# Geometric Group Theory Research

Several members of our pure mathematics research group carry out work in this area.

## About geometric group theory

An association between geometry and group theory can be traced back to Klein's crucial observation in the late 19th century. Klein observed that we can understand geometric constructions in terms of their symmetry groups.

It soon emerged that the reverse process could also be useful. For example, early in the 20th century, Dehn used techniques of topology to answer questions of logic arising in group theory. In the 1970s and 80s such ideas were unified and extended by Thurston, Cannon and particularly Gromov. Gromov used the geometric properties of spaces on which groups act to study groups, and vice-versa. This interplay between geometry and group theory, which is the subject of geometric group theory, has been very fruitful. The theory has expanded rapidly in the last few decades.

### Its applications

Notable successes are applications in:

- 3-manifold theory
- complex dynamics
- first order logic
- the theory of formal languages
- Riemannian geometry
- representation theory