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, \operatorname{Aut}(X_{m,n}) , contains incommensurable torsion-free lattices, where X_{m,n} is the combinatorial model for Baumslag–Solitar group \operatorname{BS}(m,n) . In particular, we show that \operatorname{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 . In all these cases, we construct infinitely many commensurability classes. Additionally, we show that when \operatorname{Aut}(X_{m,n}) does not contain incommensurable lattices, the cell complex X_{m,n} satisfies Leighton’s property.