Transverse groups preserving proper domains in flag manifolds

Given a semisimple Lie group G and a self-opposite flag manifold ℱ of G , we establish a necessary condition for an infinite subgroup H of G to preserve a proper domain in 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 𝐒 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 𝐒 𝐛 ⁡ ( 𝔤 ) and that preserve a proper domain in 𝐒 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.