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

Abstract

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.

Changed Question 3.13 to Conjecture 3.13 with a modified statement. Fixed some typos. To appear in Geometry & Topology

Cited by in corpus (4)

• Examples of Ricci limit spaces with non-integer Hausdorff dimension
• Nonnegative Ricci curvature and escape rate gap
• Nonnegative Ricci curvature, metric cones, and virtual abelianness
• The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature