On the denseness of horospheres in higher-rank

In this talk I will discuss a necessary and sufficient condition for denseness of horopherical orbits in the non-wandering set of a higher-rank homogeneous space $G / \Gamma$, for a Zariski dense discrete subgroup $\Gamma < G$, possibly of infinite covolume. In rank one this condition (established in this setting by Eberlein and Dal’bo) implies in particular that the horospherical subgroup acts minimally on the non-wandering set if and only if the discrete group $\Gamma$ is convex co-compact. In contrast, we show that Schottky groups in higher-rank can support non-minimal horospherical actions. This distinction between rank-one and higher-rank is due to the role that Benoist’s limit cone plays in the analysis. Based on joint work with Hee Oh.