Torsion-free lattices in Baumslag–Solitar complexes

Abstract

This paper classifies the pairs of nonzero integers $(m,n)$ for which the locally compact group of combinatorial automorphisms, $\mathrm{Aut}(X_{m,n})$, contains incommensurable torsion-free lattices, where $X_{m,n}$ is the combinatorial model for Baumslag–Solitar group $\mathrm{BS}(m,n)$. In particular, we show that $\mathrm{Aut}(X_{m,n})$ contains abstractly incommensurable torsion-free lattices if and only if there exists a prime $p\le \gcd (m,n)$ 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 $\mathrm{Aut}(X_{m,n})$ does not contain incommensurable lattices, the cell complex $X_{m,n}$ satisfies Leighton’s property.