Gromov's’ theorem on groups of polynomial growth

Abstract

The aim of this project is to prove Gromov’s theorem on groups of polynomial growth. In order to do so, we will follow the original proof from Mikhail Gromov [Gro81], in which he introduced a convergence for metric spaces, called the Gromov-Hausdorff convergence, that is now widely used in geometry. With this in mind, one of the objectives of the project will also be to study this convergence. It is worth noting that an alternative simpler proof has been found by Bruce Kleiner [Kle10], though it still relies on Tits alternative, a theorem that Gromov’s proof uses too. Later, Terence Tao and Yehuda Shalom [ST10] provided a more fundamental proof based on the work of Kleiner. However, for this project we will not study such proofs.