Volumes of Subvarieties of Complex Ball Quotients and Sparsity of Rational Points

Let $X=Γ\backslash \mathbb{B}^{n} $ be an $n$-dimensional complex ball quotient by a torsion-free non-uniform lattice $Γ$ 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 geodesi