  This article deals with topological assumptions under which the minimal
volume entropy of a closed manifold~$M$, and more generally of a finite
simplicial complex~$X$, vanishes or is positive. These topological conditions
are expressed in terms of the growth of the fundamental group of the fibers of
maps from a given finite simplicial complex~$X$ to lower dimensional simplicial
complexes~$P$. We also give examples of finite simplicial complexes with zero
simplicial volume and arbitrarily large minimal volume entropy.
