Positivity of simplicial volume for nonpositively curved manifolds with a Ricci-type curvature condition

Abstract

We show that closed manifolds supporting a nonpositively curved metric with negative $(\lfloor \frac{n}{4}\rfloor +1)$-Ricci curvature, have positive simplicial volume. This answers a special case of a conjecture of Gromov. We also establish some related results concerning bounded cohomology and volume growth entropy for this new class of manifolds.