MA4H4 Geometric group theory
Term II 2024-2025

Module Description

In this module, MA4H4 (geometric group theory), we assume as background the material from MA3F1 (introduction to topology) as well as the first and second year core material on metric spaces, groups, and linear algebra. A familiarity with euclidean and hyperbolic geometry will be extremely useful - we will quickly review the needed definitions.

As the name implies, geometric group theory is the study of groups from a geometric perspective. Mostly this means studying groups as geometric objects in their own right. However, there is also the more general notion of a "geometric action" where a group acts via isometries on a suitable metric space. After setting out the basic definitions (free groups, presentations of groups, Cayley graphs) we will develop the basic geometric notions (length, area, volume) for groups. After that we will spend significant time developing the theory of word hyperbolic groups, including their visual boundary. If time permits we will discuss other classes such as automorphism groups, right-angled Artin groups, CAT(0) groups, and/or lattices in Lie groups.

Schedule

The schedule has a planned list of topics, organised by lecture. We will change the schedule as necessary, as we work through the material. Links to (handwritten) lecture notes and example sheets will be posted week-by-week. Recorded lectures are available via lecture capture.

Instructor and TA

Class meetings

Reference materials

Professor Karen Vogtmann lectured this module previously; we will not precisely follow those topics, but the notes are still useful for self-study. They are available here: 1, 2, 3, and 4.

Other standard references for the material of this module include:

• A course on geometric group theory by Brian Bowditch;
• Géométrie et théorie des groupes; les groupes hyperboliques de Gromov by Michel Coornaert, Thomas Delzant, and Athanase Papadopoulos;
• Topics in geometric group theory by Pierre de la Harpe
• Sur les groupes hyperboliques d'après Mikhael Gromov by Etienne Ghys and Pierre de la Harpe
• Groups, languages, and automata by Derek Holt, Sarah Rees, and Claas Röver

Links to the lecture capture, the announcement forum, and the discussion forum are on the module's Moodle page.

Example sheets

See the schedule for the example sheets.

Exam

The exam will be 85% of your mark. The exam will be closed book. Here are the exam papers for this module from the last five years.

Assessed work

Assessed work will be 15% of your mark. Of this, 5% may be earned in weeks 3, 5, 7, and 9 by submitting the indicated exercise to the Moodle page. (Your mark will be the total of these four, or 15, whichever is less.) This will be marked by the TA. Please let me (Saul) know if any of the problems are unclear or have typos.

Assessed work is due Friday at 12:00noon in weeks 3, 5, 7, and 9.

On anything you submit, you should record your name, the date, the assignment number, and the module code (MA3F1) at the top of your completed work. If you collaborate with other students or AI, you must include their names and their contribution.

Solutions typeset using LaTeX are much preferred. If your solution is not readable you will lose marks. You must check the spelling and grammar of your work before turning it in. If you do not properly edit your work you will lose marks.

Mistakes

Please tell me on the forum about any errors on this website or made in class. I am especially keen to hear about mathematical errors, gaffes, or typos made in lecture or in the example sheets.