Related papers: Approximating Absolute Galois Groups
We describe the construction which takes as input a profinite group, which when applied the the absolute Galois group of a geometric field F agrees in some cases with the algebraic K-theory of F. We prove that it agrees in the case of a…
By two well-known results, one of Ax, one of Lubotzky and van den Dries, a profinite group is projective iff it is isomorphic to the absolute Galois group of a pseudo-algebraically closed field. This paper gives an analogous…
We prove that several properties of absolute Galois groups are preserved under a profinite completion.
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…
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…
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…
We study homological approximations of the profinite completion of a limit group (see Thm.~A) and obtain the analogous of Bridson and Howie's Theorem for the profinite completion of a non-abelian limit group (see Thm.~B).
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…
In this paper we produce unconditionally new instances of Galois number field extensions exhibiting strong discrepancies in the distribution of Frobenius elements among conjugacy classes of the Galois group. We first prove an inverse Galois…
Let K >= 1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A^2 is covered by K left translates of A. The main result of this paper is a qualitative…
We prove that certain fields have the property that their absolute Galois groups are free as profinite groups: the function field of a real curve with no real points; the maximal abelian extension of a 2-variable Laurent series field over a…
We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual…
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…
There are several variants of the inverse Galois problem which involve restrictions on ramification. In this paper we give sufficient conditions that a given finite group $G$ occurs infinitely often as a Galois group over the rationals…
The free profinite product of finitely many absolute Galois group is an absolute Galois group.
We generalize the notion of a projective profinite group to a projective pair of a profinite group and a closed subgroup. We establish the connection with Pseudo Algebraically Closed (PAC) extensions of PAC fields: Let M be an algebraic…
We give necessary conditions for a category fibred in pseudo-abelian additive categories over the classifying topos of a profinite group to be a stack; these conditions are sufficient when the coefficients are $\mathbf{Q}$-linear. This…
Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds…
We construct motivic $\ell$-adic representations of $\GQ$ into exceptional groups of type $E_7,E_8$ and $G_2$ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and uses the…
Our aim of this and subsequent papers is to enlighten (a part of, presumably) arithmetic structures of knots. This paper introduces a notion of profinite knots which extends topological knots and shows its various basic properties.…