2024/11/25 by Sarem, William
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2411.16620
Let Γ be a discrete and torsion-free subgroup of PU(n,1), the group of biholomorphisms of the unit ball in ℂn, denoted by ℍnℂ. We show that if Γ is Abelian, then ℍnℂ/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 ℍnℂ/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 ℍnℂ/Γ is Stein when Γ is a parabolic or geometrically finite group preserving a totally real and totally geodesic submanifold of ℍnℂ, without any hypothesis on the critical exponent.