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Suppose C is a cyclic Galois cover of the projective line branched at the three points 0, 1, and ∞ . Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of C in terms of a basis which is well-suited to studying the action of the absolute Galois group of Q . [ABSTRACT FROM AUTHOR]

Generalizing a result of Wulf-Dieter Geyer in his thesis, we prove that if $K$ is a finitely generated extension of transcendence degree $r$ of a global field and $A$ is a closed abelian subgroup of $\textrm{Gal}(K)$ , then ${\mathrm{rank}}(A)\le r+1$. Moreover, if $\mathrm{char}(K)=0$ , then ${\hat{\mathbb{Z}}}^{r+1}$ is isomorphic to a closed subgroup of $\textrm{Gal}(K)$. [ABSTRACT FROM AUTHOR]

In this paper, we associate a finite dimensional algebra, called a Brauer graph algebra, to every clean dessin d'enfant by constructing a quiver based on the monodromy of the dessin. We show that Galois conjugate dessins d'enfants give rise to derived equivalent Brauer graph algebras and that the stable Auslander-Reiten quiver and the dimension of the Brauer graph algebra are invariant under the induced action of the absolute Galois group. [ABSTRACT FROM AUTHOR]

In the present paper, we discuss the injectivity of the natural homomorphism from the automorphism group of a given field to the outer automorphism group of the associated absolute Galois group. We prove that this natural homomorphism is injective in the case where, for instance, the given field may be embedded into the field of fractions of some Noetherian local domain of mixed characteristic. [ABSTRACT FROM AUTHOR]

In the present paper, we study the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields from the point of view of anabelian geometry. Let us recall that it is well-known that the natural homomorphism from the automorphism group of a mixed-characteristic local field to the outer automorphism group of the absolute Galois group of the given mixed-characteristic local field is injective. One main result of the present paper is that if the mixed-characteristic local field satisfies certain conditions, then the set of conjugates of the image of this injective homomorphism in the outer automorphism group is infinite, which thus implies that the image of this injective homomorphism is not normal in the outer automorphism group. In particular, one may conclude that it is impossible to establish a functorial group-theoretic reconstruction, from the absolute Galois group, of the "field-theoretic" subgroup, i.e., the image of this injective homomorphism, of the outer automorphism group. [ABSTRACT FROM AUTHOR]

We use tools from combinatorial group theory in order to study actions of three types on groups acting on a curve, namely the automorphism group of a compact Riemann surface, the mapping class group acting on a surface (which now is allowed to have some points removed) and the absolute Galois group Gal (Q ¯ / Q) in the case of cyclic covers of the projective line. [ABSTRACT FROM AUTHOR]

The absolute Galois group of the cyclotomic field K = Q (ζ p) acts on the étale homology of the Fermat curve X of exponent p. We study a Galois cohomology group which is valuable for measuring an obstruction for K -rational points on X. We analyze a 2-nilpotent extension of K which contains the information needed for measuring this obstruction. We determine a large subquotient of this Galois cohomology group which arises from Heisenberg extensions of K. For p = 3 , we perform a Magma computation with ray class fields, group cohomology, and Galois cohomology which determines it completely. [ABSTRACT FROM AUTHOR]

Kodaira fibred surfaces are remarkable examples of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. Our tour through algebraic surfaces and their moduli (with results valid also for higher dimensional varieties) deals with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variation of Hodge Structures, especially a recent answer given (in joint work with Dettweiler) to a long standing question posed by Fujita. In the landscape of our tour, Galois coverings, deformations and rigid manifolds (there are by the way rigid Kodaira fibrations), projective classifying spaces, the action of the absolute Galois group on moduli spaces, stand also in the forefront. These questions lead to interesting algebraic surfaces, for instance remarkable surfaces constructed from VHS, surfaces isogenous to a product with automorphisms acting trivially on cohomology, hypersurfaces in Bagnera-de Franchis varieties, Inoue-type surfaces. [ABSTRACT FROM AUTHOR]

DAHA-Jones polynomials of torus knots T( r, s) are studied systematically for reduced root systems and in the case of $$C^\vee C_1$$ . We prove the polynomiality and evaluation conjectures from the author's previous paper on torus knots and extend the theory by the color exchange and further symmetries. The DAHA-Jones polynomials for $$C^\vee C_1$$ depend on five parameters. Their surprising connection to the DAHA-superpolynomials (type A) for the knots $$T(2p+1,2)$$ is obtained, a remarkable combination of the color exchange conditions and the author's duality conjecture (justified by Gorsky and Negut). The uncolored DAHA-superpolynomials of torus knots are expected to coincide with the Khovanov-Rozansky stable polynomials and the superpolynomials defined via rational DAHA and/or in terms of certain Hilbert schemes. We end the paper with certain arithmetic counterparts of DAHA-Jones polynomials for the absolute Galois group in the case of $$C^\vee C_1$$ , developing the author's previous results for $$A_1$$ . [ABSTRACT FROM AUTHOR]

It is well-known that various profinite groups appearing in anabelian geometry — e.g., the absolute Galois groups of p -adic local fields or number fields — satisfy distinctive group-theoretic properties such as slimness [i.e., the property that every open subgroup is center-free] and strong indecomposability [i.e., the property that every open subgroup has no nontrivial product decomposition]. In the present paper, we consider another group-theoretic property on profinite groups, which we shall refer to as strong internal indecomposability. This is a stronger property than both slimness and strong indecomposability. In the present paper, we examine basic properties of strong internal indecomposability and prove that the absolute Galois groups of Henselian discrete valuation fields with positive characteristic residue fields or Hilbertian fields [which may be regarded as generalizations of p -adic local fields or number fields] satisfy strong internal indecomposability. [ABSTRACT FROM AUTHOR]

We describe the ℓ -Sylow subgroups of Gal ( Q ) for an odd prime ℓ , by observing and studying their decomposition as F ⋊ Z ℓ , where F is a free pro- ℓ group, and Z ℓ are the ℓ -adic integers. We determine the finite Z ℓ -quotients of F and more generally show that every split embedding problem of Z ℓ -groups for F is solvable. Moreover, we analyze the Z ℓ -action on generators of F . [ABSTRACT FROM AUTHOR]

theorem of Douady says that the absolute Galois group of a rational function field F( x) in one variable over an algebraically closed field F of characteristic 0 is a free profinite group. A new method is proposed to extend Douady's theorem from the case of the complex number field F = ℂ to the case of an arbitrary field. [ABSTRACT FROM AUTHOR]

Let ℤtr be the ring of totally real integers, Gal(ℚ) the absolute Galois group of ℚ, and e a positive integer. For each σ = (σ1,...,σe) ∈ Gal(ℚ)e let ℤtr(σ) be the fixed ring in ℤtr of σ1,...,σe. Then, the theory of all first order sentences θ that are true in ℤtr(σ) for almost all σ ∈ Gal(ℚ)e (in the sense of the Haar measure) is undecidable. [ABSTRACT FROM AUTHOR]

Abstract: This paper proves a generalization of Shafarevich''s Conjecture, for fields of Laurent series in two variables over an arbitrary field. This result says that the absolute Galois group of such a field K is quasi-free of rank equal to the cardinality of K, i.e. every non-trivial finite split embedding problem for has exactly proper solutions. We also strengthen a result of Pop and Haran–Jarden on the existence of proper regular solutions to split embedding problems for curves over large fields; our strengthening concerns integral models of curves, which are two-dimensional. [Copyright &y& Elsevier]