The Structure of Euclidean Artin Groups

The Coxeter groups that act geometrically on euclidean space have long been classified and presentations for the irreducible ones are encoded in the well-known extended Dynkin diagrams. The corresponding Artin groups are called euclidean Artin groups and, despite what one might naively expect, most of them have remained fundamentally mysterious for more than forty years. Recently, my coauthors and I have resolved several longstanding conjectures about these groups, proving for the first time that every irreducible euclidean Artin group is a torsion-free centerless group with a decidable word problem and a finite-dimensional classifying space. This article surveys our results and the techniques we use to prove them. The reflection groups that act geometrically on spheres and euclidean spaces are all described by presentations of an exceptionally simple form and general Coxeter groups are defined by analogy. These spherical and euclidean Coxeter groups have long been classified and their presentations are encoded in the well-known Dynkin diagrams and extended Dynkin diagrams, respectively. Artin groups are defined by modified versions of these Coxeter presentations, and they were initially introduced to describe the fundamental group of a space constructed from the complement of the hyperplanes in a complexified version of the reflection arrangement for the corresponding spherical or euclidean Coxeter group. The most basic example of a Coxeter group is the symmetric group and the corresponding Artin group is the braid group, the fundamental group of a quotient of the complement of a complex hyperplane arrangement called the braid arrangement. The spherical Artin groups, that is the Artin groups corresponding to the Coxeter groups acting geometrically on spheres, have been well understood ever since Artin groups themselves were introduced by Pierre Deligne [Del72] and by Brieskorn and Saito [BS72] in adjacent articles in the Inventiones in 1972. One might have expected the euclidean Artin groups to be the next class of Artin groups whose Date: December 14, 2013.