Parabolic subgroups inside parabolic subgroups of Artin groups

TL;DR

The paper proves that parabolic subgroups within larger parabolic subgroups of Artin groups are themselves parabolic, resolving a previously open question using a novel set-retraction construction.

Contribution

It establishes that parabolic subgroups contained in other parabolic subgroups are themselves parabolic, providing a new proof via a set-retraction method.

Findings

Proves parabolic subgroups inside larger parabolic subgroups are parabolic.

Constructs a set-retraction from the Artin group to a parabolic subgroup.

Answers Godelle's open question about the structure of parabolic subgroups.

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 $A \to P$ 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.