Rigidity of groups of circle diffeomorphisms and Teichm\"uller spaces

We consider deformations of a group of circle diffeomorphisms with H"older continuous derivatives in the framework of quasiconformal Teichm"uller theory and show certain rigidity under conjugation by symmetric homeomorphisms of the circle. As an application, we give a condition for such a diffeomorphism group to be conjugate to a M"obius group by a diffeomorphism of the same regularity. The strategy is to find a fixed point of the group which acts isometrically on the integrable Teichm"uller space with the Weil-Petersson metric.