Commensurability in Artin groups of spherical type

Abstract

Let $A$ and $A^{\prime }$ be two Artin groups of spherical type, and let $A_1,\dots ,A_p$ (resp. $A_1^{\prime },\dots ,A_q^{\prime }$) be the irreducible components of $A$ (resp. $A^{\prime }$). 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 it will remain 6 pairs of groups to compare to get the complete classification of Artin groups of spherical type up to commensurability.

22 pages, 1 figure