  In their 2012 paper, Bobadilla and Koll\'ar studied topological conditions
which guarantee that a proper map of complex algebraic varieties is a
topological or differentiable fibration. They also asked whether a certain
finiteness property on the relative covering space can imply that a proper map
is a fibration. In this paper, we answer positively the integral homology
version of their question in the case of abelian varieties, and the rational
homology version in the case of compact ball quotients. We also propose several
conjectures in relation to the Singer-Hopf conjecture in the complex projective
setting.
