Torsion-Free Lattices in Baumslag-Solitar Complexes

This paper classifies the pairs of nonzero integers $(m,n)$ for which the locally compact group of combinatorial automorphisms, Aut$(X_{m,n})$, contains incommensurable torsion-free lattices, where $X_{m,n}$ is the combinatorial model for Baumslag-Solitar group $BS(m,n)$. In particular, we show that Aut$(X_{m,n})$ contains abstractly incommensurable torsion-free lattices if and only if there exists a prime $p \leq \mathrm{gcd}(m,n)$ such that either $\frac{m}{\mathrm{gcd}(m,n)}$ or $\frac{n}{\mathrm{gcd}(m,n)}$ is divisible by $p$. In all these cases, we construct infinitely many commensurability classes. Additionally, we show that when Aut$(X_{m,n})$ does not contain incommensurable lattices, the cell complex $X_{m,n}$ satisfies Leighton's property.