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

TL;DR

This paper establishes bounds on topological invariants like volume entropy and simplicial volume for compact manifolds with Ricci curvature nearly bounded from below, using volume comparison techniques and $L^p$ bounds.

Contribution

It introduces new bounds on topological invariants for manifolds with Ricci curvature $L^p$-bounded from below, extending previous geometric analysis results.

Findings

Bounds on volume entropy and simplicial volume derived

Comparison of mean values of functions on covers and base manifold

Implications for Betti numbers and fundamental group presentations

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.