### Abstract

Under a certain geometric assumption on a hyperbolic Riemann surface, we prove an asymptotic version of the fixed point theorem for the Teichm̈uller modular group, which asserts that every finite subgroup of the asymptotic Teichm̈uller modular group has a common fixed point in the asymptotic Teichm̈uller space. For its proof, we use a topological characterization of the asymptotically trivial mapping class group, which has been obtained in the authors' previous paper, but a simpler argument is given here. As a consequence, every finite subgroup of the asymptotic Teichm̈uller modular group is realized as a group of quasiconformal automorphisms modulo coincidence near infinity. Furthermore, every finite subgroup of a certain geometric automorphism group of the asymptotic Teichm̈uller space is realized as an automorphism group of the Royden boundary of the Riemann surface. These results can be regarded as asymptotic versions of the Nielsen realization theorem.

Original language | English |
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Pages (from-to) | 3309-3327 |

Number of pages | 19 |

Journal | Transactions of the American Mathematical Society |

Volume | 365 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2013 Apr 3 |

### Keywords

- Hyperbolic geometry
- Quasiconformal mapping class group
- Riemann surface
- Teichmüller space

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics