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  • 12 - Box Spaces: Geometry of Finite Quotients

  • Edited by Matthias Keller, Universität Potsdam, Germany, Daniel Lenz, Universität Potsdam, Germany, Radoslaw K. Wojciechowski
  • Book:
    Analysis and Geometry on Graphs and Manifolds
    Published online:
    14 August 2020
    Print publication:
    20 August 2020, pp 328-343

  • 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 $M$.