On commensurability of right-angled Artin groups I: RAAGs defined by   trees of diameter 4
  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. 
Montserrat Casals-Ruiz, Ilya Kazachkov and Alexander Zakharov
math.GR