La conjecture du $K(\pi,1)$ pour les groupes d'Artin affines (d'apr\`es Paolini et Salvetti)

TL;DR

This paper discusses the proof of the $K(\pi,1)$ conjecture for affine Artin groups, showing they have finite classifying spaces, based on recent advances by Paolini and Salvetti.

Contribution

It presents key elements of the proof of the $K(\pi,1)$ conjecture for affine Artin groups, highlighting dual Garside structures and shellability of noncrossing partitions.

Findings

Affine Artin groups have a finite classifying space.

The proof involves dual Garside structures and Euclidean isometries.

Shellability of noncrossing partitions is established.

Abstract

Consider an affine Coxeter group $W$ acting by isometries on the Euclidean space $\mathbb{R}^n$, and the arrangement of its reflection hyperplanes. The fundamental group of the complement $Y_W$ of the complexification of this arrangement in $\mathbb{C}^n$ mod out by $W$ is the affine Artin group $G_W$ associated with $W$. The $K(\pi,1)$ conjecture states that $Y_W$ is a classifying space for $G_W$. It has been recently proved by Paolini and Salvetti building on the works of McCammond and Sulway. We will present some ingredients of the proof that rests on the study of dual Garside structures for affine Artin groups, the factorisations of Euclidean isometries, and the shellability of noncrossing partitions. One consequence is that affine Artin groups, as well as braided crystallographic groups, have a finite classifying space.