Nonnegative Ricci curvature and escape rate gap

TL;DR

This paper proves that open manifolds with nonnegative Ricci curvature and sufficiently small escape rate have virtually abelian fundamental groups, extending previous results from zero escape rate to positive escape rate scenarios.

Contribution

It generalizes earlier work by showing that small positive escape rates imply virtually abelian fundamental groups in nonnegative Ricci curvature manifolds.

Findings

Manifolds with small escape rate have virtually abelian fundamental groups.

Extension of previous zero escape rate results to positive escape rate cases.

Provides a quantitative link between escape rate and algebraic properties of the fundamental group.

Abstract

Let $M$ be an open $n$-manifold of nonnegative Ricci curvature and let $p\in M$. We show that if $(M,p)$ has escape rate less than some positive constant $\epsilon(n)$, that is, minimal representing geodesic loops of $\pi_1(M,p)$ escape from any bounded balls at a small linear rate with respect to their lengths, then $\pi_1(M,p)$ is virtually abelian. This generalizes the author's previous work, where the zero escape rate is considered.