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A closed horocycle $\mathcal{U}$ on $SL_N(\mathbb{Z}) \backslashSL_N(\mathbb{R})/SO_N({\mathbb{R}})$ has many lifts to the universal cover$SL_N(\mathbb{R})/SO_N({\mathbb{R}})$. Under some conditions on the horocycle,we give a precise asymptotic count of its lifts of bounded distance away from agiven base point in the universal cover. This partially generalizes previouswork of Mohammadi--Golsefidy.

Count lifts of non-maximal closed horocycles on $SL_N(\mathbb{Z}) \backslash SL_N(\mathbb{R})/SO_N({\mathbb{R}})$