Updates:

2018-08-26: Added info about TA, office hours, and first homework.

2018-08-31: Second homework added. Future homeworks will appear on the homework page by the Friday before they are due, without an extra announcement in this space.

2018-11-08: Updated information about end of semester activities. Project now due on November 20, second prelim to be distributed November 28

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.

Homework

Homework will be due approximately weekly. LaTeX is recommended, though clearly handwritten homework will also be accepted, and drawing figures by hand is fine. You can discuss homework problems with each other but should write them up separately. Be sure to note any help you got. Looking up solutions on the internet is not allowed. Homework assignments can be found here.

Exams

There will be two take-home exams, one in early October, and one at the end of the course. You will have about a week for each exam. The second prelim will be distributed on Wednesday, November 28

Project

In addition to the homework and exams, everyone will write an expository paper on some aspect of geometric group theory not covered in the lectures. This work will also be presented in 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 November 16 November 20th in class. The last few class meetings will be set aside for student presentations.

Grades

Each exam counts for one fifth of your grade, the project/presentation together count for one quarter, and homework is the remaining thirty-five percent.

References

(The links should work on campus, or off-campus, using PassKey)

Schedule

The following schedule is incomplete and subject to change.
  Date   Topic   Reading
23 Aug Introduction to the course. Groups and graphs. This web page
28 Aug Groups acting on graphs [M], Chapter 1
30 Aug Groups acting on graphs
4 Sep Fundamental domains
6 Sep Fundamental domains, ctd.
11 Sep Reflection Groups [M], Chapter 2
13 Sep Free Groups [M], Chapter 3
18 Sep Groups acting on trees
20 Sep Groups acting on trees
25 Sep Groups acting on trees
27 Sep Groups acting on trees
2 Oct Baumslag-Solitar groups [M], Chapter 4
4 Oct The Word problem [M], Chapter 5
9-11 Oct No Class -- Fall Break
16 Oct Constructing Cayley graphs
18 Oct Constructing Cayley graphs
23 Oct An infinite torsion group [M], Chapter 6
25 Oct An infinite torsion group
30 Oct Regular languages [M], Chapter 7
1 Nov Pumping Lemma and Myhill-Nerode
6 Nov Regular languages and group theory, I
8 Nov Regular languages and group theory, II
13 Nov Lamplighter group [M], Chapter 8
15 Nov Coarse equivalence, ends [M], Chapter 11
20 Nov Ends, Student presentations: 11am Conan G.
27 Nov Student presentations: 10:10am Tom S., 10:35am Mike S., 11am Matt F.
29 Nov Student presentations: 10:10am Dongjin L., 10:35 Alex X., 11am Rohin G.
4 Dec Student presentations: 10:10am Alex R., 10:35am Michael A.
Jason Manning's home page.

Last Updated 2018-11-26