TY - JOUR

T1 - On the structure of fundamental groups of conic-line arrangements having a cycle in their graph

AU - Friedman, Michael

AU - Garber, David

PY - 2014/11/1

Y1 - 2014/11/1

N2 - 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 a direct sum of a free abelian group and free groups.

AB - 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 a direct sum of a free abelian group and free groups.

KW - Braid monodromy

KW - Conic-line arrangement

KW - Conjugation-free presentation

KW - Fundamental group

KW - Lower central series

UR - http://www.scopus.com/inward/record.url?scp=84906489563&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2014.05.013

DO - 10.1016/j.topol.2014.05.013

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AN - SCOPUS:84906489563

SN - 0166-8641

VL - 177

SP - 34

EP - 58

JO - Topology and its Applications

JF - Topology and its Applications

ER -