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

Abstract : Let $(M,g)$ be a compact manifold with Ricci curvature almost bounded from below and $\pi:\bar{M}\to M$ be a normal, Riemannian cover. We show that, for any nonnegative function $f$ on $M$, the means of $f\o\pi$ 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.
Document type :
Journal articles

https://hal.archives-ouvertes.fr/hal-00343654
Contributor : Erwann Aubry <>
Submitted on : Tuesday, December 2, 2008 - 2:20:05 PM
Last modification on : Monday, October 12, 2020 - 10:27:28 AM

### Identifiers

• HAL Id : hal-00343654, version 1
• ARXIV : 0810.0149

### Citation

Erwann Aubry. Bounds on the volume entropy and simplicial volume in Ricci curvature $L^p$ bounded from below. International Mathematics Research Notices, Oxford University Press (OUP), 2009, 2009 (10), pp.1933-1946. ⟨hal-00343654⟩

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