The Gromov-Hausdorff distance and groups of polynomial growth

Abstract

This undergraduate thesis aims to present a proof of Gromov’s Theorem on Groups of Polynomial Growth, which characterizes groups of polynomial growth as those that have nilpotent subgroups of finite index. For such, we introduce basic concepts that are essential in understanding the proof, namely metric spaces, topological spaces, manifolds, and groups. We also explore the theory of the Gromov-Hausdorff distance and convergence, which are tools from Metric Geometry that prove to be useful in the study of groups as geometric objects. Lastly, we examine group growth and the essential properties of groups of polynomial growth, culminating in a proof of Gromov’s Theorem.