Incommensurable lattices in Baumslag-Solitar complexes

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