Project 4H 2022/23 Durham University

A (countably) infinite group is a set with a group operation and where we assume that the set is countably infinite. Think of an example you know. Possibly you will think of Z^d, viewed in terms of translations of R^d. The action of Z^d by translations on a geometric object gives us a way to understand properties of the group. We have a combinatorial way to describe any group by its "presentation" -- in this way we see the group operation as concatenation of words in an equivalence class. Generally, infinite groups can be difficult to understand without a nice geometric action, and moreover there are some groups with unusual properties that you wouldn't expect from only looking a linear maps of R^d!

Geometric group theory views the group as a geometric object and seeks to characterize properties of the group law in geometric language. For example: We can show that the "word problem" is decidable for a large abundant class of groups (the free group in the figure is the prototypical example). We can characterize nilpotency of the group by the volume growth of open balls in the Cayley graph growing like a polynomial. We can express certain nice decompositions of the group in terms of how it looks at infinity.

The aim of this project is to learn the foundational concepts in geometric group theory and some influential examples, with a view studying a particular group property.

Pre-requisites: Algebra 2.

Useful to have: Geometry 3.

Some references:

Office hours with a geometric group theorist, M. Clay and D. Margalit.

Introduction to Group Theory, O. Bogopolski.