Action of $\mathbb{R}$-Fuchsian groups on $\mathbb{P}_\mathbb{C}^n$

Abstract: We consider discrete subgroups of the group of orientation preserving isometries of the $m$-dimensional hyperbolic space, whose limit set is a $(m-1)$-dimensional real sphere, acting on the $n$-dimensional complex projective space for $n\geq m$, via an embedding from the group of orientation preserving isometries of the $m$-dimensional hyperbolic space to the group of holomorphic isometries of the $n$-dimensional complex hyperbolic space. We describe the Kulkarni limit set of any of these subgroups under the embedding as a real semi-algebraic set. Also, we show that the Kulkarni region of discontinuity can only have one or three connected components. We use the Sylvester's law of inertia when $n=m$. In the other cases, we use some suitable projections of the the $n$-dimensional complex projective space to the $m$-dimensional complex projective space.