On groups definable in $p$-adically closed fields
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911404354699264 |
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| author | Pillay, Anand Yao, Ningyuan Zhang, Zhentao |
| author_facet | Pillay, Anand Yao, Ningyuan Zhang, Zhentao |
| contents | This paper is about the $dfg$/$fsg$ decomposition for groups $G$ definable in $p$-adically closed fields. It is proved that for $G$ definably amenable, $G$ has a definable normal $dfg$ subgroup $H$ such that the quotient $G/H$ is a definable $fsg$ group. The result was known for groups definable in $o$-minimal expansions of real closed fields (see \cite{C-P-o-mini}). We also give a version for arbitrary (not necessarily definably amenable) groups $G$ definable in $p$-adically closed fields: there is a definable $dfg$ subgroup $H$ of $G$ such that the homogeneous space $G/H$ is definable and definably compact. (In the $o$-minimal case this is Fact 3.25 of \cite{Peterzil-Starchenko-mutypes}).
Note that $dfg$ stands for ``has a definable $f$-generic type", and $fsg$ for ``has finitely satisfiable generics", which will be discussed together with various equivalences.
We will need to understand something about groups of the form $G(k)$ where $k$ is a $p$-adically closed field and $G$ a semisimple algebraic group over $k$, and as part of the analysis we will prove the Kneser-Tits conjecture over $p$-adically closed fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_20565 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On groups definable in $p$-adically closed fields Pillay, Anand Yao, Ningyuan Zhang, Zhentao Logic 03C65, 03C98, 20G25, 22E35 This paper is about the $dfg$/$fsg$ decomposition for groups $G$ definable in $p$-adically closed fields. It is proved that for $G$ definably amenable, $G$ has a definable normal $dfg$ subgroup $H$ such that the quotient $G/H$ is a definable $fsg$ group. The result was known for groups definable in $o$-minimal expansions of real closed fields (see \cite{C-P-o-mini}). We also give a version for arbitrary (not necessarily definably amenable) groups $G$ definable in $p$-adically closed fields: there is a definable $dfg$ subgroup $H$ of $G$ such that the homogeneous space $G/H$ is definable and definably compact. (In the $o$-minimal case this is Fact 3.25 of \cite{Peterzil-Starchenko-mutypes}). Note that $dfg$ stands for ``has a definable $f$-generic type", and $fsg$ for ``has finitely satisfiable generics", which will be discussed together with various equivalences. We will need to understand something about groups of the form $G(k)$ where $k$ is a $p$-adically closed field and $G$ a semisimple algebraic group over $k$, and as part of the analysis we will prove the Kneser-Tits conjecture over $p$-adically closed fields. |
| title | On groups definable in $p$-adically closed fields |
| topic | Logic 03C65, 03C98, 20G25, 22E35 |
| url | https://arxiv.org/abs/2601.20565 |