Freiheitssatz and phase transition for the density model of random groups
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909749733228544 |
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| author | Tsai, Tsung-Hsuan |
| author_facet | Tsai, Tsung-Hsuan |
| contents | Magnus' Freiheitssatz states that if a group is defined by a presentation with $m$ generators and a single relator containing the last generating letter, then the first $m-1$ letters freely generate a free subgroup. We study an analogue of this theorem in the Gromov density model of random groups, showing a phase transition phenomenon at density $d_r = \min\{\frac{1}{2}, 1-\log_{2m-1}(2r-1)\}$ with $1\leq r\leq m-1$: we prove that for a random group with $m$ generators at density $d$, if $d < d_r$ then the first $r$ letters freely generate a free subgroup; whereas if $d > d_r$ then the first $r$ letters generate the whole group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_08958 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Freiheitssatz and phase transition for the density model of random groups Tsai, Tsung-Hsuan Group Theory Probability 2020: 20F05, 20F06 Magnus' Freiheitssatz states that if a group is defined by a presentation with $m$ generators and a single relator containing the last generating letter, then the first $m-1$ letters freely generate a free subgroup. We study an analogue of this theorem in the Gromov density model of random groups, showing a phase transition phenomenon at density $d_r = \min\{\frac{1}{2}, 1-\log_{2m-1}(2r-1)\}$ with $1\leq r\leq m-1$: we prove that for a random group with $m$ generators at density $d$, if $d < d_r$ then the first $r$ letters freely generate a free subgroup; whereas if $d > d_r$ then the first $r$ letters generate the whole group. |
| title | Freiheitssatz and phase transition for the density model of random groups |
| topic | Group Theory Probability 2020: 20F05, 20F06 |
| url | https://arxiv.org/abs/2111.08958 |