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

Abstract

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