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 …
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 $\log_pq(G_i)\to \infty$.
The free profinite product of finitely many absolute Galois group is an absolute Galois group.
The free profinite product of finitely many absolute Galois group is an absolute Galois group.
This paper proves that if $E$ is a field, such that the Galois group $\mathcal{G}(E(p)/E)$ of the maximal $p$-extension $E(p)/E$ is a Demushkin group of finite rank $r(p)_{E} \ge 3$, …
This paper proves that if $E$ is a field, such that the Galois group $\mathcal{G}(E(p)/E)$ of the maximal $p$-extension $E(p)/E$ is a Demushkin group of finite rank $r(p)_{E} \ge 3$, for some prime number $p$, then $\mathcal{G}(E(p)/E)$ does not possess nontrivial proper decomposition groups. When $r(p)_{E} = 2$, it describes the decomposition groups of $\mathcal{G}(E(p)/E)$. The paper shows that if $(K, v)$ is a $p$-Henselian valued field with $r(p)_{K} \in \mathbb N$ and a residue field of characteristic $p$, then $P \cong \widetilde P$ or $P$ is presentable as a semidirect product $\mathbb Z_{p}^τ \rtimes \widetilde P$, for some $τ\in \mathbb N$, where $\widetilde P$ is a Demushkin group of rank $\ge 3$ or a free pro-$p$-group. It also proves that when $\widetilde P$ is of the former type, it is continuously isomorphic to $\mathcal{G}(K ^{\prime}(p)/K ^{\prime})$, for some local field $K ^{\prime}$ containing a primitive $p$-th root of unity.
Abstract We address the question of when a free profinite product of infinitely many absolute Galois groups of fields is also an absolute Galois group of some field.
Abstract We address the question of when a free profinite product of infinitely many absolute Galois groups of fields is also an absolute Galois group of some field.
A result of Baumslag and Roseblade says that a finitely presented subgroup of a direct product of two free groups is either free or virtually a direct product of free …
A result of Baumslag and Roseblade says that a finitely presented subgroup of a direct product of two free groups is either free or virtually a direct product of free groups. We give a particularly straightforward proof of this result using an argument of a more general character. Additional applications of the method are given.
With a fixed prime power $q>1$, define the ring of polynomials $A=\mathbb{F}_q[t]$ and its fraction field $F=\mathbb{F}_q(t)$. For each pair $a=(a_1,a_2) \in A^2$ with $a_2$ nonzero, let $\phi(a)\colon A\to F\{\tau\}$ …
With a fixed prime power $q>1$, define the ring of polynomials $A=\mathbb{F}_q[t]$ and its fraction field $F=\mathbb{F}_q(t)$. For each pair $a=(a_1,a_2) \in A^2$ with $a_2$ nonzero, let $\phi(a)\colon A\to F\{\tau\}$ be the Drinfeld $A$-module of rank $2$ satisfying $t\mapsto t+a_1\tau+a_2\tau^2$. The Galois action on the torsion of $\phi(a)$ gives rise to a Galois representation $\rho_{\phi(a)}\colon \operatorname{Gal}(F^{\operatorname{sep}}/F)\to \operatorname{GL}_2(\widehat{A})$, where $\widehat{A}$ is the profinite completion of $A$. We show that the image of $\rho_{\phi(a)}$ is large for random $a$. More precisely, for all $a\in A^2$ away from a set of density $0$, we prove that the index $[\operatorname{GL}_2(\widehat{A}):\rho_{\phi(a)}(\operatorname{Gal}(F^{\operatorname{sep}}/F))]$ divides $q-1$ when $q>2$ and divides $4$ when $q=2$. We also show that the representation $\rho_{\phi(a)}$ is surjective for a positive density set of $a\in A^2$.
For a list $\cal{L}$ of finite groups and for a profinite group $G$, we consider the intersection $T(G)$ of all open normal subgroups $N$ of $G$ with $G/N$ in $\cal{L}$. …
For a list $\cal{L}$ of finite groups and for a profinite group $G$, we consider the intersection $T(G)$ of all open normal subgroups $N$ of $G$ with $G/N$ in $\cal{L}$. We give a cohomological characterization of the epimorphisms $\pi\colon S\to G$ of profinite groups (satisfying some additional requirements) such that $\pi[T(S)]=T(G)$. For $p$ prime, this is used to describe cohomologically the profinite groups $G$ whose $n$th term $G_{(n,p)}$ (resp., $G^{(n,p)}$) in the $p$-Zassenhaus filtration (resp., lower $p$-central filtration) is an intersection of this form. When $G=G_F$ is the absolute Galois group of a field $F$ containing a root of unity of order $p$, we recover as special cases results by Minac, Spira and the author, describing $G_{(3,p)}$ and $G^{(3,p)}$ as $T(G)$ for appropriate lists $\cal{L}$.
We classify the pro-p subdirect products of type FP m of Demushkin or free pro-p groups, in both cases of non-trivial Euler characteristic.
We classify the pro-p subdirect products of type FP m of Demushkin or free pro-p groups, in both cases of non-trivial Euler characteristic.
We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), …
We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichmuller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation came in part from his desire to understand the absolute Galois group. But he was also interested in Thurston's work on surfaces, and he expressed this in his Esquisse d'un programme, his Recoltes et semailles and on other occasions. He introduced the notions of dessin d'enfant, Teichmuller tower, and other related objects, he considered the actions of Γ Q on them or on their etale fundamental groups, and he made conjectures on some natural homomorphisms between the absolute Galois group and the automor-phism groups (or outer automorphism groups) of these objects. We mention several ramifications of these ideas, due to various authors. We also report on the works of Sullivan and others on nonlinear actions of Γ Q , in particular in homotopy theory. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichmuller theory. This volume is dedicated to the memory of Alexander Grothendieck.
We review some ideas of Grothendieck and others on actions of the absolute Galois group {\Gamma} Q of Q (the automorphism group of the tower of finite extensions of Q), …
We review some ideas of Grothendieck and others on actions of the absolute Galois group {\Gamma} Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichm{\"u}ller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation came in part from his desire to understand the absolute Galois group. But he was also interested in Thurston's work on surfaces, and he expressed this in his Esquisse d'un programme, his R{\'e}coltes et semailles and on other occasions. He introduced the notions of dessin d'enfant, Teichm{\"u}ller tower, and other related objects, he considered the actions of {\Gamma} Q on them or on their etale fundamental groups, and he made conjectures on some natural homomorphisms between the absolute Galois group and the automor-phism groups (or outer automorphism groups) of these objects. We mention several ramifications of these ideas, due to various authors. We also report on the works of Sullivan and others on nonlinear actions of {\Gamma} Q , in particular in homotopy theory. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm{\"u}ller theory. This volume is dedicated to the memory of Alexander Grothendieck.
In [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917. [Google Scholar], the author gave a positive solution to the problem in the …
In [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917. [Google Scholar], the author gave a positive solution to the problem in the survey of Jarden [2] Jarden, M. 1996. “Infinite Galois Theory”. In Handbook of Algebra I Amsterdam: Elsevier Sci.. [Crossref] , [Google Scholar] on the closedness of the class of profinite groups that are isomorphic to absolute Galois groups of fields with respect to finite free products. In [3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar], O. V. Mel'nikov solved this problem for separable profinite groups ([3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar] was done earlier than [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917. [Google Scholar]). In the same case, a more exact result on the absolute Galois groups of fields of fixed characteristic was obtained there. The proof proposed in 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups. is simpler than that in [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917. [Google Scholar] and, in addition, provides the results of Mel'nikov. On February, 2000, the author (knowing nothing about 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups. ) found one more proof of these results. In the author opinion, this proof is the simplest and the construction used in the proof, as well as its properties (cf. Propositio n 1) can have other applications.
A pro- p -group G is said to be a Demushkin group if (1) dim F p H 1 ( G , Z/ p Z) < ∞, (2) dim F …
A pro- p -group G is said to be a Demushkin group if (1) dim F p H 1 ( G , Z/ p Z) < ∞, (2) dim F p H 2 ( G , Z/ p Z) = 1, (3) the cup product H 1 (G, Z/pZ) × H 1 (G, Z/pZ) → H 2 (G, Z/pZ) is a non-degenerate bilinear form. Here F P denotes the field with p elements. If G is a Demushkin group, then G is a finitely generated topological group with n(G) = dim H 1 (G, Z/pZ) as the minimal number of topological generators; cf. §1.3. Condition (2) means that there is only one relation among a minimal system of generators for G; that is, G is isomorphic to a quotient F/(r), where F is a free pro- p -group of rank n = n(G) and (r) is the closed normal subgroup of F generated by an element r ∈ F 9 (F, F); cf. §1.4.
If the absolute Galois group GK of a field K is a direct product GK=G1 × G2, then one of the factors is prosolvable and either G1 and G2 have …
If the absolute Galois group GK of a field K is a direct product GK=G1 × G2, then one of the factors is prosolvable and either G1 and G2 have coprime order or K is Henselian and the direct product decomposition reflects the ramification structure of GK. So, typically, the direct product of two absolute Galois groups is not an absolute Galois group. In contrast, free (profinite) products of absolute Galois groups are known to be absolute Galois groups. The same is true about free pro-p products of absolute Galois groups which are pro-p groups. We show that, conversely, if C is a class of finite groups closed under forming subgroups, quotients, and extensions, and if the class of pro-C absolute Galois groups is closed under free pro-C products, then C is either the class of all finite groups or the class of all finite p-groups. As a tool, we prove a generalization of an old theorem of Neukirch which is of interest in its own right: if K is a non-Henselian field, then every finite group is a subquotient of GK, unless all decomposition subgroups of GK are pro-p groups for a fixed prime p.
Abstract 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 …
Abstract 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.