# On commensurability of some right-angled Artin groups II: RAAGs defined by paths

## TL;DR

This paper investigates the commensurability of right-angled Artin groups defined by paths, establishing conditions under which they are not commensurable and linking the classification to linear integer programming.

## Contribution

It characterizes when RAAGs defined by paths are commensurable to those defined by trees of diameter 4, introducing a novel connection to linear integer programming.

## Key findings

- RAAGs defined by different paths of length > 3 are not commensurable.
- RAAGs defined by paths of length n > 4 are commensurable to those defined by trees of diameter 4 if n ≡ 2 (mod 4).
- Classification of RAAGs up to commensurability is connected to linear integer programming.

## Abstract

In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [CKZ]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs defined by paths are commensurable to RAAGs defined by trees of diameter 4. More precisely, we show that a RAAG defined by a path of length $n>4$ is commensurable to a RAAG defined by a tree of diameter 4 if and only if $n$ is 2 modulo 4. These results follow from the connection that we establish between the classification of RAAGs up to commensurability and linear integer-programming.

---
Source: https://tomesphere.com/paper/1803.00971