Bounded Geodesics on Locally Symmetric Spaces

Abstract

Let $\Gamma$ be a torsion-free subgroup of $SL_3(R)$ commensurable with $SL_3(Z)$, and $Y=SO_3(R)\backslash SL_3(R)/\Gamma$ 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.