Transverse groups preserving proper domains in flag manifolds

Given a semisimple Lie group $G$ and a self-opposite flag manifold $\mathcal{F}$ of $G$, we establish a necessary condition for an infinite subgroup $H$ of $G$ to preserve a proper domain in $\mathcal{F}$. In the case where $G$ is a Hermitian Lie group of tube type, we introduce and study a notion of causal convexity in the Shilov boundary $\mathbf{S}\mathbf{b}(G)$ of the symmetric space of $G$, inspired by the one already existing in conformal Lorentzian geometry. We show that subgroups $H$ of $G$ that are transverse with respect to a parabolic subgroup of $G$ defining $\mathbf{S}\mathbf{b}(G)$ and that preserve a proper domain in $\mathbf{S}\mathbf{b}(G)$ satisfy a geometric property with respect to this causal convexity, close to the strong projective convex cocompactness defined by Danciger–Guéritaud–Kassel. This result highlights the spatial nature of the dynamics of $H$. We construct Zariski-dense examples of such transverse subgroups.