Parabolic subgroups inside parabolic subgroups of Artin groups

Abstract

We prove that a parabolic subgroup $P$ contained in another parabolic subgroup $P^{\prime }$ of an Artin group $A$ is a parabolic subgroup of $P^{\prime }$. This answers a question of Godelle which is not obvious despite appearances. In order to achieve our result we construct a set-retraction $A\to P$ 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.

8 pages. Accepted for publication in Proc. Amer. Math. Soc