Gromov's Simplicial Volume Vanishing Conjecture for Positive Scalar Curvature and Fundamental Group Decay

A conjecture of Gromov predicts that every closed oriented manifold admitting a metric of positive scalar curvature has vanishing simplicial volume. In this paper, we prove this conjecture under a mild decay condition on the fundamental group. As a geometric application, we show that, in every dimension, there exists a universal negative constant such that any closed oriented manifold whose scalar curvature is bounded below by a normalized positive constant and whose Ricci curvature is bounded below by the universal constant has vanishing simplicial volume.