UC Santa Barbara
The Structure of Fundamental Groups of Smooth Metric Measure Spaces
- Author(s): Jaramillo, Maree Trisha Afaga
- Advisor(s): Wei, Guofang
- et al.
In this dissertation, we investigate the structure of fundamental groups of smooth metric measure spaces with Bakry-Emery Ricci tensor bounded from below. In particular, we generalize a result of Gabjin Yun to show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci tensor and a lower bound on volume, then its fundamental group is almost abelian. We also generalize a result of Vitali Kapovitch and Burkhard Wilking to show that there is a uniform bound on the number of generators of the fundamental groups of smooth metric measure spaces with Bakry-Emery Ricci tensor bounded from below. In order to utilize the proof techniques of Yun and Kapovitch-Wilking, we extend many valuable tools for studying Riemannian manifolds with Ricci curvature bounded from below to the smooth metric measure space setting. In particular, we extend Jeff Cheeger and Tobias Colding's Splitting Theorem, which plays a key role in the proofs of our results on fundamental groups.