Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.16196 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909610670030848 |
|---|---|
| author | Verma, Maya |
| author_facet | Verma, Maya |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16196 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torsion-Free Lattices in Baumslag-Solitar Complexes Verma, Maya Geometric Topology 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. |
| title | Torsion-Free Lattices in Baumslag-Solitar Complexes |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2406.16196 |