## Abstract

A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators x_{1},..,x _{n} and the cyclic relations: $$ x-{i-k}x-{i-{k-1}} \cdots x_{i}1} = x_{ik-1}} \cdots x_{i}1} x_{i}k} = \cdots = x_{i}1} xik} \cdots xi2} $$ with no conjugations on the generators. We have already proved in [13] that if the graph of the arrangement is a disjoint union of cycles, then its fundamental group has a conjugation-free geometric presentation. In this paper, we extend this property to arrangements whose graphs are a disjoint union of cycle-tree graphs. Moreover, we study some properties of this type of presentations for a fundamental group of a line arrangement's complement. We show that these presentations satisfy a completeness property in the sense of Dehornoy, if the corresponding graph of the arrangement is triangle-free. The completeness property is a powerful property which leads to many nice properties concerning the presentation (such as the left-cancellativity of the associated monoid and yields some simple criterion for the solvability of the word problem in the group).

Original language | English |
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Pages (from-to) | 775-792 |

Number of pages | 18 |

Journal | International Journal of Algebra and Computation |

Volume | 21 |

Issue number | 5 |

DOIs | |

State | Published - Aug 2011 |

### Bibliographical note

Funding Information:‖Partially supported by the Israeli Ministry of Science and Technology.

### Funding

‖Partially supported by the Israeli Ministry of Science and Technology.

Funders | Funder number |
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Israeli ministry of science and technology |

## Keywords

- Conjugation-free presentation
- complemented presentation
- complete presentation
- fundamental group