Classifying spaces and geometric obstructions for Artin groups. The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin $K(\pi,1)$ 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.
released 2026-09-23 | 5 theorems · 36 lemmas · 49 proofs · 31,611 words |
PLAY LEVEL 1 »(pdf)
We prove that the standard Salvetti complex of every Artin group with finitely many standard generators is aspherical. This resolves the Artin $K(\pi,1)$ conjecture in finite rank.
released 2026-09-23 | 1 theorem · 19 lemmas · 23 proofs · 12,980 words |
PLAY LEVEL 2 »(pdf)
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.
released 2026-09-23 | 3 theorems · 36 lemmas · 55 proofs · 27,261 words |
PLAY LEVEL 3 »(pdf)
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.