This post details a short minicourse that I wll be giving at the University of Geneva on the topic of volume growth in finitly-generated groups. This course consists of 2 talks, on 13 and 20 December 2023. The target audience for these talks is Master’s students in mathematics and related fields.
In geometric group theory an important observation is that every finitely-generated group can be viewed as a metric space (up to a natural equivalence relation known as quasi-isometry). It is then of great interest to characterise the asymptotics of how the volume of a closed ball in this space grow with respect to its radius. In 1968, Milnor asked if there is a characterisation of groups whose growth is bounded above by a polynomial, and if there is any group whose growth is neither polynomial nor exponential. The first of these questions was answered by Gromov in 1981 who showed that all groups with polynomial growth are virtually nilpotent; and the second question was answered by Grigorchuk in 1985 who contracted an example of a group with so-called intermediate growth. These results sparked a great interest in the study of growth.
The focus of these talks will be on understanding these results, and their impact. In our first talk, we cover the definitions, some examples, and some important results in the area, including, the Švarc-Milnor lemma and Gromov’s characterisation of polynomial growth. Our second talk will be devoted to the example, provided by Grigorchuk, of a group with intermediate growth.
You can find the LaTeX and PDFs for my notes here on GitHub.