On groups interpretable in various valued fields
Fuente:
arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866909162472996864 |
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| author | Halevi, Yatir Hasson, Assaf Peterzil, Ya'acov |
| author_facet | Halevi, Yatir Hasson, Assaf Peterzil, Ya'acov |
| contents | We study infinite groups interpretable in three families of valued fields: $V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show that every such group $G$ has unbounded exponent and that if $G$ is dp-minimal then it is abelian-by-finite.
Along the way, we associate with any infinite interpretable group an infinite type-definable subgroup which is definably isomorphic to a group in one of four distinguished sorts: the underlying valued field $K$, its residue field $\mathbf{k}$ (when infinite), its value group $Γ$, or $K/\mathcal{O}$, where $\mathcal{O}$ is the valuation ring.
Our work uses and extends techniques developed in [11] to circumvent elimination of imaginaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_05677 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On groups interpretable in various valued fields Halevi, Yatir Hasson, Assaf Peterzil, Ya'acov Logic Group Theory We study infinite groups interpretable in three families of valued fields: $V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show that every such group $G$ has unbounded exponent and that if $G$ is dp-minimal then it is abelian-by-finite. Along the way, we associate with any infinite interpretable group an infinite type-definable subgroup which is definably isomorphic to a group in one of four distinguished sorts: the underlying valued field $K$, its residue field $\mathbf{k}$ (when infinite), its value group $Γ$, or $K/\mathcal{O}$, where $\mathcal{O}$ is the valuation ring. Our work uses and extends techniques developed in [11] to circumvent elimination of imaginaries. |
| title | On groups interpretable in various valued fields |
| topic | Logic Group Theory |
| url | https://arxiv.org/abs/2206.05677 |