The higher-dimensional subvariety exclusion conjecture for complex-hyperbolic quotients

Let $\Gamma<\operatorname{Aut}(\mathbf B^n)$ be a discrete, torsion-free subgroup, and let $M_\Gamma=\mathbf B^n/\Gamma$. For a positive integer $k$, suppose that $\delta(\Gamma)<2k$. Higher-dimensional subvariety exclusion conjecture. The manifold $M_\Gamma$ does not contain compact complex subvarieties of dimension at least $k$. The case $k=1$ is established in the paper under the same critical-exponent bound; the conjecture proposes the analogous exclusion in every dimension.