The Group of Quasisymmetric Homeomorphisms of the Circle and Quantization of the Universal Teichmüller Space

Armen G. Sergeev

Symmetry, integrability and geometry: methods and applications, Tome 5 (2009)

Résumé

In the first part of the paper we describe the complex geometry of the universal Teichmüller space $\mathcal T$, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The quotient $\mathcal S$ of the diffeomorphism group of the circle modulo Möbius transformations may be treated as a smooth part of $\mathcal T$. In the second part we consider the quantization of universal Teichmüller space $\mathcal T$. We explain first how to quantize the smooth part $\mathcal S$ by embedding it into a Hilbert–Schmidt Siegel disc. This quantization method, however, does not apply to the whole universal Teichmüller space $\mathcal T$, for its quantization we use an approach, due to Connes.

Détail

Keywords: universal Teichmüller space; quasisymmetric homeomorphisms; Connes quantization.

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