On Parallel Lines and Free Group

TL;DR

This paper characterizes when the fundamental group of the complement of complex line arrangements is free, showing it occurs precisely when the lines are all parallel, thus linking geometric configuration to algebraic properties.

Contribution

It provides a complete characterization of arrangements with free fundamental groups, establishing a clear geometric-algebraic correspondence.

Findings

Fundamental group is free iff lines are parallel.

Parallel line arrangements have free fundamental groups.

Non-parallel arrangements do not have free fundamental groups.

Abstract

We show that the fundamental group of the complement of an arrangement of complex lines in the complex plane is a free group if and only if the arrangement is a union of parallel lines.