MATE6036 Geometric Group Theory (5 cr)
Cooperation network course
Network: Cross-institutional studies in advanced courses in mathematics and statistics
This course is offered through the Network for Advanced Studies in Mathematics. These studies are available for the following degree students:
- Bachelor's Degree Programme in Mathematics
- Master's Degree Programme in Mathematics
- Bachelor's Degree Programme in Mathematics (Subject Teacher)
- Master's Degree Programme in Mathematics (Subject Teacher)
- Bachelor's Degree Programme in Mathematics, Chemistry or Physics Subject Teacher Education and Primary Teacher Education (Specialication in Mathematics)
- Master's Degree Programme in Mathematics, Chemistry or Physics Subject Teacher Education and Primary Teacher Education (Specialication in Mathematics)
- Doctoral Programme in Mathematics and Statistics
- Doctoral Programme in Mathematics and Science (Specialication in Mathematics)
Grading scale:
0-5
Description
Geometric group theory is an approach to group theory, where geometric ideas are used to provide tools for solving algebraic problems and as a source of interesting properties and questions. Some of these methods are of a topological nature (fundamental groups and covering spaces), and some are more in the direction of large-scale / coarse geometry (Cayley graphs and growth rates of groups). Our emphasis is on the latter, and we concentrate on finitely-generated infinite groups.
We cover basic notions, such as Cayley graphs, growth rates of groups, quasi-isometry, the Svarc-Milnor theorem, free products with amalgamations, HNN extensions, presentations, and word-hyperbolic groups. Examples play a major role in geometric group theory, and we cover the main "nice" constructions (abelian, free groups, wreath products). We also overview more special constructions in the literature, such as the Grigorchuk group.
Learning outcomes
The student becomes familiar with the basic examples of finitely-generated infinite groups, and the basic concepts, constructions and tools in geometric group theory, in particular from a large-scale perspective.
Additional information
Course will be given on campus in autumn 2026.
Description of prerequisites
Basics of Algebra II (Algebran peruskurssi II) strongly recommended, Group Theory (Ryhmäteoria) recommended