A note on complex-hyperbolic Kleinian groups

Let $\mathrm{\Gamma }$ be a discrete group of isometries acting on the complex hyperbolic $n$-space ${\mathbb{H}}_{\mathbb{C}}^n$. In this note, we prove that if $\mathrm{\Gamma }$ is convex-cocompact, torsion-free, and the critical exponent $\delta (\mathrm{\Gamma })$ is strictly lesser than $2$, then the complex manifold ${\mathbb{H}}_{\mathbb{C}}^n\mathrm{/}\mathrm{\Gamma }$ is Stein. We also discuss several related conjectures.