### Abstract

For an analytically infinite Riemann surface R, we consider the action of the quasiconformal mapping class group MCG(R) on the Teichmüller space T(R), which preserves the fibers of the projection α: T(R) → AT(R) onto the asymptotic Teichmüller space AT(R). We prove that if MCG(R) has a common fixed point α(p) AT(R), then it acts discontinuously on the fiber T _{p} over α(p), which is a separable subspace of T(R). In particular, this implies that MCG(R) is a countable group. This is a generalization of a fact that MCG(R) acts discontinuously on T _{o} = T(R) for an analytically finite Riemann surface R.

Original language | English |
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Pages (from-to) | 1-28 |

Number of pages | 28 |

Journal | Journal d'Analyse Mathematique |

Volume | 102 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 Aug |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Analysis

### Cite this

**Quasiconformal mapping class groups having common fixed points on the asymptotic Teichmüller spaces.** / Matsuzaki, Katsuhiko.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Quasiconformal mapping class groups having common fixed points on the asymptotic Teichmüller spaces

AU - Matsuzaki, Katsuhiko

PY - 2007/8

Y1 - 2007/8

N2 - For an analytically infinite Riemann surface R, we consider the action of the quasiconformal mapping class group MCG(R) on the Teichmüller space T(R), which preserves the fibers of the projection α: T(R) → AT(R) onto the asymptotic Teichmüller space AT(R). We prove that if MCG(R) has a common fixed point α(p) AT(R), then it acts discontinuously on the fiber T p over α(p), which is a separable subspace of T(R). In particular, this implies that MCG(R) is a countable group. This is a generalization of a fact that MCG(R) acts discontinuously on T o = T(R) for an analytically finite Riemann surface R.

AB - For an analytically infinite Riemann surface R, we consider the action of the quasiconformal mapping class group MCG(R) on the Teichmüller space T(R), which preserves the fibers of the projection α: T(R) → AT(R) onto the asymptotic Teichmüller space AT(R). We prove that if MCG(R) has a common fixed point α(p) AT(R), then it acts discontinuously on the fiber T p over α(p), which is a separable subspace of T(R). In particular, this implies that MCG(R) is a countable group. This is a generalization of a fact that MCG(R) acts discontinuously on T o = T(R) for an analytically finite Riemann surface R.

UR - http://www.scopus.com/inward/record.url?scp=58449100711&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58449100711&partnerID=8YFLogxK

U2 - 10.1007/s11854-007-0015-z

DO - 10.1007/s11854-007-0015-z

M3 - Article

VL - 102

SP - 1

EP - 28

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

IS - 1

ER -