Groups and Dynamics Seminar

We complete the classification of Baumslag-Solitar groups up to measure equivalence by showing all Baumslag-Solitar groups with nonunimodular Bass-Serre tree are measure equivalent (i.e., BS(r,s) with 1<r<s). The proof makes critical use of combinatorial descriptive set theory tools in the measure class preserving setting and passes through the world of measure equivalence of nonunimodular locally compact groups. In particular, as an intermediate step we obtain measure equivalence couplings between all groups of the form $Aut(T_{r,s})$ for 1<r<s where $T_{r,s}$ is the directed tree with r incoming edges and s outgoing edges at each vertex. This is joint work with Damien Gaboriau, Antoine Poulin, Anush Tserunyan, and Robin Tucker-Drob.