  Let $X=\Gamma \backslash \mathbb{B}^{n} $ be an $n$-dimensional complex ball
quotient by a torsion-free non-uniform lattice $\Gamma$ whose parabolic
subgroups are unipotent. We prove that the volumes of subvarieties of $X$ are
controlled by the systole of $X,$ which is the length of the shortest closed
geodesic of $X$.
  There are a number of arithmetic and geometric consequences: the systole of
$X$ controls the growth rate of rational points on $X,$ uniformly in the field
of definition. Also, we obtain effective global generation and very ampleness
results for multiples of the canonical bundle $K_{\overline{X}},$ where
$\overline{X}$ is the toroidal compactification of $X.$ These results follow
from the bound we find for the Seshadri constant of $K_{\overline{X}}$ in terms
of the systole.
