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

TL;DR

This paper surveys the $K(1)$ conjecture for Artin groups, providing accessible explanations, historical context, and proofs, including some new results, aimed at non-experts.

Contribution

It compiles and explains key results on the $K(1)$ conjecture for Artin groups, offering new proofs and insights in a self-contained manner.

Findings

Comprehensive overview of the $K(1)$ conjecture for Artin groups

Proofs of many existing results, some new contributions

Accessible presentation for non-experts

Abstract

The purpose of this paper is to put together a large amount of results on the $K(\pi,1)$ conjecture for Artin groups, and to make them accessible to non-experts. Firstly, this is a survey, containing basic definitions, the main results, examples and an historical overview of the subject. But, it is also a reference text on the topic that contains proofs of a large part of the results on this question. Some proofs as well as few results are new. Furthermore, the text, being addressed to non-experts, is as self-contained as possible.