Geometric Group Theory

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Recognition of situations where group actions on trees or low dimensional spaces are relevant in applications
• To be able to prove simple free standing statements about group actions on trees, hyperbolic groups, CAT(0) metric spaces or Fuchsian groups.

1. Free groups and group presentations. Free products.

2. Group actions on trees.

3. The hyperbolic plane and the modular group.

4. Triangle groups, surface groups and Fushsian groups.

5. The modular group.

6. Gromov hyperbolic groups.

7. The CAT(0) condition and cube complexes.

8. Products of trees as natural examples of cube complexes. Time permitting, one or two more advanced topics may be included from the following list:

9. Artin groups and the Bestvina--Brady Theorem.

10. The Rips Complex of a hyperbolic group.

11. Amalgamated free products and HNN-extensions.

12. Automorphism groups and outer automorphism groups including GL(n, Z) and the automorphism group of a free group.

Teaching and learning methods

Lectures, weekly problem sheets and tutorials. Private study is also important for work on the

problem sheets.

Resources & Reading list

Textbooks

Bridson, Martin R.. Metric spaces of non-positive curvature. Haefliger: Andre.

Grundlehren der Mathematischen Wissenschaften (1999). Fundamental Principles of Mathematical Sciences. Berlin: Springer-Verlag.

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.

Repeat Information

Repeat type: Internal & External