Holomorphic functions on geometrically finite quotients of the ball

Let Γ be a discrete and torsion-free subgroup of PU ( n , 1 ) , the group of biholomorphisms of the unit ball in ℂ n , denoted by ℍ ℂ n . We show that if Γ is Abelian, then ℍ ℂ n / 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 ℍ ℂ n / 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 ℍ ℂ n / Γ is Stein when Γ is a parabolic or geometrically finite group preserving a totally real and totally geodesic submanifold of ℍ ℂ n , without any hypothesis on the critical exponent.