### 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 |
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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 |

### 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