The Geometry of Discrete Groups

This is an introduction to the geometric approach to the theory of infinite discrete groups. Topics will include group actions, the construction of Cayley graphs, connections to formal language theory, actions on trees, volume growth, and large-scale geometry. Theorems will be balanced by specific examples such as Baumslag-Solitar groups, the Lamplighter group, and Thompson’s groups.

More complete information about the class will be available via Canvas once the semester starts.

Homework

Problem sets will be posted approximately weekly, due a week later. LaTeX is recommended, though clearly handwritten homework will also be accepted, and drawing figures by hand is fine. Unless otherwise stated, you can discuss problems with each other but should write them up separately. Please note who you worked with. Looking up solutions on the internet is not allowed.

Exams

There will be a take-home exam handed out in the first half of October, and an in-class final exam on a date to be announced.

Project

Finally, you will write an expository paper on some aspect of geometric group theory not covered in the lectures. This work will also be presented to the class. Geometric group theory has connections to almost every area of modern mathematics. In lectures and homework we will restrict ourselves to topics which can be understood with a minimum of background, but this is an opportunity to find out t more about some connections with some other mathematics which you are interested in. The written paper will be due before the Thanksgiving break, and student presentations will happen after the break.

References

(The links should work on campus, or off-campus, using PassKey) [M] is the main text. [CM] and [L] are supplemental, and likely to be useful sources for project ideas.
Jason Manning's home page.

Last Updated 2020-08-27