Computable Scott sentences and the weak Whitehead problem for finitely presented groups
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909153067270144 |
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| author | Paolini, Gianluca |
| author_facet | Paolini, Gianluca |
| contents | We prove that if $A$ is a computable Hopfian finitely presented structure, then $A$ has a computable $d$-$Σ_2$ Scott sentence if and only if the weak Whitehead problem for $A$ is decidable. We use this to infer that every hyperbolic group as well as any polycyclic-by-finite group has a computable $d$-$Σ_2$ Scott sentence, thus covering two main classes of finitely presented groups. Our proof also implies that every weakly Hopfian finitely presented group is strongly defined by its $\exists^+$-types, a question which arose in a different context. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00079 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Computable Scott sentences and the weak Whitehead problem for finitely presented groups Paolini, Gianluca Logic 03C57, 20F10 We prove that if $A$ is a computable Hopfian finitely presented structure, then $A$ has a computable $d$-$Σ_2$ Scott sentence if and only if the weak Whitehead problem for $A$ is decidable. We use this to infer that every hyperbolic group as well as any polycyclic-by-finite group has a computable $d$-$Σ_2$ Scott sentence, thus covering two main classes of finitely presented groups. Our proof also implies that every weakly Hopfian finitely presented group is strongly defined by its $\exists^+$-types, a question which arose in a different context. |
| title | Computable Scott sentences and the weak Whitehead problem for finitely presented groups |
| topic | Logic 03C57, 20F10 |
| url | https://arxiv.org/abs/2401.00079 |