Parabolic subgroups inside parabolic subgroups of Artin groups

Abstract

We prove that a parabolic subgroup P contained in another parabolic subgroup P ′ of an Artin group A is a parabolic subgroup of P ′ . This answers a question of Godelle which is not obvious despite appearances. In order to achieve our result we construct a set-retraction A → P of the inclusion map from a parabolic subgroup P into A. This retraction was implicitly constructed in a previous paper by Charney and the second author.