Holomorphic functions on geometrically finite quotients of the ball
William Sarem
TL;DR
The paper analyzes holomorphic functions on quotients of complex hyperbolic space by discrete torsion-free subgroups of $\mathrm{PU}(n,1)$, establishing Steinness and holomorphic convexity under conditions on the critical exponent $δ(Γ)$ and geometric finiteness. It develops a detailed parabolic-subgroup analysis, providing a complete Stein criterion in terms of a totally real subspace and showing that virtually Abelian parabolic quotients are Stein. For geometrically finite groups, the authors prove holomorphic convexity and derive that $δ(Γ)<2$ or preservation of a totally real submanifold imply Stein quotients, while they classify $δ(Γ)=2$ cases, showing non-Stein behavior only for complex Fuchsian scenarios. The results extend Dey–Kapovich conjectures, connect end structure to holomorphic function theory on negatively curved Kähler–Hadamard manifolds, and offer tools applicable to broader contexts beyond complex hyperbolic geometry.
Abstract
Let $Γ$ 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 $Γ$ is Abelian, then $\mathbb{H}^{n}_{\mathbb{C}}/Γ$ is a Stein manifold. If the critical exponent $δ(Γ)$ of $Γ$ is less than 2, a conjecture of Dey and Kapovich predicts that the quotient $\mathbb{H}^{n}_{\mathbb{C}}/Γ$ is Stein. We confirm this conjecture in the case where $Γ$ is parabolic or geometrically finite. We also study the case of quotients with $δ(Γ)=2$ that contain compact complex curves and confirm another conjecture of Dey and Kapovich. We finally show that $\mathbb{H}^{n}_{\mathbb{C}}/Γ$ is Stein when $Γ$ 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.