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RESEARCH PRODUCT

Parabolic Subgroups of Artin Groups

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Abstract Let ( A , Σ) be an Artin system. For X  ⊆ Σ, we denote by A X the subgroup of A generated by X . Such a group is called a parabolic subgroup of A . We reprove Van der Lek's 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. Let A be a finite type Artin group, and let A X be a parabolic subgroup with connected associated Coxeter graph. The quasi-centralizer of A X in A is the set of β in A such that β X β −1  =  X . We prove that the commensurator of A X in A is equal to the normalizer of A X in A , and that this group is generated by A X and the quasi-centralizer of A X in A . Moreover, if A X is not of type D l ( l  ≥ 4 and l even), then this group is generated by A X and the centralizer of A X in A .