Nonnegative Ricci curvature and escape rate gap

Let M be an open n-manifold of nonnegative Ricci curvature and let p ∈ M {p\in M} . We show that if ( M , p ) {(M,p)} has escape rate less than some positive constant ϵ ⁢ ( n ) {\epsilon(n)} , that is, minimal representing geodesic loops of π 1 ⁢ ( M , p ) {\pi_{1}(M,p)} escape from any bounded balls at a small linear rate with respect to their lengths, then π 1 ⁢ ( M , p ) {\pi_{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.