Property FA for random $\ell$-gonal groups
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916732905455616 |
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| author | Clement, Emily Mackay, John M. |
| author_facet | Clement, Emily Mackay, John M. |
| contents | In the binomial $\ell$-gonal model for random groups, where the random relations all have fixed length $\ell\geq 3$ and the number of generators goes to infinity, we establish a double threshold near density $d=\frac{1}{\ell}$ where the group goes from being free to having Serre's property FA.
As a consequence, random $\ell$-gonal groups at densities $\frac{1}{\ell} < d< \frac{1}{2}$ have boundaries homeomorphic to the Menger sponge, and $\frac{1}{\ell}$ is also the threshold for finiteness of $\mathrm{Out}(G)$. We also see that the thresholds for property FA and Kazhdan's property (T) differ when $\ell \geq 4$.
Our methods are inspired by work of Antoniuk-Luczak-Świątkowski and Dahmani-Guirardel-Przytycki. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Property FA for random $\ell$-gonal groups Clement, Emily Mackay, John M. Group Theory Probability 20F65, 20F67, 20E08, 20F05, 20P05 In the binomial $\ell$-gonal model for random groups, where the random relations all have fixed length $\ell\geq 3$ and the number of generators goes to infinity, we establish a double threshold near density $d=\frac{1}{\ell}$ where the group goes from being free to having Serre's property FA. As a consequence, random $\ell$-gonal groups at densities $\frac{1}{\ell} < d< \frac{1}{2}$ have boundaries homeomorphic to the Menger sponge, and $\frac{1}{\ell}$ is also the threshold for finiteness of $\mathrm{Out}(G)$. We also see that the thresholds for property FA and Kazhdan's property (T) differ when $\ell \geq 4$. Our methods are inspired by work of Antoniuk-Luczak-Świątkowski and Dahmani-Guirardel-Przytycki. |
| title | Property FA for random $\ell$-gonal groups |
| topic | Group Theory Probability 20F65, 20F67, 20E08, 20F05, 20P05 |
| url | https://arxiv.org/abs/2505.07424 |