Holomorphic functions on geometrically finite quotients of the ball

TL;DR

This paper investigates conditions under which quotients of complex hyperbolic space by discrete groups are Stein manifolds, confirming several conjectures for specific classes of groups and geometric configurations.

Contribution

It proves that certain geometrically finite and parabolic quotients of complex hyperbolic space are Stein, confirming conjectures by Dey and Kapovich in these cases.

Findings

Quotients by Abelian groups are Stein.

Confirmed Dey and Kapovich's conjecture for geometrically finite groups with critical exponent less than 2.

Established Stein property for quotients with parabolic or geometrically finite groups preserving totally real submanifolds.

Abstract

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.