**Title:** On denseness of horospheres in higher rank homogeneous spaces

**Speaker:** Or Landesberg

**Speaker Info:** Yale

**Brief Description:**

**Special Note**:

**Abstract:**

In this talk I will discuss a necessary and sufficient condition for denseness of a horopherical orbit in the non-wandering set of a higher-rank homogeneous space $\Gamma \backslash G$, for a Zariski dense discrete subgroup $\Gamma < G$, possibly of infinite covolume.This is a higher rank analogue of a classical result of Dal’bo from rank-one: Let $G=SO^+(d,1)$ be the group of orientation preserving isometries of hyperbolic d-space and let $\Gamma < G$ be any Zariski dense discrete subgroup. 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 limit point in the Furstenberg boundary of $G$ is called horospherical with respect to $\Gamma$ if every horoball based at the limit point intersects any $\Gamma$-orbit. Dal’bo proved 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.

As it turns out, in higher rank there is a dependence on the particular Jordan projection of $\Gamma$, or more precisely on $\Gamma$’s limit cone. Connection to the rank-one geometric proof will be emphasized. Joint with Hee Oh.

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