Bounded orbits on homogeneous spaces and bounded geodesics on locally symmetric spaces

Abstract

We present a unified approach to proving the following two results: (i) Let $G$ be a Lie group, $\Gamma$ a lattice in $G$, $X=G/\Gamma$, and let $F$ be a one-parameter $\mathrm{Ad}$-diagonalizable subgroup of $G$. Then the set $$ \{x\in X:Fx\text{ is bounded in }X\} $$ is hyperplane absolute winning on $X$. This fully solves a conjecture [Conjecture 7.1]{AGK} of An-Guan-Kleinbock in 2015. (ii) Let $G$ be a connected semisimple Lie group with finite center and without compact factors, $K\leq G$ a maximal compact subgroup, $\Gamma\leq G$ a torsion-free lattice, and $M=K\backslash G/\Gamma$. Then for any $p\in M$, the set $$\{\xi\in S_pM:\gamma(p,\xi)\text{ is bounded in }M\}$$ is hyperplane absolute winning on $S_pM$, where $\gamma(p,\xi)$ denotes the geodesic ray through the base point $p$ in the direction $\xi$. This fully solves a conjecture [Conjecture 1.2]{GW26} of Guan-Wu in 2026.