Simple Expansion Sets and Non-Positive Curvature
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866929695987073024 |
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| author | Farley, Daniel |
| author_facet | Farley, Daniel |
| contents | An expansion set is a set $\mathcal{B}$ such that each $b \in \mathcal{B}$ is equipped with a set of expansions $\mathcal{E}(b)$. The theory of expansion sets offers a systematic approach to the construction of classifying spaces for generalized Thompson groups.
We say that $\mathcal{B}$ is simple if proper expansions are unique when they exist.
We will prove that any given simple expansion set determines a cubical complex with a metric of non-positive curvature. In many cases, the cubical complex will be CAT(0). We are thus able to recover proofs that Thompsons groups $F$, $T$, and $V$, Houghton's groups $H_{n}$, and groups defined by finite similarity structures all act on CAT(0) cubical complexes. We further state a sufficient condition for the cubical complex to be locally finite, and show that the latter condition is satisfied in the cases of $F$, $T$, $V$, and $H_{n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01544 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simple Expansion Sets and Non-Positive Curvature Farley, Daniel Group Theory Primary 20F65, Secondary 20F67 An expansion set is a set $\mathcal{B}$ such that each $b \in \mathcal{B}$ is equipped with a set of expansions $\mathcal{E}(b)$. The theory of expansion sets offers a systematic approach to the construction of classifying spaces for generalized Thompson groups. We say that $\mathcal{B}$ is simple if proper expansions are unique when they exist. We will prove that any given simple expansion set determines a cubical complex with a metric of non-positive curvature. In many cases, the cubical complex will be CAT(0). We are thus able to recover proofs that Thompsons groups $F$, $T$, and $V$, Houghton's groups $H_{n}$, and groups defined by finite similarity structures all act on CAT(0) cubical complexes. We further state a sufficient condition for the cubical complex to be locally finite, and show that the latter condition is satisfied in the cases of $F$, $T$, $V$, and $H_{n}$. |
| title | Simple Expansion Sets and Non-Positive Curvature |
| topic | Group Theory Primary 20F65, Secondary 20F67 |
| url | https://arxiv.org/abs/2502.01544 |