Introductory description

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 arehyperbolic'' 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.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Group presentations. Cayley graphs. Quasi-isometries. Hyperbolic groups. Dehn functions and other quasi-isometry invariants.

Also included would be a brief informal review of fundamental groups and hyperbolic geometry (for those who have not seen these before), though these serve mainly to provide a source illustrative examples and motivation, and are not essential to the logical development.

Learning outcomes

By the end of the module, students should be able to:

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

Indicative reading list

Brian H. Bowditch, A course on geometric group theory, October 2005. MSJ Mem. Vol 16 (2006).

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).

Subject specific skills

The course will expose students to a wide range of ideas in
an advanced subject at the forefront of current mathematical research.
The students will learn to familiarise themselves with abstract
concepts, and to relate them to physical intuition.

Transferable skills

They will learn to follow complex reasoning, to construct logical arguments, to think independently, and to develop research skills.