Action of ℝ-Fuchsian groups on ℙℂn

Let [Formula: see text] be the group of orientation preserving isometries of the [Formula: see text]-dimensional real hyperbolic space [Formula: see text], and let [Formula: see text] be the group of holomorphic isometries of the [Formula: see text]-dimensional complex hyperbolic space [Formula: see text]. The group [Formula: see text] is naturally a subgroup of [Formula: see text], so it acts on [Formula: see text] preserving a ball that serves as a model for [Formula: see text]. We consider discrete subgroups of [Formula: see text] with limit set an [Formula: see text]-dimensional real sphere and we look at their action on [Formula: see text] for [Formula: see text] via the natural embedding of [Formula: see text] into [Formula: see text]. We describe the Kulkarni limit set of these set of these actions and show that this is a real semi-algebraic set in [Formula: see text]. We also show that the Kulkarni region of discontinuity can only have either one or three connected components. We use Sylvester’s law of inertia when [Formula: see text]. In the other cases, we use some suitable projections of the [Formula: see text]-dimensional complex projective space to the [Formula: see text]-dimensional complex projective space.