Geometric Group Theory 2

There will be a follow up course in WiSe 2022/23 to this sequence on Geometric Group Theory. It will be a seminar course and we will follow the book "Office hours with a Geometric Group Theorist” by Matt Clay and Dan Margalit. You can find more information here.

Please contact me if you would like to participate.

This is a follow up course to Geometric Group Theory I (WiSe 2021/22) and will focus on the topic of CAT(0) cube complexes. The course will be given in English. The concepts introduced in GGT1 will be useful in this course. However, the course will be fairly self contained and will be suited for students who have some basic knowledge of group theory and group actions. The course is aimed at students in or past their 5th semester in mathematics (BSc or MSc). The prerequisites are basic courses in algebra and analysis, and some point-set topology.

The first course p.rimarily focused on groups and group actions. In this course, we will study the geometry of metric spaces through group actions, with a focus on CAT(0) cube complexes. CAT(0) is a notion of nonpositive curvature and CAT(0) cube complexes are a particularly nice class of CAT(0) spaces. On the one hand, there is a rather broad class of groups which admit non-trivial actions on CAT(0) cube complexes. On the other hand, CAT(0) cube complexes have a combinatorial nature that give them the look and feel of trees. The added structure provided by hyperplanes has allowed for much more progress in understanding their geometry. For example, the Tits Alternative is known to hold for groups acting on CAT(0) cube complexes, but is still open in the setting of CAT(0) spaces generally.

The topics planned for this course include CAT(0) metric spaces, cube complexes, hyperplanes, half space structures, properties of group actions, Cayley graphs, quasi-isometries, Artin and Coxeter groups, complexes associated to Coxeter groups and right angled Artin groups.

Contact Information -

Office: SRZ 305

e-mail: bsathaye@uni-muenster.de

The course is on Learnweb. Please contact me for the code for Learnweb.

Class hours: Tuesdays and Fridays 8:00 to 10:00

Class Location: Lecture hall M4

Tutorial Session: Fridays 12:00 to 14:00

Tutorial Location: Sr 1B (on the first floor of the Main building)

Class Material: Class notes and exercises are posted on Learnweb.

References:

• P. Schwer, Lecture notes on CAT(0) cube complexes

• M. Bridson and A. Haefliger, Metric spaces of nonpositive curvature

• W. Ballmann, Lectures on spaces of nonpositive curvature

• M. W. Davis, The Geometry and Topology of Coxeter groups

• M. Sageev, CAT(0) Cube complexes and groups

• D. T. Wise, From Riches to Raags: 3-manifolds, right-angled Artin groups, and cubical geometry

• M. Clay and D. Margalit, Office hours with a geometric group theorist