The critical-exponent gap conjecture for compact complex subvarieties

Let $\Gamma<\operatorname{PU}(n,1)$ be a discrete, torsion-free subgroup, let $\delta_\Gamma$ be its critical exponent, and let $k$ denote the dimension of a compact complex subvariety. Critical-exponent gap conjecture. If $\delta_\Gamma<2k$, then $M_\Gamma$ cannot contain a compact complex subvariety of dimension $k$. This is motivated by the Dey--Kapovich theorem, which proves the corresponding lower bound $\delta_\Gamma\ge 2$ for positive-dimensional compact complex subvarieties; the higher-dimensional assertion remains open.