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

Erwann Aubry (1)

1 JAD - Laboratoire Jean Alexandre Dieudonné

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.

Domaines: Géométrie différentielle [math.DG]

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