Sentences over Random Groups I: Existential Sentences
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914908442984448 |
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| author | Massalha, Sobhi |
| author_facet | Massalha, Sobhi |
| contents | Random groups of density d<\frac{1}{2} are infinite hyperbolic, and of density d>\frac{1}{2} are finite. We prove that for any given system of equations Σ, all the solutions of Σover a random group of density d<\frac{1}{2} are projected from solutions of Σover the free group F_{k}, with overwhelming probability, where k is the rank of the group. We conclude that any given sentence in the Boolean algebra of universal sentences, is a truth sentence over F_{k} if and only if it is a truth sentence over random groups of density d<\frac{1}{2}, with overwhelming probability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15080 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sentences over Random Groups I: Existential Sentences Massalha, Sobhi Group Theory Logic 20F06, 20F65, 03C10 Random groups of density d<\frac{1}{2} are infinite hyperbolic, and of density d>\frac{1}{2} are finite. We prove that for any given system of equations Σ, all the solutions of Σover a random group of density d<\frac{1}{2} are projected from solutions of Σover the free group F_{k}, with overwhelming probability, where k is the rank of the group. We conclude that any given sentence in the Boolean algebra of universal sentences, is a truth sentence over F_{k} if and only if it is a truth sentence over random groups of density d<\frac{1}{2}, with overwhelming probability. |
| title | Sentences over Random Groups I: Existential Sentences |
| topic | Group Theory Logic 20F06, 20F65, 03C10 |
| url | https://arxiv.org/abs/2406.15080 |