On the structure of fundamental groups of conic-line arrangements having a cycle in their graph

TL;DR

This paper studies the fundamental groups of conic-line arrangements with a cycle, revealing their algebraic structure varies with the cycle length, especially distinguishing between odd and even cases.

Contribution

It characterizes the fundamental group's structure for conic-line arrangements with a cyclic graph, highlighting differences based on cycle length parity and passing through all multiple points.

Findings

For odd n, the affine fundamental group is abelian but not conjugation-free.

For even n > 4, the fundamental group is not a direct sum of free and abelian groups.

Abstract

The fundamental group of the complement of a plane curve is a very important topological invariant. In particular, it is interesting to find out whether this group is determined by the combinatorics of the curve or not, and whether it is a direct sum of free groups and a free abelian group, or it has a conjugation-free geometric presentation. In this paper, we investigate the structure of this fundamental group when the graph of the conic-line arrangement is a unique cycle of length $n$ and the conic passes through all the multiple points of the cycle. We show that if n is odd, then the affine fundamental group is abelian but not conjugation-free. For the even case, if n>4, then using quotients of the lower central series, we show that the fundamental group is not even a direct sum of a free abelian group and free groups.