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

## TL;DR

This paper proves the long-standing $K(\pi,1)$ conjecture for affine Artin groups by demonstrating that the complexified complement of an affine reflection arrangement is a classifying space, advancing understanding of affine Coxeter and Artin groups.

## Contribution

It establishes the $K(\pi,1)$ property for affine Artin groups using new combinatorial and topological constructions, including EL-shellability of noncrossing partition posets and finite classifying spaces.

## Key findings

- Affine noncrossing partition posets are EL-shellable
- Finite classifying spaces constructed for dual affine Artin groups
- New CW models for orbit configuration spaces

## 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.