Two groups are called commensurable if they have isomorphic subgroups of finite index. In particular, finitely generated commensurable groups are quasi-isometric, and being commensurable is the main algebraic reason for groups to be quasi-isometric in the torsion-free case.
Baumslag-Solitar groups form an important class of torsion-free one-relator groups with unusual properties. While the quasi-isometry classification for them was known previously, due to Farb, Mosher and Whyte, the commensurability classification was only known in the solvable case.
In joint work with Montse Casals-Ruiz and Ilya Kazachkov (University of the Basque Country) we give a complete commensurability classification of Baumslag-Solitar groups. In the talk I will present these results and also (time permitting) mention related open problems regarding generalised Baumslag-Solitar groups.