On denseness of horospheres in higher rank homogeneous spaces

A limit point in the Furstenberg boundary of $G=SO^+(d,1)$ is called horospherical with respect to a discrete subgroup $\Gamma$ if every horoball based at the limit point intersects any $\Gamma$-orbit. Denote by $\Omega$ the non-wandering set in $T^1 \mathcal{M} = \Gamma \backslash T^1 \mathbb{H}^d$ with respect to the geodesic flow. A classical theorem of Dal’bo states that a horosphere in $T^1 \mathcal{M}$ is dense in $\Omega$ if and only if that horosphere is based at a horospherical limit point.

In this talk I will present a higher-rank analogue of the above criterion for denseness of horospheres. Connection to the rank-one geometric proof will be emphasized. Joint with Hee Oh.