Small volume of balls, large volume entropy and the Margulis constant

In his seminal work about bounded cohomology, M. Gromov showed that, under some topological conditions, every closed Riemannian manifold of small volume has large volume entropy. In this article, we strengthen some aspects of this result using an alternative approach. More precisely, we prove that, under some similar, yet different, topological assumptions, every closed Riemannian manifold whose volume of balls is small has large volume entropy. Along the proof of this result, we establish a new systolic inequality involving the commutator systole and a new curvature-free estimate relating the filling radius to the Margulis constant