### Abstract

We investigate the dynamics of the Teichmüller modular group on the Teichmüller space of a Riemann surface of infinite topological type. Since the modular group does not necessarily act discontinuously, the quotient space cannot inherit a rich geometric structure from the Teichmüller space. However, we introduce the set of points where the action of the Teichmüller modular group is stable, and we prove that this region of stability is generic in the Teichmüller space. By taking the quotient and completion with respect to the Teichmüller distance, we obtain a geometric object that we regard as an appropriate moduli space of the quasiconformally equivalent complex structures admitted on a topologically infinite Riemann surface.

Original language | English |
---|---|

Pages (from-to) | 1-64 |

Number of pages | 64 |

Journal | Groups, Geometry, and Dynamics |

Volume | 12 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2018 Jan 1 |

### Fingerprint

### Keywords

- Hyperbolic geometry
- Length spectrum
- Limit set
- Moduli space
- Quasiconformal deformation
- Region of discontinuity
- Riemann surface of infinite type
- Teichmüller modular group

### ASJC Scopus subject areas

- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

**Dynamics of teichmüller modular groups and topology of moduli spaces of Riemann surfaces of infinite type.** / Matsuzaki, Katsuhiko.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Dynamics of teichmüller modular groups and topology of moduli spaces of Riemann surfaces of infinite type

AU - Matsuzaki, Katsuhiko

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We investigate the dynamics of the Teichmüller modular group on the Teichmüller space of a Riemann surface of infinite topological type. Since the modular group does not necessarily act discontinuously, the quotient space cannot inherit a rich geometric structure from the Teichmüller space. However, we introduce the set of points where the action of the Teichmüller modular group is stable, and we prove that this region of stability is generic in the Teichmüller space. By taking the quotient and completion with respect to the Teichmüller distance, we obtain a geometric object that we regard as an appropriate moduli space of the quasiconformally equivalent complex structures admitted on a topologically infinite Riemann surface.

AB - We investigate the dynamics of the Teichmüller modular group on the Teichmüller space of a Riemann surface of infinite topological type. Since the modular group does not necessarily act discontinuously, the quotient space cannot inherit a rich geometric structure from the Teichmüller space. However, we introduce the set of points where the action of the Teichmüller modular group is stable, and we prove that this region of stability is generic in the Teichmüller space. By taking the quotient and completion with respect to the Teichmüller distance, we obtain a geometric object that we regard as an appropriate moduli space of the quasiconformally equivalent complex structures admitted on a topologically infinite Riemann surface.

KW - Hyperbolic geometry

KW - Length spectrum

KW - Limit set

KW - Moduli space

KW - Quasiconformal deformation

KW - Region of discontinuity

KW - Riemann surface of infinite type

KW - Teichmüller modular group

UR - http://www.scopus.com/inward/record.url?scp=85045926089&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85045926089&partnerID=8YFLogxK

U2 - 10.4171/GGD/437

DO - 10.4171/GGD/437

M3 - Article

VL - 12

SP - 1

EP - 64

JO - Groups, Geometry, and Dynamics

JF - Groups, Geometry, and Dynamics

SN - 1661-7207

IS - 1

ER -