The cohomology algebra conjecture for oriented right-angled Artin groups

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Let Γ\Gamma be a special graph, let GΓG_\Gamma be its associated oriented right-angled Artin group, and let F2(Γ)\mathbb F_2(\Gamma) be the quadratic F2\mathbb F_2-algebra generated by the vertices of Γ\Gamma subject to the defining relations associated with the graph. Cohomology algebra conjecture. The cohomology ring with coefficients in F2\mathbb F_2 of GΓG_\Gamma is isomorphic to F2(Γ)\mathbb F_2(\Gamma):

H∙(GΓ,F2)≅F2(Γ).H^\bullet(G_\Gamma,\mathbb F_2)\cong \mathbb F_2(\Gamma).

The algebra agrees with the usual graph algebra for non-oriented right-angled Artin groups and is proposed as a candidate for the cohomology of the groups associated with special graphs. The source does not provide evidence resolving the assertion in general.

References

Primary source

Simone Blumer, “Teoria Geometrica dei Gruppi Spazi CAT(0), Teorema di Gromov e oriented right-angled Artin groups”, arXiv:2105.04227 (2021).

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