The Mysterious Geometry of Artin Groups

Artin groups are easily defined but most of them are poorly understood. In this survey I try to highlight precisely where the problems begin. The first part reviews the close connection between Coxeter groups and Artin groups as well as the associated topological spaces used to investigate them. The second part describes the location of the border between the Artin groups we understand at a very basic level and those that remain fundamentally mysterious. The third part highlights those collections of Artin groups (and their relatives) that are not currently understood but which we are likely to understand sometime soon. Artin groups, also known as Artin-Tits groups, are easy to define via presentations but they are often very poorly understood. In this article, when I say that we “understand” a particular group, what I mean is do we know how to solve its word problem. As is well-known, this is equivalent to being able to construct arbitrarily large portions of its Cayley graph and arbitrarily large portions of the universal cover of its presentation 2-complex. It is in this sense that most Artin groups are poorly understood. For most Artin groups we do not know how to solve the word problem. In this survey I try to highlight exactly where the problems begin. It is divided into three parts and each part corresponds to one of the lectures in the short course I gave at the winter braids conference in Caen in early March 2017. The first part reviews the close connection between Coxeter groups and Artin groups with a particular focus on the associated topological spaces used to investigate them. The second part summarizes exactly which collections of Artin groups we understand at a basic level and which ones remain mysterious. And the third part highlights those collections of Artin groups (and their relatives) that are not currently understood but which we are likely to understand in the near future. In keeping with the informal nature of this survey, I have retained some of the blackboard images used in my talks. Date: October 9, 2017. 2010 Mathematics Subject Classification. 20F36, 20F55.