# Commensurability in Artin groups of spherical type

## TL;DR

This paper characterizes when two Artin groups of spherical type are commensurable, showing it depends on their irreducible components and ranks, and provides a classification for groups of fixed rank.

## Contribution

It establishes a criterion for commensurability of Artin groups of spherical type based on their irreducible components and ranks, and classifies those of fixed rank.

## Key findings

- Two Artin groups are commensurable iff their irreducible components are pairwise commensurable.
- Commensurable Artin groups of spherical type have the same rank.
- Complete classification of irreducible Artin groups of rank n commensurable with type A_n.

## Abstract

Let $A$ and $A'$ be two Artin groups of spherical type, and let $A_1,\dots,A_p$ (resp. $A'_1,\dots,A'_q$) be the irreducible components of $A$ (resp. $A'$). We show that $A$ and $A'$ are commensurable if and only if $p=q$ and, up to permutation of the indices, $A_i$ and $A'_i$ 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.