#### Volume 25, issue 2 (2021)

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

### Jiayin Pan

Geometry & Topology 25 (2021) 1059–1085
##### Abstract

A consequence of the Cheeger–Gromoll splitting theorem states that for any open manifold $\left(M,x\right)$ of nonnegative Ricci curvature, if all the minimal geodesic loops at $x$ that represent elements of ${\pi }_{1}\left(M,x\right)$ are contained in a bounded ball, then ${\pi }_{1}\left(M,x\right)$ is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of ${\pi }_{1}\left(M,x\right)$ escape from any bounded metric balls at a sublinear rate with respect to their lengths, then ${\pi }_{1}\left(M,x\right)$ is virtually abelian.

##### Keywords
Ricci curvature, fundamental groups
##### Mathematical Subject Classification 2010
Primary: 53C20, 53C23
Secondary: 53C21, 57S30