Abstract
We present an outline of the theory of universal Teichmtüller 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 article. Instead we focus on the analytic foundations of the theory necessary for applications to dynamical systems and rigidity.
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