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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.16196 |
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Table of Contents:
- 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.