On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature

A consequence of the Cheeger-Gromoll splitting theorem states that for any open manifold $(M,x)$ of nonnegative Ricci curvature, if all the minimal geodesic loops at x that represent elements of π₁(M,x) are contained in a bounded ball, then π₁(M,x) is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of π₁(M,x) escape from any bounded metric balls at a sublinear rate with respect to their lengths, then π₁(M,x) is virtually abelian.