CHASING MAXIMAL PRO- p GALOIS GROUPS VIA 1-CYCLOTOMICITY

. Let p be a prime. We prove that certain amalgamated free pro- p products of Demushkin groups with pro- p -cyclic amalgam cannot give rise to a 1-cyclotomic oriented pro- p group, and thus do not occur as maximal pro- p Galois groups of ﬁelds containing a root of 1 of order p . We show that other cohomological obstructions which are used to detect pro- p groups that are not maximal pro- p Galois groups — the quadraticity of Z /p -cohomology and the vanishing of Massey products — fail with the above pro- p groups. Finally, we prove that the Minaˇc-Tˆan pro- p group cannot give rise to a 1-cyclotomic oriented pro- p group, and we conjecture that every 1-cyclotomic oriented pro- p group satisfy the strong n -Massey vanishing property for n = 3 , 4.


Introduction
Let p be a prime number, and let 1 + pZ p denote the pro-p group of principal units of the ring of p-adic integers Z p -namely, 1 + pZ p = {1 + pλ | λ ∈ Z p }.An oriented pro-p group is a pair (G, θ) consisting of a pro-p group G and a morphism of pro-p groups θ : G → 1 + pZ p , called an orientation of G (see [30]; oriented pro-p groups were introduced by I. Efrat in [7], with the name "cyclotomic pro-p pairs").An oriented pro-p group (G, θ) gives rise to the continuous G-module Z p (θ), which is equal to Z p as an abelian pro-p group, and which is endowed with the continuous G-action defined by for all g ∈ G and λ ∈ Z p (θ).
An oriented pro-p group (G, θ) is said to be Kummerian if the following cohomological condition is satisfied: for every n ≥ 1 the natural morphism induced by the epimorphism of continuous G-modules Z p (θ)/p n Z p (θ) ։ Z/p is surjective (see [11]) -here we consider Z/p as a trivial G-module.Moreover, the oriented pro-p group (G, θ) is said to be 1-cyclotomic if the above cohomological condition is satisfied also for every closed subgroup of G -namely, the natural morphism (1.1) is surjective also with H instead of G, and the restriction θ| H : H → 1 + pZ p instead of θ for all closed subgroups H of G (in [26,27] a 1-cyclotomic oriented pro-p group is called a "1-smooth" oriented pro-p group).This cohomological condition was considered first by J. Labute, who showed ante litteram that for every Demushkin group G there exists precisely one orientation which completes G into a Kummerian oriented pro-p group, namely, the orientation induced by the dualizing module of G (see [14]).
In case of trivial orientations, 1-cyclotomicity translates into a purely group-theoretical statement.Namely, an oriented pro-p group (G, 1) -where 1 : G → 1 + pZ p denotes the orientation which is constantly equal to 1 -is 1-cyclotomic if, and only if, the abelianization of every closed subgroup of G is a free abelian pro-p group.Pro-p groups satisfying this group-theoretic condition are called absolutely torsion-free pro-p groups, and they were introduced by T. Würfel in [37].
The main goal of this work is to produce new examples of pro-p groups which no orientations can turn into a 1-cyclotomic oriented pro-p group.
(1.1.b)or p is odd and Then there are no orientations θ : G → 1 + pZ p such that the oriented pro-p group (G, θ) is 1-cyclotomic.
It is worth underlining that the pro-p groups described in Theorem 1.1 are amalgamated free pro-p products of two Demushkin groups -the subgroup generated by x, y 1 , . . ., y d1 and the subgroup generated by x, z 1 , . . ., z d2 -, with pro-p-cyclic amalgam, generated by x.Despite Demushkin groups and their free pro-p products are some of the (extremely few) examples of pro-p groups which are known to give rise to 1-cyclotomic oriented pro-p groups, the presence of a pro-p-cyclic amalgam is enough to lose 1-cyclotomicity.
Oriented pro-p groups satisfying 1-cyclotomicity have great prominence in Galois theory.Given a field K, let Ks and K(p) denote respectively the separable closure of K, and 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 maximal pro-p quotient of the absolute Galois group Gal( Ks /K) of K, and it coincides with the Galois group of the Galois extension K(p)/K.Detecting maximal pro-p Galois groups among pro-p groups, are crucial problems in Galois theory.Already the pursuit of concrete examples of pro-p groups which do not occur as maximal pro-p Galois groups of fields is already considered a very remarkable challenge (see [12, § 25.16], and, e.g., [1,3,4,25,34]).
The maximal pro-p Galois group G K (p) of a field K containing a root of 1 of order p gives rise to the oriented pro-p group (G K (p), θ K ), where θ K : G K (p) −→ 1 + pZ p denotes the pro-p cyclotomic character (see Example 2.4 below).By Kummer theory, the oriented pro-p group (G K (p), θ K ) is 1-cyclotomic (see [14, p. 131] and [11, § 4])in case p = 2 we need to assume further that √ −1 ∈ K. Therefore, a pro-p group which cannot complete into a 1-cyclotomic oriented pro-p group does not occur as the maximal pro-p group of a field containing a root of 1 of order p -and hence neither as the absolute Galois group of any field (see, e.g., [25,Rem. 3.3]).Hence, the following corollary may be deduced directly from Theorem 1.1.

Corollary 1.2. A pro-p group G as in Theorem 1.1 does not occur as the maximal pro-p Galois group of any field containing a root of 1 of order p (and also
√ −1 if p = 2).Hence, G does not occur as the absolute Galois group of any field.
In the recent past, other cohomological properties have been used to study maximal pro-p Galois groups -and to find examples of pro-p groups which do not occur as maximal pro-p Galois groups.By the Norm Residue Theorem -proved by M. Rost and V. Voevodsky, with the contribution by Ch.Weibel, see [13,35] -one knows that if K is a field containing a root of 1 of order p, the Z/p-cohomology algebra H • (G K (p), Z/pZ), endowed with the cup-product is quadratic, i.e., its ring structure is completely determined by the 1st and the 2nd cohomology groups (see, e.g., [23, § 2]).Moreover, it was shown by E. Matzri that if K is a field containing a root of 1 of order p, then G K (p) satisfies the triple Massey vanishing property (see [9] and references therein) -for an overview on Massey products in Galois cohomology see [20].These two cohomological properties were used to find examples of pro-p groups which do not occur as maximal pro-p Galois groups of fields containing a root of 1 of order p, for example in [4, § 8] and in [20, § 7].
We prove that the pro-p groups described in Theorems 1.1 cannot be ruled out as maximal pro-p Galois groups employing the above two cohomological obstructions.(We recall the basic notions on Massey products in Galois cohomology in § 6.1 below.)Hence, Corollary 1.2 provides brand new examples of pro-p groups which do not occur as maximal pro-p Galois groups of fields containing a root of 1 of order p, and as absolute Galois groups.Moreover, we remark that the relations which define the pro-p groups described in Theorem 1.1 are rather "elementary" -just elementary commutators of generator times, possibly, the p-power of a generator -, unlike the examples provided in [1,4,20,25], where the relations involve higher commutators.
Finally, we focus on the Minač-Tân pro-p group, i.e., the pro-p group G with pro-p presentation In [20, § 7], J. Minač and N.D. Tân showed that G does not satisfy the 3-Massey vanishing property, and thus it does not occur as the maximal pro-p Galois group of any field containing a root of 1 of order p.We prove that G cannot complete into a 1-cyclotomic oriented pro-p group.
Theorem 1.4.Let p be an odd prime.Then there are no orientations turning the Minač-Tân pro-p group into a 1-cyclotomic oriented pro-p group.
Theorem 1.4 has been proved independently by I. Snopce and P. Zalesskiȋ (unpublished).Theorem 1.4 provides a negative answer to the question posed in [30,Rem. 3.7] -namely, the Minač-Tân pro-p group may be ruled out as a maximal pro-p Galois group of a field containing a root of 1 of order p (and thus as an absolute Galois group) in a "Massey-free" way.
Altogether, 1-cyclotomicity of oriented pro-p groups provides a rather powerful tool studying maximal pro-p Galois groups, and it succeeds in detecting pro-p groups which are not maximal pro-p Galois groups when other methods fail, as underlined above.We believe that further investigations in this direction will lead to new obstructions for the realization of pro-p groups as maximal pro-p Galois group.
Actually, Theorem 1.4, and the main result in [34] (see in particular [34, p. 1907]), may lead to the suspect that 1-cyclotomicity is a more restrictive condition in comparison with the vanishing of Massey products.Thus, we formulate the following conjecture.
Conjecture 1.5.Let (G, θ) be an oriented pro-p group, such that Im(θ) then the pro-p group G satisfies the 3-Massey vanishing property; if moreover G is finitely generated, then G satisfies the strong n-Massey vanishing property for every n ≥ 3.
22", Sept. 2022, hosted by the Riemann International School of Mathematics (Varese, Italy).So, the author is gratheful to the organizers and the hosting institutions of these events.Last, but not least, the author thanks the anonymous referee, for several helpful comments.
2. Oriented pro-p groups and cohomology 2.1.Notation and preliminaries.Throughout the paper, every subgroup of a pro-p group is tacitly assumed to be closed with respect to the pro-p topology.Therefore, sets of generators of pro-p groups, and presentations, are to be intended in the topological sense.
Given a pro-p group G, we denote the closed commutator subgroup of G by G ′namely, G ′ is the closed normal subgroup generated by commutators The Frattini subgroup of G is denoted by Φ(G) -namely, Φ(G) is the closed normal subgroup generated by G ′ and by p-powers g p , g ∈ G (cf., e.g., [5,Prop. 1.13]).A minimal generating set of G gives rise to a basis of the Z/pZ-vector space G/Φ(G), and conversely (cf., e.g., [5,Prop. 1.9]).Finally, we denote the abelianization G/G ′ of G by G ab .Throughout the paper, we will make use of the following straightforward fact.Fact 2.1.Let G be a finitely generated pro-p group.Then a subset {x 1 , . . ., x d } of G is a minimal generating set of G if, and only if, the subset {x 1 G ′ , . . ., x d G ′ } of G ab is a minimal generating set of the abelian pro-p group G ab .

Oriented pro-p groups.
Let G be a pro-p group.An orientation θ : G → 1 + pZ p is said to be torsion-free if p is odd, or if p = 2 and Im(θ) ⊆ 1 + 4Z 2 .Observe that one may have an oriented pro-p group (G, θ) where G has non-trivial torsion and θ torsion-free (e.g., if G ≃ Z/p and Im(θ) = {1}).
A morphism of oriented pro-p groups (G 1 , θ 1 ) → (G 2 , θ 2 ), is a homomorphism of pro-p groups φ : Within the family of oriented pro-p groups one has the following constructions.Let (G, θ) be an oriented pro-p group.
(a) If N is a normal subgroup of G contained in Ker(θ), one has the oriented prop group (G/N, θ /N ), where θ /N : G/N → 1 + pZ p is the orientation such that θ /N • π = θ, with π : G → G/N the canonical projection.(b) If A is an abelian pro-p group (written multiplicatively), one has the oriented pro-p group A⋊(G, θ) = (A⋊G, θ), with action given by gag −1 = a θ(g) for every g ∈ G, a ∈ A, where the orientation θ : 2.3.Kummerianity and 1-cyclotomicity.Let (G, θ) be an oriented pro-p group.
Observe that the G-action on the G-module Z p (θ)/pZ p (θ) is trivial, as θ(g) ≡ 1 mod p for all g ∈ G. Thus, Z p (θ)/pZ p (θ) is isomorphic to Z/p as a trivial G-module.
If an oriented pro-p group (G, θ) with torsion-free orientation satisfies the above two equivalent properties, then it is said to be Kummerian.Moreover, (G, θ) is said to be 1-cyclotomic if (H, θ| H ) is Kummerian for every subgroup H ⊆ G.
Remark 2.3.The original definition of 1-cyclotomic oriented pro-p group requires only that for every open subgroup U of G, the oriented pro-p group (U, θ| U ) is Kummerian (cf.[30, § 1]).By a continuity argument, this is enough to imply that the oriented pro-p group (H, θ| H ) is Kummerian also for every subgroup H ⊆ G (cf. [30,Cor. 3.2]).

Examples.
Example 2.4.Let K be a field containing a root of 1 of order p, and also √ −1 if p = 2. Then the pro-p cyclotomic character θ K of G K (p) -induced by the action of G K (p) on the roots of 1 of p-power order contained in K(p) -has image contained in 1 + pZ p .Observe that Im(θ K ) = 1 + p f Z p , where f ∈ N ∪ {∞} is maximal such that K contains a root of 1 of order p f (if f = ∞, we set p ∞ = 0).In particular, θ K is a torsion-free orientation.The module Z p (θ K ) is called the 1st Tate twist of Z p (cf., e.g., [21,Def. 7.3.6]).For the convenience of the reader, here we recall J. Labute's argument to show that the oriented pro-p group (G K (p), θ K ) is Kummerian -and thus also 1-cyclotomic, as every subgroup H ⊆ G K (p) is the maximal pro-p Galois group of an extension of K, with pro-p cyclotomic character θ K | H -, as it is presented in [14, p. 131] (where the module Z p (θ K ) is denoted by I = I(χ ′ )).For every n ≥ 1 one has an isomorphism of continuous G K (p)-modules Let K × and K(p) × denote the multiplicative groups of units of K and K(p) respectively.By Hilbert 90, the short exact sequence of continuous G K (p)-modules where the left-side vertical arrow π n and the central vertical arrow are induced by the p n−1 -th power map p n : K(p) × → K(p) × , and the right-side vertical arrow is induced by the epimorphism of continuous G K (p)-modules Z p (θ K )/p n Z p (θ K ) ։ Z/pZ.Since the map π n is surjective, also the other vertical arrows are surjective.
Example 2.5.Let G be a free pro-p group.Then the oriented pro-p group (G, θ) is 1-cyclotomic for any orientation θ : Example 2.6.Let G be an infinite Demushkin group (cf., e.g., [21, Def.3.9.9]).By [14,Thm. 4], G comes endowed with a canonical orientation χ : G → 1 + pZ p which is the only one completing G into a 1-cyclotomic oriented pro-p group.In particular, Example 2.7.Let (G, θ) be an oriented pro-p group, with θ a torsion-free orientation.The oriented pro-p group (G, θ) is said to be θ-abelian if the subgroup K θ (G) is trivial and if Ker(θ) is a free abelian pro-p group -in this case G is a free abelian-by-cyclic pro-p group, i.e., for some set of indices I, and θ(x i ) = 1 for all i ∈ I (cf.[23,Prop. 3.4] Proposition 2.8.Let (G, θ) be an oriented pro-p group, with θ a torsion-free orientation.Then (G, θ) is Kummerian if, and only if, there exists a normal subgroup N of G such that N ⊆ Ker(θ) ∩ Φ(G), and the quotient (G/N, θ /N ), is a θ /N -abelian oriented pro-p group.If such a normal subgroup N exists, then N = K θ (G).
Lemma 2.9.Let (G, θ) be a finitely generated oriented pro-p group with torsion-free orientation, and let X = {x 1 , . . ., x d } be a minimal generating set of G.The following are equivalent.
Proposition 2.10.Let G be a finitely generated pro-p group, and let (G, θ) be a Kummerian oriented pro-p group with torsion-free orientation.If N is a normal subgroup of G such that N ⊆ Ker(θ) and the restriction map In order to prove Proposition 2.10 we need the following fact, whose proof -rather straightforward -is left to the reader.Fact 2.11.Let G be a finitely generated pro-p group, and let (G, θ) be an oriented pro-p group with torsion-free orientation.
Proof of Proposition 2.10.Set Ḡ = G/N and θ = θ /N .For every n ≥ 1, the canonical projection π : G → Ḡ induces the inflation maps which are injective by [21,Prop. 1.6.7].Also, the epimorphisms (respectively of continuous Ḡ-modules and continuous G-modules) Z p ( θ)/p n Z p ( θ) → Z/pZ and Z p (θ)/p n → Z/pZ induce, respectively, the morphisms induced by the inclusion N ֒→ G, and dual to res 1 G,N , is injective.Thus, one may find a minimal generating set X of G such that Y = X ∩ N generates N as a normal subgroup of G.By Lemma 2.9, there exists a continuous 1-cocycle c : denotes the cohomology class of c -, and moreover c(x) = 0 for every x ∈ Y. Therefore, by Fact 2.11-(i), the restriction is the map constantly equal to 0. By Fact 2.11-(ii), c induces a continuous 1-cocycle denotes the cohomology class of c.Altogether, one has Remark 2.12.Proposition 2.10 may be proved also in a purely group-theoretic way, see [3, Rem.3.9].

The Z/pZ-cohomology of G
The purpose of this section is to prove the first statement of Proposition 1.3, and more in general to describe the Z/pZ-cohomology algebra H • (G, Z/pZ) with G as in Theorem 1.1.
We describe now more in detail the structure of H • (X, Z/pZ).By duality -cf.
Remark 3.2.It is well-known that if a pro-p group has non-trivial torsion, then its n-th Z/p-cohomology group is non trivial for every n > 0; hence, G is torsion-free.
Without loss of generality, we may assume that d 1 ≥ 3. .Conversely, suppose that (G, θ) is Kummerian.Let N 1 and N 2 denote the normal subgroups of G generated as normal subgroups by z 1 , . . ., z d2 and y 1 , . . ., y d1 respectively.Then G/N 1 ≃ G 1 and G/N 2 ≃ G 2 .Moreover, Proposition 2.10 implies that (G/N i , θ /Ni ) is Kummerian for both i = 1, 2. Since G/N i ≃ G i for both i, Example 2.6 and the argument before the statement of the proposition imply that the torsion-free orientations θ /N1 and θ /N2 are constantly equal to 1. Hence, also θ is constantly equal to 1, as θ(w) = θ /N1 (wN 1 ) for every w ∈ G 1 , and analogously θ(w) = θ /N2 (wN 2 ) for every w ∈ G 2 .
Therefore, if G may complete into a 1-cyclotomic oriented pro-p group, then necessarily G is absolutely torsion-free.In order to prove Theorem 1.1 in case (1.1.a),we aim at exhibiting an open subgroup H of G, of index p 2 , whose abelianization H ab has non-trivial torsion.
Let φ G : G → Z/p be the homomorphism of pro-p groups defined by φ G (y  Proof.Since U is normally generated by X and G/U = {U, . . ., y p−1 3 U }, U is generated as a pro-p group by the set {w y h 3 | w ∈ X , h = 0, . . ., p − 1}.Also, U is subject to the relations Consider the abelianization U ab .Since the only factor in (4.2) which does not lie in for all h = 0, . . ., p − 1.
Analogously, the only factor in (4.3) which does not lie in U ′ is [x y h 3 , z and so on.Thus Altogether, U ab is the free abelian pro-p group generated by the cosets {wU ′ | w ∈ Y U }, so that Fact 2.1 yields the claim.
are open subgroups of G 1 , G 2 respectively of index p, and thus they are again Demushkin groups, on 2 + p(d 1 − 1) and 2 + p(d 2 − 1) generators respectively (cf.[6]).In particular, the defining relation of x y h 3 , y Also, from the relations (4.4)-(4.5)and from (4.1), one computes and so on.In fact, the two relations (4.4)-(4.5)-with the x y h 3 's replaced using (4.6) -are all the defining relations we need to get U , as shown in the following.
and subject to the 2p relations s v h 1 = 1 and s v h 2 = 1, with h = 0, . . ., p − 1.We claim that the abelianization H ab yields non-trivial torsion.
Proposition 4.4.The abelian pro-p group H ab is not torsion-free.
Proof.Since all the elements of Y U showing up in the last terms of the equalities (4.6) belong to H, one deduces that x y h 3 ≡ x mod H ′ for all h = 0, . . ., p − 1.Now, each factor of s 2 -cf.(4.5) -is a commutator of elements of H, and thus the relations s v h 2 = 1 yield trivial relations in H ab .On the other hand, every factor of s 1 -cf.(4.4) -, but [x, y 1 ] and [x y3 , y y3  1 ], is a commutator of elements of H. From (4.4) one obtains Altogether, H ab is the abelian pro-p group (non-minimally) generated by the set X H ab = {wH ′ | w ∈ X H }, and subject to the p relations as U/H = {H, vH, . . ., v p−1 H}.From these relations one deduces the equivalences: . . .
But x v p ≡ x mod H ′ , as v p ∈ H, and thus from the last of the above equivalences one obtains (4.9) Altogether, H ab is the abelian pro-p group minimally generated by where w ∈ Y U {v, y y3 1 , x}, and subject to the relation (
Henceforth, θ : G → 1 + pZ p will denote the orientation as in Proposition 5.1.
Let H be the subgroup of G generated by U 1 , U 2 and T , and let M be the subgroup of H generated by N 1 , N 2 and T .Observe that M ⊆ Ker(θ).Our aim is to show that the oriented pro-p group (H, θ| H ) is not Kummerian.For this purpose, we need the following.
Lemma 5.2. (i (ii) M is a normal subgroup of H, and H ≃ M ⋊ X p (iii) One has an isomorphism of p-elementary abelian groups Proof.Consider the pro-p tree T associated to the amalgamated free pro-p product (3.3).Namely, T consists of a set vertices V and a set of edges E, where and it comes endowed with a natural G-action, i.e., (5.3) Pick g ∈ M and hX ∈ E. Then g.hX = hX if, and only if, g ∈ hXh −1 , i.e., g = hx λ h −1 for some λ ∈ Z p .Since M ⊆ Ker(θ), it follows that and therefore λ = 0, as 1+pZ p is torsion-free.Hence, the subgroup M intersects trivially the stabilizer Stab G (hX) of every edge hX ∈ E. By [15, Thm.5.6], M decomposes as free pro-p product as follows: (5.5) where F is a free pro-p group, and V ′ ⊆ V is a continuous set of representatives of the space of orbits M \V.Clearly, the vertices G 1 and G 2 belong to different orbits, thus in the decomposition (5.5) one finds the two factors

and analogously
Stab M (G 2 ) = N 2 .Therefore, from (5.5) one obtains It is straightforward to see that t / ∈ N 1 ∐ N 2 .Since M is generated as pro-p group by N 1 , N 2 and t, the right-side factor in (5.6) is necessarily T , and this proves (i).
In order to prove (ii), we need only to show that uM u −1 = M , as  [8,20,36].
Remark 6.1.Given a sequence α 1 , . . ., α n of elements of H 1 (G, Z/pZ), if an element ω of H 2 (G, Z/pZ) is a value of the n-fiold Massey product α 1 , . . ., α n , then In [19,Thm. 8.1], J. Minač and N.D. Tân proved that the maximal pro-p Galois group of a field K containing a root of 1 of order p (and also √ −1 if p = 2) satisfies the cyclic p-Massey vanishing property.The proof of the last property for a pro-p group G as in Theorem 1.1 is rather immediate.
Proof of .By Proposition 4.1 and Proposition 5.1, G may complete into a Kummerian oriented pro-p group with torsion-free orientation.Hence, G satisfies the cyclic p-Massey vanishing property by [28,Thm. 3.10].6.2.Massey products and unipotent upper-triangular matrices.Massey products for a pro-p group G may be translated in terms of unipotent upper-triangular representations of G as follows.For n ≥ 2 let be the group of unipotent upper-triangular (n + 1) × (n + 1)-matrices over Z/p.Then U n+1 is a finite p-group.Moreover, for 1 ≤ h, l ≤ n+1 let E h,l denote the (n+1)×(n+1) matrix with the (h, l)-entry equal to 1, and all the other entries equal to 0. Now let ρ : G → U n+1 be a homomorphism of pro-p groups.Observe that for every h = 1, . . ., n, the projection ρ h,h+1 : G → Z/p of ρ onto the (h, h + 1)-entry is a homomorphism, and thus it may be considered as an element of H 1 (G, Z/pZ).One has the following "pro-p translation" of a result of W. Dwyer which interprets Massey product in terms of unipotent upper-triangular representations (cf., e.g., [11,Lemma 9.3]).Proposition 6.2.Let G be a pro-p group, and let α 1 , . . ., α n be a sequence of elements of H 1 (G, Z/pZ), with n ≥ 2. Then the n-fold Massey product α 1 , . . ., α n : (i) is not empty if, and only if, there exists a morphism of pro-p groups ρ : G → U n+1 /Z(U n+1 ) such that ρh,h+1 = α h for every h = 1, . . ., n; (ii) vanishes if, and only if, there exists a morphism of pro-p groups ρ : G → U n+1 such that ρ h,h+1 = α h for every h = 1, . . ., n.
We recall that We use this fact to prove statements (iii.a)-(iii.b) of Proposition 1.3.First of all, let G be as in Theorem 1.1, and let α 1 , . . ., α n be a sequence of elements of H 1 (G, Z/pZ).

Proposition 6.3. A pro-p group G satisfies the 3-Massey vanishing property in the following cases:
(a Proof.Let α 1 , α 2 , α 3 be a sequence of elements of H 1 (G, Z/pZ) satisfying (6.2).Then For every w ∈ X set where I denotes the 4 × 4 identity matrix.If G is as in (1.1.a),then one computes while if G is as in (1.1.b),then one computes -observe that the exponent of U 4 is p, as p > 4, and thus A(y 1 ) p = A(z 1 ) p = I.
Let I denote the identity matrix of the group U 5 .For every w ∈ X = {x, y 1 , . . ., z d2 } set Moreover, put ] .We will consider the matrix C as a function of the matrices A(x), . . ., A(y d1 ), and the matrix C ′ as a function of the matrices A(x), A(z 1 ), . . ., A(z d2 ).
Since p ≥ 5, the exponent of the p-group U 5 is p, and thus A(y 1 ) p = A(z 1 ) p = I.Moreover, for every w, w ′ ∈ X , the (h, h + 1)-entry of [A(w), A(w ′ )] is 0 for every h = 1, . . ., 4, and thus also c h,h+1 = c ′ h,h+1 = 0.Moreover, for h = 1, 2, 3 one has c h,h+2 = S h and c ′ h,h+2 = S ′ h -which are equal to 0 by (6.2).We split the proof in the analysis of the following three cases.Our aim is to modify suitably the matrices A(w) -without modifying the (h, h + 1)-entries with h = 1, . . ., 4 -in order to obtain C = C ′ = I.
An analogous argument yields C ′ = I -after replacing suitably the matrix This completes the analysis of case 2.
and replace A(y 1 ) with A(y 1 ) Ã, if w = x, or A(y i−1 ) with A(y i−1 ) Ã if w = y i with i odd, or A(y i+1 ) with A(y i+1 ) Ã, if w = y with i even.After the replacement, one has c hl = 0 for h < l ≤ h + 2, and for (h, l) = (1, 4).Then, set and replace A(y 1 ) with A(y 1 ) Ã′ , if w = x, or A(y i−1 ) with A(y i−1 ) Ã′ if w = y i with i odd, or A(y i+1 ) with A(y i+1 ) Ã′ , if w = y with i even.After this further replacement, one has c hl = 0 for h < l ≤ h + 3. Finally, set and replace A(y 1 ) with A(y 1 ) Ã′′ , if w = x, or A(y i−1 ) with A(y i−1 ) Ã′′ if w = y i with i odd, or A(y i+1 ) with A(y i+1 ) Ã′′ , if w = y with i even.After this last replacement, one has C = I.Now suppose we are in case (3.b).If w = x or w = y i with i odd, set and replace A(y 1 ) with A(y 1 ) Ã, if w = x, or A(y i−1 ) with A(y i−1 ) Ã if w = y i with i odd, or A(y i+1 ) with A(y i+1 ) Ã, if w = y with i even.After the replacement, one has c hl = 0 for h < l ≤ h + 2, and for (h, l) = (2, 5).Then, set and replace A(y 1 ) with A(y 1 ) Ã′ , if w = x, or A(y i−1 ) with A(y i−1 ) Ã′ if w = y i with i odd, or A(y i+1 ) with A(y i+1 ) Ã′ , if w = y with i even.After this further replacement, one has c hl = 0 for h < l ≤ h + 3. Finally, set and replace A(y 1 ) with A(y 1 ) Ã′′ , if w = x, or A(y i−1 ) with A(y i−1 ) Ã′′ if w = y i with i odd, or A(y i+1 ) with A(y i+1 ) Ã′′ , if w = y with i even.After this last replacement, one has C = I.Moreover, if none of the above two assumptions on the triviality of the values α h (x) and α h (z j ), with 2 ≤ j ≤ d 2 , hold true, the same argument produces suitable matrices A(z 1 ), . . ., A(z d2 ) such that the matrix C ′ is the identity matrix.This concludes the analysis of case 3.
Altogether, the assignment w → A(x) for every w ∈ X -with the matrices A(w)'s suitably modified in case of need -yields a homomorphism of pro-p groups ρ : G → U 5 with the desired properties.
Conversely, suppose that x 4 , x 5 ∈ Ker(θ), and at least one of the hypothesis (i)-(ii) holds true.Then for any choice for λ 4 , λ 5 , by ( 7 On the one hand, from (7.4) one deduces that the coset x p ≡ 1 mod U ′ , as x u 2 ≡ x 2 mod U ′ ; therefore they yield equivalent relations in U ab .Altogether, U ab is the abelian pro-p group minimally generated by X U ab and subject to the relation 2 U ′ p = 1.Hence U ab is not torsion-free, and Y U is a minimal generating set of U by Fact 2.1.From Proposition 7.3, one deduces that G is not absolutely torsion-free, and thus the oriented pro-p group (G, 1) is not 1-cyclotomic.Proof of Theorem 1.4.Suppose for contradiction that there exists a torsion free orientation θ : G → 1 + pZ p such that the oriented pro-p group (G, θ) is 1-cyclotomic.Then by § 7.1, we may assume without loss of generality that x 2 , . . ., x 5 ∈ Ker(θ), while θ(x 1 ) = 1 by § 7.2.Set λ ∈ pZ p {0} such that θ(x 1 ) = 1 + λ.

Proposition 1 . 3 .
Let G be a pro-p group as in Theorem 1.1.(i) The Z/p-cohomology algebra H • (G, Z/pZ) is quadratic.(ii) The pro-p group G satisfies the cyclic p-Massey vanishing property -namely, the p-fold Massey product α, . . ., α p times contains 0 for every α ∈ H 1 (G, Z/pZ).(iii.a)If G is as in (1.1.a),then G satisfies the 3-and the strong 4-Massey vanishing property.(iii.b)If G is as in (1.1.b)and p > 3 then G satisfies the 3-and the strong 4-Massey vanishing property.

Question 6. 6 . 7 .
(a) Let G be as in (1.1.a).Does G satisfy the strong n-Massey vanishing property for every n ≥ 3? (b) Let G be as in (1.1.b).Does G satisfy the strong n-Massey vanishing property for every 3 ≤ n < p?The Minač-Tân pro-p group

x h 1 2U
′ is generated by x 2 U ′ and x x1 2 U ′ for every h = 2, . . ., p − 1, so that Y U ab = {wU ′ | w ∈ Y U } generates U ab as an abelian pro-p group.On the other hand, from the equivalences with h = p − 2 and h = p − 1 one deduces that