Course

Geometric Group Theory

Winter term 21/22

In this course we will consider the structure of (infinite) groups. For this, we will study actions of groups on interesting topological and metric spaces. The geometry of these spaces can be used to obtain some information about the structure of the groups. For example, a group that acts freely on a tree is itself a free group. We will use methods from geometry, analysis and topology. The course can be used as a first step towards topology, geometry, or algebra. Besides this, it offers a solid basis in group theory. In the summer term 2022, a continuation Geometric Group Theory 2 is planned.

The course is aimed at students in or past their 5th semester in mathematics (BSc or MSc). It can be used as part of a specialization in Algebra, Topology, or Geometric Structures - contact me if you have questions about this.

The planned topics of the course include group actions, free groups, free products, Hopfian groups, residually finite groups, amalgams, HNN extensions, presentations, fundamental groups, covering spaces, the Seifert-Van Kampen Theorem, and subgroups of free groups.

Prerequisites include the material from the basic courses in analysis and algebra, and some point-set topology.

The class takes place Monday and Thursday 8 - 10 in lecture hall M4. It starts on Monday, Oct 11 at 8:15. As long as the university permits this, the classes will be in-person. The 4 lectures Dec 13 - 23 can be found as videos in the Learnweb. There you will also find additional material. In January the class will be given in a hybrid format (in person and as a zoom live stream).

The course is given in English.

We have weekly exercise sessions. Solving the exercises and participating in the exercise sessions is a vital part of the course. The exercise sessions take place on Monday afternoon in lecture hall M6, 16:00 - 18:00 or via zoom, depending on the pandemic situation. You may hand in your solutions in letterbox 162 or send them by email to Leon Pernak.

We will have oral exams in February for this course. The exact dates have yet to be fixed.

Literature (you will find these books in the math library):

Magnus, Karras, Solitar, Combinatorial group theory
Camps, Rebel, Rosenberger, Einführung in die kombinatorische und geometrische Gruppentheorie
Lyndon, Schupp, Combinatorial group theory
Robinson, A course in the theory of groups
Geoghegan, Topological methods in group theory
de la Harpe, Topics in geometric group theory
Bogopolski, Introduction to group theory
Stallings, Topology of finite graphs
Fine, Rosenberger, Wienke, Topics in infinite group theory
MacLane, Categories for the working mathematician
Spanier, Algebraic topology
Hatcher, Algebraic topology

Exercise problems:

In Exercise 4 we need that g_i∈G_i\CExercise 3.2 had to be changed.

Exercise sheet	from	hand in by
Sheet 1	14.10.2021	21.10.2021
Sheet 2	21.10.2021	28.10.2021
Sheet 3	28.10.2021	04.11.2021
Sheet 4	04.11.2021	11.11.2021
Sheet 5	11.11.2021	18.11.2021
Sheet 6	18.11.2021	25.11.2021
Sheet 7	25.11.2021	02.12.2021
Sheet 8	02.12.2021	09.12.2021	There was a typo in Exercise 3
Sheet 9	09.12.2021	16.12.2021
Sheet 10	16.12.2021	23.12.2021
Sheet 11	03.01.2021	13.01.2022
Sheet 12	13.01.2022	20.01.2022