Nonnegative Ricci curvature, escape rate, and virtual abelianness

Abstract: A consequence of 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 set, then $\pi_1(M,x)$ is virtually abelian. However, it is prevalent for these loops to escape from any bounded sets. In this talk, we introduce a quantity, escape rate, to measure how fast these loops escape. Then we prove that if the escape rate is less than some positive constant $\epsilon(n)$, which only depends on the dimension $n$, then $\pi_1(M,x)$ is virtually abelian. The main tools are equivariant Gromov-Hausdorff convergence and Cheeger-Colding theory on Ricci limit spaces.

differential geometrymetric geometry

Audience: researchers in the topic

( video )

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Riemannian geometry.

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