### Abstract

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.

Original language | English |
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Pages (from-to) | 637-675 |

Number of pages | 39 |

Journal | American Journal of Mathematics |

Volume | 133 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2011 Jun |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*American Journal of Mathematics*,

*133*(3), 637-675. https://doi.org/10.1353/ajm.2011.0017

**Stable quasiconformal mapping class groups and asymptotic teichmü ller spaces.** / Fujikawa, Ege; Matsuzaki, Katsuhiko.

Research output: Contribution to journal › Article

*American Journal of Mathematics*, vol. 133, no. 3, pp. 637-675. https://doi.org/10.1353/ajm.2011.0017

}

TY - JOUR

T1 - Stable quasiconformal mapping class groups and asymptotic teichmü ller spaces

AU - Fujikawa, Ege

AU - Matsuzaki, Katsuhiko

PY - 2011/6

Y1 - 2011/6

N2 - 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.

AB - 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.

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U2 - 10.1353/ajm.2011.0017

DO - 10.1353/ajm.2011.0017

M3 - Article

VL - 133

SP - 637

EP - 675

JO - American Journal of Mathematics

JF - American Journal of Mathematics

SN - 0002-9327

IS - 3

ER -