Advanced Seminar on Group Theory

Hyperbolic Groups

The main idea of geometric group theory is that groups can be thought of and studied as geometric objects. One way to do this is by introducing a metric structure on groups via word metrics on their Cayley graphs, which then allows us to study the large scale geometry of groups with respect to this metric structure.

In particular, one can introduce a notion of negative curvature to large scale geometry via slim triangles, which can be applied to Cayley graphs. This then leads to the definition of hyperbolic groups as those finitely generated groups with negative curvature.

The study of hyperbolic groups will be the main focus of our Seminar. In the first half (Talks 1-4) we will introduce all the basic notions required to define hyperbolic groups. In the second half of the seminar (Talks 5-9) we will look at some interesting properties and applications, including the Rips construction and the solvability of the word problem for hyperbolic groups.

More details can be found in the program.

References:

1. C. Löh. Geometric group theory. An introduction, Universitext. Springer, 2017. Springer
2. R. Lyndon, P. Schupp. Combinatorial Group Theory, Springer, Berlin, 1977. Springer
3. E. Ghys, P. de la Harpe (eds.). Sur les groupes hyperboliques d’après Mikhael Gromov, Birkhäuser, 1990. Springer
4. M. Hull. Hyperbolic Groups, Lecture Notes. PDF
5. E. Rips. Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982), 45–47. DOI