A conjugation-free geometric presentation of fundamental groups of arrangements II: Expansion and some properties

A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators x₁,..,x _n and the cyclic relations: $$ x-{i-k}x-{i-{k-1}} \cdots x_i1} = x_ik-1} \cdots x_i1} x_ik} = \cdots = x_i1} xik} \cdots xi2} $$ 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).