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 [Bull. Lond. Math. Soc. 46 (2014), pp. 1248–1255].