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Let $(M,g)$ be a compact manifold with Ricci curvature almost bounded frombelow and $\pi:\bar{M}\to M$ be a normal, Riemannian cover. We show that, forany nonnegative function $f$ on $M$, the means of $f\o\pi$ on the geodesicballs of $\bar{M}$ are comparable to the mean of $f$ on $M$. Combined withlogarithmic volume estimates, this implies bounds on several topologicalinvariants (volume entropy, simplicial volume, first Betti number andpresentations of the fundamental group) in Ricci curvature $L^p$-bounded frombelow.

Bounds on the volume entropy and simplicial volume in Ricci curvature $L^p$ bounded from below