Groups of polynomial growth

Abstract

In this thesis, we prove Gromov's theorem: if a finitely generated group has polynomial growth then it contains a nilpotent subgroup of finite index. We follow Gromov's original proof from his 1981 paper, "Groups of polynomial growth and expanding maps". This paper introduced revolutionary ideas in metric geometry. However, the original paper is not very approachable, especially for undergraduate students. Therefore, this thesis aims to provide a more accessible version of the paper by providing additional background and filling in omitted details.