2020/05/04 by Ilir Snopce, Snopce, Ilir, Zalesskii, Pavel · 2 citations
Mathematics · #12F10 (Primary) 20F36 #12G05 (Secondary) #20E06 #20E08 #20E18 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2005.01685
openalex publication_date 2020/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p be a prime. The right-angled Artin pro-p group GΓ associated to a fnite simplicial graph Γ is the pro-p completion of the right-angled Artin group associated to Γ. We prove that the following assertions are equivalent: (i) no induced subgraph of Γ is a square or a line with four vertices (a path of length 3); (ii) every closed subgroup of GΓ is itself a right-angled Artin pro-p group (possibly infinitely generated); (iii) GΓ is a Bloch-Kato pro-p group; (iv) every closed subgroup of GΓ has torsion free abelianization; (v) GΓ occurs as the maximal pro-p Galois group GK(p) of some field K containing a primitive pth root of unity; (vi) GΓ can be constructed from ℤp by iterating two group theoretic operations, namely, direct products with ℤp 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 GΓ is coherent if and only if each circuit of Γ of length greater than three has a chord.