Bounded Geodesics on Locally Symmetric Spaces

Abstract

Let $\hat{I}“$ be a torsion-free subgroup of $S{L}_3(R)$ commensurable with $S{L}_3(Z)$, and $Y=S{O}_3(R)\backslash S{L}_3(R)/\hat{I}“$ be endowed with the natural locally symmetric space structure. We prove that for any point y in Y, the set of directions in which the geodesic ray starting from y is bounded in Y, is hyperplane absolute winning.