Parallel geodesics and minimal stable length of random groups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908499613581312 |
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| author | Tsai, Tsung-Hsuan |
| author_facet | Tsai, Tsung-Hsuan |
| contents | We show that for any pair of long enough parallel geodesics in a random group $G_\ell(m,d)$ with $m$ generators at density $d<1/6$, there is a van Kampen diagram having only one layer of faces. Using this result, we give an upper bound, depending only on $d$, of the number of pairwise parallel geodesics in $G_\ell(m,d)$ when $d<1/6$. As an application, we show that the minimal stable length of $G_\ell(m,d)$ at $d<1/6$ is exactly $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07859 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parallel geodesics and minimal stable length of random groups Tsai, Tsung-Hsuan Group Theory 20P05, 20F65, 20F06 We show that for any pair of long enough parallel geodesics in a random group $G_\ell(m,d)$ with $m$ generators at density $d<1/6$, there is a van Kampen diagram having only one layer of faces. Using this result, we give an upper bound, depending only on $d$, of the number of pairwise parallel geodesics in $G_\ell(m,d)$ when $d<1/6$. As an application, we show that the minimal stable length of $G_\ell(m,d)$ at $d<1/6$ is exactly $1$. |
| title | Parallel geodesics and minimal stable length of random groups |
| topic | Group Theory 20P05, 20F65, 20F06 |
| url | https://arxiv.org/abs/2410.07859 |