Open Access
VOL. 72 | 2017 Circle diffeomorphisms, rigidity of symmetric conjugation and affine foliation of the universal Teichmüller space
Katsuhiko Matsuzaki

Editor(s) Taro Asuke, Shigenori Matsumoto, Yoshihiko Mitsumatsu

Adv. Stud. Pure Math., 2017: 145-180 (2017) DOI: 10.2969/aspm/07210145


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.


Published: 1 January 2017
First available in Project Euclid: 4 October 2018

zbMATH: 1387.30064

Digital Object Identifier: 10.2969/aspm/07210145

Primary: 30F60 , 58B20 , 58C30
Secondary: 37E10 , 58D05

Keywords: Beltrami coefficients , conformally natural extension , fixed point property , isometric action , Quasiconformal , quasisymmetric , Schwarzian derivative , uniformly convex , Weil–Petersson metric

Rights: Copyright © 2017 Mathematical Society of Japan


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