On subvarieties of singular quotients of bounded domains

Let (Formula presented.) be a quotient of a bounded domain in (Formula presented.). Under suitable assumptions, we prove that every subvariety of (Formula presented.) not included in the branch locus of the quotient map is of log-general type in some orbifold sense. This generalizes a recent result by Boucksom and Diverio, which treated the case of compact, étale quotients. Finally, in the case where (Formula presented.) is compact, we give a sufficient condition under which there exists a proper analytic subset of (Formula presented.) containing all entire curves and all subvarieties not of general type (meant this time in in the usual sense as opposed to the orbifold sense).