Commensurability in Artin groups of spherical type

Abstract

We give an almost complete classification of Artin groups of spherical type up to commensurability. Let $A$ and $A^{\prime }$ be two Artin groups of spherical type, and let $A_1,\dots ,A_p$ (respectively, $A_1^{\prime },\dots ,A_q^{\prime }$) be the irreducible components of $A$ (respectively, $A^{\prime }$). We show that $A$ and $A^{\prime }$ are commensurable if and only if $p=q$ and, up to permutation of the indices, $A_i$ and $A_i^{\prime }$ are commensurable for every $i$. We prove that, if two Artin groups of spherical type are commensurable, then they have the same rank. For a fixed $n$, we give a complete classification of the irreducible Artin groups of rank $n$ that are commensurable with the group of type $A_n$. Note that there are six remaining comparisons of pairs of groups to get the complete classification of Artin groups of spherical type up to commensurability, two of which have been done by Ignat Soroko after the first version of the present paper.