In 2001, K. Whyte demonstrated that all Baumslag-Solitar groups BS(r,s), with |r|,|s|≠1 and |r|≠|s|, are quasi-isometric, thus completing the quasi-isometric classification of Baumslag-Solitar groups initiated by B. Farb and L. Mosher. Since then, the question of their measure equivalence has remained an open and intriguing problem. With A. Poulin, A. Tserunyan, R. Tucker-Drob, and K. Wrobel, we have solved this problem, thus establishing the counterpart of Whyte's theorem in terms of measured equivalence. To do this we are led to a detour through the world of non-unimodular groups and actions which preserve only a class of measure (type III) and we reduce the question to a problem of the theory of measured graphs in this framework.
