Artin groups of types $F_4$ and $H_4$ are not commensurable with that of type $D_4$

TL;DR

This paper proves that Artin groups of types F_4 and H_4 are not commensurable with type D_4, resolving two open cases by analyzing their automorphism and commensurator groups.

Contribution

It establishes non-commensurability of F_4 and H_4 Artin groups with D_4 and characterizes automorphisms and torsion elements in spherical Artin groups.

Findings

F_4 and H_4 Artin groups are not commensurable with D_4.

Automorphism group of D_4 Artin group is identified as an extended mapping class group.

Torsion elements and their conjugacy classes are classified in spherical Artin groups.

Abstract

In a recent article, Cumplido and Paris studied the question of commensurability between Artin groups of spherical type. Their analysis left six cases undecided, for the following pairs of Artin groups: $(F_4,D_4)$, $(H_4,D_4)$, $(F_4,H_4)$, $(E_6,D_6)$, $(E_7,D_7)$, and $(E_8,D_8)$. In this note we resolve the first two of these cases, namely, we show that the Artin groups of types $F_4$ and $H_4$ are not commensurable with that of type $D_4$. As a key step, we realize the abstract commensurator of the Artin group of type $D_4$ as the extended mapping class group of the torus with three punctures. We also find the automorphism group of the Artin group of type $D_4$ and obtain a description of torsion elements, their orders and conjugacy classes in all irreducible Artin groups of spherical type modulo their centers.