where the words on each side of these relations are sequences of mij letters where ai and aj alternate in the sequence. The matrix of values mij is a Coxeter matrix M = (mij)i,j∈I on I. These groups generalize the braid groups established in 1925 by E. Artin in a natural way and therefore we suggest naming them Artin groups. If one adds the relations ai = 1 to the relations in the presentation ...

We define a right-angled mock reflection group to be a group G acting combinatorially on a CAT(0) cubical complex such that the action is simply-transitive on the vertex set and all edge-stabilizers are Z2. We give a combinatorial characterization of these groups in terms of graphs with local involutions. Any such graph Γ not only determines a mock reflection group, but it also determines a rig...

We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we are able to develop a Nielsen–Thurston classification for elements in the rig...

We provide a new presentation for the annular braid group. The annular braid group is known to be isomorphic to the finite type Artin group with Coxeter graph Bn. Using our presentation, we show that the annular braid group is a semidirect product of an infinite cyclic group and the affine Artin group with Coxeter graph Ãn−1. This provides a new example of an infinite type Artin group which inj...

The braid group Bn, endowed with Artin’s presentation, admits two distinguished involutions. One is the anti-automorphism rev : Bn → Bn, v 7→ v̄, defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation τ : x 7→ ∆x∆ by the generalized half-twist (Garside element). More generally, the involution rev is defined for all Artin group...

In [12], I. Kapovich and P. Schupp showed that certain 2-dimensional Artin groups (those with all relator indices at least 7) are hyperbolic relative (in the sense of Farb) to their non-free rank 2 parabolic subgroups. This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of the associated Deligne complex. We prove a relative version of the Mi...

In this paper, a connection between rewriting systems and embedding of monoids in groups is found. We show that if a group with a positive presentation has a complete rewriting system R that satisfies the condition that each rule in R with positive left-hand side has a positive right-hand side, then the monoid presented by the subset of positive rules from R embeds in the group. As an example, ...