Stability for boundary actions of cocompact lattices in Euclidean buildings

When $X$ is a locally compact Euclidean building, the isometry group of $X$ acts by homeomorphisms on the space of $k$-simplices in the visual boundary of $X$. We consider perturbations of these actions for discrete groups of isometries acting with compact quotient on $X$, showing that all small enough perturbations are semi-conjugate to the original action. This proves in particular that, when $Q$ is any parabolic subgroup in a semisimple $p$-adic Lie group $G$, the induced action of a cocompact lattice in $G$ has a topologically stable action on $G\mathrm{/}Q$.