Stein property of the complex-hyperbolic manifolds

Let $M$ be a complex-hyperbolic $n$-manifold, i.e. a quotient of the complex-hyperbolic $n$-space $\mathbb{H}^n_\mathbb{C}$ by a torsion-free discrete group of isometries, $\Gamma = \pi_1(M)$. Suppose that $M$ is {\em convex-cocompact}, i.e. the convex core of $M$ is a nonempty compact subset. In this talk, we will discuss a sufficient condition on $\Gamma$ in terms of the growth-rate of its orbits in $\mathbb{H}^n_\mathbb{C}$ for which $M$ is a Stein manifold. We will also talk about some interesting questions related to this result. This is a joint work with Misha Kapovich.