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Bounded orbits on homogeneous spaces and bounded geodesics on locally symmetric spaces
We present a unified approach to proving the following two results: (i) Let $G$ be a Lie group, $Γ$ a lattice in $G$, $X=G/Γ$, 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 $X$.
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