Incommensurable lattices in Baumslag-Solitar complexes

This paper concerns locally finite 2-complexes Xm,n which are combinatorial models for the Baumslag-Solitar groups BS(m,n). We show that, in many cases, the locally compact group Aut(Xm,n) contains incommensurable uniform lattices. The lattices we construct also admit isomorphic Cayley graphs and are finitely presented, torsion-free, and coherent.