# The $K(\pi, 1)$ conjecture for affine Artin groups

## TL;DR

This paper summarizes the recent proof of the $K(, 1)$ conjecture for affine Artin groups, showing that certain hyperplane arrangement complements are aspherical, and discusses key combinatorial and topological ideas involved.

## Contribution

It provides a comprehensive overview of the proof of the $K(, 1)$ conjecture for affine Artin groups, highlighting new insights into hyperplane arrangements and Coxeter group combinatorics.

## Key findings

- Affine Artin group complements are aspherical.
- Noncrossing partition posets relate to Coxeter groups.
- Dual Artin groups are isomorphic to standard Artin groups.

## Abstract

In this summary paper, we present the key ideas behind the recent proof of the $K(\pi, 1)$ conjecture for affine Artin groups, which states that complements of locally finite affine hyperplane arrangements with real equations and stable under orthogonal reflections are aspherical. We survey three facets of the argument: the combinatorics of noncrossing partition posets associated with Coxeter groups; the appearance of dual Artin groups and the question of their isomorphism with standard Artin groups; the topological models and their interplay in the proof.