Commensurability of Baumslag-Solitar groups

Two groups are called (abstractly) commensurable if they have isomorphic subgroups of finite index. In particular, finitely generated commensurable groups are quasi-isometric. Baumslag-Solitar groups form an interesting class of one-relator groups with unusual properties. The quasi-isometry classification for these groups was known previously, due to Farb, Mosher and Whyte. We give a complete commensurability classification of Baumslag-Solitar groups. I will also mention related open questions about generalized Baumslag-Solitar groups. This is joint work with Montse Casals-Ruiz and Ilya Kazachkov.