Conjecture on the Moussong metric for virtual Deligne complexes

Let Γ\Gamma be a Coxeter graph, and let vDΓ\mathrm{vD}_{\Gamma} be the virtual Deligne complex associated with the virtual Artin group VA[Γ]\mathrm{VA}[\Gamma]. Equip vDΓ\mathrm{vD}_{\Gamma} with its Moussong metric. Virtual Deligne CAT(0) conjecture. For any Coxeter graph Γ\Gamma, the virtual Deligne complex vDΓ\mathrm{vD}_{\Gamma}, equipped with the Moussong metric, is CAT(0). The conjecture is proved in this paper for locally reducible Coxeter graphs, including 2-dimensional graphs and graphs with no label equal to 33, but remains open for general Coxeter graphs.

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Primary source

Federica Gavazzi and Alexandre Martin, “A Deligne complex for virtual Artin groups”, arXiv:2607.24367 (2026).

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