Torsion-Free Lattices in Baumslag-Solitar Complexes

TL;DR

This paper classifies when the automorphism group of Baumslag-Solitar complexes contains incommensurable torsion-free lattices, providing a criterion based on divisibility conditions and constructing infinitely many classes.

Contribution

It provides a complete classification of the existence of incommensurable torsion-free lattices in automorphism groups of Baumslag-Solitar complexes, including explicit construction methods.

Findings

Incommensurable torsion-free lattices exist iff a prime divides certain ratios of m and n.

Constructs infinitely many distinct commensurability classes in these cases.

When such lattices do not exist, the complex satisfies Leighton's property.

Abstract

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 $p \leq \mathrm{gcd}(m,n)$ such that either $\frac{m}{\mathrm{gcd}(m,n)}$ or $\frac{n}{\mathrm{gcd}(m,n)}$ is divisible by $p$. In all these cases, we construct infinitely many commensurability classes. Additionally, we show that when Aut$(X_{m,n})$ does not contain incommensurable lattices, the cell complex $X_{m,n}$ satisfies Leighton's property.