  This article deals with topological assumptions under which the minimal
volume entropy of a closed manifold, and more generally of a finite simplicial
complex, vanishes or is positive. In the first part of the article, we present
complementing topological conditions expressed in terms of the growth of the
fundamental group of the fibers of maps from a given finite simplicial complex
to simplicial complexes of lower dimension which ensure that the minimal volume
entropy of the simplicial complex either vanishes or is positive. We also give
examples of finite simplicial complexes with zero simplicial volume and
arbitrarily large minimal volume entropy. In the second part of the article, we
present topological assumptions related to the exponential growth of certain
subgroups in the fundamental group of a finite simplicial complex and to the
topology of the loop space of its classifying space under which the minimal
volume entropy is positive. Several examples are presented throughout the text.
