Advanced Studies in Pure Mathematics

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

Katsuhiko Matsuzaki

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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.

Article information

Source
Geometry, Dynamics, and Foliations 2013: In honor of Steven Hurder and Takashi Tsuboi on the occasion of their 60th birthdays, T. Asuke, S. Matsumoto and Y. Mitsumatsu, eds. (Tokyo: Mathematical Society of Japan, 2017), 145-180

Dates
Received: 30 June 2014
Revised: 29 September 2014
First available in Project Euclid: 4 October 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1538671765

Digital Object Identifier
doi:10.2969/aspm/07210145

Zentralblatt MATH identifier
1387.30064

Subjects
Primary: 30F60: Teichmüller theory [See also 32G15] 58B20: Riemannian, Finsler and other geometric structures [See also 53C20, 53C60] 58C30: Fixed point theorems on manifolds [See also 47H10]
Secondary: 37E10: Maps of the circle 58D05: Groups of diffeomorphisms and homeomorphisms as manifolds [See also 22E65, 57S05]

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

Citation

Matsuzaki, Katsuhiko. Circle diffeomorphisms, rigidity of symmetric conjugation and affine foliation of the universal Teichmüller space. Geometry, Dynamics, and Foliations 2013: In honor of Steven Hurder and Takashi Tsuboi on the occasion of their 60th birthdays, 145--180, Mathematical Society of Japan, Tokyo, Japan, 2017. doi:10.2969/aspm/07210145. https://projecteuclid.org/euclid.aspm/1538671765


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