Parabolic subgroups inside parabolic subgroups of Artin groups

Abstract

We prove that a parabolic subgroup P P contained in another parabolic subgroup P ′ P’ of an Artin group A A is a parabolic subgroup of P ′ 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 A \to P of the inclusion map from a parabolic subgroup P P into A 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].