Classification of Artin groups admitting retractions onto their parabolic subgroups

TL;DR

This paper classifies Artin groups that can be retracted onto all their parabolic subgroups, revealing structural properties and establishing conditions for such retractions to exist.

Contribution

It provides a complete classification of Artin groups with retractions onto parabolic subgroups, including a detailed analysis of homomorphisms between dihedral Artin groups.

Findings

Artin groups admitting retractions also admit ordinary retractions.

Classification of homomorphisms between dihedral Artin groups.

Characterization of triangular subgroups in Artin groups.

Abstract

We classify the Artin groups that admit retractions onto all of their parabolic subgroups. Our approach relies on a detailed analysis of triangular subgroups, with a key ingredient being the classification of homomorphisms between dihedral Artin groups that map one of the standard generators to a standard generator. As a consequence, we show that whenever an Artin group admits retractions to parabolic subgroups, it also admits ordinary ones - that is, retractions that send each standard generator either to a standard generator or to the identity.