Kwai Man Fan proved that if the intersection lattice of a line arrangement does not contain a cycle, then the fundamental group of its complement is a direct sum of infinite and cyclic free groups. He also conjectured that the converse is true as well. The main purpose of this paper is to prove this conjecture.
Meital Eliyahu. Eran Liberman. Malka Schaps. Mina Teicher. "The characterization of a line arrangement whose fundamental group of the complement is a direct sum of free groups." Algebr. Geom. Topol. 10 (3) 1285 - 1304, 2010. https://doi.org/10.2140/agt.2010.10.1285