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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.11202 |
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| _version_ | 1866916949071495168 |
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| author | Ciobotaru, Corina |
| author_facet | Ciobotaru, Corina |
| contents | We study Chabauty limits of the fixed-point group of $k$-points $H_k$ associated with an involutive $k$-automorphism $θ$ of a connected linear reductive group $G$ defined over a non-Archimedean local field $k$ of characteristic zero. Leveraging the geometry of the Bruhat--Tits building, the structure of $(θ,k)$-split tori, and the $K\mathcal{B}_kH_k$ decomposition of $G_k$, we establish that any nontrivial Chabauty limit $L$ of $H_k$ is $G_k$-conjugate to a subgroup of $$U_{σ_+}(k) \rtimes (Ker(α)^0 \cdot (H_k \cap M_{σ_{\pm}})) \leq P_{σ_+}(k),$$ where $α$ is a projection map arising from a Levi factor $M_{σ_{\pm}}$ of a parabolic subgroup $P_{σ_+} \subset G$, and $Ker(α)^0$ denotes the subgroup of elliptic elements in the kernel of $α$. Our analysis distinguishes between elliptic and hyperbolic elements and constructs explicit unipotent elements in the limit group $L$ using the Moufang property of $G_k$. Furthermore, we show that $L$ acts transitively on the set of ideal simplices opposite to $σ_+$. These results yield a detailed description of the Chabauty compactification of $H_k$, and provide new insights into its interaction with the non-Archimedean geometry of $G_k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11202 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chabauty limits of fixed point groups of $p$-adic involutions Ciobotaru, Corina Representation Theory Group Theory Metric Geometry We study Chabauty limits of the fixed-point group of $k$-points $H_k$ associated with an involutive $k$-automorphism $θ$ of a connected linear reductive group $G$ defined over a non-Archimedean local field $k$ of characteristic zero. Leveraging the geometry of the Bruhat--Tits building, the structure of $(θ,k)$-split tori, and the $K\mathcal{B}_kH_k$ decomposition of $G_k$, we establish that any nontrivial Chabauty limit $L$ of $H_k$ is $G_k$-conjugate to a subgroup of $$U_{σ_+}(k) \rtimes (Ker(α)^0 \cdot (H_k \cap M_{σ_{\pm}})) \leq P_{σ_+}(k),$$ where $α$ is a projection map arising from a Levi factor $M_{σ_{\pm}}$ of a parabolic subgroup $P_{σ_+} \subset G$, and $Ker(α)^0$ denotes the subgroup of elliptic elements in the kernel of $α$. Our analysis distinguishes between elliptic and hyperbolic elements and constructs explicit unipotent elements in the limit group $L$ using the Moufang property of $G_k$. Furthermore, we show that $L$ acts transitively on the set of ideal simplices opposite to $σ_+$. These results yield a detailed description of the Chabauty compactification of $H_k$, and provide new insights into its interaction with the non-Archimedean geometry of $G_k$. |
| title | Chabauty limits of fixed point groups of $p$-adic involutions |
| topic | Representation Theory Group Theory Metric Geometry |
| url | https://arxiv.org/abs/2509.11202 |