# Nonnegative Ricci curvature, escape rate, and virtual abelianness

# Nonnegative Ricci curvature, escape rate, and virtual abelianness

**Zoom link**: **https://princeton.zoom.us/j/594605776**

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 M has nonnegative Ricci curvature and escape rate 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.