arXiv subjects

Claudio Quadrelli

Publications and source records attributed to Claudio Quadrelli.

At least 19 recordsLinked to original sources

arXiv2026

A cohomological translation of the Kaplansky radical for profinite groups

The Kaplansky radical of a field consists of the nonzero elements represented by every norm quadratic form in two variables. D. Kijima and M. Nishi conjectured that, for quadratic extensions, the Kaplansky radicals are related by the norm map in a manner analogous to Hilbert's Theorem 90. Although this H-conjecture was disproved by K.J. Becher and D.B. Leep, it is known to hold for several important classes of fields. We introduce a cohomological analogue of the Kaplansky radical for arbitrary profinite groups and primes $p$, defined as the orthogonal of $\mathrm{H}^1(G,\mathbb{F}_p)$ with respect to the cup product with itself. For absolute Galois groups, this recovers the classical Kaplansky radical when $p=2$ and the $p$-radical of Dario-Engler for arbitrary p. We also formulate a group-theoretic analogue of the H-conjecture, proving that, for fields, it is equivalent to the original conjectural property and depends only on the maximal pro-$2$ quotient of the absolute Galois group. We establish this property for broad classes of fields, including local and global fields, rational function fields, and all fields whose maximal pro-$p$ Galois group is of elementary type. Beyond its arithmetic origins, we investigate the property for general pro-$p$ groups, proving its stability under several natural group-theoretic constructions and obtaining new examples, including generalized right-angled Artin pro-$p$ groups and fundamental pro-$p$ groups of suitable graphs of groups, many of which cannot occur as maximal pro-$p$ Galois groups.

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arXiv2026

Variations of Demushkin Groups that are not Absolute Galois Groups

We construct two families of examples of pro-p groups, with rather elementary presentations, that do not complete into 1-cyclotomic oriented pro-p groups. These provide brand new examples of pro-p groups that do not occur as maximal pro-p Galois groups of fields containing a root of unity of order p - and thus, as absolute Galois groups. Moreover, we show that these pro-p groups may not be ruled out as maximal pro-p Galois groups employing other cohomological properties that are known to hold for all maximal pro-p Galois groups, such as the triple Massey vanishing property, or the quadraticity of Fp-cohomology.

math.GR↗

arXiv2025

Droms Theorems for twisted right-angled Artin groups

We characterize twisted right-angled Artin groups whose finitely generated subgroups are also twisted right-angled Artin groups. Additionally, we give a classification of coherence within this class of groups in terms of the defining graph. Furthermore, we provide a solution to the isomorphism problem for a notable subclass of these groups.

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arXiv2024

Directed graphs, Frattini-resistance, and maximal pro-$p$ Galois groups

Let $p$ be a prime. Following Snopce-Tanushevski, a pro-$p$ group $G$ is called Frattini-resistant if the function $H\mapsto\Phi(H)$, from the poset of all closed finitely-generated subgroups of $G$ into itself, is a poset embedding. We prove that for an oriented right-angled Artin pro-$p$ group (oriented pro-$p$ RAAG) $G$ associated to a directed graph the following four conditions are equivalent: the associated directed graph is of elementary type; $G$ is Frattini-resistant; every topologically finitely generated closed subgroup of $G$ is an oriented pro-$p$ RAAG; $G$ is the maximal pro-$p$ Galois group of a field containing a root of 1 of order $p$. Also, we conjecture that in the $\mathbb{Z}/p$-cohomology of a Frattini-resistant pro-$p$ group there are no essential triple Massey products.

math.GR↗

arXiv2024

Digraphs, pro-$p$ groups and Massey products in Galois cohomology

Let $p$ be a prime. We characterize the oriented right-angled Artin pro-$p$ groups whose $\mathbb{F}_p$-cohomology algebra yields no essential $n$-fold Massey products for every $n>2$, in terms of the associated digraph. Moreover, we show that the $\mathbb{F}_p$-cohomology algebra of such oriented right-angled Artin pro-$p$ groups is isomorphic to the exterior Stanley-Reisner $\mathbb{F}_p$-algebra associated to the same digraph. This work also aims at providing a concrete and group-theoretic introduction to the study of Massey products in Galois cohomology for non-specialists, especially graduate students working in profinite group theory.

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arXiv2023

Massey products in Galois cohomology and Pythagorean fields

We prove that a strengthened version of Minac-Tan's Massey Vanishing Conjecture holds true for fields with a finite number of square classes whose maximal pro-$2$ Galois group is of elementary type (as defined by I. Efrat). In particular, this proves Minac-Tan's Massey Vanishing Conjecture for Pythagorean fields with a finite number of square classes and their finite extensions.

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arXiv2023

Oriented right-angled Artin pro-$\ell$ groups and maximal pro-$\ell$ Galois groups

For a prime number $\ell$ we introduce and study oriented right-angled Artin pro-$\ell$ groups $G_{\Gamma,\lambda}$(oriented pro-$\ell$ RAAGs for short) associated to a finite oriented graph $\Gamma$ and a continuous group homomorphism $\lambda\colon\mathbb Z_\ell\to\mathbb Z_\ell^\times$. We show that an oriented pro-$\ell$ RAAG $G_{\Gamma,\lambda}$ is a Bloch-Kato pro-$\ell$ group if, and only if, $(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ is an oriented pro-$\ell$ group of elementary type generalizing a recent result of I. Snopche and P. Zalesskii. Here $\theta_{\Gamma,\lambda}\colon G_{\Gamma,\lambda}\to\mathbb Z_p^\times$ denotes the canonical $\ell$-orientation on $G_{\Gamma,\lambda}$. We invest some effort in order to show that oriented right-angled Artin pro-$\ell$ groups share many properties with right-angled Artin pro-$\ell$-groups or even discrete RAAG's, e.g., if $\Gamma$ is a specially oriented chordal graph, then $G_{\Gamma,\lambda}$ is coherent, generalizing a result of C. Droms. Moreover, in this case $(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ has the Positselski-Bogomolov property generalizing a result of H. Servatius, C. Droms and B. Servatius for discrete RAAG's. If $\Gamma$ is a specially oriented chordal graph and ${\rm Im}(\lambda)\subseteq 1+4\mathbb Z_2$ in case that $\ell=2$, then ${\rm H}^\bullet(G_{\Gamma,\lambda},\mathbb F_\ell) \simeq \Lambda^\bullet(\ddot{\Gamma}^{\rm op})$ generalizing a well known result of M. Salvetti.

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arXiv2022

Massey products in Galois cohomology and the Elementary Type Conjecture

Let $p$ be a prime. We prove that a positive solution to Efrat's Elementary Type Conjecture implies a positive solution to the strengthened version of Mina\v{c}--T\^an's Massey Vanishing Conjecture in the case of finitely generated maximal pro-$p$ Galois groups whose pro-$p$ cyclotomic character has torsion-free image. Consequently, the maximal pro-$p$ Galois group of a field $\mathbb{K}$ containing a root of 1 of order $p$ (and also \sqrt{-1} if $p=2$) satisfies the strong $n$-Massey vanishing property for every $n>2$ (which is equivalent to the cup-defining $n$-Massey product property for every $n>2$, as defined by Mina\v{c}--T\^an) in several relevant cases.

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arXiv2021

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

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.

math.GR↗

arXiv2021

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

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 $θ\colon G\to\mathrm{GL}_1(\mathbb{Z}_p)$ such that every open subgroup $H$ of $G$, together with the restriction $θ\vert_H$, 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".

math.GR↗

arXiv2021

Mild pro-p groups and the Koszulity conjectures

Let $p$ be a prime, and $\mathbb{F}_p$ the field with $p$ elements. We prove that if $G$ is a mild pro-$p$ group with quadratic $\mathbb{F}_p$-cohomology algebra $H^\bullet(G,\mathbb{F}_p)$, then the algebras $H^\bullet(G,\mathbb{F}_p)$ and $\mathrm{gr}\mathbb{F}_p[\![G]\!]$ - the latter being induced by the quotients of consecutive terms of the $p$-Zassenhaus filtration of $G$ - are both Koszul, and they are quadratically dual to each other. Consequently, if the maximal pro-$p$ Galois group of a field is mild, then Positselski's and Weigel's Koszulity conjectures hold true for such a field.

math.GR↗

arXiv2021

Chasing maximal pro-p Galois groups via 1-cyclotomicity

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 $\mathbb{Z}/p\mathbb{Z}$-cohomology and the vanishing of Massey products - fail with the above pro-$p$ groups. Finally, we prove that the Mina\v{c}-T\^an 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 $n>2$.

math.GR↗

arXiv2021

Two families of pro-p groups that are not absolute Galois groups

Let $p$ be a prime. We produce two new families of pro-$p$ groups which are not realizable as absolute Galois groups of fields. To prove this we use the 1-smoothness property of absolute Galois pro-$p$ groups. Moreover, we show in these families one has one-relator pro-$p$ groups which may not be ruled out as absolute Galois groups employing the quadraticity of Galois cohomology (a consequence of Rost-Voevodsky Theorem), or the vanishing of Massey products in Galois cohomology.

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arXiv2021

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

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

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arXiv2020

Koszul algebras and quadratic duals in Galois cohomology

We investigate the Galois cohomology of finitely generated maximal pro-$p$ quotients of absolute Galois groups. Assuming the well-known conjectural description of these groups, we show that Galois cohomology has the PBW property. Hence in particular it is a Koszul algebra. This answers positively a conjecture by Positselski in this case. We also provide an analogous unconditional result about Pythagorean fields. Moreover, we establish some results that relate the quadratic dual of Galois cohomology with $p$-Zassenhaus filtration on the group. This paper also contains a survey of Koszul property in Galois cohomology and its relation with absolute Galois groups.

math.NT↗

arXiv2020

Profinite groups with a cyclotomic $p$-orientation

Profinite groups with a cyclotomic $p$-orientation are introduced and studied. The special interest in this class of groups arises from the fact that any absolute Galois group $G_{K}$ of a field $K$ is indeed a profinite group with a cyclotomic $p$-orientation $θ_{K,p}\colon G_{K}\to\mathbb{Z}_p^\times$ which is even Bloch-Kato. The same is true for its maximal pro-$p$ quotient $G_{K}(p)$ provided the field $K$ contains a primitive $p^{th}$-root of unity. The class of cyclotomically $p$-oriented profinite groups (resp. pro-$p$ groups) which are Bloch-Kato is closed with respect to inverse limits, free product and certain fibre products. For profinite groups with a cyclotomic $p$-orientation the classical Artin-Schreier theorem holds. Moreover, Bloch-Kato pro-$p$ groups with a cyclotomic orientation satisfy a strong form of Tits' alternative, and the elementary type conjecture formulated by I. Efrat can be restated that the only finitely generated indecomposable torsion free Bloch-Kato pro-$p$ groups with a cyclotomic orientation should be Poincaré duality pro-$p$ groups of dimension less or equal to $2$.

math.GR↗

arXiv2020

Integrals of groups II

An $integral$ of a group $G$ is a group $H$ whose commutator subgroup is isomorphic to $G$. This paper continues the investigation on integrals of groups started in the work arXiv:1803.10179. We study: (1) A sufficient condition for a bound on the order of an integral for a finite integrable group and a necessary condition for a group to be integrable. (2) The existence of integrals that are $p$-groups for abelian $p$-groups, and of nilpotent integrals for all abelian groups. (3) Integrals of (finite or infinite) abelian groups, including nilpotent integrals, groups with finite index in some integral, periodic groups, torsion-free groups and finitely generated groups. (4) The variety of integrals of groups from a given variety, varieties of integrable groups and classes of groups whose integrals (when they exist) still belong to such a class. (5) Integrals of profinite groups and a characterization for integrability for finitely generated profinite centreless groups. (6) Integrals of Cartesian products, which are then used to construct examples of integrable profinite groups without a profinite integral. We end the paper with a number of open problems.

math.GR↗

arXiv2020

Right-angled Artin groups and enhanced Koszul properties

Let F be a finite field. We prove that the cohomology algebra with coefficients in F of a right-angled Artin group is a strongly Koszul algebra for every finite graph $Γ$. Moreover, the same algebra is a universally Koszul algebra if, and only if, the graph $Γ$ associated to the right-angled Artin group has the diagonal property. From this we obtain several new examples of pro-p groups, for a prime number p, whose continuous cochain cohomology algebra with coefficients in the field of p elements is strongly and universally (or strongly and non-universally) Koszul. This provides new support to a conjecture on Galois cohomology of maximal prop Galois groups of fields formulated by J. Mináč et al.

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