[Paper Review] Conjugacy stability of parabolic subgroups of Artin-Tits groups of spherical type

Matthieu Calvez, Bruno Aarón Cisneros de la Cruz+1 more|arXiv (Cornell University)|Jul 18, 2018

Geometric and Algebraic Topology12 references5 citations

TL;DR

This paper provides a complete classification of conjugacy stable parabolic subgroups in Artin-Tits groups of spherical type, resolving a question posed by Ivan Marin. It proves that standard parabolic subgroups are conjugacy stable unless they are of specific types (e.g., $D_5$ in $E_6$, $H_3$ in $H_4$), and establishes that reducible parabolic subgroups are conjugacy stable only under special conditions, using ribbon conjugations and graph-theoretic properties of Coxeter diagrams.

ABSTRACT

We give a complete classification of conjugacy stable parabolic subgroups of Artin-Tits groups of spherical type. This answers a question posed by Ivan Marin and generalizes a theorem obtained by Juan Gonz\\'alez-Meneses in the specific case of Artin braid groups.

Motivation & Objective

• To determine which parabolic subgroups of Artin-Tits groups of spherical type are conjugacy stable.
• To generalize González-Meneses' result on braid groups to all Artin-Tits groups of spherical type.
• To resolve a question posed by Ivan Marin regarding the conjugacy stability of standard parabolic subgroups.
• To classify all cases where conjugacy classes of parabolic subgroups merge in the ambient Artin-Tits group.

Proposed method

• Using the structure of Coxeter graphs and their subgraphs to classify standard parabolic subgroups by type.
• Applying ribbon conjugations—specific elements derived from adjacent ribbons in the Coxeter graph—to analyze conjugacy relations.
• Employing the property that two generators are conjugate in the Artin-Tits group if and only if they are connected by a path of odd-labeled edges in the Coxeter graph.
• Analyzing the action of specific conjugating elements on generators to detect when conjugacy classes merge across subgroups.
• Using the concept of Property $\star$ to characterize conjugacy stability in reducible cases.
• Reducing the problem to irreducible components and analyzing each case via known classification of spherical type Artin-Tits groups.

Experimental results

Research questions

• RQ1 Under what conditions is a standard parabolic subgroup of an Artin-Tits group of spherical type conjugacy stable?
• RQ2 Which embeddings of parabolic subgroups into larger Artin-Tits groups cause conjugacy classes to merge?
• RQ3 When is a reducible parabolic subgroup conjugacy stable in its ambient Artin-Tits group of spherical type?
• RQ4 What role do odd-labeled paths in the Coxeter graph play in determining conjugacy of generators?
• RQ5 Is there a uniform criterion—such as Property $\star$—that characterizes conjugacy stability in these groups?

Key findings

• A standard parabolic subgroup $A_X$ is conjugacy stable in $A_S$ unless $A_X$ is of type $D_5$ and $A_S$ is of type $E_6$, $E_7$, or $E_8$, or of type $D_7$ in $E_8$, or of type $E_7$ in $E_8$, or of type $D_{2k}$, or of type $H_3$ in $H_4$.
• For reducible parabolic subgroups, conjugacy stability holds only when $A_S$ is of type $B_n$ ($n \geq 3$) and $A_X = A_{\{s_1\}} \times A_Z$, where $A_{\{s_1\}}$ is cyclic and $A_Z$ is a braid group.
• Conjugacy classes of $A_X$ do not merge in $A_S$ if and only if the pair $(A_X, A_S)$ satisfies Property $\star$, which is equivalent to conjugacy stability.
• The proof shows that in the exceptional cases, specific conjugating elements (ribbons) exist that conjugate elements in $A_X$ across the ambient group but not within $A_X$, proving class merging.
• The authors establish that two generators are conjugate in $A_S$ if and only if they are connected by a path of odd-labeled edges in the Coxeter graph, a key tool in the analysis.
• The classification is complete and applies to all Artin-Tits groups of spherical type, resolving Marin’s question in full generality.