Chabauty limits of algebraic groups acting on trees: the quasi-split case

Given a locally finite leafless tree $T$ , various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$ . We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$. This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) haracteristic. In particular, we show that in residue characteristic 2, the Tits index of simple algebraic subgroups of $\operatorname{Aut}(T)$ is not always preserved under Chabauty limits