The Kummerian property and maximal pro-p Galois groups

Efrat, Ido; Quadrelli, Claudio

doi:10.1016/j.jalgebra.2019.01.015

Cited by 28 publications

(62 citation statements)

References 22 publications

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“…Proof For G finitely generated the equivalence between (i) and (ii) was shown in [10,Theorem 5.6], and the equivalence between (ii) and (iii) follows from Remark 2.2. For general G the equivalences were shown in [24,…”

Section: Fact 25 Let a Be An Abelian Pro-group And Let B Be A Closed ...mentioning

confidence: 83%

Oriented pro-$$\ell $$ groups with the Bogomolov–Positselski property

Quadrelli

Weigel

2022

Res. number theory

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For a prime number $$\ell $$ ℓ we say that an oriented pro-$$\ell $$ ℓ group $$(G,\theta )$$ ( G , θ ) has the Bogomolov–Positselski property if the kernel of the canonical projection on its maximal $$\theta $$ θ -abelian quotient $$\pi ^{\mathrm {ab}}_{G,\theta }:G\rightarrow G(\theta )$$ π G , θ ab : G → G ( θ ) is a free pro-$$\ell $$ ℓ group contained in the Frattini subgroup of G. We show that oriented pro-$$\ell $$ ℓ groups of elementary type have the Bogomolov–Positselski property (cf. Theorem 1.2). This shows that Efrat’s Elementary Type Conjecture implies a positive answer to Positselski’s version of Bogomolov’s Conjecture on maximal pro-$$\ell $$ ℓ Galois groups of a field $$\mathbb {K}$$ K in case that $$\mathbb {K}^\times /(\mathbb {K}^\times )^\ell $$ K × / ( K × ) ℓ is finite. Secondly, it is shown that for an $$H^\bullet $$ H ∙ -quadratic oriented pro-$$\ell $$ ℓ group $$(G,\theta )$$ ( G , θ ) the Bogomolov–Positselski property can be expressed by the injectivity of the transgression map $$d_2^{2,1}$$ d 2 2 , 1 in the Hochschild–Serre spectral sequence (cf. Theorem 1.4).

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Section: Fact 25 Let a Be An Abelian Pro-group And Let B Be A Closed ...mentioning

confidence: 83%

Oriented pro-$$\ell $$ groups with the Bogomolov–Positselski property

Quadrelli

Weigel

2022

Res. number theory

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“…Pro-p RAAGs and maximal pro-p Galois group. The following notions were introduced respectively in [34] and in [14,38]. Definition 2.9.…”

Section: 3mentioning

confidence: 99%

“…3.3]). For this reason, the pursue of obstructions which detect effectively pro-p groups which do not occur as absolute Galois groups has great prominence in current research in Galois theory (see, e.g., [1,5,14,36]).…”

Section: Introductionmentioning

confidence: 99%

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Groups of p-absolute Galois type that are not absolute Galois groups

Blumer¹,

Cassella²,

Quadrelli³

2021

Preprint

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Let p be a prime. We study pro-p groups of p-absolute Galois type, as defined by Lam-Liu-Sharifi-Wake-Wang. We prove that the pro-p completion of the right-angled Artin group associated to a chordal simplicial graph is of p-absolute Galois type, and moreover it satisfies a strong version of the Massey vanishing property. Also, we prove that Demushkin groups are of p-absolute Galois type, and that the free pro-p product -and, under certain conditions, the direct product -of two pro-p groups of p-absolute Galois type satisfying the Massey vanishing property, is again a pro-p group of p-absolute Galois type satisfying the Massey vanishing property. Consequently, there is a plethora of pro-p groups of p-absolute Galois type satisfying the Massey vanishing property that do not occur as absolute Galois groups.

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“…(b) A cyclotomic pro-p pair (G, θ) with G an infinite Demushkin pro-p group is 1-smooth if and only if θ : G → 1 + pZ p is defined as in[14, Thm. 4] (cf [9, . Thm.…”

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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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The Kummerian property and maximal pro-p Galois groups

Cited by 28 publications

References 22 publications

Oriented pro-$$\ell $$ groups with the Bogomolov–Positselski property

Oriented pro-$$\ell $$ groups with the Bogomolov–Positselski property

Groups of p-absolute Galois type that are not absolute Galois groups

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

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