Nonnegative Ricci curvature and escape rate gap

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.

Fixed some typos. Slightly modified some wordings in the proof of Theorem 0.1 to fix a minor issue. To appear in Crelle's Journal

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