Margulis' Inequality for translates of horospherical orbits and applications to equidistribution

TL;DR

This paper develops a quantitative non-divergence theorem for horospherical orbit translates using Margulis' inequality, leading to effective equidistribution results for lattices over number fields without relying on spectral gap assumptions.

Contribution

It introduces a new non-divergence theorem based on Margulis' inequality that does not depend on spectral gap, with applications to equidistribution of horospherical orbits.

Findings

Proves a quantitative non-divergence theorem for horospherical translates.

Shows effective equidistribution of certain lattice orbits.

Results are independent of spectral gap assumptions.

Abstract

In this paper we develope a quantitative non-divergence theorem for translates of horospherical orbits, using the technique of Margulis' inequality as developed by Eskin-Margulis-Mozes and Eskin-Margulis. As we use the Margulis' inequality, our results do not depend on the spectral gap of the action. As an application of our techniques, we show that given a horospherical flow over the space of lattices, the horospherical orbit of every lattice defined over a number field, not contained in a proper rational parabolic subgroup is equidistributed with an effective rate.