# 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 will be an introduction to the basic ideas of geometric group theory. The main aim of subject is to apply geometric constructions to understand finitely generated groups. Although many of the ideas can be traced back a century or more, the modern subject has its origins in the 1980s and has rapidly grown into a major field in its own right. It draws on ideas from many subjects, though two particular sources of inspiration are low dimensional topology and hyperbolic geometry. A significant insight is that ``most'' finitely presented groups are ``hyperbolic'' in a broad sense. This has many profound applications. Some familiarity with group presentations will be useful. Beyond that, geometric or topological background is probably more relevant than algebraic background.

**Learning outcomes:** An understanding of the main notions of quasi-isometry, quasi-isometry invariants, and hyperbolic groups. To be able to apply these in particular examples.

**Books**:

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).

B. H. Bowditch, *A course on geometric group theory* : MSJ Memoirs, Vol 16, Mathematical Society of Japan (2006).