For each Artin group we compute the reduced `cohomology of (the universal cover of) its “Salvetti complex”. This is a CW-complex which is conjectured to be a model for the classifying space of the Artin group. In the many cases when this conjecture is known to hold our calculation describes the reduced `-cohomology of the Artin group.

In this paper, we consider the kernel of the natural projection from the Artin group of dihedral type I2(k) to the corresponding Coxeter group, which we call the pure Artin group of dihedral type. We present a rational function expression for the spherical growth series of the pure Artin group of dihedral type with respect to a natural generating set.

Recently, right-angled Artin groups have attracted much attention in geometric group theory. They have a rich structure of subgroups and nice algorithmic properties, and they give rise to cubical complexes with a variety of applications. This survey article is meant to introduce readers to these groups and to give an overview of the relevant literature. Artin groups span a wide range of groups ...

For a local field of characteristic p > 0, K, the combination of local class field theory and Artin-Schreier-Witt theory yield what is known as the Schmid-Witt symbol. The symbol encodes interesting data about the ramification theory of p-extensions of K and we can, for example, use it to compute the higher ramification groups of such extensions. In 1936, Schmid discovered an explicit formula f...

We construct finite complete rewriting systems for two large classes of Artin groups: those of finite type, and those whose defining graphs are based on trees. The constructions in the two cases are quite different; while the construction for Artin groups of finite type uses normal forms introduced through work on complex hyperplane arrangements, the rewriting systems for Artin groups based on ...

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 ver...

In the present work, we give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Also, we show that right-angled Artin groups are conjugacy separable. Moreover, we investigate the profinite topology of F 2 × F 2 and of the group L in [22], which are the only obstructions for the subgroup separability of th...

We prove that the natural homomorphism from an Artin monoid to its associated Artin group is always injective. AMS Subject Classification: Primary 20F36; Secondary 20F55.

The irreducible euclidean Coxeter groups that naturally act geometrically on euclidean space are classified by the well-known extended Dynkin diagrams and these diagrams also encode the modified presentations that define the irreducible euclidean Artin groups. These Artin groups have remained mysterious with some exceptions until very recently. Craig Squier clarified the structure of the three ...

In this handout we shall discuss a result that was conjectured by Emil Artin in 1935 and proved shortly thereafter by Claude Chevalley. A refinement was given by Artin’s graduate student Ewald Warning, who, as the story goes, was the one whom Artin had intended to prove the theorem before Chevalley came visiting Göttingen and got Artin to say a little too much about the mathematics his student ...