The dual approach to the $K(\pi, 1)$ conjecture

TL;DR

This paper discusses the dual approach to Coxeter groups that led to proving the $K( ext{pi}, 1)$ conjecture for affine Artin groups, exploring its potential extension to broader classes.

Contribution

It details the techniques used to prove the $K( ext{pi}, 1)$ conjecture for affine Artin groups and raises open questions about extending these methods.

Findings

Proof of the $K( ext{pi}, 1)$ conjecture for affine Artin groups

Techniques for dual presentations of Coxeter groups

Open questions on generalizing the approach

Abstract

Dual presentations of Coxeter groups have recently led to breakthroughs in our understanding of affine Artin groups. In particular, they led to the proof of the $K(\pi, 1)$ conjecture and to the solution of the word problem. Will the "dual approach" extend to more general classes of Coxeter and Artin groups? In this paper, we describe the techniques used to prove the $K(\pi, 1)$ conjecture for affine Artin groups and we ask a series of questions that are mostly open beyond the spherical and affine cases.