Right-angled Artin groups and enhanced Koszul properties

Cassella, Alberto; Quadrelli, Claudio

doi:10.48550/arxiv.1907.03824

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2020

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“…(vi) =⇒ (v). This implication is proved implicitly in the proof of [4,Proposition 5.8]. Indeed, it follows from the following two facts.…”

Section: Resultsmentioning

confidence: 81%

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Right-angled Artin pro-$p$ groups

Snopce¹,

Zalesskiĭ²

2020

Preprint

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Let p be a prime. The right-angled Artin pro-p group G Γ 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 G Γ is itself a right-angled Artin pro-p group (possibly infinitely generated); (iii) G Γ is a Bloch-Kato pro-p group; (iv) every closed subgroup of G Γ has torsion free abelianization; (v) G Γ occurs as the maximal pro-p Galois group G K (p) of some field K containing a primitive pth root of unity; (vi) G Γ can be constructed from Z p by iterating two group theoretic operations, namely, direct products with 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 rightangled Artin pro-p groups. Moreover, we prove that G Γ is coherent if and only if each circuit of Γ of length greater than three has a chord.

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“…(vi) =⇒ (v). This implication is proved implicitly in the proof of [4,Proposition 5.8]. Indeed, it follows from the following two facts.…”

Section: Resultsmentioning

confidence: 81%

“…Quadrelli and Weigel conjectured that G L 3 does not occur as G K (p) as well (cf. [4]); clearly, this would follow from Conjecture 1.1.…”

Section: Introductionmentioning

confidence: 93%