The conjugacy stability problem for parabolic subgroups in Artin groups

TL;DR

This paper investigates the conjugacy stability problem for parabolic subgroups in Artin groups, providing algorithms and solutions under certain conditions, and establishing uniqueness of minimal parabolic subgroups in specific cases.

Contribution

It introduces an algorithm to decide conjugacy stability in Artin groups satisfying certain conjectured properties and fully solves the problem for free products of spherical Artin groups.

Findings

An algorithm for conjugacy decision problem under specific properties.

Partial solutions for FC-type Artin groups.

Complete solution for free products of spherical Artin groups.

Abstract

Given an Artin group $A$ and a parabolic subgroup $P$, we study if every two elements of $P$ that are conjugate in $A$, are also conjugate in $P$. We provide an algorithm to solve this decision problem if $A$ satisfies three properties that are conjectured to be true for every Artin group. We partially solve the problem if $A$ has $FC$-type, and we totally solve it if $A$ is isomorphic to a free product of spherical Artin groups. In particular, we show that in this latter case, every element of $A$ is contained in a unique minimal (by inclusion) parabolic subgroup.