The conjugacy stability problem for parabolic subgroups in 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.

18 pages, 1 figure. About replacement: Details added, little mistakes corrected, recent application of the main result cited. About second replacement: Some proofs needed more explanation

References in corpus (2)

• On parabolic subgroups of Artin-Tits groups of spherical type
• Conjugacy stability of parabolic subgroups of Artin-Tits groups of spherical type

Cited by in corpus (1)

• Lattices, injective metrics and the $K(\pi ,1)$ conjecture