A retraction from a group onto a subgroup is a homomorphism that acts as the identity on the subgroup — it “projects” the whole group onto the subgroup. For Artin groups, parabolic subgroups are generated by subsets of the standard generators, and the question is: which Artin groups admit retractions onto all their parabolic subgroups?
Cisneros de la Cruz, Cumplido, Foniqi, and Paris classify these completely. The analysis centers on “triangular subgroups” — rank-3 parabolic subgroups generated by three standard generators — because these are where the constraints become visible. The existence of a retraction onto a dihedral (rank-2) parabolic subgroup forces specific relationships between generators, and these relationships compound when three generators interact.
The key finding is a collapse: when an Artin group admits retractions onto all parabolic subgroups (allowing the most general homomorphisms), it automatically admits “ordinary” retractions — those that send each standard generator either to another standard generator or to the identity. The generality of the retraction buys you nothing. The most permissive definition of retraction and the most restrictive definition give the same class of groups.
This is a rigidity result. You might expect that allowing retractions to send generators to arbitrary group elements would expand the class of groups admitting them — more flexibility in the homomorphism should mean more groups qualify. Instead, the constraints propagate through the triangular subgroups and force the retraction into the rigid form. The group's presentation, specifically the braid relations between generators, is so constraining that the only way to retract onto a parabolic subgroup is the obvious way.