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