Dey–Kapovich's Steinness conjecture for low-critical-exponent complex hyperbolic quotients

Let $n\geq 1$ and let $\Gamma$ be a discrete and torsion-free subgroup of $\mathrm{PU}(n,1)$ acting on the complex hyperbolic space $\mathbb{H}^{n}_{\mathbb{C}}$. Define the critical exponent by

$$\delta(\Gamma):=\inf\left\{s\in\mathbb{R}_{+}\ \middle|\ \sum_{\gamma\in\Gamma}e^{-s d(o,\gamma o)}<\infty\right\},$$

where $o\in\mathbb{H}^{n}_{\mathbb{C}}$ and $d$ is the complex hyperbolic distance. Dey–Kapovich's Steinness conjecture. If $\delta(\Gamma)<2$, then

$$\mathbb{H}^{n}_{\mathbb{C}}/\Gamma$$

is a Stein manifold.

The conjecture concerns the existence of non-constant holomorphic functions and Stein structures on quotients of complex hyperbolic space. It was known for convex-cocompact subgroups and is confirmed in the paper for parabolic and geometrically finite groups.