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.

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.