### 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 |
---|---|

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 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**The nielsen realization problem for asymptotic teichm̈uller modular groups.** / Fujikawa, Ege; Matsuzaki, Katsuhiko.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 365, no. 6, pp. 3309-3327. https://doi.org/10.1090/S0002-9947-2013-05767-4

}

TY - JOUR

T1 - The nielsen realization problem for asymptotic teichm̈uller modular groups

AU - Fujikawa, Ege

AU - Matsuzaki, Katsuhiko

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Hyperbolic geometry

KW - Quasiconformal mapping class group

KW - Riemann surface

KW - Teichmüller space

UR - http://www.scopus.com/inward/record.url?scp=84875495796&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84875495796&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2013-05767-4

DO - 10.1090/S0002-9947-2013-05767-4

M3 - Article

AN - SCOPUS:84875495796

VL - 365

SP - 3309

EP - 3327

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 6

ER -