A Garside monoid is a cancellative monoid with a finite lattice generating set; a Garside group is the group of fractions of a Garside monoid. The family of Garside groups contains the Artin-Tits groups of spherical type. We generalise the well-known notion of a parabolic subgroup of an Artin-Tits group into that of a parabolic subgroup of a Garside group. We also define the more general notion...

The braid group of n-strings is the group of homotopy types of movements of n distinct points in the 2-plane R. It was introduced by E. Artin [1] in 1926 in order to study knots in R. He gave a presentation of the braid group by generators and relations, which are, nowadays, called the Artin braid relations. Since then, not only in the study of knots, the braid groups appear in several contexts...

The result of this paper is the determination of the cohomology of Artin groups of type An, Bn and Ãn with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type Bn with coefficients over the module Q[q±1, t±1]. Here the first n− 1 standard generators of the group act by (−q)-multiplication, while the last one acts by (−t)-multipl...

We prove that any Coxeter group that is not virtually free contains a surface group. In particular if the Coxeter group is word hyperbolic and not virtually free this establishes the existence of a hyperbolic surface group, and answers in the affirmative a question of Gromov in this setting. We also discuss when Artin groups contain hyperbolic surface groups. 2000 Mathematics Subject Classifica...

The action dimension of a discrete group G is the minimum dimension of a contractible manifold, which admits a proper G-action. In this dissertation, we study the action dimension of general Artin groups. The main result is that the action dimension of an Artin group with the nerve L of dimension n for n 6= 2 is less than or equal to (2n+1) if the Artin group satisfies the K(π, 1)-Conjecture an...

These are notes for a course offered at Yale University in the spring semester of 2013.

In this paper we construct a gathering process by the means of which we obtain new normal forms in braid groups. The new normal forms generalise Artin-Markoff normal forms and possess an extremely natural geometric description. In the two last sections of the paper we discuss the implementation of the introduced gathering process and the questions that arose in our work. This discussion leads u...

The purpose of this paper is to put together a large amount of results on the K(π, 1) conjecture for Artin groups, and to make them accessible to non-experts. Firstly, this is a survey, containing basic definitions, the main results, examples and an historical overview of the subject. But, it is also a reference text on the topic that contains proofs of a large part of the results on this quest...

Recent results on the linearity of braid groups are extended in two ways. We generalize the Lawrence Krammer representation as well as Krammer’s faithfulness proof for this linear representation to Artin groups of finite type.

1. Introduction. By a labeled graph we shall mean a finite (non-empty) graph Γ, without loops or multiple edges, each of whose edges is labeled by an integer greater than or equal to 2. Let the vertices of Γ be s 1 , s 2 ,. .. , s n , and let the label on an edge with endpoints s i and s j be m ij ≥ 2. Define ab m to be the word abab. .. of length m. Then the Artin group AΓ associated with the ...