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₌, ₍), contains incommensurable torsion-free lattices, where X₌, ₍ is the combinatorial model for Baumslag-Solitar group BS (m, n). In particular, we show that Aut (X₌, ₍) 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 (X₌, ₍) does not contain incommensurable lattices, the cell complex X₌, ₍ satisfies Leighton's property.