  In this article, we study mappings acting between domains of two factor
spaces by certain groups of M\"{o}bius automorphisms of the unit ball that act
discontinuously and do not have fixed points. For such mappings, we have
established estimates for the distortion of the modulus of families of paths,
which are similar to the well-known Poletsky and V\"{a}is\"{a}l\"{a}
inequalities. As applications, we have obtained several important results on
the local and boundary behavior of mappings.
