Chabauty limits of fixed point groups of $p$-adic involutions

Abstract

We study Chabauty limits of the fixed-point group of $k$-points $H_k$ associated with an involutive $k$-automorphism $\theta$ 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 $(\theta,k)$-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 $$U_{\sigma_+}(k) \rtimes (Ker(\alpha)^0 \cdot (H_k \cap M_{\sigma_{\pm}})) \leq P_{\sigma_+}(k),$$ where $\alpha$ is a projection map arising from a Levi factor $M_{\sigma_{\pm}}$ of a parabolic subgroup $P_{\sigma_+} \subset G$, and $Ker(\alpha)^0$ denotes the subgroup of elliptic elements in the kernel of $\alpha$. 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 $\sigma_+$. 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 $G_k$.