On topology of line arrangements

In the talk I shall give some background on the question of topology vs combinatorics of line arrangements, the application to classification of algebraic surfaces and will report on two results.The first one is a roof of a conjecture by Fan — characterization of arrangements with no cycles and the second on conjugation-free geometric presentation of fundamental groups of arrangements.
Fan proved that Line arrangements whose graph has no cycles then the fundamental group is a sum of free groups each in the order of one the multiple points minus 1 (no of groups equal to number of multiple points) plus a commutative free to the power that will make the total group of base l minus 1 when l is the number of lines. We proved the opposite — from fundamental group to a property of the conjugation. (Joint work with Eliyahu, Liberman and Schaps.)
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 study some properties of this type of presentations for a fundamental group of a line arrangement's complement. We actually show that a large family of these presentations satisfy a completeness property in the sense of Dehornoy. The completeness property is a powerful property which leads to many nice properties concerning the presentation (as the left-cancellativity in the associated monoid and yields some simple criterion for the solvability of the word problem in the group). (Joint work with Eliyahu and Garber.)