Geometric Group Theory
Wintersemester 2024/25
Links: heiCO (lecture), heiCO (exercise class), Müsli
This lecture is classified as a "Grundmodul".
Weekly schedule
Lectures: Wednesday 09:15 - 10:45 in SR 4, Friday 09:15 - 10:45 in SR 3Exercise classes: Monday 09:15 - 10:45 in SR 3
Each week on Friday, an exercise sheet will be published, which is to be handed in on the following Friday, and will be discussed in the ensuing exercise class on Monday. In the first week there will be no exercise class, and in the second week we will have in-class exercises.
Evaluation
Evaluation is through an oral examination at the end of the semester. Here are the evaluation criteria: Bewertungskriterien für mündliche Prüfungen.
Attendance of the lectures and exercise classes, as well as homework submission, are optional. Students are however encouraged to work on the problem sheets and submit their work individually or in groups, and to present their solutions in class. Homework can be submitted by e-mail to jquintanilha [at] mathi [dot] uni-heidelberg [dot] de.
Program
In geometric group theory, we study the interactions between groups and geometry. The goal is to investigate groups with the help of geometric tools. To that end, we endow them with a metric, or consider their actions on suitable metric spaces. The techniques are thus often rich in visual imagery. The following quote conveys the flavor of the subject:
We often think by analogies. We have pictures in small dimensions and must try to decide how much of the picture remains accurate in higher dimensions and how much has to change. This visualization is very different from just manipulating a string of symbols.
(John Milnor, 2011)
Pre-requisites: Basic group theory and linear algebra.
Problem sheet 0 (in-class exercises)
- Woche 01 (KW42): Gruppen und metrische Räume
Notizen: 01 - Gruppen und Räume, 02 - Gruppenwirkungen
Problem sheet 1 - Woche 02 (KW43): Freie Gruppen I
Notizen: 03 - Freie Gruppen I
Problem sheet 2 - Woche 03 (KW44): Freie Gruppen II
03 - Freie Gruppen II
Problem sheet 3
Notizen: - Woche 04 (KW45): Endlich präsentierte Gruppen & Cayleygraphen
Notizen: 04 - Endlich präsentierte Gruppen
Problem sheet 4 - Woche 05 (KW46): Satz von Milnor-Svarc
Notizen: 05 - Cayleygraphen und Milnor-Svarc
Problem sheet 5 - Woche 06 (KW47): Satz von Milnor-Svarc und QI-Invarianten
Notizen: 06 - Quasi-Isometrie
- Woche 07 (KW48): Rest QI-Invarianten & Hyperbolische Räume
- Woche 08 (KW49): Rest hyperbolische Räume und Das Wortproblem
- Woche 09 (KW50): Das Wortproblem - Teil 2
Literature
It will not be necessary to consult literature beyond the lecture notes in order successfully take part in the course. Nevertheless, here are some additional resources:
- Bridson, M., Haefliger, A. - Metric spaces of non-positive curvature, Springer, 1999.
- Löh, C. - Geometric group theory. An introduction, Springer International Publishing, 2017. (SpringerLink)
- Bridson, M. - Geometric and combinatorial group theory. (Übersichtsartikel)
- Rosebrock, S. - Geometrische Gruppentheorie. Ein Einstieg mit dem Computer Basiswissen für Studium und Mathematikunterricht, Vieweg+Teubner Verlag, Wiesbaden, 2010. (SpringerLink)
- Stillwell, J. - Classical topology and combinatorial group theory, 2nd ed, Springer, 1993.
- Lyndon, R., Schupp, P. - Combinatorial group theory, Springer, 1977.
- Serre, J.-P. - Trees, corrected 2nd print, Springer, 2003.