Semisimple groups interpretable in various valued fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909722226982912 |
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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 power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups) then, up to a finite index subgroup, it is definably isogenous to a group $G_1\times G_2$, where $G_1$ is a $K$-linear group and $G_2$ is a $\mathbf{k}$-linear group. The analysis is carried out by studying the interaction of $G$ with four distinguished sorts: the valued field $K$, the residue field $\mathbf{k}$, the value group $Γ$, and the closed $0$-balls $K/\mathcal{O}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02727 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Semisimple groups interpretable in various valued fields Halevi, Yatir Hasson, Assaf Peterzil, Ya'acov Logic Group Theory We study infinite groups interpretable in power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups) then, up to a finite index subgroup, it is definably isogenous to a group $G_1\times G_2$, where $G_1$ is a $K$-linear group and $G_2$ is a $\mathbf{k}$-linear group. The analysis is carried out by studying the interaction of $G$ with four distinguished sorts: the valued field $K$, the residue field $\mathbf{k}$, the value group $Γ$, and the closed $0$-balls $K/\mathcal{O}$. |
| title | Semisimple groups interpretable in various valued fields |
| topic | Logic Group Theory |
| url | https://arxiv.org/abs/2309.02727 |