Bounds on the Volume Entropy and Simplicial Volume in Ricci Curvature Lp-Bounded from Below

摘要

Let (M, g) be a compact manifold with Ricci curvature almost bounded from below and be a normal, Riemannian cover. We show that, for any non-negative function f on M, the means of f○π on the geodesic balls of are comparable to the mean of f on M. Combined with logarithmic volume estimates, this implies bounds on several topological invariants (volume entropy, simplicial volume, first Betti number, and presentations of the fundamental group) in Ricci curvature Lp-bounded from below.