  A Riemannian manifold resp. a complex space $X$ is called Liouville if it
carries no nonconstant bounded harmonic resp. holomorphic functions. It is
called Carath\'eodory, or Carath\'eodory 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 Carath\'eodory hyperbolic?
