A 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 θabelian. (See Thm B.) In case that G is also finitely generated, this implies that G is powerful, p-adic analytic with d(G) = cd(G), and its Fp-cohomology ring is an exterior algebra (see Cor. 4.8). These results will be obtained by studying locally powerful groups (see Thm A). 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 (see Corollary 4.9). Finally, we study certain closure operations of the class of Bloch-Kato pro-p groups, connected with the Elementary type conjecture.
Let 𝔽 be a finite field. We prove that the cohomology algebra H^{\bullet}(G_{\Gamma},\mathbb{F}) with coefficients in 𝔽 of a right-angled Artin group G_{\Gamma} is a strongly Koszul algebra for every finite graph Γ. Moreover, H^{\bullet}(G_{\Gamma},\mathbb{F}) is a universally Koszul algebra if, and only if, the graph Γ associated to the group G_{\Gamma} 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 pro-𝑝 Galois groups of fields formulated by J. Mináč et al.
Fix an odd prime p p , and let F F be a field containing a primitive p p th root of unity. It is known that a p p -rigid field F F is characterized by the property that the Galois group G F ( p ) G_F(p) of the maximal p p -extension F ( p ) / F F(p)/F is a solvable group. We give a new characterization of p p -rigidity which says that a field F F is p p -rigid precisely when two fundamental canonical quotients of the absolute Galois groups coincide. This condition is further related to analytic p p -adic groups and to some Galois modules. When F F is p p -rigid, we also show that it is possible to solve for the roots of any irreducible polynomials in F [ X ] F[X] whose splitting field over F F has a p p -power degree via non-nested radicals. We provide new direct proofs for hereditary p p -rigidity, together with some characterizations for G F ( p ) G_F(p) – including a complete description for such a group and for the action of it on F ( p ) F(p) – in the case F F is p p -rigid.
The main purpose of this article is to study pro-p groups with quadratic Fp-cohomology algebra, i.e. H • -quadratic pro-p groups. Prime examples of such groups are the maximal Galois pro-p groups of fields containing a primitive root of unity of order p.We show that the amalgamated free product and HNN-extension of H • -quadratic pro-p groups is H • -quadratic, under certain necessary conditions. Moreover, we introduce and investigate a new family of pro-p groups that yields many new examples of H • -quadratic groups: p-RAAGs. These examples generalise right angled Artin groups in the category of pro-p groups. Finally, we explore "Tits alternative behaviour" of H • -quadratic pro-p groups.
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