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

Ignat Soroko

doi:10.1016/j.topol.2021.107770

paper

Artin groups of types $F_{4}$ and $H_{4}$ are not commensurable with that of type $D_{4}$

arXiv:2009.10471 · doi:10.1016/j.topol.2021.107770

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.

v4: a shortened version for the journal publication, to appear in "Topology and its Applications", 15 pages, 5 figures

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Artin groups of type B and D

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

Abstract

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Artin groups of types $F_{4}$ and $H_{4}$ are not commensurable with that of type $D_{4}$