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

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 $\{u_{t}x{\}}_{t\in [0,T]}$, 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.

23 pages

Cited by in corpus (3)

• Expanding cone and applications to homogeneous dynamics
• A Shrinking Target Problem with Target at Infinity in Rank One Homogeneous Spaces
• Quantitative disjointness of nilflows from horospherical flows