arXiv subjects

Simone Blumer

Publications and source records attributed to Simone Blumer.

10 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.

math.NT↗

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

Restricted graph Lie algebras in characteristic two

We investigate restricted Lie algebras arising as analogues of (twisted) right-angled Artin groups and right-angled Coxeter groups over fields of characteristic two. These algebras are defined via quadratic relations determined by decorated graphs. We compute their cohomology rings with trivial coefficients and uncover phenomena specific to characteristic two: unlike in zero/odd characteristics, where quadratically defined ordinary and restricted Lie algebras have equivalent cohomology theories, the characteristic two case exhibits dependence on the base field. In particular, we prove that the ground field being the prime field $\mathbb F_2$ characterizes when a Lie-theoretic analogue of the twisted Droms theorem holds. Generalizations of graph Lie algebras are also discussed.

math.RA↗

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.

math.GR↗

arXiv2025

Subgroups of Bestvina-Brady groups

In "Subgroups of Graph Groups", 1987, J. Alg., Droms proved that all the subgroups of a right-angled Artin group (RAAG) defined by a finite simplicial graph $\Gamma$ are themselves RAAGs if, and only if, $\Gamma$ has no induced square graph nor line-graph of length $3$. The present work provides a similar result for specific normal subgroups of RAAGs, called Bestvina-Brady groups: We characterize those graphs in which every subgroup of such a group is itself a RAAG. In turn, we confirm several Galois theoretic conjectures for the pro-$p$ completions of these groups.

math.GR↗

arXiv2024

Koszul Lie algebras and their subalgebras

This paper examines (restricted) Koszul Lie algebras, a class of positively graded Lie algebras with a quadratic presentation and specific cohomological properties. The study employs HNN-extensions as a key tool for decomposing and analysing these algebras. Building on a previous work on Koszul Lie algebras ("Kurosh theorem for certain Koszul Lie algebras", S. Blumer), this paper also deals with Bloch-Kato Lie algebras, which constitute a distinguished subclass of that of Koszul Lie algebras where all subalgebras generated by elements of degree $1$ have a quadratic presentation. It is shown that Bloch-Kato Lie algebras satisfy a version of the Levi decomposition theorem and that they satisfy the Toral Rank Conjecture. Two new families of such Lie algebras are introduced, including all graded Lie algebras generated in degree $1$ and defined by two quadratic relations. Throughout the paper, we show many properties of right-angled Artin graded (RAAG) Lie algebras, which form a large class of Koszul Lie algebras.

math.RA↗

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.

math.NT↗

arXiv2022

Kurosh theorem for certain Koszul Lie algebras

The Kurosh theorem for groups provides the structure of any subgroup of a free product of groups and its proof relies on Bass-Serre theory of groups acting on trees. In the case of Lie algebras, such a general theory does not exists and the Kurosh theorem is false in general, as it was first noticed by Shirshov. However, we prove that, for a class of positively graded Lie algebras satisfying certain local properties in cohomology, such a structure theorem holds true for subalgebras generated in degree 1. Such class consists of Lie algebras, which have all the subalgebras generated in degree $1$ that are Koszul.

math.RA↗

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

Teoria Geometrica dei Gruppi Spazi CAT(0), Teorema di Gromov e oriented right-angled Artin groups

The aim of this thesis is to present the notion of spaces whose curvature is bounded above, and to give some of its application in the context of Combinatorial Algebra. The thesis is made of two parts, one of theoretic purpose, and the other applicative. In particular, we present an application to the notion of $CAT(0)$-space and one to the Gromov Theorem for cubical complexes. In the final part we present a new class of discrete groups generalizing the well-known right-angled Artin groups. We concoct a cubical CAT(0)-complex on which the group acts, from whom we extract some algebraic properties of such groups.

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