A conjugation-free geometric presentation of fundamental groups of arrangements II: Expansion and some properties
arXiv:1009.1349
Abstract
A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators and the cyclic relations: with no conjugations on the generators. We have already proved 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 has no edges. 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).
17 pages, 9 figures; final version, which corrects a mistake in the published version