Huang's relative Artin complex conjecture

Let AΓA_\Gamma be an Artin group with defining graph Γ\Gamma, and let ΔS,S\Delta_{S,S'} denote the relative Artin complex spanned by vertices of types indexed by SS'. Suppose Γ\Gamma is a complete graph with vertex set SS, and TST\subset S is such that ATA_T is almost spherical. If Γ\Gamma and the subgraph spanned by TT are both connected and free of infinity, Huang's conjecture.

ΔS,S.\Delta_{S,S'}\simeq *.

Here ATA_T being almost spherical means that its defining graph is free of infinity, ATA_T is not finite type, and every proper standard subgroup AUA_U with UTU\subsetneq T is finite type. The conjecture is stated as equivalent to the K(π,1)K(\pi,1) conjecture in the cited work. Elias and Williamson initially claimed that it had been proved, but retracted that claim, so it remains open.

Sources & referencesView supporting material

Primary source

Rachael Boyd, “An introduction to the geometric and combinatorial group theory of Artin groups”, arXiv:2601.08658 (2026).

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