Kurosh Subgroup Theorem in Free Pro-p Products
We provide a sufficient condition under which a closed subgroup of a restricted free pro-p product is itself a free pro-p product.
math.GR↗
arXiv subjects
Tamar Bar-On
Publications and source records attributed to Tamar Bar-On.
Cofinality of Galois Cohomology within Purely Quadratic Graded Algebras
Let $p$ be a prime number. For a field $F$ containing a root of unity of order $p$, let $H^\bullet(F)=H^\bullet(F,\mathbb{F}_p)$ be the mod-$p$ Galois cohomology graded $\mathbb{F}_p$-algebra of $F$. By the Norm Residue Theorem, $H^\bullet(F)$ is a purely quadratic graded-commutative algebra, and is therefore determined by the cup product $\cup\colon H^1(F)\times H^1(F)\to H^2(F)$. We prove that the class of all Galois cohomology algebras $H^\bullet(F)$ is cofinal in the class of all purely quadratic graded-commutative $\mathbb{F}_p$-algebras $A_\bullet$, in the following sense: For every $A_\bullet$ there exists $F$ such that the bilinear map $A_1\times A_1\to A_2$, which determines $A_\bullet$, embeds in the cup product bilinear map $\cup\colon H^1(F)\times H^1(F)\to H^2(F)$. We further provide examples of $\mathbb{F}_p$-bilinear maps which are not realizable by fields $F$ in this way. These are related to recent results by Snopce-Zalesskii and Blumer-Quadrelli-Weigel on the Galois theory of pro-$p$ right-angled Artin groups, as well as to a conjecture by Marshall on the possible axiomatization of quadratic form theory of fields.
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Jordan-Holder Theorem for profinite groups and applications
We generalize the notions of composition series and composition factors for profinite groups, and prove a profinite version of the Jordan-Holder Theorem. We apply this to prove a Galois Theorem for infinite prosolvable extensions. In addition, we investigate the connection between the abstract and topological composition factors of a nonstrongly complete profinite group.
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Krull-Schmidt Theorem for small profinite groups
We prove that every small profinite group can be decomposed into a direct product of indecomposable profinite groups, and that such a decomposition is unique up to order and isomorphisms of the components. We also investigate the cancellation property of some free pro-$\mathcal{C}$ groups, and give a new criterion for a profinite group to be small.
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Free product of Demushkin groups as absolute Galois group
We prove that a free profinite (pro-$p$) product over a set converging to 1 of countably many Demushkin groups of rank $\aleph_0$, $G_i$, that can be realized as absolute Galois groups, is isomorphic to an absolute Galois group if and only if $\log_pq(G_i)\to \infty$.
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Groups of profinite type and profinite rigidity
We say that a group $G$ is of \textit{profinite type} if it can be realized as a Galois group of some field extension. Using Krull's theory, this is equivalent to the ability of $G$ to be equipped with a profinite topology. We also say that a group of profinite type is \textit{profinitely rigid} if it admits a unique profinite topology. In this paper we study when abelian groups and some group extensions are of profinite type or profinitely rigid. We also discuss the connection between the properties of profinite type and profinite rigidity to the injectivity and surjectivity of the cohomology comparison maps, which were studied by Sury and other authors.
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Demushkin groups of uncountable rank
We extend the theory of countably generated Demushkin groups to Demushkin groups of arbitrary rank. We investigate their algebraic properties and invariants, count their isomorphism classes and study their realization as absolute Galois group. At the end, we compute their profinite completion and conclude with some results on profinite completion of absolute Galois groups.
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Pro-$p$ Poincar\'{e}-Duality groups of infinite rank
We introduce the class of \textit{Generalized Poincar\'{e}-Duality groups}: i.e, pro-$p$ groups of infinite rank which satisfy a Poincar\'{e}-duality. We prove some basic properties of Generalized Poincar\'{e}-Duality groups, and show that under some conditions, a group that fits into an exact sequence of GPD groups must be a Demushkin group.
math.GR↗
The algebra of supernatural matrices
The algebra of supernatural matrices is a key example in the theory of locally finite central simple algebras, which studied in a previous paper of the authors (\cite{Local}). It is also a stand-alone object admits a rich study and various connections to other fields. The goal of this paper is to expose some new information about supernatural matrices, mainly in terms of the "inner" ways to identify such algebras, and their appearance as minimal solutions to equations of the form $M_n(A)\cong A$. Viewing a natural representation of this algebra, we show that supernatural matrices generalize both McCrimmon's deep matrices algebra and $m$-petal Leavitt path algebra. We also study their simple representations.
math.RA↗
Cohomological properties of absolute Galois groups and their profinite completion
We prove that several properties of absolute Galois groups are preserved under a profinite completion.
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Profinite Completion of Free Pro-$\mathcal{C}$ Groups
We investigate the ability of a free pro-$\CC$ group of infinite rank to abstractly solve abstract embedding problems, and conclude that for some varieties $\CC$, the profinite completion of any order, of a free pro-$\CC$ group of infinite rank, is a free pro-$\CC$ group as well, of the corresponding rank.
math.GR↗