2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/148400A Riemannian manifold resp. a complex space $X$ is called Liouville if it carries no nonconstant bounded harmonic resp. holomorphic functions. It is called Carathéodory, or Carathéodory hyperbolic, if bounded harmonic resp. holomorphic functions separate the points of $X$. The problems which we discuss in this paper arise from the following question: When a Galois covering $X$ with Galois group $G$ over a Liouville base $Y$ is Liouville or, at least, is not Carathéodory hyperbolic?20 pages, AMSTeX. A revised version. The proof of Theorem 3.1 has been completed, and some other minor correction has been doneAlgebraic Geometry14E20 (Primary) 32H25, 53B35 (Secondary)text