Right-angled Artin groups: commensurability classification and subgroup intersection problem

The class of right-angled Artin groups (RAAGs) is a prominent class of groups which is widely studied in different branches of mathematics and computer science. Two groups are called commensurable if they have isomorphic subgroups of finite index. In the first part of the talk I will present our results with Montse Casals-Ruiz and Ilya Kazachkov on the commensurability classification of RAAGs defined by trees, which answer a question of J. Behrstock and W. Neumann. In the second part of the talk I will mention some algorithmic properties of RAAGs and present our results with Jordi Delgado and Enric Ventura on the subgroup intersection problem for Droms RAAGs.