On commensurability of right-angled Artin groups I: RAAGs defined by trees of diameter 4

TL;DR

This paper classifies right-angled Artin groups defined by trees of diameter 4, demonstrating there are infinitely many commensurability classes and providing examples of groups that are quasi-isometric but not commensurable.

Contribution

It characterizes the commensurability classes of RAAGs from trees of diameter 4 and confirms the conjecture of infinitely many classes.

Findings

Infinitely many commensurability classes of RAAGs from trees of diameter 4

Existence of RAAGs that are quasi-isometric but not commensurable

Confirmation of Behrstock and Neumann's conjecture

Abstract

In this paper we study the classification of right-angled Artin groups up to commensurability. We characterise the commensurability classes of RAAGs defined by trees of diameter 4. In particular, we prove a conjecture of Behrstock and Neumann that there are infinitely many commensurability classes. Hence, we give first examples of RAAGs that are quasi-isometric but not commensurable.