On pro-$p$ groups with quadratic cohomology

Quadrelli, Claudio; Snopce, Ilir; Vannacci, Matteo

doi:10.48550/arxiv.1906.04789

Cited by 8 publications

(14 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof By [21, Theorem A], U is a uniformly powerful pro-group, and therefore [25,Proposition 5.22] implies that G = G 1 ˆ U G 2 is a proper amalgam. Moreover, by hypoth-esis one has the monomorphisms of -elementary abelian groups ῑi and ῑU,i , with i = 1, 2 [cf.…”

Section: Free Constructionsmentioning

confidence: 99%

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

Quadrelli

Weigel

2022

Res. number theory

Self Cite

View full text Add to dashboard Cite

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).

show abstract

Section: Free Constructionsmentioning

confidence: 99%

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

Quadrelli

Weigel

2022

Res. number theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…The iterated procedure to construct chordal simplicial graphs (cf. Proposition 2.6) makes them -and the associated pro-p RAAGs, also in the generalized version (see [37, § 5.1] for the definition of generalized pro-p RAAG) -rather special: indeed, by [37,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: Massey Productsmentioning

confidence: 99%

“…On the one hand, this property is crucial in the proof of Proposition 4.5; 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. [37,Rem. 5.25]) -notice that, unlike pro-p RAAGs, a generalized pro-p RAAG may yield a non-quadratic Z/p-cohomology algebra (see [37,Ex.…”

Section: Massey Productsmentioning

confidence: 99%

See 1 more Smart Citation

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

Blumer¹,

Cassella²,

Quadrelli³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…The pro-p completion of Grigorchuk, Gupta-Sidki groups and other branch groups are FA pro-p groups as well as the Nottingham prop group. Splittings as amalgamated free products of Fab analytic pro-p groups occur naturally in the study of generalized RAAG pro-p groups [13,Subsection 5.5] where it is also proved that an amalgamated free pro-p product of uniformly powerful pro-p groups is always proper. Thus Theorem 2 applies to these splittings of generalized RAAG pro-p groups.…”

Section: Introductionmentioning

confidence: 99%

Generalized Stallings' decomposition theorem for pro-$p$ groups

Mattheus¹,

Zalesski²

2021

Preprint

View full text Add to dashboard Cite

The celebrated Stallings' decomposition theorem states that the splitting of a finite index subgroup H of a finitely generated group G as an amalgamated free product or an HNN-extension over a finite group implies the same for G. We generalize the pro-p version of it proved by Weigel and the second author in [25] to splittings over infinite pro-p groups. This generalization does not have any abstract analogs. We also prove that generalized accessibility of finitely generated pro-p groups is closed for commensurability.

show abstract

On pro-$p$ groups with quadratic cohomology

Cited by 8 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

Generalized Stallings' decomposition theorem for pro-$p$ groups

Contact Info

Product

Resources

About