Proof of the $K(\pi ,1)$ conjecture for affine Artin groups

Abstract

We prove the $K(\pi ,1)$ conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space. This is a long-standing problem, due to Arnol'd, Pham, and Thom. Our proof is based on recent advancements in the theory of dual Coxeter and Artin groups, as well as on several new results and constructions. In particular: we show that all affine noncrossing partition posets are EL-shellable; we use these posets to construct finite classifying spaces for dual affine Artin groups; we introduce new CW models for the orbit configuration spaces associated with arbitrary Coxeter groups; we construct finite classifying spaces for the braided crystallographic groups introduced by McCammond and Sulway.

References in corpus (6)

• The K(π, 1) problem for the affine Artin group of type \widetilde{B}_n and its cohomology
• Weighted sheaves and homology of Artin groups
• Cohomology of Artin groups of type tilde{A}_n, B_n and applications
• Some topological problems on the configuration spaces of Artin and Coxeter groups
• On the classifying space of Artin monoids
• On the local homology of Artin groups of finite and affine type

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• Curve graphs for Artin-Tits groups of type $B$, $\tilde{A}$ and $\tilde{C}$ are hyperbolic
• CAT(0) and cubulated Shephard groups
• Modular flats of oriented matroids and poset quasi-fibrations
• A Deligne complex for Artin Monoids
• On aspherical configuration Lie groupoids
• The $K(\pi ,1)$ conjecture for Artin groups of spherical type
• Some virtually poly-free Artin groups
• The $K(\pi ,1)$ conjecture and acylindrical hyperbolicity for relatively extra-large Artin groups