Conjugation-free geometric presentations of fundamental groups of arrangements

TL;DR

This paper introduces conjugation-free geometric presentations for fundamental groups of line arrangement complements, providing explicit structures for certain arrangements, including a detailed example of a 6-line configuration.

Contribution

It defines conjugation-free geometric presentations and demonstrates their existence for a class of arrangements, also computing the exact group structure for a specific 6-line case.

Findings

Fundamental groups of certain arrangements have conjugation-free presentations.

Explicit group structure computed for a 6-line arrangement with a cycle graph.

Arrangement conditions ensure conjugation-free presentations.

Abstract

We introduce the notion of a conjugation-free geometric presentation for a fundamental group of a line arrangement's complement, and we show that the fundamental groups of the following family of arrangements have a conjugation-free geometric presentation: A real arrangement L, whose graph of multiple points is a union of disjoint cycles, has no line with more than two multiple points, and where the multiplicities of the multiple points are arbitrary. We also compute the exact group structure (by means of a semi-direct product of groups) of the arrangement of 6 lines whose graph consists of a cycle of length 3, and all the multiple points have multiplicity 3.