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.Comment: 23 page