Holomorphic functions on geometrically finite quotients of the ball

Let $\mathrm{\Gamma }$ be a discrete and torsion-free subgroup of $\mathrm{P}\mathrm{U}(n,1)$, the group of biholomorphisms of the unit ball in ${\mathbb{C}}^n$, denoted by ${\mathbb{H}}_{\mathbb{C}}^n$. We show that if $\mathrm{\Gamma }$ is Abelian, then ${\mathbb{H}}_{\mathbb{C}}^n\mathrm{/}\mathrm{\Gamma }$ is a Stein manifold. If the critical exponent $\delta (\mathrm{\Gamma })$ of $\mathrm{\Gamma }$ is less than 2, a conjecture of Dey and Kapovich predicts that the quotient ${\mathbb{H}}_{\mathbb{C}}^n\mathrm{/}\mathrm{\Gamma }$ is Stein. We confirm this conjecture in the case where $\mathrm{\Gamma }$ is parabolic or geometrically finite. We also study the case of quotients with $\delta (\mathrm{\Gamma })=2$ that contain compact complex curves and confirm another conjecture of Dey and Kapovich. We finally show that ${\mathbb{H}}_{\mathbb{C}}^n\mathrm{/}\mathrm{\Gamma }$ is Stein when $\mathrm{\Gamma }$ is a parabolic or geometrically finite group preserving a totally real and totally geodesic submanifold of ${\mathbb{H}}_{\mathbb{C}}^n$, without any hypothesis on the critical exponent.