Circle diffeomorphisms, rigidity of symmetric conjugation and affine foliation of the universal Teichmüller space

Abstract

The little Teichmüller space of symmetric homeomorphisms of the circle defines a Banach foliated structure of the universal Teichmüller space. First we consider rigidity of Möbius representations given by symmetric conjugation and failure of the fixed point property for isometric group action on the little Teichmüller space. This space includes the Teichmüller space of circle diffeomorphisms with Hölder continuous derivatives. Then we characterize these diffeomorphisms by Beltrami coefficients of quasiconformal extensions and Schwarzian derivatives of their Bers embeddings. This is used for proving certain rigidity of representations by symmetric conjugation in the group of circle diffeomorphisms. We also consider Teichmüller spaces of integrable symmetric homeomorphisms, which induce another Banach foliated structure and the generalized Weil–Petersson metric on the universal Teichmüller space. As an application, we investigate the fixed point property for isometric group action on these spaces and give a condition for a group of circle diffeomorphisms with Hölder continuous derivatives to be conjugate to a Möbius group in the same class.