Artin groups of types F4 and H4 are not commensurable with that of type D4

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: (F4,D4), (H4,D4), (F4,H4), (E6,D6), (E7,D7), and (E8,D8). In this note we resolve the first two of these cases, namely, we show that the Artin groups of types F4 and H4 are not commensurable with that of type D4. As a key step, we realize the abstract commensurator of the Artin group of type D4 as the extended mapping class group of the torus with three punctures. We also find the automorphism group of the Artin group of type D4 and obtain a description of torsion elements, their orders and conjugacy classes in all irreducible Artin groups of spherical type modulo their centers.