### Abstract

We consider a random walk on the mapping class group of a surface of finite type. We assume that the random walk is determined by a probability measure whose support is finite and generates a non-elementary subgroup H. We further assume that H is not consisting only of lifts with respect to any one covering. Then we prove that the probability that such a random walk gives a non-minimal mapping class in its fibered commensurability class decays exponentially. As an application of theminimality,we prove that for the case where a surface has at least one puncture, the probability that a random walk gives mapping classes with arithmetic mapping tori decays exponentially. We also prove that a random walk gives rise to asymmetric mapping tori with exponentially high probability for closed case.

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

Pages (from-to) | 1253-1279 |

Number of pages | 27 |

Journal | Groups, Geometry, and Dynamics |

Volume | 11 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2017 Jan 1 |

### Keywords

- Arithmetic 3-manifold
- Fibered commensurability
- Mapping class group
- Random walk

### ASJC Scopus subject areas

- Geometry and Topology
- Discrete Mathematics and Combinatorics

## Fingerprint Dive into the research topics of 'Fibered commensurability and arithmeticity of random mapping tori'. Together they form a unique fingerprint.

## Cite this

*Groups, Geometry, and Dynamics*,

*11*(4), 1253-1279. https://doi.org/10.4171/GGD/428