Abstract
We determine conditions under which the quotient of a Kähler-Hadamard manifold by a torsion-free, discrete group of isometries admits a big set of nontrivial holomorphic functions. We also generalize a theorem of G. A. Margulis on centerless of cocompact discrete groups of automorphisms of a Carathéodory hyperbolic manifold by a different approach based on the Carathéodory pseudodistance.
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B.-Y. Chen was supported by Chinese NSF grant No. 11031008 and Fok Ying Tung Education Foundation grant No. 111004. Partially supported by Chinese NSF grant No. 11171255.