An introduction to the geometric and combinatorial group theory of Artin groups

TL;DR

This paper provides an overview of the geometric and combinatorial aspects of Artin groups, discussing the $K(,1)$ conjecture and summarizing recent results as of January 2024.

Contribution

It introduces the $K(,1)$ conjecture for Artin groups and surveys the current state of research, serving as a comprehensive introductory resource.

Findings

Survey of known results as of January 2024

Introduction of the $K(,1)$ conjecture

Connection to triangulated categories and stability conditions

Abstract

We give a brief introduction to the geometric and combinatorial group theory of Artin groups. In particular we introduce the $K(\pi,1)$ conjecture for Artin groups and survey known results as of January 2024. These notes were written as companion notes for the MFO mini-workshop 2405a "Artin groups meet triangulated categories" alongside Edmund Heng's notes "Introduction to stability conditions and its relation to the $K(\pi,1)$ conjecture for Artin groups".