On Artin groups admitting retractions to parabolic subgroups

TL;DR

This paper extends the concept of retractions to broader classes of Artin groups, enabling new insights into their subgroup intersections and coherence properties.

Contribution

It generalizes retractions to FC-type and other Artin groups, and applies these to analyze subgroup intersections and coherence.

Findings

Retractions uniquely extend to all parabolic subgroups.

Reduced intersection problems to weaker conditions.

Characterized coherence for FC-type Artin groups.

Abstract

We generalize the retractions to standard parabolic subgroups for even Artin groups to FC-type Artin groups and other more general families. We prove that these retractions uniquely extend to any parabolic subgroup. We use retractions to generalize the results of Antol\'in and Foniqi that reduce the problem of intersection of parabolic subgroups to weaker conditions. As a corollary, we characterize coherence for the FC case.