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(Xm,n), contains incommensurable torsion-free lattices, where Xm,n is the combinatorial model for Baumslag-Solitar group $BS(m, n)$. In particular, we show that Aut(Xm,n) contains abstractly incommensurable torsion-free lattices if and only if there exists a prime p ≤ 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(Xm,n) does not contain incommensurable lattices, the cell complex Xm,n satisfies Leighton$'$s property.