Galois-theoretic features for 1-smooth pro-<i>p</i> groups

Quadrelli, Claudio

doi:10.4153/s0008439521000461

Cited by 6 publications

(3 citation statements)

References 27 publications

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“…The proof of the following lemma is straightforward (see for instance [33, Remark 2.4]). Lemma Let

G

be a pro‐

p

group and let

(G, 1)

be the

p

‐oriented pro‐

p

group

(G, \theta )

with

\theta

the trivial homomorphism.…”

Section: Smoothness Conjecturementioning

confidence: 99%

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Right‐angled Artin pro‐p$p$ groups

Snopce

Zalesskiĭ

2022

Bulletin of London Math Soc

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

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“…The proof of the following lemma is straightforward (see for instance [33, Remark 2.4]). Lemma Let

G

be a pro‐

p

group and let

(G, 1)

be the

p

‐oriented pro‐

p

group

(G, \theta )

with

\theta

the trivial homomorphism.…”

Section: Smoothness Conjecturementioning

confidence: 99%

“…[21] and [32]); in particular, if 𝜃 is the trivial homomorphism, then 𝐺∕𝐾() is a free Abelian pro-𝑝 group. The proof of the following lemma is straightforward (see for instance [33,Remark 2.4]).…”

Section: Smoothness Conjecturementioning

confidence: 99%

Right‐angled Artin pro‐p$p$ groups

Snopce

Zalesskiĭ

2022

Bulletin of London Math Soc

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“…G is absolutely torsion-free, i.e., H/H ′ is a free abelian pro-p group for every subgroup H of G (cf. [35,Rem. 2.3]).…”

Section: Cyclotomic and The Action On The Associated G-module M Is Tr...mentioning

confidence: 99%

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

Cited by 6 publications

References 27 publications

Right‐angled Artin pro‐p$p$ groups

Right‐angled Artin pro‐p$p$ groups

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

Chasing Maximal Pro-p Galois Groups via 1-Cyclotomicity

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