Retraction to a parabolic subgroup and applications

TL;DR

This paper advances the understanding of retractions in Artin groups, providing new insights into subgroup structures, intersections, and conjugacy classes through the use of right and left retractions.

Contribution

It introduces novel results on minimal parabolic subgroups, intersections, and conjugacy classes in Artin groups using retraction techniques.

Findings

New results on minimal parabolic subgroups

Insights into intersection of parabolic subgroups

Analysis of conjugacy classes in Artin groups

Abstract

We continue the study of the retraction from an Artin group to a standard parabolic subgroup introduced by Blufstein, Charney, Paris and the second author. Using right and left retractions we obtain new results on minimal parabolic subgroups, intersection of parabolic subgroups, double cosets with respect to parabolic subgroups and conjugacy classes in Artin groups.