Topological proof of Benoist-Quint's orbit closure theorem for $ \boldsymbol{ \operatorname{SO}(d, 1)} $

We present a new proof of the following theorem of Benoist-Quint: Let \begin{document}$ G: = \operatorname{SO}^\circ(d, 1) $\end{document} , \begin{document}$ d\ge 2 $\end{document} and \begin{document}$ \Delta a cocompact lattice. Any orbit of a Zariski dense subgroup \begin{document}$ \Gamma $\end{document} of \begin{document}$ G $\end{document} is either finite or dense in \begin{document}$ \Delta \backslash G $\end{document} . While Benoist and Quint's proof is based on the classification of stationary measures, our proof is topological, using ideas from the study of dynamics of unipotent flows on the infinite volume homogeneous space \begin{document}$ \Gamma \backslash G $\end{document} .