Sentences over Random Groups II: Sentences of Minimal Rank
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917746003935232 |
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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 the existence of a uniform quantifier elimination procedure for formulas of minimal rank (probably the superstable part of the theory). Namely, given a minimal rank formula V(p), we prove the existence of a formula φ(p) that belongs to the Boolean algebra of two quantifiers, so that the two formulas V(p) and φ(p) define the same set over the free group F_{k} and over a random group of density d<\frac{1}{2}. We conclude that any given sentence of minimal rank is a truth sentence over the free group F_{k} if and only if it is a truth sentence over random groups of density d<\frac{1}{2}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15089 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sentences over Random Groups II: Sentences of Minimal Rank 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 the existence of a uniform quantifier elimination procedure for formulas of minimal rank (probably the superstable part of the theory). Namely, given a minimal rank formula V(p), we prove the existence of a formula φ(p) that belongs to the Boolean algebra of two quantifiers, so that the two formulas V(p) and φ(p) define the same set over the free group F_{k} and over a random group of density d<\frac{1}{2}. We conclude that any given sentence of minimal rank is a truth sentence over the free group F_{k} if and only if it is a truth sentence over random groups of density d<\frac{1}{2}. |
| title | Sentences over Random Groups II: Sentences of Minimal Rank |
| topic | Group Theory Logic 20F06, 20F65, 03C10 |
| url | https://arxiv.org/abs/2406.15089 |