Parabolic Subgroups of Artin Groups Original Research Article

Let (A, Σ) be an Artin system. ForX subset of or equal to Σ, we denote byAXthe subgroup ofAgenerated byX. Such a group is called a parabolic subgroup ofA. We reprove Van der Lekʹs theorem: “a parabolic subgroup of an Artin group is an Artin group.” We give an algorithm which decides whether two parabolic subgroups of an Artin group are conjugate. LetAbe a finite type Artin group, and letAXbe a parabolic subgroup with connected associated Coxeter graph. The quasi-centralizer ofAXinAis the set of β inAsuch that βXβ−1 = X. We prove that the commensurator ofAXinAis equal to the normalizer ofAXinA, and that this group is generated byAXand the quasi-centralizer ofAXinA. Moreover, ifAXis not of typeDl(l ≥ 4 andleven), then this group is generated byAXand the centralizer ofAXinA.