Manchester Geometry Seminar 2002/2003

It is probably best to start with the ordinary two-dimensional torus. Any complex structure on the torus is given by a lattice in C, which can be normalised to be of the form Z+α Z from some α in C with Im α >0 . The Teichmüller space of the torus is the upper half-plane, that is, the set of all such α . Each α represents a torus with a complex structure, together with two marked loops. For a general compact surface, or more generally, a "finite type" surface, Teichmüller space is a very classical object, and is always a finite-dimensional Euclidean space, up to homeomorphism. There is a distance on Teichmüller space, called the Teichmüller distance, which is large if the distortion between the corresponding surfaces with marked loop sets is large. The Teichmüller distance for the torus is simply the Poincaré distance on the upper half-plane. For the other compact surfaces, Teichmüller distance is not a Riemannian metric. But unique shortest paths - geodesics - do exist between any two points. These geodesics have somewhat special features and have been the object of study in recent years. One reason for dynamicists to be interested in them is that there is a family of very basic dynamical systems - interval exchanges - for which the dynamical behaviour which has strong connections with the geometry of Teichmüller space, and with the behaviour of Teichmüller geodesics. I shall also try to mention applications - to the classification and structure of hyperbolic three-manifolds and to automatic properties of the Mapping Class group.