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

摘要

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. (C) 2021 Elsevier B.V. All rights reserved.