Chabauty limits of fixed point groups of p-adic involutions

We study Chabauty limits of the fixed-point group of k-points Hₖ 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 KBₖHₖ decomposition of Gₖ, we establish that any nontrivial Chabauty limit L of Hₖ is Gₖ-conjugate to a subgroup of Uσ₊(k) (Ker(α)⁰ · (Hₖ ∩ M_σ±)) ≤ Pσ₊(k), where $α$ is a projection map arising from a Levi factor M_σ± of a parabolic subgroup Pσ₊ ⊂ 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 Gₖ. 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 Hₖ, and provide new insights into its interaction with the non-Archimedean geometry of Gₖ.