References & Citations
Mathematics > Logic
Title: Computable Scott sentences and the weak Whitehead problem for finitely presented groups
(Submitted on 29 Dec 2023 (v1), last revised 27 Mar 2024 (this version, v3))
Abstract: We prove that if $A$ is a computable Hopfian finitely presented structure, then $A$ has a computable $d$-$\Sigma_2$ 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 $d$-$\Sigma_2$ 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.
Submission history
From: Gianluca Paolini [view email][v1] Fri, 29 Dec 2023 21:44:47 GMT (12kb)
[v2] Mon, 4 Mar 2024 21:27:58 GMT (12kb)
[v3] Wed, 27 Mar 2024 17:18:16 GMT (12kb)
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