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

TL;DR

This paper provides an asymptotic count of lifts of non-maximal closed horocycles on certain symmetric spaces, extending previous results to a broader class of horocycles in higher rank groups.

Contribution

It offers a precise asymptotic count of lifts of non-maximal closed horocycles in $SL_N$ spaces, generalizing earlier work by Mohammadi and Golsefidy.

Findings

Derived asymptotic formulas for lift counts

Extended previous results to non-maximal horocycles

Applicable to higher rank symmetric spaces

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.