First-order theory of torsion-free Tarski monsters
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908672661127168 |
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| author | Coulon, Rémi Fournier-Facio, Francesco Ho, Meng-Che "Turbo" |
| author_facet | Coulon, Rémi Fournier-Facio, Francesco Ho, Meng-Che "Turbo" |
| contents | We develop methods to control the first-order theory of groups arising as certain direct limits of torsion-free hyperbolic groups, answering several questions in the literature. We construct simple torsion-free Tarski monsters $Γ$ (non-abelian groups whose non-trivial, proper subgroups are infinite cyclic) that are $\exists \forall \exists$-elementarily embedded into $Γ\ast \mathbf{Z}$. In particular, such $Γ$ have the same two-quantifier theory as $Γ\ast \mathbf{Z}$, and hence the same positive theory as a non-abelian free group. All previously known examples of groups with the same positive theory as the free group admit a non-elementary action on a hyperbolic space, while our examples cannot act on a hyperbolic space with a loxodromic element. Along the way, we solve the one-quantifier Knight conjecture for random quotients of arbitrary torsion-free, non-elementary, hyperbolic groups in the few-relator model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21244 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | First-order theory of torsion-free Tarski monsters Coulon, Rémi Fournier-Facio, Francesco Ho, Meng-Che "Turbo" Group Theory We develop methods to control the first-order theory of groups arising as certain direct limits of torsion-free hyperbolic groups, answering several questions in the literature. We construct simple torsion-free Tarski monsters $Γ$ (non-abelian groups whose non-trivial, proper subgroups are infinite cyclic) that are $\exists \forall \exists$-elementarily embedded into $Γ\ast \mathbf{Z}$. In particular, such $Γ$ have the same two-quantifier theory as $Γ\ast \mathbf{Z}$, and hence the same positive theory as a non-abelian free group. All previously known examples of groups with the same positive theory as the free group admit a non-elementary action on a hyperbolic space, while our examples cannot act on a hyperbolic space with a loxodromic element. Along the way, we solve the one-quantifier Knight conjecture for random quotients of arbitrary torsion-free, non-elementary, hyperbolic groups in the few-relator model. |
| title | First-order theory of torsion-free Tarski monsters |
| topic | Group Theory |
| url | https://arxiv.org/abs/2508.21244 |