Geometric Group Theory WS25
In geometric group theory, we study the interactions between groups and geometry. We view groups themselves as geometric objects or as symmetries of intereresting spaces.
We endow groups with a metric, or consider their actions on suitable metric spaces. The techniques are thus often rich in visual imagery.
Lecturer: Prof. Dr. Petra Schwer
Teaching assistant: Dr. José Pedro Quintanilha
Links: Muesli, HeiCO (lecture), Heico (exercise class)
This lecture is classified as a "Grundmodul".
Weekly schedule
Lecture: Tuesday and Thursday 8:30 -- 10:00, SR 4
Exercise class: Monday 9:15 -- 10:45, SR 4
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. To take the exam, students need to register for the lecture on HeiCO, and to contact Prof. Schwer for scheduling the exam.
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
Pre-requisites: Basic group theory and linear algebra.
Woche 01: Gruppen und metrische Räume
Inhalt: Wiederholung einiger wichtiger Themen zu Gruppen und metrischen Räumen
Lernziele:
- Was besagt der Satz von Cayley?
- Wie können wir neue Gruppen aus alten konstruieren?
- Was sind Erzeugendensysteme?
- Einige wichtige Beispiele von Gruppen und metrischen Räumen
Woche 02: Cayleygraphen
Inhalt: Gruppenwirkungen auf Graphen, Cayleygraphen
Lernziele:
- Was sind (freie, treue, transitive) Gruppenwirkungen? + Beispiele dafür?
- Was sind Gruppenwirkungen auf Graphen?
- Eigenschaften von Cayleygraphen
Woche 03: Freie Gruppen und Bäume
Inhalt: Präsentation und Konstruktion freier Gruppen, Wirkung auf Bäumen
Lernziele:
- Konstruktion und Definition von freien Gruppen
- Universelle Eigenschaft; Eindeutige reduzierte Form für Elemente
- Zusammenhang Bäume und freie Gruppen
Woche 04: Ping-Pong Lemma
Inhalt: Satz von Nielsen Schreier, Ping-Pong Lemma
Lernziele:
- Was ist das Ping-Pong Lemma?
- Was sagt der Satz von Nielsen-Schreier?
More information will appear soon. To get an idea you may want to consult the course website of the same lecture held in the winter term 2024/25: GGT Winter Semester 2024
Lecture notes and further reading
Typed lecture notes (in German) will be made available as the semester unfolds:
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.
