Time: 14:00-15:00, Monday, December 30

Venue: Lecture Hall

Speaker: Xingyu Zhu (SLMath/Michigan State University)

Title: Fundamental groups and Ricci curvature

Abstract: Understanding the topological obstruction of curvature lower bounds is a central theme in geometry. For complete manifolds of nonnegative Ricci curvature, it is previously known that when the manifold is compact its fundamental group is virtually abelian and when non-compact, any finitely generated subgroup of the fundamental group is virtually nilpotent. Moreover any finitely generated virtually nilpotent group can be realized a fundamental group of some manifold of nonnegative Ricci curvature. A natural question arising in this context is, how to recognize the (virtually) abelian fundamental group among the (virtually) nilpotent ones. In this talk we present a result that under extra linear volume growth condition, the fundamental group will always be virtually abelian and if one further assumes that the Ricci curvature is strictly positive, then the fundamental group is finite. We  highlight that, in the proof, the calculus on non-smooth metric measure spaces with lower Ricci curvature bounds (RCD spaces) plays a crucial role. This is joint work with Dimitri Navarro and Jiayin Pan.