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

Let (M, g) be a compact manifold with Ricci curvature almost bounded from below and $\pi:\bar M\rightarrow M$ 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 $\bar M$ 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 L^p-bounded from below.