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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., Comment: Final version, as it will appear on JPAA

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., Comment: In this version we use the notion of "digraph" instead of the notion of "oriented graph" previously (and uncorrectly) used, and adapt the content suitably

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., Comment: Final version, to appear on "Communications in Algebra". Minor changes from the previous version. arXiv admin note: text overlap with arXiv:2203.16232

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., Comment: The differences between the 1st version (Apr'23) and the 2nd are: correction of a couple of minor misprints, dedication to the memory of Avinoam Mann

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., Comment: Final version, published on the Journal of Number Theory, 258 (2024), 40-65

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., Comment: Final version, as it will appear on JPAA

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., Comment: Final revised version, to be published on {\guillemotleft}Expositiones Mathematic{\ae}{\guillemotright}

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 $n>2$., Comment: Thorough revision of the old version (v3), revision of the previous version following the referee's comments (v4-v6). Conjecture 1.5 (relating Massey products and 1-cyclotomicity) has been extended to all n>2

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., Comment: The title of the old version of this paper was {\guillemotleft}Oriented pro-$\ell$ groups with the Bogomolov property{\guillemotright}: following te referee's advice, we changed the name of the property we study from {\guillemotleft}Bogomolov property{\guillemotright} to {\guillemotleft}Bogomolov-Positselski property{\guillemotright}

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.

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., Comment: 44 pages, no figures. Revised version with a new co-author (Claudio Quadrelli). The proof of Theorem 4.1(c) has been revised and Section 8 has been rewritten to address the comments of an anonymous referee

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 $\theta\colon G\to\mathrm{GL}_1(\mathbb{Z}_p)$ such that every open subgroup $H$ of $G$, together with the restriction $\theta\vert_H$, 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"., Comment: To appear on "Canadian Math. Bull"

Let $p$ be a prime. We show that if a pro-$p$ group with at most 2 defining relations has quadratic $\mathbb{F}_p$-cohomology, then such algebra is universally Koszul. This proves the "Universal Koszulity Conjecture" formulated by J. Min\'a\v{c} et al. in the case of maximal pro-$p$ Galois groups of fields with at most 2 defining relations., Comment: To appear on "Math. Scandinavica"

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 ${\Gamma}$. Moreover, the same algebra is a universally Koszul algebra if, and only if, the graph ${\Gamma}$ 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. Min\'a\v{c} et al., Comment: To appear in J. Group Theory

The main purpose of this article is to study pro-$p$ groups with quadratic $\mathbb{F}_p$-cohomology algebra, i.e. $H^\bullet$-quadratic pro-$p$ groups. Prime examples of such groups are the maximal Galois pro-$p$ groups of fields containing a primitive root of unity of order $p$. We show that the amalgamated free product and HNN-extension of $H^\bullet$-quadratic pro-$p$ groups is $H^\bullet$-quadratic, under certain necessary conditions. Moreover, we introduce and investigate a new family of pro-$p$ groups that yields many new examples of $H^\bullet$-quadratic groups: $p$-RAAGs. These examples generalise right angled Artin groups in the category of pro-$p$ groups. Finally, we explore "Tits alternative behaviour" of $H^\bullet$-quadratic pro-$p$ groups., Comment: 40 pages

Let $p$ be a prime. A pro-$p$ group $G$ is said to be 1-smooth if it can be endowed with a homomorphism of pro-$p$ groups $G\to1+p\mathbb{Z}_p$ satisfying a formal version of Hilbert 90. By Kummer theory, maximal pro-$p$ Galois groups of fields containing a root of 1 of order $p$, together with the cyclotomic character, are 1-smooth. We prove that a finitely generated $p$-adic analytic pro-$p$ group is 1-smooth if, and only if, it occurs as the maximal pro-$p$ Galois group of a field containing a root of 1 of order $p$. This gives a positive answer to De Clerq-Florence's "Smoothness Conjecture" - which states that the Rost-Voevodsky Theorem (a.k.a. Bloch-Kato Conjecture) follows from 1-smoothness - for the class of finitely generated $p$-adic analytic pro-$p$ groups., Comment: To appear on 'Homology, Homotopy and Applications'. The current version of this paper is only one half of the original paper (versions v1--v4), as the author decided to split the original paper after its rejection from the referred journal which it was submitted to, more than 2 years ago (see Remark 1.2 for details)

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 $\theta_{K,p}\colon G_{K}\to\mathbb{Z}_p^\times$ 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 Poincar\'e duality pro-$p$ groups of dimension less or equal to $2$., Comment: To appear on "Doc. Math"

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., Comment: 39 pages

For a prime number $p$, we give a new restriction on pro-$p$ groups $G$ which are realizable as the maximal pro-$p$ Galois group $G_F(p)$ for a field $F$ containing a root of unity of order $p$. This restriction arises from Kummer Theory and the structure of the maximal $p$-radical extension of $F$. We study it in the abstract context of pro-$p$ groups $G$ with a continuous homomorphism $\theta\colon G\to1+p\mathbb{Z}_p$, and characterize it cohomologically, and in terms of 1-cocycles on $G$. This is used to produce new examples of pro-$p$ groups which do not occur as maximal pro-$p$ Galois groups of fields as above., Comment: Final revised version. To appear in the Journal of Algebra

Let $p$ be a prime number and let ${K}$ be a field containing a root of 1 of order $p$. If the absolute Galois group $G_{K}$ satisfies $\dim H^1(G_{K},\mathbb{F}_p)<\infty$ and $\dim H^2(G_{K},\mathbb{F}_p)=1$, we show that L.~Positselski's and T.~Weigel's Koszulity conjectures are true for ${K}$. Also, under the above hypothesis we show that the $\mathbb{F}_p$-cohomology algebra of $G_{K}$ is the quadratic dual of the graded algebra $\mathrm{gr}_\bullet\mathbb{F}_p[G_{K}]$, induced by the powers of the augmentation ideal of the group algebra $\mathbb{F}_p[G_{K}]$, and these two algebras decompose as products of elementary quadratic algebras. Finally, we propose a refinement of the Koszulity conjectures, analogous to I. Efrat's Elementary Type Conjecture., Comment: Updated version

For a prime number $p$, we show that if two certain canonical finite quotients of a finitely generated Bloch-Kato pro-$p$ group $G$ coincide, then $G$ has a very simple structure, i.e., $G$ is a $p$-adic analytic pro-$p$ group. This result has a remarkable Galois-theoretic consequence: if the two corresponding canonical finite extensions $F^{(3)}/F$ and $F^{\{3\}}/F$ of a field $F$ -- with $F$ containing a primitive $p$-th root of unity -- coincide, then $F$ is $p$-rigid. The proof relies only on group-theoretic tools, and on certain properties of Bloch-Kato pro-$p$ groups., Comment: 8 pages, to appear on the Annales math\'ematiques du Qu\'ebec

It is shown that for a normal subgroup $N$ of a group $G$, $G/N$ cyclic, the kernel of the map $N^{\mathrm{ab}}\to G^{\mathrm{ab}}$ satisfies the classical Hilbert 90 property (cf. Thm. A). As a consequence, if $G$ is finitely generated, $|G:N|<\infty$, and all abelian groups $H^{\mathrm{ab}}$, $N\subseteq H\subseteq G$, are torsion free, then $N^{\mathrm{ab}}$ must be a pseudo permutation module for $G/N$ (cf. Thm. B). From Theorem A one also deduces a non-trivial relation between the order of the transfer kernel and co-kernel which determines the Hilbert-Suzuki multiplier (cf. Thm. C). Translated into a number theoretic context one obtains a strong form of Hilbert's theorem 94. In case that $G$ is finitely generated and $N$ has prime index $p$ in $G$ there holds a "generalized Schreier formula" involving the torsion free ranks of $G$ and $N$ and the ratio of the order of the transfer kernel and co-kernel (cf. Thm. D).

The main problem this thesis deals with is the characterization of profinite groups which are realizable as absolute Galois groups of fields: this is currently one of the major problems in Galois theory. Usually one reduces the problem to the pro-$p$ case, i.e., one would like to know which pro-$p$ groups occur as maximal pro-$p$ Galois groups, i.e., maximal pro-$p$ quotients of absolute Galois groups. Indeed, pro-$p$ groups are easier to deal with than general profinite groups, yet they carry a lot of information on the whole absolute Galois group. We define a new class of pro-$p$ groups, called Bloch-Kato pro-$p$ group, whose Galois cohomology satisfies the consequences of the Bloch-Kato conjecture. Also we introduce the notion of cyclotomic orientation for a pro-$p$ group. With this approach, we are able to recover new substantial information about the structure of maximal pro-$p$ Galois groups, and in particular on $\theta$-abelian pro-$p$ groups, which represent the "upper bound" of such groups. Also, we study the restricted Lie algebra and the universal envelope induced by the Zassenhaus filtration of a maximal pro-$p$ Galois group, and their relations with Galois cohomology via Koszul duality. Altogether, this thesis provides a rather new approach to maximal pro-$p$ Galois groups, besides new substantial results., Comment: Ph.D. thesis at Western University (Canada) and Universit\`a di Milano-Bicocca (Italy), 103 pages, 1 figure

Fix an odd prime $p$, and let $F$ be a field containing a primitive $p$th root of unity. It is known that a $p$-rigid field $F$ is characterized by the property that the Galois group $G_F(p)$ of the maximal $p$-extension $F(p)/F$ is a solvable group. We give a new characterization of $p$-rigidity which says that a field $F$ is $p$-rigid precisely when two fundamental canonical quotients of the absolute Galois groups coincide. This condition is further related to analytic $p$-adic groups and to some Galois modules. When $F$ is $p$-rigid, we also show that it is possible to solve for the roots of any irreducible polynomials in $F[X]$ whose splitting field over $F$ has a $p$-power degree via non-nested radicals. We provide new direct proofs for hereditary $p$-rigidity, together with some characterizations for $G_F(p)$ -- including a complete description for such a group and for the action of it on $F(p)$ -- in the case $F$ is $p$-rigid., Comment: 25 pages, to appear in "Transactions of the AMS"

A Bloch-Kato pro-p group G is a pro-p group with the property that the F_p-cohomology ring of every closed subgroup of G is quadratic. It is shown that either such a pro-p group G contains no closed free pro-p groups of infinite rank, or there exists an orientation $\theta\colon G\rightarrow \Z_p^\times$ such that G is theta-abelian. In case that G is also finitely generated, this implies that G is powerful, p-adic analytic with d(G)=cd(G), and its \F_p-cohomology ring is an exterior algebra. These results will be obtained by studying locally powerful groups (see Theorem A). There are certain Galois-theoretical implications, since Bloch-Kato pro-p groups arise naturally as maximal pro-p quotients and pro-p Sylow subgroups of absolute Galois groups (see Corollary 4.9). Finally, we study certain closure operations of the class of Bloch-Kato pro-p groups, connected with the Elementary type conjecture., Comment: 17 pages, to appear on Forum Math