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

Rachael Boyd

TL;DR

This work surveys the geometric and combinatorial framework surrounding Artin groups, focusing on the central K(π,1) conjecture and its many reformulations via hyperplane arrangements, the Salvetti complex, Deligne and extended Deligne complexes, and the classifying spaces of Artin monoids. It catalogues the major progress across families (finite type, FC-type, 2D/locally reducible, large type, Euclidean/affine) and highlights recent dual-Garside approaches that underpin affine and rank-3 results, as well as newer methods involving relative Artin complexes. The notes also outline foundational techniques—Garside theory, dual Garside structures, and union-of-chambers arguments—that drive proofs of asphericity and contractibility of key complexes. Together, these elements provide a cohesive map of how algebraic, combinatorial, and topological methods interlink to establish (and in some cases, approach) the K(π,1) conjecture for broad classes of Artin groups, with implications for torsion, centers, and algorithmic questions. The work emphasizes that resolving any of the equivalent formulations yields broad structural consequences and clarifies the landscape of remaining open problems and conjectural directions.

Abstract

We give a brief introduction to the geometric and combinatorial group theory of Artin groups. In particular we introduce the $K(π,1)$ 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 $K(π,1)$ conjecture for Artin groups".

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

TL;DR

This work surveys the geometric and combinatorial framework surrounding Artin groups, focusing on the central K(π,1) conjecture and its many reformulations via hyperplane arrangements, the Salvetti complex, Deligne and extended Deligne complexes, and the classifying spaces of Artin monoids. It catalogues the major progress across families (finite type, FC-type, 2D/locally reducible, large type, Euclidean/affine) and highlights recent dual-Garside approaches that underpin affine and rank-3 results, as well as newer methods involving relative Artin complexes. The notes also outline foundational techniques—Garside theory, dual Garside structures, and union-of-chambers arguments—that drive proofs of asphericity and contractibility of key complexes. Together, these elements provide a cohesive map of how algebraic, combinatorial, and topological methods interlink to establish (and in some cases, approach) the K(π,1) conjecture for broad classes of Artin groups, with implications for torsion, centers, and algorithmic questions. The work emphasizes that resolving any of the equivalent formulations yields broad structural consequences and clarifies the landscape of remaining open problems and conjectural directions.

Abstract

We give a brief introduction to the geometric and combinatorial group theory of Artin groups. In particular we introduce the 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 conjecture for Artin groups".
Paper Structure (23 sections, 9 theorems, 19 equations)

This paper contains 23 sections, 9 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Open questions
  3. The $K(\pi,1)$ conjecture
  4. Statement of the conjecture
  5. The canonical representation
  6. Alternative restatements of the conjecture
  7. The Salvetti complex
  8. The Deligne complex
  9. The extended Deligne complex
  10. The classifying space of the Artin monoid
  11. Overview
  12. Progress on the conjecture
  13. Finite type Artin groups
  14. FC-type Artin groups
  15. 2-Dimensional and locally reducible Artin groups
  16. ...and 8 more sections

Key Result

theorem 1.1

A Coxeter group $W_\Gamma$ is finite if and only if $\Gamma$ is a disjoint union of finitely many of the following connected graphs. \xymatrix@R=3mm@C=2mm{ & \textrm{Infinite families} & & & \textrm{Exceptional groups}\\ {\bf A}_n \,\, (n\geq 1) & \begin{tikzpicture}[scale=0.15, baseline=0

Theorems & Definitions (29)

  • theorem 1.1: Classification of finite Coxeter groups, Coxeter Coxeter1935
  • definition 2.1
  • remark 2.3
  • theorem 2.5: Tits1961
  • remark 2.6
  • definition 2.7: Non-singular Tits cone
  • definition 2.8: Tits cone; general case
  • remark 2.9
  • theorem 2.11: Van der Lek VanderLek1983
  • conjecture 2.12: Arnol'd, Brieskorn, Pham, Thom
  • ...and 19 more