Fundamental Groups of Spaces with Bakry–Emery Ricci Tensor Bounded Below

Abstract

We first extend Cheeger–Colding’s Almost Splitting Theorem (Ann Math 144:189–237, 1996) to smooth metric measure spaces. Arguments utilizing this extension show that if a smooth metric measure space has almost nonnegative Bakry–Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abelian. Second, if the smooth metric measure space has Bakry–Emery Ricci curvature bounded from below then the number of generators of the fundamental group is uniformly bounded. These results are extensions of theorems which hold for Riemannian manifolds with Ricci curvature bounded from below. The first result extends a result of Yun (Proc Amer Math Soc 125:1517–1522, 1997), while the second extends a result of Kapovitch and Wilking (Structure of fundamental groups of manifolds with Ricci curvature bounded below, 2011).