Stable quasiconformal mapping class groups and asymptotic teichmü ller spaces

Ege Fujikawa*, Katsuhiko Matsuzaki

*この研究の対応する著者

研究成果査読

4 被引用数 (Scopus)

抄録

The stable quasiconformal mapping class group is a group of quasiconformal mapping classes of a Riemann surface that are homotopic to the identity outside some topologically finite subsurface. Its analytic counterpart is a group of mapping classes that act on the asymptotic Teichm üller space trivially. We prove that the stable quasiconformal mapping class group is coincident with the asymptotically trivial mapping class group for every Riemann surface satisfying a certain geometric condition. Consequently, the intermediate Teichmüller space, which is the quotient space of the Teichmüller space by the asymptotically trivial mapping class group, has a complex manifold structure, and its automorphism group is geometrically isomorphic to the asymptotic Teichmüllermodular group. The proof utilizes a condition for an asymptotic Teichmüller modular transformation to be of finite order, and this is given by the consideration of hyperbolic geometry of topologically infinite surfaces and its deformation under quasiconformal homeomorphisms. Also these arguments enable us to show that every asymptotic Teichmüller modular transformation of finite order has a fixed point on the asymptotic Teichmüller space, which can be regarded as an asymptotic version of the Nielsen theorem.

本文言語English
ページ(範囲)637-675
ページ数39
ジャーナルAmerican Journal of Mathematics
133
3
DOI
出版ステータスPublished - 2011 6

ASJC Scopus subject areas

  • 数学 (全般)

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