Nonnegative Ricci curvature and escape rate gap (Q2065871)

In this paper, the author generalizes his previous work on the \(escape\; rate\) of an open \(n\)-manifold of nonegative Ricci curvature, a notion whose definition he also previously prposed, and which measures how fast the representing loops of the fundamental group escape from any bounded balls. More precisely, he shows in Theorem A that if a pointed \(n\)-manifold \((M,p)\) has escape rate less than some positive constant \(\epsilon(n)\), then \(\pi(M,p)\) is virtually abelian. An essential tool in the proof of this main theorem is the use of the pointed Gromov-Hausdorff metric, and more specifically of preparatory result (Theorem 0.1.) which he also proves.