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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

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.

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Ricci curvature, fundamental groups
Mathematical Subject Classification 2010
Primary: 53C20, 53C23
Secondary: 53C21, 57S30
Received: 6 March 2020
Revised: 17 May 2020
Accepted: 17 June 2020
Published: 27 April 2021
Proposed: Tobias H Colding
Seconded: András I Stipsicz, David M Fisher
Jiayin Pan
Department of Mathematics
University of California, Santa Barbara
Santa Barbara, CA
United States