On pro-p groups with quadratic cohomology

Quadrelli, Claudio; Snopce, Ilir; Vannacci, Matteo

doi:10.1016/j.jalgebra.2022.08.023

Cited by 6 publications

(5 citation statements)

References 61 publications

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“…Finally, note that

1 = [z^{t_1}, y] = [[z,t_1], y]

. Hence,

H

is not quadratic by [28, Theorem 7.3] (see also [34, Proposition 2.4]).

\Box

…”

Section: Resultsmentioning

confidence: 99%

“…A particular challenge is to determine what special properties

G_K(p)

have among all pro‐

p

groups; so far only a few properties have been found (cf., for example[2, 15, 29, 35] and references therein). A pro‐

p

group

G

is called

H^\bullet

‐quadratic (or simply quadratic) if the graded algebra

H^\bullet (G,\mathbb {F}_p) = \bigoplus _{n\geqslant 0}H^n(G,\mathbb {F}_p)

, endowed with the cup product and with

\mathbb {F}_p

as a trivial

G

‐module, is a quadratic algebra over

\mathbb {F}_p

, that is, all its elements of positive degree are combinations of products of elements of degree 1, and its defining relations are homogeneous relations of degree 2 (see [34]). A pro‐

p

group

G

is called a Bloch–Kato pro‐

p

group if every closed subgroup

U

G

is quadratic (see [32]).…”

Section: Introductionmentioning

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.

show abstract

“…Finally, note that

1 = [z^{t_1}, y] = [[z,t_1], y]

. Hence,

H

is not quadratic by [28, Theorem 7.3] (see also [34, Proposition 2.4]).

\Box

…”

Section: Resultsmentioning

confidence: 99%

“…A particular challenge is to determine what special properties

G_K(p)

have among all pro‐

p

groups; so far only a few properties have been found (cf., for example[2, 15, 29, 35] and references therein). A pro‐

p

group

G

is called

H^\bullet

‐quadratic (or simply quadratic) if the graded algebra

H^\bullet (G,\mathbb {F}_p) = \bigoplus _{n\geqslant 0}H^n(G,\mathbb {F}_p)

, endowed with the cup product and with

\mathbb {F}_p

as a trivial

G

‐module, is a quadratic algebra over

\mathbb {F}_p

, that is, all its elements of positive degree are combinations of products of elements of degree 1, and its defining relations are homogeneous relations of degree 2 (see [34]). A pro‐

p

group

G

is called a Bloch–Kato pro‐

p

group if every closed subgroup

U

G

is quadratic (see [32]).…”

Section: Introductionmentioning

confidence: 99%

Right‐angled Artin pro‐p$p$ groups

Snopce

Zalesskiĭ

2022

Bulletin of London Math Soc

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“…The iterated procedure to construct chordal simplicial graphs (cf. Proposition 2.5) makes them -and the associated pro-p RAAGs, also in the generalized version (see [40, § 5.1] for the definition of generalized pro-p RAAG) -rather special: indeed, by [40,Prop. 5.22] a generalized pro-p RAAG associated to a chordal simplicial graph may be constructed by iterating proper amalgamated free pro-p products over uniformly powerful (in some cases, free abelian) subgroups.…”

Section: Proof Of Theorem 12 First We Prove Theorem 12-(i)mentioning

confidence: 99%

“…On the one hand, this property is crucial in the proof of Proposition 4.4; on the other hand this implies that the Z/p-cohomology algebra of a generalized pro-p RAAG associated to a chordal simplicial graph is quadratic (cf. [40,Rem. 5.25]) -notice that, unlike pro-p RAAGs, a generalized pro-p RAAG may yield a non-quadratic Z/p-cohomology algebra (see [40,Ex.…”

Section: Proof Of Theorem 12 First We Prove Theorem 12-(i)mentioning

confidence: 99%

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

Blumer

Cassella

Quadrelli

2023

Journal of Pure and Applied Algebra

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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 .

show abstract

On pro-p groups with quadratic cohomology

Cited by 6 publications

References 61 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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