Non-Divergence of Unipotent Flows on Quotients of Rank One Semisimple Groups

TL;DR

This paper proves that unipotent flows on quotients of rank one semisimple groups mostly stay in regions with large injectivity radius, extending non-divergence results to more general discrete subgroups and product groups.

Contribution

It generalizes Dani's quantitative nondivergence theorem to finitely generated subgroups and product groups, providing new bounds on unipotent flow trajectories.

Findings

Unipotent trajectories spend most of their time in regions with large injectivity radius.

The results extend to finitely generated subgroups and product groups of rank-one semisimple groups.

Either trajectories stay mostly in large injectivity regions or relate to small covolume lattices in abelian subgroups.

Abstract

Let $G$ be a semisimple Lie group of rank $1$ and $\Gamma$ be a torsion free discrete subgroup of $G$. We show that in $G/\Gamma$, given $\epsilon>0$, any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than $ \delta$ for $1-\epsilon$ proportion of the time for some $\delta>0$. The result also holds for any finitely generated discrete subgroup $\Gamma$ and this generalizes Dani's quantitative nondivergence theorem \cite{D} for lattices of rank one semisimple groups. Furthermore, for a fixed $\epsilon>0$ there exists an injectivity radius $\delta$ such that for any unipotent trajectory $\{u_tx\}_{t\in [0,T]}$, either it spends at least $1-\epsilon$ proportion of the time in the set with injectivity radius larger than $\delta$ for all large $T>0$ or there exists a $\{u_t\}_{t\in\mathbb{R}}$-normalized abelian subgroup $L$ of $G$ which intersects $g\Gamma g^{-1}$ in a small covolume lattice. We also extend these results when $G$ is the product of rank-$1$ semisimple groups and $\Gamma$ a discrete subgroup of $G$ whose projection onto each nontrivial factor is torsion free.