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

Abstract

A closed horocycle $$\mathcal {U}$$ on $$SL_N(\mathbb {Z}) \backslash SL_N(\mathbb {R})/SO_N({\mathbb {R}})$$ 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 a given base point in the universal cover. This partially generalizes previous work of Mohammadi–Golsefidy.