Right-angled Artin groups and enhanced Koszul properties

Abstract

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.

Groups of p-absolute Galois type that are not absolute Galois groups

“…. Q(i q(Qα) ) are the subsquares of Q α , the list of q(Q α ) relations above satisfies the hypothesis of Proposition 2.1, as their images modulo F (3) give rise to the set 3) .…”

“…Therefore, since a set of defining relations of G gives rise to a basis of the Z/p-vector space R/R p [R, F ], it yields a basis of H 2 (G, Z/p), via the isomorphism trg, as well. Let F (3) be the third term of the descending p-central series of F , i.e.,…”

“…for some a ij ∈ Z/p and y ∈ F (3) , and the exponents a ij are uniquely determined [32, Prop. 3.9.13-(i)].…”

“…Now let G Γ be the pro-p RAAG associated to Γ. By (2.1), H 1 (G Γ , Z/p) has a basis [33, § 3.2] and [3,Thm. 5.1]).…”

Pro-$p$ groups with few relations and universal Koszulity

“…A two-relator pro-p group G is quadratically defined if the cupproduct induces an epimorphism H 1 (G) ⊗2 → H 2 (G), and also a • a = 0 for every a ∈ H 1 (G) in the case p = 2. By Proposition 3.2, G is quadratically defined if and only if r 1 , r 2 ∈ F (2) F (3) for any set of defining relations {r 1 , r 2 } ⊆ F (2) , and also α ii = 0 for every i = 1, . .…”