  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 $p \leq gcd(m, n)$ such that either $m/gcd(m,n)$
or $n/gcd(m,n)$ is divisible by $p$. Additionally, we show that when
Aut$(X_{m,n})$ does not contain incommensurable lattices, the cell complex
$X_{m,n}$ satisfies Leighton$'$s property.
