  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 $\pi_1(M,x)$ are contained in
a bounded ball, then $\pi_1(M,x)$ is virtually abelian. We generalize the above
result: if these minimal representing geodesic loops of $\pi_1(M,x)$ escape
from any bounded metric balls at a sublinear rate with respect to their
lengths, then $\pi_1(M,x)$ is virtually abelian.
