12 - Box Spaces: Geometry of Finite Quotients

Summary

This set of notes is an introductory survey on box spaces and their connections with Riemannian and spectral geometry. After defining box spaces of finitely generated residually finite groups $G$, we give a brief introduction to coarse geometry, then the link between box spaces and expanders, and the dictionary between coarse geometric properties of box spaces and analytic properties of $G$. When $G$ is the fundamental group of a closed Riemannian manifold $M$, there are deep connections between the geometry of a box space of $G$ and the geometry of the associated tower of Riemannian covers of $M$.