Abstract

Hyperbolic (also called non-Euclidean) geometry is like geometry on a leaf of kale. We shall discuss some of its crucial features: for example, walking across the diagonal of an arbitrarily big field is not much quicker than going round the edge. Hyperbolic geometry has been used to study chaotic dynamics on surfaces, while in recent years it has revolutionised the study of three dimensional manifolds (the three dimensional analogue of surfaces).

Abstracting some special features of hyperbolic geometry led Mikhail Gromov to the simple but profound idea of what is now called a Gromov hyperbolic space. This has in turn fed back into hyperbolic geometry itself, being a crucial ingredient of some recent major advances which have given us a more or less complete picture of the geometry of all hyperbolic 3-manifolds.