# MA4H4 Geometric Group Theory

**Lecturer:** Karen Vogtmann

**Term(s):** Term 1

**Status for Mathematics students:** List C

**Commitment:** 30 lectures

**Assessment:** 3-hour exam 85%, coursework 15%

**Formal registration prerequisites: **None

**Assumed knowledge: **Group theory, Euclidean and hyperbolic geometry, Fundamental group and covering spaces. These subjects are covered by Warwick courses MA136 Introduction to Abstract Algebra, MA243 Geometry and MA3F1 Introduction to Topology. Note that MA249 Algebra II: Groups and Rings and MA260 Norms, Metrics and Topologies or MA222 Metric Spaces will also be assumed as they are prerequisites for MA3F1 Introduction to Topology.

**Useful background: **Any course in algebra, geometry or topology, in particular,

- MA251 Algebra I: Advanced Linear Algebra
- MA3K4 Introduction to Group Theory or MA442 Group Theory (taken last year)

**Synergies:**

The following modules go well together with Geometric Group Theory:

- MA3H6 Algebraic Topology
- MA3H5 Manifolds
- MA3D9 Geometry of Curves and Surfaces
- MA3E1 Groups and Representations
- MA475 Riemann Surfaces
- MA448 Hyperbolic Geometry
- MA473: Reflection Groups

**Content**: This module is an introduction to the field of geometric group theory. The basic premise of this field is that topological and geometric methods can be applied to the study of finitely generated groups by studying actions of these groups on various spaces. For example restrictions on the topology and curvature of the space can have strong algebraic consequences for groups that act "properly." Although this basic idea can be traced back a century or more, the subject exploded in the 1980s with work of Thurston and Gromov, and has become a major area of current research. Prominent roles in geometric group theory are played by low dimensional topology and hyperbolic geometry, but it has points of contact with and borrows techniques from a wide range of mathematical subjects**.**

**Learning outcomes:** Familiarity with classes of groups commonly studied in geometric group theory, the spaces they act on and ability to use features of these spaces to extract information about the groups.

**Books**:

Löh, *Geometric Group Theory, An Introduction* : Universitext, Springer (2017)

P. de la Harpe, *Topics in Geometric Group Theory* : Chicago lectures in mathematics, University of Chicago Press (2000)

M. Bridson, A. Haefliger, *Metric Spaces of Non-Positive Curvature* : Grundlehren der Math. Wiss. No. 319, Springer (1999)

C. Druţu, M. Kapovich, *Geometric Group Theory* : Colloquium publications, Vol. 63, American Mathematics Society (2018)

A. Casson and S. Bleiler, *Automorphisms of Surfaces after Thurston*. Cambridge University Press (1988)