Nonnegative Ricci curvature and escape rate gap

Abstract Let M be an open n -manifold of nonnegative Ricci curvature and let p ∈ M p M. We show that if (M, p) (M, p) has escape rate less than some positive constant ϵ ⁢ (n) (n), that is, minimal representing geodesic loops of π 1 ⁢ (M, p) ₁ (M, p) escape from any bounded balls at a small linear rate with respect to their lengths, then π 1 ⁢ (M, p) ₁ (M, p) is virtually abelian. This generalizes the author’s previous work J. Pan, On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature, Geom. Topol. 25 2021, 2, 1059–1085, where the zero escape rate is considered.