal {G}=(G,\theta )$ is said to be $\theta $ -abelian if the following equivalent conditions hold:

1. (i) $\operatorname {\mathrm {Ker}}(\theta )$ is a free abelian pro-p group, and $\mathcal {G}\simeq \operatorname {\mathrm {Ker}}(\theta )\rtimes (\mathcal {G}/\operatorname {\mathrm {Ker}}(\theta ))$ ;

2. (ii) $Z(\mathcal {G})$ is a free abelian pro-p group, and $Z(\mathcal {G})=\operatorname {\mathrm {Ker}}(\theta )$ ;

3. (iii) $\mathcal {G}$ is Kummerian and $K(\mathcal {G})=\{1\}$

(cf. [Reference Quadrelli17, Proposition 3.4] and [Reference Quadrelli20, Section 2.3]). Explicitly, a torsion-free pro-p pair $\mathcal {G}=(G,\theta )$ is $\theta $ -abelian if and only if G has a minimal presentation

(2.5) $$ \begin{align} G=\left\langle x_0,x_i,i\in I\mid [x_0,x_i]=x_i^{q},[x_i,x_j]=1\:\forall\:i,j\in I\right\rangle\simeq\mathbb{Z}_p^I\rtimes \mathbb{Z}_p \end{align} $$

for some set I and some p-power q (possibly $q=p^\infty =0$ ), and in this case $\operatorname {\mathrm {Im}}(\theta )=1+q\mathbb {Z}_p$ . In particular, a $\theta $ -abelian pro-p pair is also 1-smooth, as every open subgroup U of G is again isomorphic to $\mathbb {Z}_p^I\rtimes \mathbb {Z}_p$ , with action induced by $\theta \vert _U$ , and therefore $\operatorname {\mathrm {Res}}_U(\mathcal {G})$ is $\theta \vert _U$ -abelian.

Remark 2.7 If $G\simeq \mathbb {Z}_p$ , then the pair $(G,\theta )$ is $\theta $ -abelian, and thus also 1-smooth, for any orientation $\theta \colon G\to 1+p\mathbb {Z}_p$ .

On the other hand, if $\mathcal {G}=(G,\theta )$ is a $\theta $ -abelian pro-p pair with $d(G)\geq 2$ , then $\theta $ is the only orientation which may complete G into a 1-smooth pro-p pair. Indeed, let $\mathcal {G}'=(G,\theta ')$ be a cyclotomic pro-p pair, with $\theta '\colon G\to 1+p\mathbb {Z}_p$ different to $\theta $ , and let $\{x_0,x_i,i\in I\}$ be a minimal generating set of G as in the presentation (2.5)—thus, $\theta (x_i)=1$ for all $i\in I$ , and $\theta (x_0)\in 1+q\mathbb {Z}_p$ . Then for some $i\in I$ one has $\theta '\vert _H\not \equiv \theta \vert _H$ , with H the subgroup of G generated by the two elements $x_0$ and $x_i$ . In particular, one has $\theta ([x_0,x_i])=\theta '([x_0,x_i])=1$ .

Suppose that $\mathcal {G}'$ is 1-smooth. If $\theta '(x_i)\neq 1$ , then

$$ \begin{align*} x_i^q=x_i\cdot x_i^q\cdot x_i^{-1}=(x_i^q)^{\theta'(x_i)}=x_i^{q\theta'(x_i)}, \end{align*} $$

hence $x_i^{q(1-\theta '(x_i))}=1$ , a contradiction as G is torsion-free by Remark 2.5. If $\theta '(x_i)=1$ then necessarily $\theta '(x_0)\neq \theta (x_0)$ , and thus

$$ \begin{align*} x_i^{\theta(x_0)}=x_0\cdot x_i\cdot x_0^{-1}=x_i^{\theta'(x_0)}, \end{align*} $$

hence $x_i^{\theta (x_0)-\theta '(x_0)}=1$ , again a contradiction as G is torsion-free. (See also [Reference Quadrelli and Weigel21, Corollary 3.4].)

2.2 The Galois case

Let $\mathbb {K}$ be a field containing a root of 1 of order p, and let $\mu _{p^\infty }$ denote the group of roots of 1 of order a p-power contained in the separable closure of $\mathbb {K}$ . Then $\mu _{p^\infty }\subseteq \mathbb {K}(p)$ , and the action of the maximal pro-p Galois group $G_{\mathbb {K}}(p)=\operatorname {\mathrm {Gal}}(\mathbb {K}(p)/\mathbb {K})$ on $\mu _{p^\infty }$ induces a continuous homomorphism

$$ \begin{align*}\theta_{\mathbb{K}}\colon G_{\mathbb{K}}(p)\longrightarrow1+p\mathbb{Z}_p\end{align*} $$

—called the pro- p cyclotomic character of $G_{\mathbb {K}}(p)$ —as the group of the automorphisms of $\mu _{p^{\infty }}$ which fix the roots of order p is isomorphic to $1+p\mathbb {Z}_p$ (see, e.g., [Reference Efrat8, p. 202] and [Reference Efrat and Quadrelli9, Section 4]). In particular, if $\mathbb {K}$ contains a root of 1 of order $p^k$ for $k\geq 1$ , then $\operatorname {\mathrm {Im}}(\theta _{\mathbb {K}})\subseteq 1+p^k\mathbb {Z}_p$ .

Set $\mathcal {G}_{\mathbb {K}}=(G_{\mathbb {K}}(p),\theta _{\mathbb {K}})$ . Then by Kummer theory one has the following (see, e.g., [Reference Efrat and Quadrelli9, Theorem 4.2]).

1-smooth pro-p pairs share the following properties with maximal pro-p Galois groups of fields.

2.3 Bloch–Kato and the Smoothness Conjecture

A non-negatively graded algebra $A_\bullet =\bigoplus _{n\geq 0}A_n$ over a field $\mathbb {F}$ , with $A_0=\mathbb {F}$ , is called a quadratic algebra if it is one-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. [Reference De Clercq and Florence5, Definition 14.21] and [Reference Quadrelli17, Section 1]).

Definition 2.2 Let G be a pro-p group, and let $n\geq 1$ . Cohomology classes in the image of the natural cup-product

$$ \begin{align*} H^1(G,\mathbb{Z}/p)\times\ldots\times H^1(G,\mathbb{Z}/p)\overset{\cup}{\longrightarrow} H^n(G,\mathbb{Z}/p) \end{align*} $$

are called symbols (relative to $\mathbb {Z}/p$ , wieved as trivial G-module).

1. (i) If for every open subgroup $U\subseteq G$ every element $\alpha \in H^n(U,\mathbb {Z}/p)$ , for every $n\geq 1$ , can be written as

   $$ \begin{align*}\alpha=\mathrm{cor}_{V_1,U}^n(\alpha_1)+\cdots+\mathrm{cor}^n_{V_r,U}(\alpha_r),\end{align*} $$
   with $r\geq 1$ , where $\alpha _i\in H^n(V_i,\mathbb {Z}/p)$ is a symbol and
   $$ \begin{align*}\mathrm{cor}_{V_i,U}^n\colon H^n(V_i,\mathbb{Z}/p)\longrightarrow H^n(U,\mathbb{Z}/p)\end{align*} $$
   is the corestriction map (cf. [Reference Neukirch, Schmidt and Wingberg16, Chapter I, Section 5]), for some open subgroups $V_i\subseteq U$ , then G is called a weakly Bloch–Kato pro- p group.

2. (ii) If for every closed subgroup $H\subseteq G$ the $\mathbb {Z}/p$ -cohomology algebra

   $$ \begin{align*}H^\bullet(H,\mathbb{Z}/p)=\bigoplus_{n\geq0}H^n(H,\mathbb{Z}/p),\end{align*} $$
   endowed with the cup-product, is a quadratic algebra over $\mathbb {Z}/p$ , 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 $\mathbb {K}$ contains a root of unity of order p, then the maximal pro-p Galois group $G_{\mathbb {K}}(p)$ 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. [Reference De Clercq and Florence5, Conjugation 14.25]).

In the case of $\mathcal {G}=\mathcal {G}_{\mathbb {K}}$ for some field $\mathbb {K}$ containing a root of 1 of order p, using Milnor K-theory one may show that the weak Bloch–Kato condition implies that $H^\bullet (G,\mathbb {Z}/p)$ is one-generated (cf. [Reference De Clercq and Florence5, Rem. 14.26]). In view of Theorem 2.8, a positive answer to the Smoothness Conjecture would provide a new proof of the surjectivity of the norm residue isomorphism, i.e., the “surjectivity” half of the Bloch–Kato conjecture (cf. [Reference De Clercq and Florence5, Section 1.1]).

Conjecture 2.10 has been settled positively for the following classes of pro-p groups.

1. (a) Finite p-groups: indeed, if $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair with G a finite (nontrivial) p-group, then by Example 2.4–(e) $p=2$ , G is a cyclic group of order two and $\theta \colon G\twoheadrightarrow \{\pm 1\}$ , so that $\mathcal {G}\simeq (\operatorname {\mathrm {Gal}}(\mathbb {C}/\mathbb {R}),\theta _{\mathbb {R}})$ , and G is Bloch–Kato.

2. (b) Analytic pro-p groups: indeed if $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair with G a p-adic analytic pro-p group, then by [Reference Quadrelli18, Theorem 1.1] G is locally uniformly powerful and thus Bloch–Kato (see § 3.1 below).

3. (c) Pro-p completions of right-angled Artin groups: indeed, in [Reference Snopce and Zalesskii25], it is shown that if $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair with G the pro-p completion of a right-angled Artin group induced by a simplicial graph $\Gamma $ , then necessarily $\theta $ is trivial and $\Gamma $ 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 $\mathbb {Z}_p$ —and thus G is Bloch–Kato (cf. [Reference Snopce and Zalesskii25, Theorem 1.2]).

3.1 Powerful pro-p groups

For the properties of powerful and uniformly powerful pro-p groups, we refer to [Reference Dixon, du Sautoy, Mann and Segal6, Chapter 4].

A pro-p group whose finitely generated subgroups are uniformly powerful, is said to be locally uniformly powerful. As mentioned in Section 1, 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 $\theta \colon G\to 1+p\mathbb {Z}_p$ such that $(G,\theta )$ is a torsion-free $\theta $ -abelian pro-p pair (cf. [Reference Quadrelli17, Theorem A] and [Reference Chebolu, Mináč and Quadrelli3, Proposition 3.5]).

Therefore, a locally uniformly powerful pro-p group G comes endowed automatically with an orientation $\theta \colon G\to 1+p\mathbb {Z}_p$ such that $\mathcal {G}=(G,\theta )$ 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. [Reference Quadrelli18, Proposition 4.3]).

It is well-known that the $\mathbb {Z}/p$ -cohomology algebra of a pro-p group G with minimal presentation (2.5) is the exterior $\mathbb {Z}/p$ -algebra

$$ \begin{align*} H^\bullet(H,\mathbb{Z}/p)\simeq \bigwedge_{n\geq0} H^1(H,\mathbb{Z}/p) \end{align*} $$

—if $p=2$ then $\bigwedge _{n\geq 0} V$ is defined to be the quotient of the tensor algebra over $\mathbb {Z}/p$ generated by V by the two-sided ideal generated by the elements $v\otimes v$ , $v\in V$ —so that $H^\bullet (G,\mathbb {Z}/p)$ is quadratic. Moreover, every subgroup $H\subseteq G$ is again locally uniformly powerful, and thus also $H^\bullet (H,\mathbb {Z}/p)$ 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 $\mathbb {K}$ be a field containing a root of 1 of order p (and also $\sqrt {-1}$ if $p=2$ ). In Galois theory, one has the following result, due to Engler et al. (cf. [Reference Engler and Nogueira11] and [Reference Engler and Koenigsmann10]).

By [Reference Quadrelli and Weigel21, Theorem 7.7], such a maximal abelian normal subgroup coincides with the $\theta _{\mathbb {K}}$ -center $Z(\mathcal {G}_{\mathbb {K}})$ of the pro-p pair $\mathcal {G}_{\mathbb {K}}=(G_{\mathbb {K}}(p),\theta _{\mathbb {K}})$ induced by the pro-p cyclotomic character $\theta _{\mathbb {K}}$ (cf. § 2.2). Moreover, the field $\mathbb {K}$ admits a p-Henselian valuation with residue characteristic not p and non-p-divisible value group, such that the residue field $\kappa $ of such a valuation gives rise to the cyclotomic pro-p pair $\mathcal {G}_{\kappa }$ isomorphic to $\mathcal {G}_{\mathbb {K}}/Z(\mathcal {G}_{\mathbb {K}})$ , and the induced short exact sequence of pro-p groups

(3.1)

splits (cf. [Reference Engler and Koenigsmann10, Section 1] and [Reference Efrat8, Example 22.1.6]—for the definitions related to p-henselian valuations of fields, we direct the reader to [Reference Efrat8, Section 15.3]). In particular, $G_{\mathbb {K}}(p)/Z(\mathcal {G}_{\mathbb {K}})$ is torsion-free.

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.

Proposition 3.4 Let $\mathcal {G}=(G,\theta )$ be a torsion-free 1-smooth pro-p pair, with $d(G)=2$ and $G=\langle x,y\rangle $ . If $[[x,y],y]=1$ , then $\operatorname {\mathrm {Ker}}(\theta )=\langle y\rangle $ and

$$ \begin{align*}xyx^{-1}=y^{\theta(x)}.\end{align*} $$

Proof Let H be the subgroup of G generated by y and $[x,y]$ . Recall that by Remark 2.5, G (and hence also H) is torsion-free.

If $d(H)=1$ then $H\simeq \mathbb {Z}_p$ , as H is torsion-free. Moreover, H is generated by y and $x^{-1}yx$ , and thus $xHx^{-1}\subseteq H$ . Therefore, x acts on $H\simeq \mathbb {Z}_p$ by multiplication by $1+p\lambda $ for some $\lambda \in \mathbb {Z}_p$ . If $\lambda =0$ then G is abelian, and thus $G\simeq \mathbb {Z}_p^2$ as it is absolutely torsion-free, and $\theta \equiv \mathbf {1}$ by Remark 2.7. If $\lambda \neq 0$ then x acts nontrivially on the elements of H, and thus $\langle x\rangle \cap H=\{1\}$ and $G=H\rtimes \langle x\rangle $ : by (2.5), $(G,\theta ')$ is a $\theta '$ -abelian pro-p pair, with $\theta '\colon G\to 1+p\mathbb {Z}_p$ defined by $\theta '(x)=1+p\lambda $ and $\theta '(y)=1$ . By Remark 2.7, one has $\theta '\equiv \theta $ , and thus $\theta (x)=1+p\lambda $ and $\theta (y)=1$ .

If $d(H)=2$ , then H is abelian by hypothesis, and torsion-free, and thus $(H,\theta ')$ is $\theta '$ -abelian, with $\theta '\equiv {\mathbf {1}}\colon H\to 1+p\mathbb {Z}_p$ trivial. By Remark 2.7, one has $\theta '=\theta \vert _H$ , and thus $y,[x,y]\in \operatorname {\mathrm {Ker}}(\theta )$ . Now put $z=[x,y]$ and $t=y^p$ , and let U be the open subgroup of G generated by $x,z,t$ . Clearly, $\operatorname {\mathrm {Res}}_U(\mathcal {G})$ is again 1-smooth. By hypothesis one has $z^y=z$ , and hence commutator calculus yields

(3.2) $$ \begin{align} [x,t]=[x,y^p]=z\cdot z^y\cdots z^{y^{p-1}}=z^p. \end{align} $$

Put $\lambda =1-\theta (x)^{-1}\in p\mathbb {Z}_p$ . Since $t\in \operatorname {\mathrm {Ker}}(\theta )$ , by (2.1) $[x,t]\cdot t^{-\lambda }$ lies in $K(\operatorname {\mathrm {Res}}_U(\mathcal {G}))$ . Since t and z commute, from (3.2) one deduces

(3.3) $$ \begin{align} [x,t]t^{-\lambda}=z^pt^{-\lambda}=z^pt^{-\frac{\lambda}{p}p}=\left(zt^{-\lambda/p}\right)^p\in K(\operatorname{\mathrm{Res}}_U(\mathcal{G})). \end{align} $$

Moreover, $zt^{-\lambda /p}\in \operatorname {\mathrm {Ker}}(\theta \vert _U)$ . Since $\operatorname {\mathrm {Res}}_U(\mathcal {G})$ is 1-smooth, by Proposition 2.2, the quotient $\operatorname {\mathrm {Ker}}(\theta \vert _U)/K(\operatorname {\mathrm {Res}}_U(\mathcal {G}))$ is a free abelian pro-p group, and therefore (3.3) implies that also $zt^{-\lambda /p}$ is an element of $K(\operatorname {\mathrm {Res}}_U(\mathcal {G}))$ .

Since $K(\operatorname {\mathrm {Res}}_U(\mathcal {G}))\subseteq \Phi (U)$ , one has $z\equiv t^{\lambda /p}\bmod \Phi (U)$ . Then by [Reference Dixon, du Sautoy, Mann and Segal6, Proposition 1.9] $d(U)=2$ and U is generated by x and t. Since $[x,t]\in U^p$ by (3.2), the pro-p group U is powerful. Therefore, $\operatorname {\mathrm {Res}}_U(\mathcal {G})$ is $\theta \vert _U$ -abelian by Proposition 3.1. In particular, the subgroup $K(\operatorname {\mathrm {Res}}_U(\mathcal {G}))$ is trivial, and thus

$$ \begin{align*}[x,y]=z=t^{\lambda/p}=y^{1-\theta(x)^{-1}},\end{align*} $$

and the claim follows.▪

Proposition 3.4 is a generalization of [Reference Quadrelli18, Proposition 5.6].

Proof Recall that G is torsion-free by Remark 2.5. Since $Z(\mathcal {G})$ 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 $A\subseteq Z(\mathcal {G})$ .

First, we show that $A\subseteq \operatorname {\mathrm {Ker}}(\theta )$ . If $A\simeq \mathbb {Z}_p$ , let y be a generator of A. For every $x\in G$ one has $xyx^{-1}\in A$ , and thus $xyx^{-1}=y^{\lambda }$ , for some $\lambda \in 1+p\mathbb {Z}_p$ . Let H be the subgroup of G generated by x and y, for some $x\in G$ such that $d(H)=2$ . Then the pair $(H,\theta ')$ is $\theta '$ -abelian for some orientation $\theta '\colon H\to 1+p\mathbb {Z}_p$ such that $y\in \operatorname {\mathrm {Ker}}(\theta ')$ , as H has a presentation as in (2.5). Since both $\operatorname {\mathrm {Res}}_H(\mathcal {G})$ and $(H,\theta ')$ are 1-smooth pro-p pairs, by Remark 2.7, one has $\theta '=\theta \vert _H$ , and thus $A\subseteq \operatorname {\mathrm {Ker}}(\theta )$ .

If $A\not \simeq \mathbb {Z}_p$ , then A is a free abelian pro-p group with $d(A)\geq 2$ , as G is torsion-free. Therefore, by Remark 2.3 the pro-p pair $(A,\mathbf {1})$ is 1-smooth. Since also $\operatorname {\mathrm {Res}}_A(\mathcal {G})$ is 1-smooth, Remark 2.7 implies that $\theta \vert _A=\mathbf {1}$ , and hence $A\subseteq \operatorname {\mathrm {Ker}}(\theta )$ .

Now, for arbitrary elements $x\in G$ and $y\in A$ , put $z=[x,y]$ . Since A is normal in G, one has $z\in A$ , and since A is abelian, one has $[z,y]=1$ . Then Proposition 3.4 applied to the subgroup of G generated by $\{x,y\}$ yields $xyx^{-1}=x^{\theta (x)}$ , and this completes the proof of statement (i).

In order to prove statement (ii), suppose that $y^p\in Z(\mathcal {G})$ for some $y\in G$ . Then $y^p\in \operatorname {\mathrm {Ker}}(\theta )$ , and since $\operatorname {\mathrm {Im}}(\theta )$ has no nontrivial torsion, also y lies in $\operatorname {\mathrm {Ker}}(\theta )$ . Since G is torsion-free by Remark 2.5, $y^p\neq 1$ . Let H be the subgroup of G generated by y and x, for some $x\in G$ such that $d(H)\geq 2$ . Since $xy^px^{-1}=(y^p)^{\theta (x)}$ , commutator calculus yields

(3.4) $$ \begin{align} y^{p(1-\theta(x)^{-1})}=[x,y^p]=[x,y]\cdot[x,y]^y\cdots[x,y]^{y^{p-1}}. \end{align} $$

Put $z=[x,y]$ , and let S be the subgroup of H generated by $y,z$ . Clearly, $\operatorname {\mathrm {Res}}_S(\mathcal {G})$ is 1-smooth, and since $y,z\in \operatorname {\mathrm {Ker}}(\theta )$ , one has $\theta \vert _S=\mathbf {1}$ , and thus $S/S'$ is a free abelian pro-p group by Remark 2.3. From (3.4) one deduces

(3.5) $$ \begin{align} y^{p(1-\theta(x)^{-1})}\cdot z^{-p}\equiv \left(y^{1-\theta(x)^{-1}}\cdot z^{-1}\right)^p\equiv 1\ \mod S'. \end{align} $$

Since $S/S'$ is torsion-free, (3.5) implies that $z\equiv y^{1-\theta (x)^{-1}}\bmod \Phi (S)$ , so that S is generated by y, and $S\simeq \mathbb {Z}_p$ , as G is torsion-free. Therefore, $S'=\{1\}$ , and (3.5) yields $[x,y]=y^{1-\theta (x)^{-1}}$ , and this completes the proof of statement (ii).▪

In view of the splitting of (3.1) (and in view of Remark 3.3), it seems natural to ask the following question.

Question 3.7 Let $\mathcal {G}=(G,\theta )$ be a torsion-free 1-smooth pro-p pair, with $d(G)\geq 2$ . Is the pro-p pair $\mathcal {G}/Z(\mathcal {G})=(G/Z(\mathcal {G}),\bar {\theta })$ 1-smooth? Does the short exact sequence of pro-p groups

split?

If $\mathcal {G}=(G,\theta )$ is a torsion-free pro-p pair, then either $\operatorname {\mathrm {Ker}}(\theta )=G$ , or $\operatorname {\mathrm {Im}}(\theta )\simeq \mathbb {Z}_p$ , hence in the former case one has $G\simeq \operatorname {\mathrm {Ker}}(\theta )\rtimes (G/\operatorname {\mathrm {Ker}}(\theta ))$ , as the right-side factor is isomorphic to $\mathbb {Z}_p$ , and thus p-projective (cf. [Reference Neukirch, Schmidt and Wingberg16, Chapter III, Section 5]). Since $Z(\mathcal {G})\subseteq Z(\operatorname {\mathrm {Ker}}(\theta ))$ (and $Z(\mathcal {G})= Z(G)$ if $\operatorname {\mathrm {Ker}}(\theta )=G$ ), and since $\operatorname {\mathrm {Ker}}(\theta )$ is absolutely torsion-free if $\mathcal {G}$ is 1-smooth, Question 3.7 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

$$ \begin{align*}G\simeq Z(G)\times (G/Z(G))\:\text{?}\end{align*} $$

One has the following partial answer (cf. [Reference Würfel30, Proposition 5]): if G is absolutely torsion-free, and $Z(G)$ is finitely generated, then $\Phi _n(G)=Z(\Phi _n(G))\times H$ , for some $n\geq 1 $ and some subgroup $H\subseteq \Phi _n(G)$ (here $\Phi _n(G)$ denotes the iterated Frattini series of G, i.e., $\Phi _1(G)=G$ and $\Phi _{n+1}(G)=\Phi (\Phi _n(G))$ for $n\geq 1$ ).

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 and $\bar G$ are abelian; or, equivalently, if the commutator subgroup $G'$ is abelian. Moreover, a pro-p group G is solvable if the derived series $(G^{(n)})_{n\geq 1}$ 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.

In Galois theory, one has the following result by Ware (cf. [Reference Ware28, Theorem 3], see also [Reference Koenigsmann13] and [Reference Quadrelli17, Theorem 4.6]).

4.2 Proof of Theorem 1.2 and Corollary 1.3

In order to prove Theorem 1.2, we prove first the following intermediate results—a consequence of Würfel’s result [Reference Würfel30, Proposition 2] —, which may be seen as the “1-smooth analogue” of [Reference Ware28, Theorem 2].

Note that Proposition 4.3 generalizes [Reference Würfel30, Proposition 2] from absolutely torsion-free pro-p groups to 1-smooth pro-p groups. From Proposition 4.3, we may deduce Theorem 1.2.

Proposition 4.4 may be seen as the 1-smooth analogue of Ware’s Theorem 4.2. Corollary 1.3 follows from Proposition 4.4 and from the fact that a locally uniformly powerful pro-p group is Bloch–Kato (cf. § 3.1).

This settles the Smoothness Conjecture for the class of solvable pro-p groups.

4.3 A Tits’ alternative for 1-smooth pro-p groups

For maximal pro-p Galois groups of fields one has the following Tits’ alternative (cf. [Reference Ware28, Corollary 1]).

Actually, the above Tits’ alternative holds also for the class of Bloch–Kato pro-p groups, with p odd: if a Bloch–Kato pro-p group G does not contain any free nonabelian subgroups, then it can complete into a $\theta $ -abelian pro-p pair $\mathcal {G}=(G,\theta )$ (cf. [Reference Quadrelli17, Theorem B], this Tits’ alternative holds also for $p=2$ under the further assumption that the Bockstein morphism $\beta \colon H^1(G,\mathbb {Z}/2)\to H^2(G,\mathbb {Z}/2)$ is trivial, see [Reference Quadrelli17, Theorem 4.11]).

Clearly, a solvable pro-p group contains no free nonabelian subgroups.

A pro-p group is p-adic analytic if it is a p-adic analytic manifold and the map $(x,y)\mapsto x^{-1} y$ is analytic, or, equivalently, if it contains an open uniformly powerful subgroup (cf. [Reference Dixon, du Sautoy, Mann and Segal6, Theorem 8.32])—e.g., the Heisenberg pro-p group is analytic. Similarly to solvable pro-p groups, a p-adic analytic pro-p group does not contain a free nonabelian subgroup (cf. [Reference Dixon, du Sautoy, Mann and Segal6, Corollary 8.34]).

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.,

1. (a) the wreath product $\mathbb {Z}_p\wr \mathbb {Z}_p\simeq \mathbb {Z}_p^{\mathbb {Z}_p}\rtimes \mathbb {Z}_p$ is a meta-abelian pro-p group, but it is not p-adic analytic (cf. [Reference Shalev23]) and

2. (b) if G is a pro-p-Sylow subgroup of $\mathrm {SL}_2(\mathbb {Z}_p)$ , 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 nonabelian 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 [Reference Quadrelli18], 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 [Reference Quadrelli18] and [Reference Snopce and Zalesskii25], 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 1-smooth pro-p pairs.

In other words, we are asking whether there exists torsion-free 1-smooth pro-p pairs $\mathcal {G}=(G,\theta )$ 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 [Reference Quadrelli17, Theorem B], a positive answer to Question 4.7 would corroborate the Smoothness Conjecture.

Observe that—analogously to Question 3.7—Question 4.7 is equivalent to asking whether an absolutely torsion-free pro-p group which is not abelian contains a closed nonabelian free subgroup. Indeed, by Proposition 3.4 (in fact, just by [Reference Quadrelli18, Proposition 5.6]), if $\mathcal {G}=(G,\theta )$ is a torsion-free 1-smooth pro-p pair and $\operatorname {\mathrm {Ker}}(\theta )$ is abelian, then $\mathcal {G}$ is $\theta $ -abelian.