On a class of pro-p groups occurring in Galois theory

Würfel, Tilmann

doi:10.1016/0022-4049(85)90063-5

Cited by 13 publications

(18 citation statements)

References 2 publications

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“…Hence, by [24,Theorem 5.6], G C 4 is not a Bloch-Kato pro-p group, and therefore, G Γ is not as well. Moreover, G Γ is not absolutely torsion free by [37,Proposition 4]. On the other hand, if Γ contains L 3 as an induced subgraph, then G Γ is not a Bloch-Kato pro-p group by Theorem 3.1, and G Γ is not absolutely torsion free by Theorem 3.3.…”

Section: Resultsmentioning

confidence: 99%

“…A pro-p group G is called absolutely torsion free if for every closed subgroup H of G the abelianization H ab is torsion free. This property was introduced and studied by Würfel in [37]. Free pro-p groups, free abelian pro-p groups and pro-p completions of surface groups are examples of absolutely torsion free pro-p groups.…”

Section: Introductionmentioning

confidence: 99%

“…Free pro-p groups, free abelian pro-p groups and pro-p completions of surface groups are examples of absolutely torsion free pro-p groups. Würfel proved that the class of absolutely torsion free pro-p groups is closed under forming inverse limits and free pro-p products (see [37,Proposition 3]); moreover, he proved that a direct product of a free abelian pro-p group and an absolutely torsion free pro-p group is absolutely torsion free. One may ask are there any other pro-p groups that are absolutely torsion free.…”

Section: Introductionmentioning

confidence: 99%

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Right-angled Artin pro-$p$ groups

Snopce¹,

Zalesskiĭ²

2020

Preprint

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Let p be a prime. The right-angled Artin pro-p group G Γ associated to a fnite simplicial graph Γ is the pro-p completion of the right-angled Artin group associated to Γ. We prove that the following assertions are equivalent: (i) no induced subgraph of Γ is a square or a line with four vertices (a path of length 3); (ii) every closed subgroup of G Γ is itself a right-angled Artin pro-p group (possibly infinitely generated); (iii) G Γ is a Bloch-Kato pro-p group; (iv) every closed subgroup of G Γ has torsion free abelianization; (v) G Γ occurs as the maximal pro-p Galois group G K (p) of some field K containing a primitive pth root of unity; (vi) G Γ can be constructed from Z p by iterating two group theoretic operations, namely, direct products with Z p and free pro-p products. This settles in the affirmative a conjecture of Quadrelli and Weigel. Also, we show that the Smoothness Conjecture of De Clercq and Florens holds for rightangled Artin pro-p groups. Moreover, we prove that G Γ is coherent if and only if each circuit of Γ of length greater than three has a chord.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Right-angled Artin pro-$p$ groups

Snopce¹,

Zalesskiĭ²

2020

Preprint

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“…Moreover, Ker(θ) ⊆ G , and since the latter is abelian, also Ker(θ) is abelian, i.e., Ker(θ) is meta-abelian. Thus Ker(θ) is a free abelian pro-p group by [30,Prop. 2].…”

Section: Example 41mentioning

confidence: 99%

“…Galois-theoretic features for 1-smooth pro-p groups 13 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).…”

Section: Example 41mentioning

confidence: 99%

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

Quadrelli

2021

Can. Math. Bull.

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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 θ : G → GL 1 (Z p ) such that every open subgroup H of G, together with the restriction θ | 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".

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Chasing Maximal Pro-p Galois Groups via 1-Cyclotomicity

Quadrelli

2024

Mediterr. J. Math.

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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 $$\mathbb {Z}/p\mathbb {Z}$$ Z / p Z -cohomology and the vanishing of Massey products—fail with the above pro-p groups. Finally, we prove that the Minač–Tân 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 $$n=3,4$$ n = 3 , 4 .

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On a class of pro-p groups occurring in Galois theory

Cited by 13 publications

References 2 publications

Right-angled Artin pro-$p$ groups

Right-angled Artin pro-$p$ groups

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

Chasing Maximal Pro-p Galois Groups via 1-Cyclotomicity

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