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

We prove the $$K(\pi ,1)$$ K ( π , 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.

Combinatorics of extended affine root systems (type A1)

We establish extensions of some important features of affine theory to affine reflection systems (extended affine root systems) of type [Formula: see text]. We present a positivity theory which decomposes in a natural way the nonisotropic roots into positive and negative roots, then using that, we give an extended version of the well-known exchange condition for the corresponding Weyl group, and finally give an extended version of the Bruhat ordering and the [Formula: see text]-Lemma. Furthermore, a new presentation of the Weyl group in terms of the parity permutations is given, this in turn leads to a parity theorem which gives a characterization of the reduced words in the Weyl group. All root systems involved in this work appear as the root systems of certain well-studied Lie algebras.

ROOT SYSTEMS ARISING FROM AUTOMORPHISMS

We study a combinatorial approach of producing new root systems from the old ones in the context of affine root systems and their new generalizations. The appearance of this approach in the literature goes back to the outstanding work of Kac in the realization of affine Kac–Moody Lie algebras. In recent years, this approach has been appeared in many other works, including the study of affinization of extended affine Lie algebras and invariant affine reflection algebras.