## Abstract

The twin group T
_{n}
is a Coxeter group generated by n- 1 involutions and the pure twin group PT
_{n}
is the kernel of the natural surjection of T
_{n}
onto the symmetric group on n letters. In this paper, we investigate structural aspects of twin and pure twin groups. We prove that the twin group T
_{n}
decomposes into a free product with amalgamation for n> 4. It is shown that the pure twin group PT
_{n}
is free for n= 3 , 4 , and not free for n≥ 6. We determine a generating set for PT
_{n}
, and give an upper bound for its rank. We also construct a natural faithful representation of T
_{4}
into Aut (F
_{7}
). In the end, we propose virtual and welded analogues of these groups and some directions for future work.

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

Pages (from-to) | 135-154 |

Number of pages | 20 |

Journal | Geometriae Dedicata |

Volume | 203 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Dec 2019 |

## Keywords

- Coxeter group
- Doodle
- Eilenberg–Maclane space
- Free group
- Hyperbolic plane
- Pure twin group
- Twin group