Hereditarily-phthagorean fields and orderings of higher level

The main purpose of this article is to study pro-$p$ groups with quadratic $\mathbb{F}_p$-cohomology algebra, i.e. $H^\bullet$-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 $H^\bullet$-quadratic pro-$p$ groups is $H^\bullet$-quadratic, under certain necessary conditions. Moreover, we introduce and investigate a new family of pro-$p$ groups that yields many new examples of $H^\bullet$-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 $H^\bullet$-quadratic pro-$p$ groups.

On pro-$p$ groups with quadratic cohomology

Let $p$ be a prime. The right-angled Artin pro-$p$ group $G_{\Gamma}$ associated to a fnite simplicial graph $\Gamma$ is the pro-$p$ completion of the right-angled Artin group associated to $\Gamma$. We prove that the following assertions are equivalent: (i) no induced subgraph of $\Gamma$ is a square or a line with four vertices (a path of length 3); (ii) every closed subgroup of $G_{\Gamma}$ is itself a right-angled Artin pro-$p$ group (possibly infinitely generated); (iii) $G_{\Gamma}$ is a Bloch-Kato pro-$p$ group; (iv) every closed subgroup of $G_{\Gamma}$ has torsion free abelianization; (v) $G_{\Gamma}$ occurs as the maximal pro-$p$ Galois group $G_K(p)$ of some field $K$ containing a primitive $p$th root of unity; (vi) $G_{\Gamma}$ can be constructed from $\mathbb{Z}_p$ by iterating two group theoretic operations, namely, direct products with $\mathbb{Z}_p$ and free pro-$p$ products. This settles in the affirmative a conjecture of Quadrelli and Weigel. Also, we show that the Smoothness Conjecture of De Clercq and Florens holds for right-angled Artin pro-$p$ groups. Moreover, we prove that $G_{\Gamma}$ is coherent if and only if each circuit of $\Gamma$ of length greater than three has a chord.

Right-angled Artin pro-$p$ groups

We prove that Galois cohomology satisfies several surprisingly strong versions of Koszul properties, under a well known conjecture, in the finitely generated case. In fact, these versions of Koszulity hold for all finitely generated maximal pro-p quotients of absolute Galois groups which are currently understood. We point out several of these unconditional results which follow from our work. We show how these enhanced versions are preserved under certain natural operations on algebras, generalising several results that were previously established only in the commutative case. The subject matter in this paper contains topics which are used in various branches of algebra and computer science.

Enhanced Koszul properties in Galois cohomology

Let p$p$ be a prime. The right‐angled Artin pro‐ p$p$ group GΓ$G_{\Gamma }$ associated to a finite simplicial graph Γ$\Gamma$ is the pro‐ p$p$ completion of the right‐angled Artin group associated to Γ$\Gamma$ . We prove that the following assertions are equivalent: (i) no induced subgraph of Γ$\Gamma$ is a square or a line with four vertices (a path of length 3); (ii) every closed subgroup of GΓ$G_{\Gamma }$ is itself a right‐angled Artin pro‐ p$p$ group (possibly infinitely generated); (iii) GΓ$G_{\Gamma }$ is a Bloch–Kato pro‐ p$p$ group; (iv) every closed subgroup of GΓ$G_{\Gamma }$ has torsion free Abelianization; (v) GΓ$G_{\Gamma }$ occurs as the maximal pro‐ p$p$ Galois group GK(p)$G_K(p)$ of some field K$K$ containing a primitive p$p$ th root of unity; (vi) GΓ$G_{\Gamma }$ can be constructed from Zp$\mathbb {Z}_p$ by iterating two group theoretic operations, namely, direct products with Zp$\mathbb {Z}_p$ and free pro‐ p$p$ products. This settles in the affirmative a conjecture of Quadrelli and Weigel. Also, we show that the Smoothness Conjecture of De Clercq and Florence holds for right‐angled Artin pro‐ p$p$ groups. Moreover, we prove that GΓ$G_{\Gamma }$ is coherent if and only if each circuit of Γ$\Gamma$ of length greater than three has a chord.

Right‐angled Artin pro‐ p$p$ groups

We show that a finitely generated abelian group G of torsion free rank n ≥ 1 admits a n + r dimensional model for EFrG, where Fr is the family of subgroups of torsion-free rank less than or equal to r ≥ 0.

https://www.intlpress.com/site/pub/files/_fulltext/journals/hha/2017/0019/0002/HHA-2017-0019-0002-a005.pdf

On the dimension of classifying spaces for families of abelian subgroups

On the joins of group rings

We study the spectrum of the join of several circulant matrices. We apply our results to compute explicitly the spectrum of certain graphs obtained by joining several circulant graphs.

Joins of circulant matrices

. We study the spectrum of the join of several normal matrices with constant row sums. We apply our results to compute the characteristic polynomial of the join of several regular graphs. We then use this calculation to study several problems in spectral graph theory. In particular, we provide some simple constructions of Ramanujan graphs and give new proofs for some theorems in [4].

On the spectrum of the joins of normal matrices and applications

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