Topological proof of Benoist-Quint's orbit closure theorem for SO(d,1)

TL;DR

This paper provides a new topological proof of Benoist-Quint's orbit closure theorem for SO(d,1), showing that Zariski dense subgroup orbits are either finite or dense, using unipotent flow dynamics.

Contribution

The authors introduce a topological approach to prove Benoist-Quint's theorem, differing from the original measure classification method.

Findings

Proves that Zariski dense subgroup orbits are either finite or dense.

Uses topological methods from unipotent flow dynamics.

Offers an alternative proof to Benoist-Quint's measure-based approach.

Abstract

We present a new proof of the following theorem of Benoist-Quint: Let $G:=SO^\circ(d,1)$, $d\ge 2$ and $\Delta<G$ a cocompact lattice. Any orbit of a Zariski dense subgroup $\Gamma$ of $G$ is either finite or dense in $\Delta\backslash G$. While Benoist and Quint's proof is based on the classification of stationary measures, our proof is topological, using ideas from the study of dynamics of unipotent flows on the infinite volume homogeneous space $\Gamma\backslash G$.