Hyperbolic structures on Houghton groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915157826863104 |
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| author | Genevois, Anthony Tournier, Geoffrey |
| author_facet | Genevois, Anthony Tournier, Geoffrey |
| contents | Given a group $G$, its poset of hyperbolic structures $\mathcal{H}(G)$ encodes all the possible cobounded actions of $G$ on hyperbolic spaces. In this article, we describe the poset $\mathcal{H}(H_n)$ for every Houghton group $H_n$, $n \geq 2$. In particular, we show that $H_n$ admits exactly $n$ focal hyperbolic structures. As an application, we construct the first example of a group admitting exactly one focal hyperbolic structure, answering a question of Abbott, Balasubramanya, and Osin. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_12590 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hyperbolic structures on Houghton groups Genevois, Anthony Tournier, Geoffrey Group Theory 20F65, 20F67 Given a group $G$, its poset of hyperbolic structures $\mathcal{H}(G)$ encodes all the possible cobounded actions of $G$ on hyperbolic spaces. In this article, we describe the poset $\mathcal{H}(H_n)$ for every Houghton group $H_n$, $n \geq 2$. In particular, we show that $H_n$ admits exactly $n$ focal hyperbolic structures. As an application, we construct the first example of a group admitting exactly one focal hyperbolic structure, answering a question of Abbott, Balasubramanya, and Osin. |
| title | Hyperbolic structures on Houghton groups |
| topic | Group Theory 20F65, 20F67 |
| url | https://arxiv.org/abs/2502.12590 |