OpenAI research collection · 3 manuscripts · Source checked 2026-10-19
The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin K(π,1) conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture. An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT(0) space.
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We prove that the standard Salvetti complex of every Artin group with finitely many standard generators is aspherical. This resolves the Artin K(π,1) conjecture in finite rank.
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We construct an Artin group on 116 generators that admits no proper, cocompact isometric action on a nonempty proper CAT(0) space. This refutes the CAT(0) conjecture for Artin groups.
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We prove that every intersection of parabolic subgroups of any finite-rank Artin group, with arbitrary finite or infinite Coxeter labels, is parabolic. This resolves the Parabolic Intersection Conjecture affirmatively.
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