The infinitesimal subgroup of interpretable groups in some dp-minimal valued fields
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912230407143424 |
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| author | Halevi, Yatir Hasson, Assaf Peterzil, Ya'acov |
| author_facet | Halevi, Yatir Hasson, Assaf Peterzil, Ya'acov |
| contents | We continue our local analysis of groups interpretable in various dp-minimal valued fields, as introduced in [8]. We associate with every infinite group $G$ interpretable in those fields an infinite type-definable infinitesimal subgroup $ν(G)$, generated by the four infinitesimal subgroups $ν_D(G)$ associated with the distinguished sorts $K$, $\textbf{k}$, $Γ$ and $K/\mathcal{O}$.
To show that $ν(G)$ is type-definable, we show that the resulting subgroups $ν_D(G)$ commute with each other as $D$ ranges over the four distinguished sorts. We then study the basic properties of $ν(G)$. Among others, we show that $ν(G_1\times G_2)=ν(G_1)\times ν(G_2)$ and that if $G_1\le G$ is a definable subgroup then $ν(G_1)$ is relatively definable in $ν(G)$. We also discuss possible connections between $\mathrm{dp\text{-}rk}(ν(G))$ and elimination of imaginaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08163 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The infinitesimal subgroup of interpretable groups in some dp-minimal valued fields Halevi, Yatir Hasson, Assaf Peterzil, Ya'acov Logic We continue our local analysis of groups interpretable in various dp-minimal valued fields, as introduced in [8]. We associate with every infinite group $G$ interpretable in those fields an infinite type-definable infinitesimal subgroup $ν(G)$, generated by the four infinitesimal subgroups $ν_D(G)$ associated with the distinguished sorts $K$, $\textbf{k}$, $Γ$ and $K/\mathcal{O}$. To show that $ν(G)$ is type-definable, we show that the resulting subgroups $ν_D(G)$ commute with each other as $D$ ranges over the four distinguished sorts. We then study the basic properties of $ν(G)$. Among others, we show that $ν(G_1\times G_2)=ν(G_1)\times ν(G_2)$ and that if $G_1\le G$ is a definable subgroup then $ν(G_1)$ is relatively definable in $ν(G)$. We also discuss possible connections between $\mathrm{dp\text{-}rk}(ν(G))$ and elimination of imaginaries. |
| title | The infinitesimal subgroup of interpretable groups in some dp-minimal valued fields |
| topic | Logic |
| url | https://arxiv.org/abs/2502.08163 |