Crystallographic actions on Lie groups and post-Lie algebra structures

This survey on crystallographic groups, geometric structures on Lie groups and associated algebraic structures is based on a lecture given in the Ostrava research seminar in 2017.

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.