Torsion-Free Lattices in Baumslag-Solitar Complexes

TL;DR

This work classifies when the locally compact group Aut$(X_{m,n})$ of combinatorial automorphisms for the Baumslag-Solitar complex $X_{m,n}$ contains torsion-free lattices that are abstractly incommensurable. The authors employ the GBS framework, construct $p$-unimodular admissible covers, and use depth-profile and CRKZ invariants to distinguish commensurability classes, showing that for coprime $m,n$ the existence of incommensurable lattices in Aut$(X_{dm,dn})$ is controlled by a prime $p \le d$ with $p \mid m$ or $p \mid n$, yielding infinitely many classes when such $p$ exists. Conversely, when no such prime exists, Leighton's property holds for $X_{dm,dn}$, ensuring a common finite cover for compact spaces covered by $X_{dm,dn}$. The results extend prior work on incommensurability and provide new tools (e.g., depth profiles, CRKZ invariants) to analyze lattices in automorphism groups of GBS-type complexes.

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.

Figures (1)

• Figure 1: The cell structure for $Z_{m,n}$ when $m=3k$, $n=2k$.

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 3.1
• Theorem 3.2
• Theorem 3.3
• Remark 3.4