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Galois-theoretic features for 1-smooth pro-p groups
No full textAbstract. Let p be a prime. A pro-p group G is said to be 1-smooth if it can be endowed with a continuous representation θ : G → GL1 (Z p ) such that every open subgroup H of G, together with the restriction θ |H , satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-p group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-p Galois groups of fields, and that if a 1-smooth pro-p group is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-p Galois groups of fields. Finally we ask whether 1-smooth pro-p groups satisfy a “Tits’ alternative”
Two families of pro-p groups that are not Absolute Galois Groups
No full textLet p be a prime. We produce two new families of pro-p groups which are not realizable as absolute Galois groups of fields. To prove this we use the 1-smoothness property of absolute Galois pro-p groups. Moreover, we show in these families one has one-relator pro-p groups which may not be ruled out as absolute Galois groups employing the quadraticity of Galois cohomology (a consequence of Rost-Voevodsky Theorem), or the vanishing of Massey products in Galois cohomology
One-relator maximal pro-p Galois groups and the Koszulity conjectures
No full textLet p be a prime number and let K be a field containing a root of 1 of order p. If the absolute Galois group G_K satisfies dim H^1(G_K, F_p) < ∞ and dim H^2(G_K ,F_p) = 1, we show that L. Positselski’s and T. Weigel’s Koszulity conjectures are true for K. Also, under the above hypothesis we show that the F_p -cohomology algebra of G_K is the quadratic dual of the graded algebra gr_*F_p[G_K], induced by the powers of the augmentation ideal of the group algebra F_p[G_K], and these two algebras decompose as products of elementary quadratic algebras. Finally, we propose a refinement of the Koszulity conjectures, analogous to I. Efrat’s Elementary Type Conjecture
Massey products in Galois cohomology and Pythagorean fields
Full text linkWe prove that a strengthened version of Minac-Tan's Massey Vanishing Conjecture holds true for fields with a finite number of square classes whose maximal pro-2 Galois group is of elementary type (as defined by I. Efrat). In particular, this proves Minac-Tan's Massey Vanishing Conjecture for Pythagorean fields with a finite number of square classes and their finite extensions
Pro-p groups with few relations and Universal Koszulity
No full textLet p be a prime. We show that if a pro-p group with at most 2 defining relations has quadratic F_p-cohomology, then such algebra is universally Koszul. This proves the "Universal Koszulity Conjecture" formulated by J. Mináč et al. in the case of maximal pro-p Galois groups of fields with at most 2 defining relations
Bloch-Kato pro-p groups and locally powerful groups
No full textA Bloch-Kato pro-p group G is a pro-p group with the property that the Fp-cohomology ring of every closed subgroup of G is quadratic. It is shown that either such a pro-p group G contains no closed free pro-p groups of infinite rank, or there exists an orientation θ:G ! Z×p such that G is θ-abelian. In case that G is also finitely generated, this implies that G is powerful, p-adic analytic with d.G/ D cd.G/, and its Fp-cohomology ring is an exterior algebra. These results will be obtained by studying locally powerful groups. There are certain Galois-theoretical implications, since Bloch-Kato pro-p groups arise naturally as maximal pro-p quotients and pro-p Sylow subgroups of absolute Galois groups. Finally, we study certain closure operations of the class of Bloch-Kato pro-p groups, connected with the Elementary Type Conjecture
1-smooth pro-p groups and Bloch-Kato pro-p groups
Full text linkLet be a prime. A pro- group is said to be 1-smooth if it can be
endowed with a homomorphism of pro- groups satisfying
a formal version of Hilbert 90. By Kummer theory, maximal pro- Galois groups
of fields containing a root of 1 of order , together with the cyclotomic
character, are 1-smooth. We prove that a finitely generated -adic analytic
pro- group is 1-smooth if, and only if, it occurs as the maximal pro-
Galois group of a field containing a root of 1 of order . This gives a
positive answer to De Clerq-Florence's "Smoothness Conjecture" - which states
that the Rost-Voevodsky Theorem (a.k.a. Bloch-Kato Conjecture) follows from
1-smoothness - for the class of finitely generated -adic analytic pro-
groups.Comment: To appear on 'Homology, Homotopy and Applications'. The current
version of this paper is only one half of the original paper (versions
v1--v4), as the author decided to split the original paper after its
rejection from the referred journal which it was submitted to, more than 2
years ago (see Remark 1.2 for details
Directed graphs, Frattini-resistance, and maximal pro- Galois groups
Full text linkLet be a prime. Following Snopce-Tanushevski, a pro- group is called Frattini-resistant if the function , from the poset of all closed finitely-generated subgroups of into itself, is a poset embedding. We prove that for an oriented right-angled Artin pro- group (oriented pro- RAAG) associated to a directed graph the following four conditions are equivalent: the associated directed graph is of elementary type; is Frattini-resistant; every topologically finitely generated closed subgroup of is an oriented pro- RAAG; is the maximal pro- Galois group of a field containing a root of 1 of order . Also, we conjecture that in the -cohomology of a Frattini-resistant pro- group there are no essential triple Massey products.Final version, as it will appear on JPA
Right-angled Artin groups and enhanced Koszul properties
No full textLet F be a finite field. We prove that the cohomology algebra with coefficients in F of a right-angled Artin group is a strongly Koszul algebra for every finite graph Γ. Moreover, the same algebra is a universally Koszul algebra if, and only if, the graph Γ associated to the right-angled Artin group has the diagonal property. From this we obtain several new examples of pro-p groups, for a prime number p, whose continuous cochain cohomology algebra with coefficients in the field of p elements is strongly and universally (or strongly and non-universally) Koszul. This provides new support to a conjecture on Galois cohomology of maximal prop Galois groups of fields formulated by J. Minac et al
Profinite Groups with a Cyclotomic p-Orientation
Full text linkProfinite 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 GK of a field K is indeed a profinite group with a cyclotomic p-orientation θK,p:GK→Z×p which is even Bloch-Kato. The same is true for its maximal pro-p quotient GK(p) provided the field K contains a primitive pth-root of unity. The class of cyclotomically p-oriented profinite groups (resp. pro-p groups) which are Bloch-Kato is closed with respect to inverse limits, free product and certain fibre products. For profinite groups with a cyclotomic p-orientation the classical Artin-Schreier theorem holds. Moreover, Bloch-Kato pro-p groups with a cyclotomic orientation satisfy a strong form of Tits' alternative, and the elementary type conjecture formulated by I. Efrat can be restated that the only finitely generated indecomposable torsion free Bloch-Kato pro-p groups with a cyclotomic orientation should be Poincaré duality pro-p groups of dimension less or equal to 2
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