Universal Teichmüller Space

We present an outline of the theory of universal Teichmüller space, viewed as part of the theory of QS, the space of quasisymmetric homeomorphisms of a circle. Although elements of QS act in one dimension, most results about QS depend on a two-dimensional proof. QS has a manifold structure modelled on a Banach space, and after factorization by PSL(2,R) it becomes a complex manifold. In applications, QS is seen to contain many deformation spaces for dynamical systems acting in one, two and three dimensions; it also contains deformation spaces of every hyperbolic Riemann surface, and in this naive sense it is universal. The deformation spaces are complex submanifolds and often have certain universal properties themselves, but those properties are not the object of this paper. Instead we focus on the analytic foundations of the theory necessary for applications to dynamical systems and rigidity. We divide the paper into two parts. The first part concerns the real theory of QS and results that can be stated purely in real terms; the basic properties are given mostly without proof, except in certain cases when an easy real-variable proof is available. The second part of the paper brings in the complex analysis and promotes the view that properties of quasisymmetric maps are most easily understood by consideration of their possible two-dimensional quasiconformal extensions.