The fundamental group of the complement of a line arrangement plays an important role in studying these arrangements. We introduce the notion of a conjugation-free geometric presentation for such fundamental groups.
The main motivation for defining this presentation is due to the fact that for arrangements having such a presentation, its lattice determines the fundamental group. Moreover, these presentations have very interesting algebraic properties, as completeness (in the sense of Dehornoy) and complementedness, which imply cancellativity and existence of least common multiples of pairs of elements.
We construct some families of line arrangements have a conjugation-free geometric presentation.
In the second part, we generalize these ideas to the case of conic-line arrangements, and we prove that once the graph associated to conic-line arrangements (defined slightly different than the corresponding graph for line arrangements) has no cycles, then the fundamental group of its complement has a conjugation-free geometric presentation and can be written as a direct sum of free groups and a free abelian group. Moreover, if there is a cycle in the graph but the conic does not pass through all the multiple points, we have that its fundamental group has a conjugation-free geometric presentation as well. On the other hand, if the conic passes through all the multiple points of the cycle, the presentation is not conjugation-free anymore, but if length of the cycle is odd, the group still can be written as a direct sum of free groups and a free abelian group.