The University of Southampton
Courses

# MATH6138 Geometric Group Theory

## Module Overview

Students will be able to apply logic and reason to prove basic results in geometric group theory. They will be able to use the geometry of the hyperbolic plane and other related simple geometric artefacts to demonstrate many of the standard features of geometric group theory. Students will also be able to work with free groups and group presentations. While developing theoretical understanding, students will be able to use examples to illustrate different aspects of geometry and group theory.

### Aims and Objectives

#### Module Aims

Students will be able to apply logic and reason to prove basic results in geometric group theory. They will be able to use the geometry of the hyperbolic plane and other related simple geometric artefacts to demonstrate many of the standard features of geometric group theory. Students will also be able to work with free groups and group presentations. While developing theoretical understanding, students will be able to use examples to illustrate different aspects of geometry and 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.

### Syllabus

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.

### Learning and Teaching

#### Teaching and learning methods

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

TypeHours
Independent Study150
Total study time150

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

Grundlehren der Mathematischen Wissenschaften (1999). Fundamental Principles of Mathematical Sciences.

### Assessment

#### Summative

MethodPercentage contribution
Exam  (2 hours) 100%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

Prerequisites: (MATH2003 AND MATH2049) or (MATH2003 and MATH2046)

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.